SINGLE- AND MULTIPHONON ATOM-SURFACE SCATTERING IN THE QUANTUM REGIME
Branko GUMHALTER Institute of Physics of the University, Bijenicka C.46, POB 304, 10001 Zagreb, Croatia
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 351 (2001) 1}159
Single- and multiphonon atom}surface scattering in the quantum regime Branko Gumhalter* Institute of Physics of the University, Bijenicka C. 46, POB 304, 10001 Zagreb, Croatia Received November 2000; editor: D.L. Mills
Contents 1. Introduction 2. Atom scattering as a tool for investigation of structural and dynamical properties of surfaces and adlayers 2.1. Kinematics of atom}surface scattering. Energy and parallel momentum shell. Di!raction and rainbow scattering, resonance processes 2.2. Investigations of the structural properties of ordered surfaces and adlayers by thermal energy atomic and molecular beams 2.3. Investigations of the structural properties of disordered surfaces and adlayers by thermal energy atom scattering 2.4. Investigations of the dynamical properties of surfaces, adlayers and adsorbates by noble gas atom scattering 2.5. Comparison with other techniques 3. Interactions and inelastic scattering of atoms from surface vibrations. Short overview of the achievements and shortcomings of standard theoretical descriptions 3.1. Descriptions of the vibrational dynamics of surfaces and adlayers 3.2. Particle}surface interaction potentials
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3.3. Inelastic scattering from surfaces. One phonon vs. multiphonon scattering regimes, advantages and shortcomings of particular models 3.4. The search for a uni"ed approach 4. Scattering spectrum approach in the theoretical description of inelastic inert atom scattering from surfaces 4.1. Formulation of the scattering spectrum expression and its relation to the TOF spectra 4.2. Development of the scattering spectrum formalism (SSF) 4.3. Choice of approximations 5. Scattering from #at surfaces in the SSF approach 5.1. Single-phonon scattering regime and distorted wave Born approximation (DWBA) 5.2. Multiphonon scattering regime and the exponentiated Born approximation (EBA) 6. Examples of application of the developed formalism: Debye}Waller factors and scattering spectra of selected benchmark systems 6.1. Scattering of He atoms from (1 1 1) surface of condensed Xe
* Tel.: #385-1-469-8805 (direct); #385-1-469-8888; fax: #385-1-469-8889. E-mail address:
[email protected] (B. Gumhalter). 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 4 3 - 5
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6.2. Scattering of He atoms from Cu surfaces 6.3. Scattering of He atoms from monolayers of Xe atoms 6.4. Debye}Waller factors for scattering of heavier noble gas atoms from surfaces 7. Energy transfer in gas}surface collisions 7.1. Angular resolved vs. angular integrated energy transfer
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7.2. Quantum vs. classical results for energy transfer in benchmark systems 8. Concluding remarks and protocol for the use of EBA-formalism in interpretations of atom}surface scattering experiments Acknowledgements References
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Abstract Recent developments and achievements in the theoretical interpretation of inelastic scattering of thermal energy beams of He and other noble gas atoms from surfaces are reviewed, with a special emphasis on the successful interpretation of multiphonon He atom scattering (HAS) experiments. These developments have been stimulated by the remarkable successes of HAS time-of-#ight spectroscopy in revealing information on the low-energy dynamics of the various surfaces, adlayers and isolated adsorbates. The diversity of the developed theoretical approaches re#ects also the diversity of the various observables that have been assessed under the di!erent experimental conditions. To aid the systematization and cross-correlation among the di!erent model descriptions we "rst present a short outline of their characteristics and main achievements. Although many of these theories have been improved and re"ned in the course of time, a uni"ed approach was required for a fully quantum treatment of elastic (di!ractive or di!use) and inelastic (single- and multiphonon) atom}surface scattering processes on an equivalent footing. A substantial progress towards this end has been made in recent years by going beyond the standard semiclassical and perturbation methods in the analyses of HAS experiments. The present review focuses on the development of one such approach based on the so-called scattering spectrum formalism in which the quantum scattering amplitudes are calculated by using cumulant or linked cluster expansion in terms of the correlated and uncorrelated scattering events. This formalism is equally well suited for making a passage to perturbative quantummechanical and nonperturbative semiclassical treatments of inelastic atom}surface scattering. Using the developed formalism we "rst establish the relevant approximations for calculating the scattering spectra and examine their validity for the scattering conditions typical of HAS. In the next step the formalism is applied to benchmark systems to interpret the scattering data which intermingledly depend on the vibrational dynamics of the investigated surfaces per se and on the projectile}surface interaction potentials. A very good agreement between experimental results and theoretical predictions for HAS from surfaces characteristic of the di!erent types of surface vibrational dynamics is obtained in all studied scattering regimes. This demonstrates a broad applicability of the developed formalism in the interpretations of inelastic HAS experiments and in the assessments of phonon-mediated energy transfer in gas}surface collisions. 2001 Elsevier Science B.V. All rights reserved. PACS: 34.50.Dy; 47.45.Md; 61.10.Dp; 61.18.Bn; 63.22.#m; 68.35.Ja; 79.20.Rf Keywords: Gas}surface interactions; Molecular beam}surface scattering; He atom scattering; Surface phonons; Multiphonon excitations; Surface accommodation
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1. Introduction The earliest gas}surface scattering experiments can be traced back to the 19th century when Kundt and Warburg [1] attempted to test Maxwell's predictions [2] on the properties of gas viscosity, and to the beginning of the 20th century when Knudsen [3] and Smoluchowski [4] carried out "rst measurements of the accommodation coe$cients. A brief overview of these and related measurements pertaining to the gas}surface dynamics phenomena can be found in the book of Goodman and Wachman [5]. However, in the modern sense of de"nition of a laboratory tool, atomic and molecular beam scattering has been in use as an experimental technique in the studies of structural and dynamic properties of surfaces since the advent of quantum mechanics. In the late 1920s and early 1930s Stern and collaborators [6] and Johnson [7] carried out "rst gas}surface di!raction experiments with the goal of demonstrating the wave nature of atomic particles. In these studies the beams of He, H and H from the e!usive sources were scattered from (0 0 1) surfaces of alkali-halides (LiF, NaF and NaCl). The scattered beams have been found to exhibit di!raction, thereby verifying the basic quantum principles contained in the de Broglie relations. Thermal energy He atom beams proved particularly convenient in this respect, mainly for two reasons. First, with the low He atom mass and the energy in thermal range the associated de Broglie wavelength can easily match the di!raction conditions imposed by the surface crystallography, and second, due to the inert electronic structure and low polarizability of the projectile atoms the collisions with the surface are dominantly nonreactive and elastic. The investigations of inelastic gas particle interactions with well-de"ned and monocrystal surfaces were at that time to a large extent hampered by the absence of ultra-high-vacuum (UHV) technology and adequate analytical surface science techniques needed to maintain and control the microscopic structure and cleanliness of surfaces. Owing to this, the studies of inelastic gas}surface interactions were in this period mainly focused on the integrated or global quantities (like the accommodation and sticking coe$cients) characteristic of technical surfaces, the more so as for many years the main motivation for carrying out such experiments was coming from aerospace research. The development of modern UHV technology gave a strong impetus to the studies of gas}surface dynamics and scattering. About 30 years ago, and concomitant with the trends in development of surface science, the thermal energy atomic and molecular beam scattering technique has emerged as one of the most sensitive and universal experimental methods for investigations of the structural, dynamical and even electronic properties of surfaces. This particularly applies to He atom scattering (HAS) as an analytical technique whose rapid development was mainly due to the progress achieved by combining the UHV techniques with the high-pressure nozzle beam production and time-of-#ight (TOF) methods for analyzing the energy of scattered particles. Thus, at present it is possible to maintain highly collimated and intense monoenergetic primary beams with kinetic energies in the range of 8}150 meV per particle, and combine it with high-resolution energy analyses of the scattered beams in the range of 0.1 meV. With the high angular resolution of the direction of motion of the scattered particles, this enables the investigations of dynamical processes at surfaces like surface vibrations, di!usion, surface phase transformations, growth phenomena, etc., with unrivaled precision in many aspects. These characteristics, as well as the nondestructiveness and absolute surface sensitivity of HAS, make this technique one of the most versatile and universal tools of surface science research, and complementary to the electron-based spectroscopies like low-energy electron di!raction (LEED), high-resolution
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electron energy loss spectroscopy (HREELS) and scanning tunnelling microscopy (STM). Owing to all these characteristics the data acquired by HAS usually constitute an indispensable component of information needed for establishing a complete microscopic, quantum-mechanical picture of surface properties of the studied system. The various aspects and modes of utilization of thermal energy atom scattering from surfaces, with a particular emphasis on HAS, have been described in several reviews and books [8}18]. The high angular and energy resolution that can be achieved in HAS from surfaces (but also in similar scattering experiments utilizing the beams of other inert atoms and molecules) have proved particularly advantageous in the studies of processes a!ected and controlled by the low-energy dynamics characteristic of and taking place at surfaces. Typical and important examples of such processes are the excitations of surface, adlayer and adsorbate vibrational modes or phonons (excitation energies of few meV) and di!usion of atomic particles (mass transport) at surfaces. The studies of these processes by molecular beam scattering techniques have provided some of the most valuable information pertaining to the microscopic properties of surfaces. In order to obtain clear "ngerprints of surface phonons from atom}surface scattering, i.e. to reveal their dispersion over the corresponding surface Brillouin zone (SBZ), the experiments have to be carried out in the single-phonon scattering regime. Only in this case it is possible to precisely and unambiguously determine within the limits of experimental resolution the direction of the momentum and the amount of energy exchanged between the projectile and a phonon, which is needed to restore the dispersion relations. With state-of-the-art technology this regime can be routinely achieved in the scattering of light particles at su$ciently low incoming energy and by keeping the substrate temperature relatively low. The concrete borderline between a single- and multiphonon scattering regime depends on the various parameters of the collision system but it is most sensitive to the beam energy, projectile mass, maximum phonon frequency and substrate temperature. Again, the vast majority of the data has been accumulated from HAS measurements. Combined with the complementary EELS data, which were also available for a number of systems, it has been possible to reveal the dispersion curves of the various surface projected phonon branches of the investigated materials over the entire SBZ, and in some cases also beyond. Thanks to the extensive work carried out in a number of laboratories in the past two decades the presently existing database on surface phonons of the various materials is already rich [19,20]. This provides a solid prerequisite for our understanding of the surface dynamics and thermodynamics at the microscopic and fundamental level. The interpretation of the results of electron, atom and molecular beam scattering from surfaces, and in particular of the beam di!raction and phonon excitation intensities, has strongly motivated and stimulated the development of quantum-mechanical descriptions of elastic and inelastic projectile}surface scattering. Substantial e!orts have been devoted towards this end simultaneously with the development and implementation of the various scattering-based techniques in surface science research. Particularly illustrative and instructive in this respect are the developments in the theory of HAS because the conditions prevailing in the majority of experiments necessitate a fully quantum approach. The quantum theory of HAS from surfaces has followed two main directions of development. One of them is the description of elastic scattering phenomena like di!raction, selective adsorption and di!use scattering by defects and impurities. The other is the description of inelastic phenomena like single- and multiphonon atom}surface scattering and quasielastic scattering by di!using
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Fig. 1. Schematic diagram showing the di!erent collision processes which can occur in the nonreactive scattering from a crystal surface of a light atom with a de Broglie wavelength comparable to the lattice spacing (after Ref. [32]).
scatterers on surfaces. Of course, in a scattering experiment all these scattering channels may be open simultaneously (see Fig. 1) and theoretical approaches accounting for the e!ects of multichannel scattering have also been developed at various stages of sophistication. This review will be primarily concerned with the exposition of novel theoretical developments in the descriptions of inelastic scattering of thermal energy atomic beams from surface vibrations, with a particular emphasis on He atom scattering as the most widely used technique for revealing surface phonon dynamics. Recent experimental activities in this "eld have stimulated the development of quantum models capable of describing both the one-phonon and multiphonon He atom scattering on an equivalent footing. In this respect the various forms of the quantum one-phonon theory of HAS in the distorted wave Born approximation (DWBA), which have been perfected following the pioneering works of the Cambridge school in the 1930s [21] and later reviewed by Goodman [22], represent a solid basis and a prerequisite also for the development of multiphonon HAS theory. An obvious requirement on more complete theories is that they should naturally encompass the DWBA results as a special limit. On the other hand, the various classical and semiclassical theories of multiphonon scattering should also provide a useful guideline in the development of their quantum counterparts because upon approaching the classical multiphonon scattering regime one expects that the two descriptions should consistently give similar results. This particularly applies to the semiclassical trajectory approximation (TA) for the description of projectile dynamics which has for a long time been in the focus of applications and often with very good results. Although the range of the validity of TA in atom}surface scattering theory could not until recently be rigorously estimated and justi"ed on theoretical grounds [23}25], recent theoretical developments enable the assessment of validity of the various quasiclassical approximations employed in the multiphonon atom}surface scattering regime. These topics will be also addressed in the present review. The developments and applications of the scattering theory in interpreting the results of He atom scattering from surfaces available by the beginning of the 1980s were described by Levi [26] in 1979, and Celli [27] in 1982, those available by the mid-1980s in a comprehensive review by Bortolani and Levi [28] in 1986, and by the beginning of the 1990s by Manson [29], and Santoro and Bortolani [30]. Hence, the present review will focus mainly on the later developments, with
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a brief reference to the earlier material in the cases in which it becomes instrumental to the development of a uni"ed single- and multiphonon quantum scattering formalism described in subsequent sections. With this goal in mind the outline of this review is as follows. In Section 2 we present a short survey of the use of atom scattering, with the emphasis on HAS, in the investigations of structural and dynamical properties of surfaces and adlayers, and relate the information available from these measurements with the information obtainable from other, complementary techniques. In Section 3 we present an overview of the various theoretical models employed in the interpretations of atom}surface scattering experiments, describe their main achievements but also pinpoint the shortcomings that have ultimately motivated the development of new, nonstandard theoretical models of inelastic atom}surface collisions. In Section 4 a formal development of such a new approach } the scattering spectrum formalism (SSF) } for description of inelastic HAS from surfaces is presented. This formalism enables a quantum mechanical treatment of single- and multiphonon scattering processes on the same footing and the obtained results can be directly related to the measured inelastic HAS TOF intensities. Illustrations of the applications of the scattering spectrum formalism to the calculations of inelastic scattering intensities and Debye}Waller factors in the various regimes of He atom}surface scattering are presented in Section 5. Interpretations of experimental HAS data for #at surfaces based on the SSF are elaborated in Section 6. Here we have adopted the approach to concentrate on comprehensive descriptions of applications of the formalism to a restricted number of benchmark or prototype systems characteristic of the various types of phonon dynamics, rather than to present only brief overviews of the applications to all the systems studied so far. Section 7 is devoted to a discussion of the energy (heat) transfer in atom}surface scattering with a particular emphasis on some applied problems of accommodation in gas}surface collisions. Finally, in Section 8 we reiterate the main achievements and advantages of using the scattering spectrum formalism in the interpretations of inelastic atom}surface scattering by phonons, and present a summary of the most important derived formulae in the form of a brief protocol for their concrete applications.
2. Atom scattering as a tool for investigation of structural and dynamical properties of surfaces and adlayers Helium atom scattering plays a role in investigations of the properties and processes on surfaces similar to what thermal neutron scattering plays in the investigations of bulk solids and liquids. The high resolution and intensity of low-energy thermal He beams, in combination with the inert chemical structure of He atoms, make the HAS an extremely surface-sensitive and nondestructive method and as such ideally suited for investigations of the structure and dynamics of and phase transitions and di!usion on surfaces. Using HAS it is also possible to probe the processes characterized by large wavevector transfer (e.g. excitation of large wavevector phonons) which is inaccessible to optical spectroscopies, and to resolve closely spaced modes which may not be possible by EELS. To fully exploit the potentiality and advantages of HAS, many requirements
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have to be met in the design and operation of the HAS apparatuses in order to achieve optimal performance and results. The TOF technique enables simultaneous measurement of the energy and wavevector transfer in atom}surface collisions which in turn makes possible the studies of low-energy dynamics of surfaces, adlayers and isolated adsorbates with unrivaled energy resolution and surface sensitivity. However, the combination of high resolution and intensity of He beams is subject to experimental limitations. Instrumental broadening in energy and momentum measurements depends on many factors, as does the signal intensity, and there is a trade o! between the two characteristics. These problems have been explained and discussed in several reviews and books [12,14,18,32,34,35]. Here it is worth pointing out that at present the monochromaticity of the source beams which can be routinely achieved in modern HAS apparatuses is in the range of v/v41% where v is the velocity of He atoms, and the major limitation comes from the TOF technique, i.e. from the energy resolution. For a liquid-nitrogen-cooled nozzle the attainable energy resolution in phonon creation events is of the order of 0.1 meV at an incoming energy of 10 meV. The angular resolution, which a!ects the wavevector resolution, depends also on several factors and at present [34] is in the range of 0.13. For the theoretical interpretations of HAS experiments these resolution characteristics, together with the negligible e!ects of the "nite size of the beam source [18], are of utmost importance. The magnitude and direction of the momentum of atoms in He beams, collimated from free-molecular#ow atom trajectories emanating from the nozzle and impinging on the target, should be determinable with su$cient precision so that the incoming He atom wavefunction can be described by a plane wave which is well de"ned or coherent over a su$ciently large volume. At present it is possible to attain a coherence volume of about (300) As and the corresponding coherence length of 300 As de"nes the surface distance from which scattering from individual surface atoms interferes coherently [33]. Such a high coherence enables a straightforward application of the parallel momentum conservation conditions and the use of simple kinematics of HAS experiments that is a prerequisite for interpretation of the collision dynamics. This is brie#y discussed in the next subsection. 2.1. Kinematics of atom}surface scattering. Energy and parallel momentum shell. Diwraction and rainbow scattering, resonance processes Apart from the above-mentioned characteristics that distinguish light noble gas atom scattering from the neutron scattering technique, an additional large di!erence exists in the form and properties of the projectile}target interaction potentials. Whereas in the case of neutron scattering one can safely assume that contact pseudopotentials can be used to describe the interaction of the projectile (neutron) with atomic nuclei of the target, the projectile}surface interaction in He and other atom or molecule scattering from surfaces is far more complicated. Due to this, it is common to start the interpretations of experimental results and developments of the theories of HAS with the discussion of the properties of projectile}surface potentials that determine the dynamics of collisions. This line of thought will be also followed in the present review, and a brief discussion of the projectile}surface potentials is the "rst subject of the next section in which the standard HAS theories are brie#y reviewed. However, some basic notions needed to understand the general concepts and results which will be outlined in this introductory section can be explained also in
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terms of the much simpler kinematics of atom}surface scattering events. In fact, the mere application of kinematics of HAS in surface studies, which is possible due to the high coherence of the beams, enables very often a quick and straightforward analysis of the data on the structural and dynamical properties of surfaces, like the reciprocal surface lattice wavevectors G, amount of disorder on the surface, phonon dispersion curves, etc. To "x the notation we shall denote from now on the components of the vectors parallel to the surface by capital bold letters, e.g. for the components of the radius vector r"(R, z)
(1)
with the direction of z perpendicular to the surface. Accordingly, the components of the wavevectors are denoted as k"(K, k ) X
(2)
and analogously for the other vector quantities unless explicitly stated otherwise. In the case of translationally invariant surfaces or surfaces with well-de"ned periodicity (such as corrugated surfaces) the natural choice of quantum numbers for the description of unperturbed motion of the projectile particles comprises the particle momentum P" K or wavevector K"(K , K ) parallel (lateral) to the surface, and the particle total energy E. It should be observed V W that the particle}surface potential ;(r) is periodic only in the directions R along the surface since the latter introduces a breakdown of translational symmetry along the z-direction. Due to this the perpendicular particle momentum is not a good quantum number. However, the perpendicular wavevector of the particle at large distance z from the surface (outside the range of the particle}surface potential) is again a good quantum number, k , and in combination with K can be X employed instead of E in the description of unperturbed particle motion. With this nomenclature we can set up the basic kinematics needed for description of an atom}surface scattering event. We introduce a coordinate system with the (x, y) plane coinciding with the surface of the crystal and the z-axis pointing outward, and denote by and the corresponding polar and azimuthal angles, respectively. In what follows, we shall use the subscripts i and f to denote the asymptotic initial and "nal values, respectively, of the scattering particle energy, momentum, etc. The scattering geometry in terms of these symbols is illustrated in Fig. 2. Thus we have k "(K , k )"(k , , ) , G G XG G G G
(3)
k "(K , k )"(k , , ) , D D XD D D D
(4)
(K#k ) G XG E" G 2M
(5)
(K #k ) D XD , E " D 2M
(6)
and
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Fig. 2. (a) General de"nitions of scattering geometry. Q denotes the in-surface plane wavevector of a phonon. The in-sagittal-plane scattering geometry is typi"ed by " , usually with # " "const. (b) The surface Ewald D G G D 1" diagram is shown for the case of elastic scattering. Elastic events are limited to speci"c discrete directions, as indicated, e.g. by I for specular and by I for a possible di!raction peak (after Ref. [32]).
where M is the mass of the projectile. For ease of measurements the He atom scattering experiments are usually carried out with "xed scattering geometry in which the angle between the incoming and outgoing scattered beams in the apparatus, # " , is "xed. The sampling of G D 1" the various scattering angles is then achieved by varying the incident angle relative to the normal G of the sample surface, and the tilt angle between the surface normal and the plane containing the incident and outgoing beams. If the tilt angle is zero, the scattered particles are detected in the sagittal plane which is de"ned by the surface normal and the incoming projectile wavevector. Then, " because in this case the scattered particle wavevector is also con"ned to the sagittal plane. D G An important notion in the terminology of atom}surface scattering is that of specular scattering direction relative to the direction of the incident beam. The polar and azimuthal angles of the specularly scattered beam are the same as for the incident beam, i.e. " and " . The D G D G elastic specular scattering from a planar surface is then characterized by K "K and k "k , D G XD XG and takes place in the sagittal plane. In a general case of atom}surface scattering the change of the projectile parallel momentum K"K !K , D G
(7)
is nonzero and depends on the structural and dynamical properties of the target. The interaction between thermal energy He atoms and the "rst surface layer of atoms is such that they are re#ected either elastically or inelastically without penetrating the surface layer. This is so because the repulsive component of the interaction potential is by far the dominant one at distances of few atomic radii (bohrs) outside the surface (cf. Section 3.2). Due to this, the classical turning points for motion of thermal He atoms are located in the outer region of the surface electronic density. The conditions for conservation of energy and parallel momentum in surface scattering of nonpenetrating particles can be combined to yield a relation between the change of
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the parallel momentum and energy E of the projectile in the collision:
K #K G E"E !E " !k . D G 2M sin G D
(8)
This expression is usually referred to as the scan curve and represents a quadratic function E(K). In elastic scattering from a static periodic surface potential the parallel momentum of the projectile is conserved up to a two-dimensional wavevector G of the reciprocal surface lattice: K"K !K "G . D G
(9)
From the conservation of the projectile energy in elastic collisions, E"0, we have E(K )"E(K ) D G
(10)
and hence the z-component k of the scattered particle wavevector k (G) can be determined as XD D a function of G. For the set of G-values the solutions of Eqs. (9) and (10) give a discrete set of possible k values or directions of diwraction peaks in the angular distribution spectrum of the D scattered particles. The set of possible k values can be obtained from the Ewald construction D schematically illustrated in Fig. 2 (see also Fig. 2 of Ref. [17]). The intensities of these peaks, however, cannot be obtained solely from the kinematic considerations because they depend on the details of the projectile}surface interaction potential. An interesting phenomenon in elastic surface scattering that is closely related to di!raction but also embodies a classical e!ect is the rainbow scattering. The phenomenon of classical surface rainbow pertains to the occurrence of a pair of strong maxima in the classical probability of scattering from a corrugated surface as a function of the scattering angle [36]. A simple visualization of this phenomenon is usually presented in a two-dimensional scattering from one-dimensional surface corrugation. In this case the rainbow maxima appear at the minimum and maximum scattering angles at which the intensity is enhanced due to the coalescence of several classical scattering paths (cf. Fig. 7.4 in Ref. [5] and Fig. 16 in Ref. [59]). Under special conditions the two rainbow maxima may reduce to a single one. Garibaldi et al. [60] have demonstrated and interpreted another, strictly quantum or wave-like feature of rainbow scattering, the so-called supernumerary rainbow, which may occur between the rainbow maximum and the specular beam in the case of strongly corrugated surfaces. The quantum e!ect arises if the phase shift between the trajectories with the same scattering angle but from di!erent impact parameters within the unit cell approaches 2, thus causing an oscillating pattern in which the intense and weak di!raction peaks alternate. These authors have shown within the eikonal approximation for the scattering amplitudes (cf. Section 2.2) that rainbow and di!raction are, in a sense, one and the same phenomenon: the rainbow pattern is the envelope of the di!raction peak intensities. The same conclusion has been arrived at by Berry [61] in a comprehensive study of He atom scattering from a vibrating corrugated hard wall surface, also called a `rippling mirrora. Some novel aspects of kinematical rainbow and focusing e!ects emerging from the singular structure of "nal density of states of scattered particles have been discussed by Miret-ArteH s and Manson [62]. Experimental aspects of surface rainbow scattering, relevant experimental references, and the modelling and interpretation of experimental data have been recently reviewed by FarmH as and Rieder [17].
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As the static projectile}surface potential also supports bound states for particle motion perpendicular to the surface, a special case of elastic scattering may arise if the projectile after exchanging momentum G with the surface ends up with an energy of the perpendicular motion equal to the energy of one of the bound states, , viz., L
k (K #G) G" G ! . L 2M 2M
(11)
Such a process in the entrance channel has been termed by Lennard-Jones and Devonshire as selective adsorption [37,38] because after this event the projectile continues its motion parallel to the surface. Of course, this can take place only for `selecteda values of k and G and in this special G case of elastic scattering the particle is not recorded by the detector which is placed far away from the surface. Without further scattering these processes would lead to minima in the intensity of the specularly re#ected beam. However, the particle during its motion parallel to the surface may further undergo elastic or inelastic scattering. It may again exchange momentum G with the lattice and acquire positive energy of perpendicular motion to leave the surface region. After this selective desorption process the angular distribution of scattered particles would show pronounced variations in the intensity. Alternatively, the particle may exit from the bound state in an inelastic process by exchanging a vibrational quantum (phonon) of "nite energy and momentum with the surface (see below). A reverse process is also possible, by exchanging a vibrational quantum with the surface the particle may "rst be temporarily trapped at the surface with perpendicular energy equal to a bound state energy (0. After this it may make a transition into a "nal state with LY momentum ( K , k '0) by exchanging the reciprocal lattice momentum G in an elastic D XD desorption process. The energy and parallel momentum conservation in the exit channel gives the selective desorption condition:
k (K !G) D D" ! , LY 2M 2M
(12)
which when combined with Eq. (11) yields the condition for elastic selective adsorption/desorption resonance " and no vibrational quanta exchanged with the surface. L LY Another possible resonant process combines selective adsorption into a bound state with energy , inelastic transition from this state into another bound state of energy by exchanging L LY a vibrational quantum with the surface, and "nally a selective desorption process from the surface. Invoking the argument of the larger available phase space for this type of scattering event, the corresponding scattering intensity may be largely enhanced (`supernovaa) relative to the case in which inelastic transitions in the intermediate state are absent [63]. Measurements of the selective adsorption/desorption resonances make possible the experimental determination of bound state energies of the atom}surface potentials and this method has been used in the past for empirical reconstructions of the potential properties. Also, at the early stage of implementing HAS for surface studies, the selective adsorption/desorption processes in combination with the inelastic transitions were proposed as energy and momentum analyzing events because of their selectiveness [39].
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The third type of elastic scattering may occur from the various kinds of defects (random adsorbates or clusters of adsorbates, vacancies, steps, kinks, etc.) present on the surface. Defects introduce disorder in the regular structure of the surface and hence destroy its periodicity and translational invariance. The scattering from defects is incoherent and there is no special condition on the "nal momentum of the scattered particles. Hence, the angular distributions of the scattered particles are di!use and superimposed on the specular peak of correspondingly reduced intensity. In the above descriptions of kinematics of the various scattering events we have somewhat arti"cially separated the elastic from inelastic processes and their possible interference. In reality the scattering events are not `single-hita processes and may encompass any combination of elastic and inelastic scattering. In particular, inelastic scattering processes in HAS may involve any of the types of elastic processes quoted above in combination with the exchange of energy and momentum with the dynamical degrees of freedom characteristic of the surface. Moreover, in the case of inelastic scattering of projectiles with internal degrees of freedom (molecules) the situation becomes even more complex because due to the more complicated projectile}surface interaction all the projectile degrees of freedom (translational, rotational, vibrational, etc.) may participate in the energy and momentum exchange with the heat bath of the target. In such collisions the conversion of energy and momentum between intraparticle degrees of freedom may also take place. Another complication arises in connection with the degrees of freedom constituting the heatbath. In the low, thermal energy scattering regime neutral inert atoms couple most strongly to surface vibrations [40] whereas at higher scattering energies the coupling to electronic charge density #uctuations may open up additional inelastic scattering channels [41}44]. Taking into account all possible scattering channels and multiple scattering processes which may take place during the collision, we observe that the total parallel momentum and energy conservation connect only the initial and "nal scattering states between which the system evolves during the collision event. In this context we say that the full scattering matrix describing evolution of the system is constrained to the parallel (lateral) momentum and energy shell. This is strictly a kinematic requirement stemming from the symmetry properties of the scattering system that selects (projects) the allowed scattering probabilities from the full o!-shell scattering matrix. In the scattering regime in which the multiple scattering amplitudes are nonnegligible, the total scattering matrix is given as a sum of transition amplitudes describing the transitions of the system between the various intermediate states. These transitions involve on-shell as well as o!-shell processes and only the sum of all the single and multiple scattering amplitudes is constrained to the parallel momentum and energy shell. Such sums, and hence the corresponding scattering probabilities, may exhibit strong interference e!ects which are usually referred to as scattering resonances or antiresonances, depending on whether the interference is constructive or destructive. Particularly strong resonances may appear if several scattering processes of either the same or di!erent multiplicities become degenerate with the total energy and parallel momentum conservation embodied in the full scattering matrix. This can manifest itself in the form of maxima in the scattering intensities expressed as functions of the energy and parallel momentum exchange. Such features, depending on the type of the scattering processes involved, are known in the literature as resonance-enhanced scattering and kinematic focusing e!ects. Some of them will be discussed in more detail in Section 3.3.3, and summaries of the various aspects of resonant and focusing processes in atom}surface scattering have been presented by Doak [18] and Miret-ArteH s [64,65].
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In the case of thermal energy HAS from translationally invariant surfaces the experimentally studied inelastic mechanism is the projectile coupling to atomic vibrations propagating in the surface layer. In a periodic structure bounded by a surface the vibrational degrees of freedom can be represented by normal modes or phonons. Some of these modes can be to a larger or smaller extent localized in the surface region. A phonon in such a system is characterized by the quantum number Q, which is the mode wavevector parallel to the surface (restricted to the "rst Brillouin zone of the two-dimensional surface lattice), and a discrete mode index j, which plays the role of the quantum number in the direction perpendicular to the surface. Associated with each mode are the mode frequency Q and the mode polarization vector e (Q, j) pertaining to the vibration of th H G atom from the unit cell basis. The corresponding mode (quasi)momentum and energy are then given by the de Broglie relations Q and Q , respectively. H The kinematic relations corresponding to inelastic phonon-induced projectile transition from an initial continuum state k to a "nal continuum state k are obtained by combining Eq. (8) with G D the relations expressing the conservations of parallel momentum and energy exchanged between the projectile and phonons: K"K !K "G# $Q , D G
(13)
E"E !E " $ Q , H D G
(14)
and
where in the present convention the signs #(!) refer to phonon emission (absorption) processes and Q is taken to be positive. In the case of one-phonon scattering the sums on the RHS of H Eqs. (13) and (14) reduce to single terms $Q and $ Q , respectively. H For in-the-sagittal-plane scattering geometry " with "xed total scattering angle G D " # , which are typical of the majority of TOF experiments, the combination of Eqs. (8), 1" G D (13) and (14) yields in the one-phonon scattering regime the relation
(sin #K/k ) G G !1 , sin( ! ) 1" G
(15)
K"$Q#G"k sin( ! )!k sin . D 1" G G G
(16)
E"$ Q "E H G where
The expression on the RHS of Eq. (15) is the scan curve for one-phonon scattering which is a parabolic function of the magnitude of the exchanged wavevector K. Its point of intersection with the phonon dispersion curve E"$ K plotted as a function of K in the (E, K) plane H gives the possible values of energy and momentum of the phonon mode (Q, j) which can be exchanged in the sagittal plane scattering and thus observed as a peak in the experimental TOF spectrum for "xed ( , , E ). This is illustrated in Fig. 3. Systematic plotting of these points for the G 1" G various values of K and K enables an experimental determination of the phonon dispersion H
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Fig. 3. Extended zone diagram showing scan curves for various and for "903 and in-sagittal-plane scattering G 1" along the 1 0 0 direction of LiF(1 0 0) surface with k "6.0 As . The heavy solid lines show the Rayleigh dispersion G curves in the sine approximation (after Ref. [100]).
curves provided a particular phonon peak can be traced as a distinct spectral feature across the SBZ. However, like the di!raction case, the peak intensity or the probability of such a phonon exchange event cannot be determined only from the kinematic considerations because it depends on the details of the projectile}surface interaction. In the case of out-of-sagittal-plane scattering the expression relating the change of the projectile energy with the change of its parallel momentum becomes more complex [45] and will not be elaborated here. The kinematics of phonon-mediated inelastic scattering from continuum states into bound states of the projectile}surface interaction potential, or the prompt sticking processes, can be formulated analogously as for the inelastic scattering into the continuum states described above. As such processes do not directly contribute to the measured intensities of the HAS TOF spectra (except for the modi"cation of the on-the-energy-shell Debye}Waller factor common to all the phonon exchange probabilities in a particular TOF spectrum), they will be discussed separately in Section 5.2.5. Inelastic atom scattering from defects on surfaces embodies, in addition to all the complexities of elastic scattering from defects, also the new features brought about by the inelastic e!ects. First, additional degrees of freedom associated with the defects themselves may occur (the typical examples of which are the vibrations, translations and rotations of isolated adsorbates), and second, the defects represent localized scattering centers which cause the breakdown of lateral symmetry of the surface and thereby also act as the sources of parallel momentum for the scattered particles. Hence, there is no simple relation connecting the change of energy and momentum in atom scattering from surfaces with defects and the only constant of motion characteristic of the entire scattering system is its total energy. However, even in such circumstances HAS experiments can provide extremely useful information on the structural and dynamic properties of surfaces covered with adsorbates, clusters and adlayers [14].
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2.2. Investigations of the structural properties of ordered surfaces and adlayers by thermal energy atomic and molecular beams The "rst production of an e!usive molecular beam had been demonstrated by Dunoyer [46] in 1911 and the applications of the beam methods have been developing ever since [47]. The "rst use of light atom and molecule beam scattering in the investigations of structural properties of surfaces was connected with the demonstration of the wave nature of atomic particles in the experiments of Stern and collaborators [6] in the early 1930s. They scattered He and H beams of thermal energy from (0 0 1) surfaces of NaCl, LiF and NaF and observed angular scattering distributions which could be interpreted as di!raction of matter waves from a periodic surface structure. The wavelength associated with these waves was found to be given by the de Broglie relation h h " , (17)
" p (2ME G G where M, E and p stand for the mass, the incoming energy and momentum of the gas particles, G G respectively, p" k and h"2 is the Planck constant. Analogous di!raction results were obtained by Johnson [7] in the scattering of atomic hydrogen from LiF(0 0 1) surface and thereby these works also provided the "rst demonstration of quantum regime of motion of atoms and molecules. The "rst review of the early atomic and molecular di!raction experiments was presented by Frisch and Stern [48]. Since then many new reviews have been published following the development of the atomic particle di!raction techniques and their applications to structural studies of the various surfaces and adlayers [8,10,17,49,50]. The earliest measurements of Stern and collaborators [6] and Johnson [7] have stimulated theoretical developments with the aim of interpreting the di!raction maxima and minima observed in the experimental spectra. The position of the di!raction maxima could be relatively simply explained using the kinematic arguments (9) and (10), yielding k (G)"k!K #G'0 , (18) XD G G which for in-sagittal-plane scattering can be obtained from a two-dimensional Ewald construction (cf. Fig. 2 and also Fig. 2 in Ref. [17]). On the other hand, the mechanism leading to the observed losses in di!raction intensity at certain initial scattering conditions was proposed a little later by Lennard-Jones and Devonshire [37,38] within "rst-order perturbation theory. They introduced the notion of selective adsorption processes in which the projectile can make a transition into a bound state of the projectile}surface potential if at a certain incoming angle, and energy and for a particular G the expression on the RHS of Eq. (18) becomes negative and equal to 2M where L (0 is the bound state energy (cf. Eq. (11). These conditions of bound state resonances found L extensive use in the determination of bound state energies of atom}surface potentials (cf. Section 3.2). Although we know today that "rst-order theory is inadequate for calculating di!racted beam intensities at bound state resonances (cf. Ref. [50]), the physical concepts of the Cambridge school are still valid and much of their work can be appropriately extended in the current treatments of more complicated processes involving inelastic scattering. After this early period of development of quantum atom}surface scattering theory in the 1930s, not much progress has been made until the 1970s when Cabrera et al. [52] proposed the scattering
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of He atoms as a convenient technique for studies of surfaces. This period was also marked by new experimental advances in achieving UHV conditions, the application of surface analytical tools and the development of intense monoenergetic atomic and molecular beam sources. The latter are based on the expansion of a gas from a source chamber at stagnation temperature ¹ through a supersonic nozzle. During isentropic expansion through the nozzle, the enthalpy of the gas prior to expansion is converted into the energy of one-directional motion of particles [51]. For a monoatomic gas like He this produces a beam with kinetic energy E k ¹ and very G low-velocity spread v/v40.001. These technical developments opened up the possibility of carrying out the atom}surface scattering experiments with much improved resolution and on much better prepared and de"ned samples whose structure could be controlled on the atomic scale. This has resulted in the accumulation of experimental data on light atom}surface scattering of very high quality. The need for their interpretation gave a strong impetus to the development of adequate theories going beyond the "rst-order perturbation approach of Lennard-Jones and Devonshire [22]. These new developments have been reviewed many times since, and here we shall mention only the methods introduced in the calculation of di!raction and bound state resonance e!ects which also proved relevant to the treatment of inelastic atom}surface scattering. The close coupling formalism (CCF) [53] gives, in principle, an exact and general solution to the stationary SchroK dinger equation for a static corrugated projectile}surface potential ;(r) [54]. The potential which exhibits the periodicity of the substrate crystal structure is expanded into a twodimensional Fourier transform: ;(r)" ;G (z)exp(iGR) ,
(19)
G
where r"(R, z) denotes the radius vector of the projectile. The solution for the projectile wavefunction is sought in the Bloch form K (r)" exp[i(K#G)R]G (z) . Y G Y
(20)
Substitution of expressions (19) and (20) into the stationary SchroK dinger equation yields a set of coupled ordinary di!erential equations
2M d #k (G) G (z)! ;G G (z)G (z)"0 , \ Y Y X
G dz Y
(21)
where k (G)"k!(K#G) and E"( k )/2M with k "(K, k ). This set should be solved by X G G G X obeying the boundary conditions for the outgoing scattered waves [8,29,78] given by lim G (zPR)" G 0 exp(!ik z)#AG exp[ik (G)z] GX X
(22)
for the open Bragg channels k(G)50, and X lim F (zPR)"AF exp[! (F)z] X
(23)
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for the evanescent waves with (F)"(!k(F) where k(F)(0. The di!raction probabilities X X X PG are then obtained from the di!raction amplitudes AG : k (G) PG " X AG , k GX which ful"ll the unitarity condition expressing the conservation of total current:
(24)
PG "1 . (25) G IX In practical calculations the in"nite set of G vectors is truncated to a set of size N which is large enough to ensure the desired numerical accuracy of the algorithm that is used in solving the resultant set of N di!erential equations. Thus, for instance, this formalism treats exactly and therefore on an equivalent footing the rainbow scattering and di!raction. With the advent of "rst results on molecular beam di!raction from solid surfaces, and in particular from the scattering of H and He beams from Xe adlayers on graphite surface [55], and He beams from LiF(0 0 1) surface [57], which exhibit strong corrugation, the close coupling method has been used with success to calculate the di!raction intensities using the potentials obtained from pair summation of projectile}surface atom gas-phase potentials [56,57]. However, as the close coupling procedure is in general very time consuming, many approximate methods have been developed that can much faster yield information on the surface corrugation for comparison with experiment. Some of these methods are outlined below and for a review see e.g. Refs. [17,29]. The CCGM-method [52] is based on setting up an exact ¹-matrix formalism for the description of projectile scattering by a periodic atom}surface potential and introducing an e!ective decoupling scheme by retaining only the imaginary parts of the Green's functions describing projectile propagation in the intermediate states. This yields the transition probabilities or di!raction intensities in the form of a `unitarized Born approximationa and thus corrects the de"ciencies and shortcomings of "rst-order distorted wave Born approximation (FODWBA) used by LennardJones and coworkers [22]. The e!ect of surface corrugation is usually expressed or described in terms of the corrugation function G
(R)" G exp(iGR)
(26)
G
which is de"ned as the locus of classical turning points for projectile particle motion in the surface potential for a given incident energy. The determination of this function for di!erent combinations of the projectile}surface interactions has been the subject of many experimental and theoretical studies based on the various scattering models and levels of approximation. One such frequently used model in the development of di!raction theory in the 1970s was based on the assumption of a hard corrugated wall (HCW), often in conjunction with the so-called Beeby correction [66] (see below). In this approach it is assumed that the scattering potential may be represented by
;(R, z)"
0,
z' (R) ,
R,
z( (R) .
(27)
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This assumption was "rst put forward by Lord Rayleigh [58] over a century ago in the treatment of re#ection of sound waves from corrugated surfaces. Analogous quantum theory of wavefunction re#ection from surfaces was presented by Garibaldi et al. [60]. Of course, the neglect of the potential well, which may be a good approximation only at higher projectile incident energies, eliminates all the e!ects of selective adsorption. Some e!ects of the well depth D can be accounted for by introducing the e!ective incident energy of normal motion E "E #D (Beeby correction X X [66]). Several methods of increasing complexity have been developed to yield the solutions to the ensuing scattering equations (see discussion in Ref. [17], Section 2). A relatively simple method for calculating the di!raction intensities within the HCW approximation is the so-called GR-method developed by GarcmH a [67] which, provided the unitarity of solutions is obeyed, gives a very good agreement with more complex theories. Another method introduced in the early calculations of di!raction intensities and the explanations of quantum surface rainbow e!ects is based on the eikonal approximation [28,60]. In this approach the di!raction amplitudes AG are expressed as Fourier transforms of a phase factor involving the corrugation function:
1 exp!iGR!i[k (G)#k ] (R) dR , AG "! X GX A
(28)
where A is the area of the surface unit cell. It is a `single-hita approximation which can be easily implemented but its validity is restricted to weakly corrugated systems which can be su$ciently well described only by small G vectors. Its main de"ciency is that it does not satisfy the unitarity condition, although improvements in this regard are possible, and that it cannot discriminate between (R) and ! (R) in the data analyses for surfaces with mirror symmetry plane. One of the merits of the eikonal approximation lies in the possibility of its extension to the treatments of inelastic atom}surface scattering events. A simpler form of the eikonal approximation is the sudden approximation in which the perpendicular component of the scattered projectile wavevector, k (G), X is approximated in the high-energy limit by the value of the incident perpendicular wavevector k . XG This gives the asymptotic form of the scattered projectile wavefunction which di!ers from the re#ected wavefuntion only by a phase factor depending on the lateral coordinate R. Ultimately, this phase factor can then be calculated using the WKB approximation. Recently, the R-matrix propagation technique for solving close coupled equations [68] has been applied to model di!raction of He atoms from clean NaCl(0 0 1) surface and the same surface covered by a commensurate square overlayer of Kr atoms [69]. A good agreement with experimental results [70] was obtained although the Debye}Waller e!ects associated with overlayer phonons were apparently underestimated in that work. Instead of treating the di!raction problem as a solution of the stationary SchroK dinger equation, one can exploit the time dependence of the latter and treat the scattering problem as a wavepacket propagation in space and time. Using this approach Koslo! et al. [71] have developed an e$cient numerical method that can also be extended to inelastic scattering problems. The various aspects of the time-dependent SchroK dinger equation approach to treating the problem of atom}surface scattering have been reviewed by Gerber [41], Darling and Holloway [72] and De Pristo [73]. A semiclassical method of propagating localized Gaussian wavepackets in the interaction potential approximated at every point by second-order Taylor expansion (harmonic form of the
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potential preserves the Gaussian shape of the wavepacket along the classical trajectory) has been proposed by Drolshagen and Heller [75] and applied to a number of problems because the method does not depend on the periodicity of the potential. A further development of the Gaussian wavepacket approach presented by Varga [76] is based on the application of the split operator method to the evolution operator for a three-dimensional time-dependent SchroK dinger equation. Finally, the path integral method, in which the scattering amplitude is given by summing up the exponentiated action over classically allowed and forbidden paths, has also been applied to the surface problem (see Ref. [29] for more details). However, the approximations made to reduce the number of paths in order to make the calculations tractable usually lead to the paths that are classically allowed. Less stringent approximations lead to eikonal approximation or the related ones. Hence, the usefulness of this method is in the possibility to lead to approximations at various levels, of which the classical trajectory approximation is an illustrative example. Related to the path integral approach is the de Broglie}Bohm formalism in which the collision dynamics is described by well-de"ned quantum trajectories deterministically governed by a `quantuma Hamilton}Jacobi equation. The application of this formalism to study He atom di!raction from Cu(0 0 1) surfaces has given results in good agreement with the quantum wavepacket theories [77]. The results from structural studies of clean surfaces by He, Ne and H beam scattering available since the 1970s and the analyses of the data using the methods quoted above have been hitherto reviewed several times [8,10,17,49,50,78,79]. These studies provided information on the corrugation amplitude as probed by the projectile particles and on the depth of the well of the projectile}surface potential. A number of ordered adlayer systems have also been investigated by HAS and their structures analyzed using the various methods referred to above. An updated review and exhaustive tables of these results have been presented by FarmH as and Rieder [17]. 2.3. Investigations of the structural properties of disordered surfaces and adlayers by thermal energy atom scattering Thermal energy atom scattering (TEAS), and in particular He atom scattering (HAS), have also found broad applications in the investigations of structural properties of disordered and rough surfaces [14,84] and in the studies of surface growth [14,17], roughening and surface phase transformations [82,84]. As the order and translational invariance of surfaces in such systems are no longer preserved the scattering is to a larger or smaller extent incoherent due to the presence of irregularities which act as localized sources of momentum for the scattered particles. This leads to the appearance of a di!use component in the angular distribution of the scattered particles and hence to a reduction of intensity of the specular beam. Monitoring the specular beam intensity and its variation with the parameters characterizing the scattering process is then exploited as a method for studying the structure of disordered surfaces. Surfaces can exhibit various kinds of disorder or partial order of di!erent dimensionality. Isolated point defects on otherwise ordered surfaces, such as adatoms and vacancies, can be considered as being zero dimensional, whereas line defects such as step edges are one dimensional. Examples of two-dimensional defects are terraces on monocrystal surfaces and of the threedimensional ones are big adsorbed clusters.
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The studies of point defects by TEAS is based on the large total elastic cross sections which such localized scattering centers exhibit in HAS and which many times exceed the values that would be obtained from the corresponding van der Waals radii. The di!use scattering from isolated point defects with large cross sections causes the correspondingly strong reduction of the specular beam intensity. This reduction can be measured as a function of the projectile}defect interaction potentials, scattering conditions, concentration of defects, etc. A number of systems with point defects (mainly adsorbates) have been investigated by TEAS and the results of these studies are available in several reviews [14,17]. Line- or one-dimensional defects like step edges, sharp grain boundaries and edges of adsorbate islands can a!ect light atom scattering in several ways. For instance, step edges of "nite width give rise to azimuthally dependent di!use scattering. The phase shifts between the projectile wavefunctions scattered from the upper and lower terraces of a step or adlayer edge [85] will give rise to interference patterns in the intensity of the specularly re#ected beam. Both e!ects will be a function of the step density. The same mechanisms apply also to scattering from grain boundaries and adsorbate islands. Studies of these e!ects have been reviewed by Poelsema and Comsa [14], Lahee and WoK ll [81], Lapujoulade [82] and FarmH as and Rieder [17]. He atom scattering has also been extensively and successively used in the structural and growth studies of two- and three-dimensional imperfections on surfaces like "nite-size terraces, adsorbate islands, hillocks, clusters, etc. The literature on the results of these studies is abundant and recent reviews [14,17,82] are helpful guidelines to original works. An exhaustive account of HAS studies of cluster deposition on well-de"ned surfaces has been presented by Vandoni [83]. A large number of studies of roughening, reconstructions and phase transformations at surfaces carried out by HAS have been reviewed by Poelsema and Comsa [14], Lapujoulade [82] and FarmH as and Rieder [17]. 2.4. Investigations of the dynamical properties of surfaces, adlayers and adsorbates by noble gas atom scattering The conventional spectroscopic techniques for studies of structure and low-energy dynamics in the bulk, such as neutron, X-ray or Rahman scattering, are rather insensitive to surfaces. On the other hand, the characteristics of light gas atom and speci"cally of He atom scattering, i.e. the strict surface sensitivity and high-energy resolution achieved with these beams, make them best suited to the study of low-energy dynamics at surfaces, such as surface and adsorbate vibrations and di!usion processes. In particular, in the range of energy exchange of a few tens of millielectron volts no other spectroscopy can match inelastic HAS in providing the information on low-energy surface dynamics. It is worth noting that by the early 1970s there existed ideas and plenty of theoretical material on the nature and dispersion of surface phonons in di!erent types of crystals (cf. reviews by Maradudin and Stegeman [86], Benedek and Miglio [87] and de Wette [88]), but no experimental evidence demonstrating that behavior. In this respect the theoretical studies of surface phonons preceded the experimental ones. As the theoretical demonstration of the possibility of utilizing He atom beams in surface studies and single-phonon detection, which was put forth in the works of Cabrera, Celli, Goodman and Manson [52,89,90], also coincided with the new developments in UHV and surface science technology, this gave a strong impetus to the use of HAS in surface
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phonon spectroscopy in the 1970s and 1980s. The "rst studies of surface phonons stimulated by theoretical developments were carried out in exploratory experiments by several groups [91}93], but these early works were hindered by inadequate beam velocity resolution and detection sensitivity. The "rst high-resolution measurements were made possible by the discovery of production of highly monochromatic nozzle beams [94,97] in which the velocity spread of He atoms was well under 1%, about a factor 5}10 times better than that in the earlier experiments. The high intensity and other properties of the beams obtained in free-jet expansion [95,96] enabled their use in TOF measurements required for resolving inelastic transitions in the collisions of beam atoms with the target. This has opened up the possibility of measuring the characteristics of surface dynamical processes with an unprecedented precision, as it was "rst demonstrated in 1981 by Brusdeylins et al. [99] and Doak [100] in the detection of phonons characteristic of LiF(1 0 0) crystal surface. The "rst inelastic HAS studies have concentrated on surfaces of insulators, mostly alkali-halides, because for these crystals the interatomic forces were best understood theoretically and because it was easier to prepare the clean insulator surfaces free of defects or contaminants than in the case of metals. These experiments showed, "rst, the remarkable potentiality of inelastic HAS in detecting surface phonons, and second, the ability of the theory to provide interpretation of the data recorded in the one-phonon scattering regime. However, the full potentiality of inelastic HAS in detecting surface phonons was demonstrated by extending its application also to semiconductor surfaces, metal surfaces, and surfaces covered with adsorbates (for an exhaustive list of earlier references see Table 5.5 in Ref. [18]). Since that time the HAS technique has been further developed and improved with the emphasis on increasing the resolution and signal intensity. Thus, already by the early 1980s more than a dozen of HAS apparatuses (not all of them using TOF for energy resolution) had been put in operation in the various laboratories (see Table 1 in Ref. [32]). The various experimental and technical aspects of the utilization of HAS TOF technique in the studies of phonon dispersion curves characteristic of clean #at surfaces have been discussed in a comprehensive review by Doak [18]. Systematic HAS TOF experiments carried out with these apparatuses have produced a large amount of data on the vibrational properties of clean single-crystal surfaces, adsorbates and adlayers. In these measurements attention has been focused mainly on obtaining the information on phonon dispersion and excitation intensities in the single-phonon scattering regime, and on the ubiquitous Debye}Waller factors. The capability of HAS for studying phonons characteristic of stepped surfaces, in which case the demand on experiment is severe, has also been recently demonstrated [98]. A signi"cant amount of the collected material and database has already been reviewed with references to original works in a number of publications. Early reviews by Toennies and coworkers summarize the work on phonon dynamics of insulating surfaces [12], noble metals [101] and transition metals [19,34,102], and more recent ones of layered materials and structures [103,104] and adsorbates [105]. HAS has also been successfully utilized in the studies of phonon dynamics of epitaxially grown thin metal "lms [106]. Tables with exhaustive lists of references to HAS studies of surface phonons carried out till the early 1990s have been presented by Toennies [34] and Doak [18]. A comprehensive review of all the available surface phonon data from HAS experiments is in preparation [20]. Applications of atom}surface scattering theory to the various systems investigated by the beginning of the 1980s were described by Celli [27] and by the mid-1980s in a comprehensive review by Bortolani and Levi [28]. Theoretical interpretations of the phonon dispersion curves for
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the low-index surfaces of Al, noble metals and some transition metals as obtained by HAS and EELS (for EELS studies of surface phonons see subsection below) have been reviewed by Santoro and Bortolani [30]. The application of HAS in investigations of surface phonon anomalies has been brie#y reviewed by Hulpke [107]. A short review of the genesis of HAS and its achievements is also available [108]. The use of thermal energy Ne atom scattering in investigations of dynamical surface properties has been described in Refs. [109,110]. In recent years HAS has also been employed in the studies of multiphonon processes and the corresponding Debye}Waller factors (which will be the main subject of subsequent sections), as well as of the processes of surface di!usion [111] for which theoretical descriptions in the quasiclassical scattering limit had been developed earlier [112]. Both these aspects are rather challenging from the conceptual point of view because the theoretical framework needed for their interpretation goes beyond the one developed to study single-phonon excitations. Since the multiphonon He atom scattering theory has now entered a mature stage the present review is speci"cally devoted to these new developments and their applications to the systems and processes of current interest. 2.5. Comparison with other techniques HAS is only one of the spectroscopic techniques that is at present used in the investigations of structural and dynamical properties of surfaces. Therefore, it is useful to compare it with complementary techniques in order to assess the situations and scattering regimes in which its advantageous properties over the other spectroscopies can be optimally and fully exploited. The technique of low-energy electron di!raction (LEED) and its di!use counterpart (DLEED) have become the standard laboratory tools in surface science research. They utilize electron scattering [113}116] and operate on the same basic principles as HAS. Electron beam energies from a few tens to about 100 V are commonly used in experiments since they give rise to the electron de Broglie wavelengths like those characteristic of thermal energy HAS and close to the values of crystal lattice constants. The scattered electrons produce two-dimensional images on a #uorescent screen and thus provide two-dimensional "ngerprints of the surface structure in the reciprocal space. This immediate availability of information on the structure in the reciprocal space is a de"nite advantage of LEED over HAS because in a single run HAS can provide structural information only in one direction. However, the sensitivity and resolution of HAS is much higher, in particular when it comes to low concentrations of defects on the surface. Another advantage of HAS over electron scattering spectroscopies of surfaces is the absolute surface sensitivity, backscattered electrons usually carry information on several surface layers of atoms and therefore the interpretation of the LEED intensities is more involved. Scanning tunneling microscopy (STM) has developed into one of the most powerful and popular techniques for investigations of the local surface structure. This method yields information on the surface structure directly in the real space and at present can be operated already on the level of atomic resolution. Due to this, STM has proved an ideal probe for obtaining information on the presence and morphology of surface defects, like steps, kinks, islands, adsorbates, small clusters, etc. The limitations of STM spectroscopy are in that it can be applied only to conducting surfaces and that the size of the scanned area is relatively small. Moreover, although atomic resolution can be achieved and the presence of single atoms on surfaces detected, the STM may not be able to
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distinguish a vacancy from a substitutional impurity. Also, as the STM imaging is a slow process, di!usion of light atoms on surfaces at higher temperatures can considerably blur and invalidate the STM pictures. On the other hand, the HAS data collected under the same conditions may contain indispensable information on surface dynamics. Thus, STM and HAS can be considered as truly complementary techniques for obtaining valuable information on the various aspects of surface structure. Complementary techniques to inelastic HAS are optical spectroscopies of phonons and electron energy loss spectroscopy (EELS). Optical spectroscopies can probe only the long-wavelength vibrations (phonons of zero wavevector) and as such cannot provide information on their dispersion. On the other hand, EELS operates on the same physical basis as inelastic HAS [113] and therefore the same kinematic relations should, in principle, be used in quick assessments of the data. However, due caution is needed here because electrons can be inelastically backscattered from subsurface layers, picking up information on the symmetry of the target and of the scattering events from those regions as well. Hence, the EELS spectra can contain additional information relative to HAS, both with respect to the local (spatial) phonon density of states and with respect to the multiple character of the scattering event, because the likelihood of the latter is higher in EELS. Thus, from the viewpoint of complementariness and completeness it is desirable that both techniques be used in the investigations of phonon dynamics of the same system, technical conditions permitting, and the two sets of results then be compared against each other. However, in making such comparisons and concomitant interpretations, one should take into account that surface phonon excitation intensities depend on the matrix elements of projectile}phonon interaction which are generally di!erent for the two spectroscopies. Hence, the same phonon mode can exhibit di!erent intensity in HAS and EELS even for the same intersection point (Q, Q ) of the H scan curve with the dispersion curve (see Section 6). The techniques listed in this subsection exhibit great potentiality for studying either structural or dynamical properties (and in some cases both) of the various types of surfaces. However, in the limits of low energy of surface dynamical processes (few meV), or low concentration of surface defects whose properties are studied, the HAS technique is unrivaled by virtue of its resolution and sensitivity. Recent development of the HAS techniques in the direction of producing intense focused He atom beams, with the desire to achieve conditions for `atom spectroscopya [117,118], is expected to provide additional impetus for the use of HAS as an analytical technique in the investigations of surfaces.
3. Interactions and inelastic scattering of atoms from surface vibrations. Short overview of the achievements and shortcomings of standard theoretical descriptions 3.1. Descriptions of the vibrational dynamics of surfaces and adlayers Vibrational normal modes or phonons in bulk crystals have been investigated by the various experimental techniques, of which the thermal energy neutron scattering has been in extensive use for this purpose since the 1950s [119]. This method provides the most complete information on the dispersion of phonon modes in crystals through the entire "rst Brillouin zone of the reciprocal
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lattice. However, measurements of the characteristics of the modes localized at surfaces were not possible before the advent of adequate surface-sensitive techniques, in particular of EELS and HAS. Creation of a free surface introduces a breakdown of crystal symmetry and tends to lower the normal mode frequencies and produce a class of modes called surface modes for which the displacement amplitudes are large at the surface and decrease inward the crystal. The "rst study of surface-localized modes dates back more than one century ago when Lord Rayleigh [58] demonstrated that the long-wave component of earthquakes is due to sagittal surface waves propagating in an elastic continuum. Born and von KaH rmaH n presented as early as 1912 the "rst microscopic theory of bulk phonons based on classical mechanics [120]. Surface lattice dynamics was pioneered in the 1940s by the theoretical work of Lifshitz and Rozenzweig [121] who employed the Green's function approach to study phonons in semiin"nite media. This work was continued by Wallis in 1950s [122], Wallis and coworkers [123] and Feuchtwang [124] in the 1960s, and Garcia-Moliner [125] and Armand [126] in the 1970s. The progress made in the studies of lattice dynamics by the Green's function method was reviewed by Maradudin et al. [127]. In 1971 de Wette and coworkers [128] presented their method for calculating bulk and surface phonon dispersion curves and mode polarizations in the quasiharmonic approximation starting from a dynamical matrix of a "nite slab of atoms modelling a thermally expanded lattice. The approach has been "rst applied to alkali crystals by adjusting the force constants constituting the dynamical matrix so as to reproduce well the bulk phonon data available from neutron scattering experiments. Benedek [129] then applied the Green's function method to the same alkali crystals studied by de Wette and obtained similar results, which at that time was very advantageous in view of numerical convergence requirements. Black et al. [130] applied the continuous fraction method to calculate the spectral densities of phonons localized on surface atoms and Black [131] extended this method to adsorbates. The Green's function approach also proved very convenient and useful in the studies of surface phonon lifetimes and vibrational characteristics of disordered adlayers and isolated adsorbates [132]. Lifetime broadening of the adlayer modes due to their coupling to the vibrations of the substrate described in a continuum model has been studied in the quasiharmonic approximation by Hall et al. [133]. Anharmonic linewidth broadening of phonons localized at surfaces [134] and in adlayers [135] have been studied beyond the quasiharmonic approximation in the dynamical matrix approach by using perturbation theory. Recently, the multipole expansion method [136] has been used to study phonon anomalies in metals [137]. Following these applications Benedek and coworkers have extended the electron pseudocharge multipole expansion method to construct an e!ective dynamical matrix for the surface vibrations of Cu(1 1 1) [138] and Cu(1 0 0) [139]. A large amount of the work on surface phonons using the methods referred to above has been reviewed by Wallis [140], Maradudin and Mills [141], Maradudin et al. [15], Benedek et al. [16], Bortolani and Levi [28], Santoro and Bortolani [30], Wallis [142], Benedek et al. [143], and Toennies and Benedek [20]. More recently, "rst principles approaches have been employed to obtain information on the interatomic forces and thereby on the vibrational degrees of freedom of atoms in surface layers of selected systems [148}156]. First reviews on the results of these calculations are already available [157}160], and all these results have been systematized and reviewed by Toennies and Benedek [20]. It is expected that in the near future these approaches are going to be extended to a number of
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other systems of interest and thereby provide the much sought for ab initio type of information on the dynamical properties of surfaces. The molecular dynamics approach [161] (MD) for studying crystal vibrations is based on solving numerically the classical and ab initio MD equations of motion [162] for crystal atoms subject to the total interaction potential without expanding it into a power series in small ionic displacements and retaining only the lowest order terms, as is done in the lattice dynamics approach (cf. Eq. (30) below). Of course, thereby the anharmonicity is automatically taken into account which means that at "nite temperatures the method is capable of describing the e!ects associated with anharmonicity of the interatomic potentials, of which the thermal expansion and phase transformations are typical examples. The MD approach has been applied to study the temperature dependence of phonon spectral densities, atomic mean square displacements and interlayer separations of model crystals [163], clean Si surfaces [164], surfaces of W(0 0 1) [165], Cu(1 1 0) [166], Ni(1 0 0) and Ni(1 1 0) [167], Al, Ni and Cu [168], Mo(0 0 1) [169], and Ag(1 0 0), Cu(1 0 0) and Ag(1 1 1) [170], as well as of surfaces with adsorbed layers like Xe/graphite [171] and N /graphite [172] by using diverse forms of the interaction potentials. In many cases of interest good agreement has been found between the MD and lattice dynamics results at substrate temperatures su$ciently below the onset of anharmonic e!ects causing disordering, etc. In this subsection we shall outline in some detail the approach for calculating phonon dispersion curves and polarization vectors based on the slab method and quasiharmonic approximation which reduces the problem to "nding the eigenvalues and eigenvectors of the dynamical matrix of the system. The reason for doing so is, "rst, that this is now a standard and quick method for assessment of these quantities and, second, because we shall extensively use it in the forthcoming sections to calculate the inelastic HAS intensities. In the slab method the crystal is modelled by a slab which is bounded by surfaces at, say, z"0 and !¸ and thus exhibits a two-dimensional periodicity only in the direction parallel to the X surfaces. The unit cell of this crystal extends through the entire thickness ¸ of the slab and X therefore the crystal is generated by applying the operations of a 2D translation group to the so-de"ned large unit cell with a basis. The cross section of a particular crystal plane with the unit cell forms a 2D unit cell in that plane and may generally contain di!erent atoms for di!erent planes. The instantaneous position of an atom in the slab is given by rl "rl #ul , G G G
(29)
where rl "(Rl #R , z ) gives the equilibrium position of an atom in the crystal and ul denotes G G G G its time-dependent displacement from equilibrium. The vector Rl is associated with the lth unit cell in the slab and (R , z ) is the basis vector in the cell which gives the position of the th atom in the G G layer "xed by the perpendicular coordinate z . With N being the number of unit cells and A the G area of the 2D cell in the surface plane, the crystal surface area is given by ¸"NA . Next, one assumes that the total potential energy of the slab crystal is a function of the atom positions and can be expanded in the Taylor series in powers of displacements 1 #2 , " # (l, )ul # (l, ; l, )ul ul G? 2 l l G? GY@ ? ?@ l G? G? YGY?Y
(30)
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where and denote the Cartesian components x, y, and z and
(l, )" ? ul G?
(31)
and
(l, ; l, )" ?@ ul ul G? GY@
(32)
are the force coe$cients calculated with atoms at their equilibrium positions (i.e. the potential minima) at which (l, )"0. They satisfy a number of symmetry relations which were discussed ? e.g. in Ref. [128]. In the quasiharmonic approximation one neglects all the terms except the second one on the RHS of Eq. (30). This yields the equations of motion for coupled harmonic oscillators: d , M u "! (l, ; l, )ul GY@ G dt lG? ?@ l YGY@
(33)
where M is the mass of the th particle in the unit cell with a basis. These equations can be solved G by exploiting a 2D translational invariance of the crystal due to which derivatives (32) depend only on the di!erence (l!l) in the directions parallel to the surface. The translational invariance leads to periodic solutions in terms of normal modes in the Bloch form 1
e (Q, j)exp[iQ(Rl #R )] . G (M G G
(34)
Here the quantum numbers of normal modes are the parallel wavevector Q, which is restricted to the "rst surface Brillouin zone of the reciprocal 2D lattice characterized by the reciprocal lattice vectors G, and a discrete index j which plays the role of the third quantum number (branch index) for 3D vibrations in the slab. The polarization vector e (Q, j)"(e (Q, j), e (Q, j), e (Q, j)) G GV GW GX
(35)
denotes the three-dimensional polarization of vibration of the th atom associated with the jth normal mode. Substitution of such periodic solutions into Eq. (33) gives rise to a set of coupled linear algebraic equations: D (Q,,)e (Q, j)"(Q, j)e (Q, j) , ?@ GY@ G? @GY
(36)
in which ,"x, y, z and the elements of the dynamical matrix are given by 1 exp[iQ(R #R !Rl !R )] (l, ; 0, ) . D (Q, , )" G GY ?@ ?@ (M M l G GY
(37)
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The solutions of the secular equation of (36) yield phonon eigenfrequencies Q and polarization H eigenvectors which then satisfy the orthonormality condition within the unit cell with a basis e (Q, j)e (Q, j)" . (38) G G HHY G In some cases it may occur that the symmetry of the crystal induces decoupling of the eigenvalue equation (36) for the slab modes. This leads to a division of the modes into two classes with the members of one class being orthogonal to those of the other class [128]. One then speaks of partitioning of the modes into mutually orthogonal classes. Moreover, if the surface has a complete inversion symmetry with respect to a plane perpendicular to it, then for Q parallel to this plane there will be a partitioning of modes into two mutually orthogonal classes. Two-thirds of the modes will belong to the class with polarization vector strictly in the sagittal plane, and one-third will belong to the second class with pure shear horizontal (SH) polarization, i.e. their polarization vectors will be orthogonal to the sagittal plane. These properties of the modes are commonly used in the analyses of the HAS data recorded along the high-symmetry directions of the crystal surfaces (see Sections 5 and 6). The crystal atoms displacements can be expressed in terms of quantized normal modes satisfying the Bloch condition (34) and termed phonons:
e (Q, j)exp[iQ(R #R )](aQ #aR!Q ) , (39) ul " G J G G Q 2NM Q H H G H H where aQ and aRQ are the usual phonon annihilation and creation operators, respectively, which H H satisfy boson-type commutation rules: (40) [aQ , aRQ ]" Q Q . H HY Y HHY The Hamiltonian of the unperturbed phonons is expressed in terms of phonon operators as H" Q (aRQ aQ #) . (41) H H H Q H The time dependence of ul enters through the time dependence of the phonon operators G QH aQ (t)"aQ e\ S R (42) H H and (43) aRQ (t)"aRQ e SQH R . H H A phonon system satis"es Bose}Einstein statistics and at substrate temperature ¹ the distribu tion of phonons in the state described by the quantum numbers (Q, j) is given by 1 , [aRQ aQ \"n(Q )" H H H exp( Q /k¹ )!1 H where [...\ denotes the thermal average.
(44)
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Fig. 4. Calculated phonon dispersion curves as a function of the two-dimensional wavevector Q along the boundary M PKM PM M PM of the irreducible part of the "rst surface Brillouin zone (SBZ) of an 80-layer fcc Xe slab with (1 1 1) surfaces [146]. For force constants see Table 1.
Table 1 The values of the radial, , and tangential, , force constants for the "rst (1) and second (2) nearest neighbors used in the analysis of phonon dispersion curves in the fcc xenon crystal bounded by the (1 1 1) surface. The values were obtained from the HFD-B2 potential [147] using the experimental interatomic distance of a"4.37 As obtained for substrate temperature ¹ "40 K Force constant (Xe(1 1 1) system)
Value (N/m)
6}6 6}6 6}6 } 6 6
1.636 !0.088 0.002 0.012
The knowledge of force constants (32) and the crystal structure enables the calculation of the elements of the dynamical matrix (37) and thereby of the phonon frequencies and polarizations. However, the force constants for a restricted number of systems are known from "rst principles calculations (see below) and for the majority of the systems that have been studied so far they have been obtained either by using empirical pair potentials or by "tting the potential parameters to bulk properties and phonon dispersion curves available from neutron measurements [144]. These procedures have been discussed in detail in the various reviews on surface phonons quoted above. Improvements of the quasiharmonic approximation by taking into account the anharmonicity at the various levels of accuracy have been recently demonstrated on the example of phonons localized in noble gas monolayers adsorbed on Pt(1 1 1) surface [145]. A comprehensive table of the theoretical studies of surface phonons listed by the methods and surfaces is available from Ref. [20]. As an example of the dynamical matrix calculation we present in Fig. 4 the results for phonon dispersion curves in an 80-layer thick face centered cubic (fcc) crystal slab of Xe atoms bounded by (1 1 1) surfaces [146] by using the force constants displayed in Table 1 and calculated
B. Gumhalter / Physics Reports 351 (2001) 1}159
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from Xe}Xe gas-phase pair potentials [147]. Three surface-localized modes detached from the bulk continuum are clearly discernible: the dominantly perpendicular or Z-polarized Rayleigh wave (RW) below the bulk continuum, the dominantly shear-horizontally polarized or SH-mode in the gap around the KM point (this mode becomes degenerate with the bulk continuum in the remainder of the SBZ), and the dominantly longitudinally polarized or L-mode in the gap extending from KM to M M points of the SBZ, and turning into a longitudinal resonance in the remainder of the SBZ. 3.2. Particle}surface interaction potentials The experimental data collected in thermal energy atom or molecular beam scattering from surfaces can be analyzed at several levels of complexity. The analyses based on simple kinematics of these experiments (see Section 2.1) already provide useful information on a number of important surface properties. Thus, for instance, from the two-dimensional di!raction patterns it is possible to determine the reciprocal lattice vectors of ordered surfaces, and the variation of the intensity of di!racted beams with the change of scattering parameters gives information on the bound state energies of the projectile}surface potential. This has been exploited in early gas}surface scattering experiments with e!usive beams to extract information on the structure of the static component of projectile}surface potentials and thereby on surface corrugation. On the other hand, the pronounced peaks in inelastic scattering TOF spectra yield direct information on dispersion curves of the various phonon branches de"ning surface vibrational dynamics, etc. However, the structure of all the measured scattering spectra, either angular distributions only or the energy-resolved TOF spectra, are complicated functions of the properties of the investigated surface per se (e.g. the phonon density of states), as well as of the potential which describes the interaction of the projectile with the target. Hence, for more complete analyses of these experiments a knowledge of the projectile}surface interaction is often needed, particularly in the studies of projectile scattering intensities. In the cases when one must discriminate between the various possible physical e!ects and processes taking place in the scattering event, this knowledge becomes essential. Microscopic properties of the projectile}surface interactions were implicit already in the earliest quantum-mechanical formulations of the gas}surface scattering developed by the Cambridge group in the 1930s [21]. However, the interest in gas}surface potentials was revived with the advent and wide application of the nozzle beams in scattering experiments in which the focus was on the interactions of inert atoms or molecules with surfaces. As a result of this, but also owing to the versatility of methods used to calculate the potentials, the literature on this topic has grown rich. To this end three main approaches for obtaining the potential properties have been pursued: (i) xrst principles calculations, (ii) semiempirical calculations, and (iii) reconstruction of the potentials from experimental data (mainly from diwraction and selective adsorption measurements). In the present context of nonreactive atom}surface scattering, and in particular of HAS from surfaces, one considers the gas}surface interactions and processes which do not involve any appreciable electron charge transfer or sharing between the projectile particle and the target. This is equivalent to the statement that the scattering event can be viewed as a series of transitions of the collision system between multidimensional diabatic potential energy surfaces that are una!ected by chemical bonding e!ects. Conventionally, the ground state potential energy surface is termed the physisorption potential to emphasize the absence of chemisorptive contributions to its components. The dynamics of the projectile motion is then determined by the potential ;(r, r ) which in H
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addition to the projectile coordinate r depends also on the set of nuclear coordinates of the atoms of the target, r , which are generally time-dependent. To facilitate the studies and analyses of H elastic and inelastic atom}surface scattering processes, it is convenient to discuss the static and dynamical components of ;(r, r ) separately although this may not be either justi"ed or possible H in all situations. 3.2.1. Static projectile}surface interaction. Corrugation vs. anticorrugation. Projectile}adsorbate interaction The static component ;(r) of the projectile}surface potential ;(r, r ) (i.e. of its frozen equilibH rium con"guration) exhibits a general structure that is characterized by a short-range strongly repulsive part in the vicinity of the physical surface, a long-ranged attractive part in the asymptotic region far outside the surface, and a physisorption potential well between these two regions. There is no general rule for the lateral variation of the static potential except that close to the surface it is strongly a!ected by the geometric and electronic structure of the crystal, and in particular by the outermost surface layer of atoms. Hence, the basic periodicity of the system should be also re#ected in some way in the lateral periodicity of ;(r) so that it could be represented in the form given by expression (19). The laterally averaged surface potential ; (z) is then given by the G"0 term of expansion (19). In the description of inelastic particle}surface scattering in Sections 4}7 we shall consider only the interactions taking place between solid surfaces and projectile particles with closed-shell electronic structure. In a scattering experiment the projectile approaches the surface from in"nity and therefore is "rst subject to the long-range component of the total potential. This component arises from the van der Waals (VdW) or dynamic electronic polarization interaction between the two separated (nonoverlapping) and neutral subsystems of the projectile and the solid. For a planar static surface the asymptotic expansion of the nonretarded VdW interaction potential can be expressed as [173,174] C #O(z\) . ; (r)"! 45 (z!Z )
(45)
Here the projectile coordinate z perpendicular to the surface is measured relative to the equivalent jellium background edge which is shifted from the topmost crystal plane by half the distance between two adjacent equivalent crystal planes parallel to the surface. The strength of the interaction C and the VdW reference plane position Z are obtained as integrals over imaginary frequencies of the dynamic electronic polarizabilities characterizing the projectile and the surface:
(46)
(47)
C " du (iu)R (iu) 4 and
du Z " (iu)R (iu)d (iu) , C '. 2
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31
where Z is also measured relative to the equivalent jellium edge. Here () is the dynamic electronic polarizability of the projectile particle, R ()"(1!Q ())/(1#Q ()) is the long wavelength limit (Q"0) of the Fourier transform of the surface electronic response function [175] expressed in terms of the surface dielectric function Q () [176], and d () is the frequency'. dependent centroid of the image charge introduced by Feibelman [177] and later elaborated many times in the literature (for more details see Ref. [178]). Expression (45) gives the dominant term in the asymptotic expansion of the particle}surface polarization energy in powers of z at large particle}surface separations and hence its divergent behavior for zPZ when the two electronic subsystems begin to overlap is unphysical. As the calculation of the corresponding interaction at intermediate distances becomes increasingly complicated (see also below) several empirical and heuristic schemes have been proposed to remedy this de"ciency. The most popular one is based on a modi"cation of the `damping functionsa introduced by Tang and Toennies [179] to remove the singularities from analogous potentials acting between gas-phase particles. This amounts to multiplying the "rst term on the RHS of Eq. (45) by the damping function , [I (z!Z )]L , f (I (z!Z ))"1!exp[!I (z!Z )] , n! L
(48)
where N"2 or 3, depending on the requirement that the damped VdW term either saturates at a "nite value or goes to zero for z"Z . The parameter I is chosen such that it coincides with the inverse range of the exponential potential which is frequently used (and may be justi"ed on physical grounds, see below) to model the fall-o! of the repulsive projectile}surface interaction. Another possibility to avoid divergences connected with the asymptotic potential (45) is that it is used piecewise, i.e. only beyond a certain point z at which it must match smoothly the inner part of the J potential which has been calculated following other calculational schemes [182,184]. Corrugation of the surface electronic density is not expected to a!ect much the asymptotic form of the VdW interaction (45) because the latter is obtained by integrating the long-wavelength limit of the quantities describing the electronic properties of the surface which should be insensitive to lateral variations. The values of the parameters C and Z for a number of various adparticle}surface combinations, and in particular for those involving He atoms, have been calculated and tabulated [8,186}193]. At closer distances to the surface, the overlap between the projectile and surface electronic charge densities will give rise to additional potential terms making up the total static adparticle}surface interaction. Although the use of the two-parameter (9,3) Lennard-Jones type of potential [8] (cf. Eq. (54)) with correct asymptotic behavior (45) may at "rst sight seem advantageous in modelling the total projectile}surface interaction because it does not require the damping or cutting-o! of the attractive component, it su!ers from introducing unphysical divergences in the total potential at very short distances. Hence, it is generally unacceptable for surfaces exhibiting slow variation of the electron density. Surface corrugation e!ects may a!ect the total potential in the intermediate region and are expected to be most prominent in the strongly repulsive part of the total potential in the immediate surface vicinity. To investigate these e!ects, it is common to divide the total static interaction into the surface averaged part ; (z), given by the G"0 term of expansion (19) and depending only on
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the z coordinate of the projectile, and the `corrugateda part exhibiting the dependence also on the lateral coordinates x and y. However, a uni"ed theory which could describe the long-range VdW and short-range overlap-induced components of the adiabatic projectile}surface potential on an equivalent footing is still lacking. Although very recent developments in the application of density functional theory to this problem [195}197] may soon help to remedy such a situation by enabling a `seamlessa solution, at present one has to rely on the approximate or phenomenological schemes in the calculations of the various components of the projectile}surface interaction. To obtain the potentials at intermediate and short separations a number of approaches have been used, ranging from the "rst-principle ones in the cases of simpler systems, to semiempirical and to empirical ones in the cases of more complex systems. Some of these approaches are believed to be more suited to the treatment of projectile interactions with the metal and others with the dielectric (insulating) surfaces, and only a small number of them are considered to be equally well applicable in both cases. Many of these approaches have been recently reviewed [193,194] and here we shall outline only a few of them that will prove relevant to the scattering calculations in Sections 7 and 8. 3.2.1.1. Dielectric surfaces. Dielectric, i.e. insulating surfaces were historically the "rst ones whose vibrational properties have been successfully studied by HAS. The high-resolution HAS TOF studies revealed the surface phonon dispersion curves throughout the SBZ for the "rst time in the case of alkali-halide surfaces [12,99]. As the earlier He atom di!raction and selective adsorption experiments had been focused on the studies of strong corrugation of these and other dielectric surfaces, including also graphite and semiconductors (cf. Refs. [8,10,223]), the e!orts to construct the corresponding interaction potentials have had a longer history than those for metals. Most of the dielectric substrates and inert atoms of interest are complex enough that the "rst principles calculations of the physisorption potentials are at present not capable of providing a realistic description of the projectile}surface interaction. Hence, in the majority of cases one has to rely on semiempirical constructions to obtain their characteristics. In this respect three di!erent approaches have been successful in providing expressions for the particle}surface potentials which could be used in the studies and analyses of gas}surface scattering. The "rst one is based on the empirical reconstruction of potentials (inverse scattering problem) from di!raction and selective adsorption measurements which provide information on surface corrugation and energies of bound states of the interaction potential. These procedures have been reviewed in Refs. [8,10]. The second approach developed by Beder [224] and Steele [225] starts from the models for interactions among the constituents to construct the potential functions. The third approach by Cole et al. [186,193,226] has been based on systematizing the empirical data into families that have similar shapes for the potential functions from which the so-called universal functions describing the physisorption phenomena could be deduced. A comprehensive review of empirical procedures for reconstruction of projectile}surface potentials has been presented by Hoinkes [8]. These procedures make use of the appearance of minima in the specular beam intensities that are associated with the bound state resonances (selective adsorption). The minima are measured by varying the azimuthal angle of scattering and keeping and constant, thereby scanning the possible pairs of values G and . This enables an L G L unequivocal determination of the binding energies of the interaction potential by using only the kinematic resonance condition (11). Once the experimental spectrum of binding energies is L
B. Gumhalter / Physics Reports 351 (2001) 1}159
33
known, one can construct the potential energy curve ;G 0 (z) by determining the parameters of a presumed model potential function. This `inversion problema was discussed by LeRoy [227] who has shown that it has no unique solution but the procedure may be used to yield the width of the potential as a function of its depth. LeRoy [227] has also shown that by assuming the asymptotic behavior (45) of the potential and knowing the bound state energies close to the vacuum level P0, it is also possible to estimate the value of the constant C . L Di!erent functional forms for the potentials ;G 0 (z) describing the various adsorbate}surface combinations have been "tted using the experimentally available data. The analytical expressions of the various potential functions, the values of the potential parameters obtained by the "ts, and the merits and demerits of particular model potential expressions have been summarized in Appendix C of Ref. [8]. Experimental bound state energies and resonances can also be used to determine the periodic potential terms ;GO0 (z) in expansion (19) by following the procedure proposed by Chow and Thompson [54]. The application of this procedure to dielectric surfaces and the results thereof have also been reviewed by Hoinkes [8]. Of course, it is also possible to revert the experimental information on surface corrugation back into the model expressions for corrugation functions and thus deduce the periodic potential terms from which the corrugation derives (cf. expressions (19), (66) and (72)). However, the results of this inversion procedure are not unique, as outlined in Section 2.2 and reviewed in Refs. [8,10,17,78]. The pairwise summation calculation is another frequently used approximate procedure for obtaining physisorption potentials that due to the electron charge localization may work better in the case of dielectric (insulating) surfaces than in the case of metals. Starting from the additivity assumption the potential function is obtained by summing or integrating over all binary interactions between the adsorbate atom and the lattice atoms or ions. Denoting the distance between the adsorbate atom at r and the jth solid atom at r by H (49) "r!r , H H the total potential is given by the sum ;(r)" v( ) , (50) H H where v( ) is the adsorbate}solid atom pair potential. Although this method avoids the di$culties H present in the calculations based on the electronic structure of the solid and adsorbate atom, it only reverts the problem to the determination of appropriate pair potential parameters which cannot be the same as in the case of interactions between isolated pairs. Hence, using this method di!erent problems are encountered and other types of approximations have to be introduced. The "rst concerns the distribution of atoms in the solid which in the "rst approximation is usually assumed to be the same as in the bulk. The second is connected with the parameters pertaining to the gas atom}solid atom pair potential energy curves which cannot be determined from independent measurements, e.g. gas-phase collisions. Due to this, the relevant parameters are usually evaluated by relying on some theoretical estimates. To this end, the following combination rules involving gas atom}gas atom (gg) and solid atom}solid atom (ss) parameters have often been used in the determination of gas atom}solid atom (gs) parameters: D "(D D
(51)
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for the potential well depth of the binary gas atom}solid atom potential, and # " 2
(52)
for the range of the gas atom}solid atom potential. A demerit of the described procedure is that it completely misses out the many-body e!ects on the pair potentials due to the presence of other atoms in the solid, and these problems have been discussed in detail in Ref. [225]. On the other hand, one of its merits is that it enables a clear illustration of how one proceeds from the binary potentials with correct asymptotic behavior J!\ to the gas}solid potentials with correct H asymptotic behavior (45), simply upon replacing the summations over crystal site coordinates by integrations in the asymptotic limit PR. This can be easily demonstrated in the example of H (12,6) Lennard-Jones (LJ) pair potential expressed in terms of reduced variables H"/ and zH"z/ : ;(r)"4D H
1 1 . ! H H H H
(53)
Upon integration over the crystal site coordinates, this potential produces a modi"ed (9,3) Lennard-Jones potential [8,225] corresponding to the G"0 term in expression (19):
2 1 1 2 ! ;*((z)" D zH zH 3 15
(54)
with z measured from the topmost crystal plane. However, the replacement of summations by integrations closer to the surface and in the region of the potential minimum usually yields quantitatively erroneous results (cf. Fig. 27 in Ref. [8]) as it misses out the contributions of periodic terms which fall o! exponentially with the adatom}surface distance [225]. It is expected that the pairwise summation method produces the most reliable results for He atom interactions with surfaces made of mono- or multilayers of noble gas atoms. Due to the inert electronic structure of these atoms, the modi"cation of the binary potentials is expected to be small, in which case rather accurate forms of the parametrized He atom}noble gas atom gas-phase potentials can be used [179}181,147]. Fig. 5 shows the results of the calculation [228] for the Fourier components ;G (z) (see Eq. (19)) of the He atom}Xe(1 1 1) surface interaction potential based on a summation He}Xe gas-phase pair potentials [179] over a slab of 60 Xe layers. Applications of the pairwise summation method to L-J binary potentials to yield the various G-terms in expansion (19) by carrying out lattice sums were demonstrated in Ref. [225]. Applications to generalized inverse power law, exponentially repulsive potentials and Tang-Toennies [179] damping functions were discussed in Section 2.3.1 of Ref. [193]. In the context of pairwise summation methods for obtaining the projectile}surface potential one should also mention corrections to this procedure based on supplementing the two-body interactions with the threebody interactions. This enables the removal of the discrepancy between the theoretical and experimental values of bound state energies for He atom interaction with a Kr overlayer on graphite [185].
B. Gumhalter / Physics Reports 351 (2001) 1}159
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Fig. 5. Lowest-order Fourier components ;G (z)"; (z) of the periodic He atom}Xe(1 1 1) surface interaction potential KL obtained from pairwise summation of He}Xe gas-phase potentials [179] over a slab of 60 Xe layers (after Ref. [228]).
Another empirical approach to construct the physisorption potentials starts by noting some common trends and unifying features in the extensive body of information on the laterally averaged physisorption potentials [186]. Vidali et al. [226,230] have proposed that the correlation among the data could be achieved by assuming that the uncorrugated potential term may be expressed in an empirical three-parameter universal form: ; (z)"Df (z/z ) ,
(55)
where D stands for the potential well depth and z is a characteristic length. The latter is determined from the asymptotic van der Waals behavior f (zPR)P!C /(Dz)
(56)
and hence is given by z "(C /D) ,
(57)
which reduces the scaling parameters to only two independent ones, i.e. to the strength and range of the repulsive potential contained in expression (55). One such universal potential form is given by 1 3 exp[!u(x/x !1)]! , f (x)"
\ x u!3
(58)
where the adimensional potential minimum x "(1!3/u) has one free parameter, either u or
x . Using Bohr}Sommerfeld quantization rules
1 y "a n# " dx( /D!f (x) , L L 2
(59)
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one can determine the bound state energies of this potential as a function of the parameter L a" /[z (2MD)]. By "tting the thus-obtained theoretical to the experimental 's for a given L adsorption system one can determine u"5.25. As shown in Refs. [186,226,230], the plot of y as L a function of for He, H and H adsorption on insulator, semiconductor and some metal surfaces L exhibits a universal behavior although some deviations have been noticed for noble metals. The discussion of the applications and results of the above approach is given in Ref. [193], Section 2.4. 3.2.1.2. Metal surfaces. In clear contrast to insulating surfaces, the He di!raction experiments on clean close-packed metal surfaces (like (1 1 1) and (1 0 0) planes of fcc metals) revealed weak di!raction peaks only after considerable e!ort [198}200]. The weak intensity of the measured peaks demonstrates the consequences of the valence electron smoothing e!ect and hence weak surface corrugation as probed by the He atoms. Only by using more open or stepped metal surfaces the di!raction e!ects were found to be more pronounced (cf. Table 5 in Ref. [17]). Owing to these "ndings the earliest "rst principles calculations of noble gas}metal surface potentials were carried out for the jellium model of the surface [201,202] by juxtaposing the repulsive, overlap-induced, and the attractive, VdW components of a noble gas}surface interaction. In a seminal paper on noble}gas interactions with metal surfaces Zaremba and Kohn [202] used the Hartree}Fock approximation to calculate to lowest order in the overlap the repulsive component of the interaction due to the change of the single-particle density of states of a simple metal modelled by jellium. The idea behind this approach has been followed in numerous subsequent works undertaken to improve and generalize the earliest results for noble gas}simple metal physisorption potentials [203]. Another approach based on the small overlap approximation and using the LCAO method has been developed by Goldberg et al. [204] and applied to noble gas atom adsorption on simple metals [205]. The e!ect of d-orbitals of transition metal substrates on the physisorption potentials has been examined in cluster model calculations of the repulsive He}surface interaction. Ab initio SCF method at the Hartree}Fock level was employed to calculate He atom interaction with (1 1 1) and (1 0 0) surfaces of Cu and Ni [206]. Likewise in other types of calculations the repulsive component was found to follow a nearly exponential fall-o! with the He-surface distance. In combination with the VdW potential terms obtained earlier for the same surfaces [187] this produced the total potential in a very good accord with the results available from the other approaches quoted above. One of the advantages of the LCAO and cluster methods is that the lateral variation of ;(r) due to the corrugation of the surface electronic density is directly obtained by shifting the center of the adatom from one surface site to the other. Fig. 6 illustrates the He}Cu(1 1 1) and He}Ni(1 1 1) potentials calculated by this method. The density functional theory (DFT) within the local density approximation (LDA) has been applied rather early to the problem of noble gas atom physisorption on metals [207}210]. One very important result of this approach [209,210], which has been widely quoted as the `ewective medium approximationa and often used as a universal recipe, establishes a proportionality between the repulsive part of the physisorption potential and the local charge density J (r) of the substrate surface ;(r)"A (r) .
(60)
As at distances relevant to physisorption on metallic surfaces the fall-o! of the substrate electronic charge is nearly exponential, the e!ective medium approximation also predicts the exponential fall-o! of the corresponding noble gas}surface potential.
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Fig. 6. (a) He}Cu(1 1 1) and He}Ni(1 1 1) model potentials ; (z) derived in Ref. [206] by combining the results from RMR cluster approach, yielding the short-range component, with the van der Waals type of calculations (cf. Section 3.2.1), yielding the long-range component of the total interaction. The z-axis passes perpendicularly through the center of the cluster at z"0 marked by X in the inset. (b) Same for the He}Cu(0 0 1) and He}Ni(0 0 1) interactions.
A relatively quick procedure to derive the corrugation amplitude of the atom}surface potential by exploiting the e!ective medium approximation was proposed by Bortolani et al. [211]. This amounts to making a superposition of atomic charges available from tables and applying expression (60) to the obtained density. This procedure also predicts exponential asymptotic attenuation of the repulsive component of the interaction. Hence, on the basis of the various independently obtained results referred to above, one may construct a net surface averaged model potential (i.e. the zeroth term in expansion (19)) for inert
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gas}atom interaction with a #at metal substrate in the form C f [I (z!Z )] , ; (z)"< exp(!I z)! , (61) (z!Z ) where the constants < and I are determined from best "ts to the values of the potential that have been obtained numerically in the interval of interest (usually extending from around the classical turning point to around the minimum or the in#ection point of the total potential) by using one of the above-mentioned methods. The part of the total laterally averaged potential ; (z) around the turning point and the minimum at z"z can be well approximated [270,229] by the Morse potential:
;+(z)"Dexp[!2(z!z )]!2 exp[!(z!z )] , (62)
where D, 2"I and z denote the well depth, the inverse range and the position of minimum of
the potential, respectively. An advantageous feature of this potential is that the corresponding wavefunctions and bound state energies are available in analytical form [22] which in some cases can enormously reduce the computing e!orts. The e!ective medium approximation (60) implies that the corrugated part of the total atom}surface potential ;(r) should exhibit corrugation obtained by scaling the spatial variation of the substrate electron density J (r). Starting from this assumption and the superposition of atomic electronic densities, Takada and Kohn [191] have represented the unperturbed surface electronic charge density as a two-dimensional Fourier series: (r)" (r)# G (z) exp(iGR) G 0 $ and postulated the asymptotic relation for the repulsive He}surface potential in the form
(63)
(64) ;(r)"A (r)# AG G (z) exp(iGR) . G 0 $ The coe$cients AG were calculated by reformulating the theory of Zaremba and Kohn [202] and determined for the He interaction with (1 1 0) surfaces of Ni, Cu and Ag. This reduced the task of calculating the corrugation of physisorption potentials to calculating the variations of the unperturbed substrate electron densities and the corresponding proportionality factors AG . Once these parameters are known, the corrugation function (R) and the corresponding corrugation amplitudes can be readily determined and compared with the ones available from di!raction and rainbow scattering experiments. Corrugation amplitudes have been studied for a number of metal surfaces and di!erent types of beams (He, Ne, Ar, H , D ), mainly by combining experimental and theoretical approaches. A discussion of these works with a full list of references can be found in Ref. [17] in which Table 5 summarizes the available corrugation amplitudes for the various combinations of metal surfaces and room-temperature beams. However, although the experimental evidence in support of the e!ective medium approximation can be found in the case of Ne, Ar and H beam di!raction from corrugated surfaces, the He di!raction experiments carried out by Rieder et al. [79,212] provide evidence which is at variance
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39
with the simple e!ective medium recipe embodied in expression (60). The "rst attempt to explain this "nding was made by Annett and Haydock [213] who showed using simple arguments that in the case of (1 1 0) surfaces of Ni, Cu, Pd and Ag the hybridization of He 1s occupied orbitals with the unoccupied metal states gives rise to `anticorrugationa e!ects, i.e. the corrugation of the potential in the opposite sense to the charge density. Later elaborate calculations based on the DFT [214] have demonstrated how in the case of Rh (1 1 0) surface the anticorrugation phenomenon is determined by the polarizability and hybridization of the projectile atom valence electron density with the substrate d-electrons close to the Fermi level. These e!ects are di!erent for He 1s and Ne 2p orbitals and vary as the projectile position is moved from an ontop to a bridge site, in VW agreement with the results of Rieder and coworkers [212]. In current practical applications of these "ndings to He atom scattering calculations in which, for example, the He}surface interaction potential is derived from the superposition of atomic charges, one introduces the anticorrugation e!ects through multiplying the interaction matrix elements of the thus-derived potential by the so-called anticorrugation function [215]. Instead of making a superposition of the electron charge of atoms making up a metal and then applying the e!ective medium approximation, some authors have followed the procedure of a direct summation of the pair potentials acting between a He atom and the atoms of the semiin"nite crystal. Eichenauer et al. [216] have applied this method to calculate He}Cu(1 1 1) and He}Ag(1 1 1) interactions and their results for Cu(1 1 1) surface agree well with those obtained later from combining the cluster and VdW calculations [206]. The pairwise summation approach was reviewed in Ref. [219] and discussed in more detail also above in connection with the insulating surfaces. However, due to the delocalization of metal valence electrons this usually requires the assumption of nonspherical atoms to obtain a good agreement with the He scattering data [182}184], or an ad hoc introduction of the anticorrugation function in the case of He atoms [215,217], as pointed out above. The assumption of nonspherical, i.e. oblate atoms leads to anisotropic pseudopairwise potentials v(r) which in the calculations of the HAS re#ection coe$cients have been derived from spherical potentials following the prescription [183]: (65) v(r)" v((z#( x)#( y)) , V W V W where the anisotropy parameters and are usually determined by "tting the elastic cross V W section calculated with the anisotropic potentials to the experimental data [183,184]. A useful approximate form for the repulsive component of the projectile}surface potential at large distances, which can be derived from the pairwise summation procedure, reads [28,211] ;(r)"E exp[!I (z!z ! (R))] , (66) G where E is the incoming energy of the projectile, z is the classical turning point chosen such that G the integral over the corrugation function (R) over a unit cell vanishes, and the softness parameter I can be determined, for example, by matching the logarithmic derivatives of expressions (60) and (66). In accord with approximation (66) is the assumption that the static repulsive potential can be written in the form ;(r)"AI exp[!I r!rl ]"AI exp[!I ((R!RlG)#(z!z )] . G G l l G G
(67)
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Restricting the summation of the short-range components only to the surface layer z "0, and on G noticing that at the turning point z , whose value is large for He atoms, one "nds [222] (R!Rl ) G , ((R!Rl )#zKz# G 2z
(68)
which yields
I ;(r)"AI exp[!I z] exp ! (R!Rl ) . G 2z l G
(69)
The sum on the RHS of expression (69) may then be identi"ed with the exponent of the corrugation function in expression (66), expressed here as a sum of identical Gaussians centered at the lattice points. This form of the potential is extremely convenient in calculations of the scattering matrix elements as the two-dimensional Fourier transform of a Gaussian is again a Gaussian. Further modi"cations of this approach have been discussed in Refs. [219,222]. The surface averaged component of potential (69) is then expressed as 2 AI exp(!I z) , ;(z)" QA
(70)
where Q"I /z
(71)
and A is the area of the surface unit cell over which the averaging is carried out. In the case of more strongly corrugated metal surfaces (more open and reconstructed surfaces with missing rows, etc.) Rieder and collaborators [218] have remarked that both the repulsive and attractive components of the total potential may exhibit corrugation. To model such a situation they proposed an analytical form for the corrugated potential: D exp[!I (z!z !(x, z))]! exp[!(I /)(z!z !(x, z))] . ;(x, z)"
!1
(72)
Here D denotes the potential well depth, I is the reciprocal range parameter, is the scaling factor determining the potential width (for "2 one retrieves the corrugated Morse potential) and z is
the position of the minimum of the laterally averaged potential. The corrugation dependence perpendicular to the missing rows is then given by
2j d x , (x, z)" exp[!I (z!z )] H cos H
2 a H
(73)
where I denotes the exponential decay rate of corrugation, d is twice the Fourier component of H H the potential corrugation at the minimum and a is the length of the unit cell of the reconstructed surface. Potential (72) proved very successful in coupled channel calculations of the di!raction
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intensities from the missing row reconstructed Pt(1 1 0)2;1 surface. Interestingly enough, an excellent agreement between the experimental and calculated data for the di!raction intensities was obtained for "2.06, i.e. close to the value for which the potential (72) goes over into the corrugated Morse potential. 3.2.1.3. Interactions with adsorbates. A prerequisite for the theoretical analyses of the data from thermal energy atom scattering by surface defects and adsorbates is a good knowledge of the corresponding scattering potentials. In this context of particular interest are the interactions with the various adsorbates whose vibrational properties have been studied by HAS. Whereas it may be possible to describe He atom interactions with adsorbate layers by using the methods described in this subsection, the interactions with isolated adsorbates necessitate special attention in the light of the e!ects which establish HAS as a powerful tool for the structural studies of adsorbate-covered surfaces [14], as outlined in Section 2.3. The structural HAS studies of surfaces covered by low concentration of adsorbates rest on the e!ect of large scattering cross sections induced by projectile}adsorbate interactions (cf. Section 2.3). At thermal beam energies the dominant contribution to the magnitude of these cross sections comes from the long-range van der Waals component of the interaction. However, in contrast to the short-range overlap-induced repulsion, the van der Waals component cannot be expressed in terms of a binary interaction taking place between the projectile and the adsorbate, as would be the case in the absence of the substrate. Namely, the presence of a polarizable substrate through its e!ect on the polarization properties of the projectile and the adsorbate must be taken into account. In theoretical language, the dynamic polarizability of the substrate renormalizes the bare projectile}adsorbate polarization interaction. Due to this renormalization, which also introduces the substrate coordinates into the "nal formulae, the projectile}adsorbate van der Waals interaction can be considered as a three-body anisotropic (noncentral) potential. The angular dependence of this interaction between a projectile atom and a spherically symmetric adsorbate was derived by McLachlan [231]. In a more general case of projectile interaction with uniaxially symmetric molecular adsorbates with symmetry axis perpendicular to the surface (e.g. the case of He scattering from CO and NO chemisorbed on Cu, Ni, Pt, etc.) the expressions for the corresponding van der Waals potential were derived by Gumhalter and Liu [232] using the Feynman diagrammatic approach and the relevant formulae were systematized and the parameters calculated and tabulated by LovricH and Gumhalter [187]. It has been shown that the total projectile atom}adsorbate van der Waals potential comprises the `directa gas-phase-like binary interaction (dir), the so-called `image interactiona contribution (im), and two equivalent `interferencea contributions (int), both originating from the presence of the substrate (s). The relevant expressions can be most suitably expressed in terms of the coordinates of three relative radiusvectors: "rst, of the `directa projectile}adsorbate radiusvector r"r !r "(R !R , z !z )"(R, z !z ) ,
(74)
where z '0 and z '0 are both referenced to the positive jellium background edge Z "0, second, of the `imagea radiusvector rH "(R, z #z !2Z )
(75)
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Fig. 7. Geometry of the He-adsorbate collision system and notation relevant to the de"nition of the various van der Waals interactions. The substrate occupies halfspace z(Z "0 where Z denotes the edge of the equivalent positive jellium background. The image of the adsorbate relative to the Z plane is denoted by a , and relative to the pertinent van der Waals plane at Z (either Z or Z ) by aH.
and, third, of the `interferencea radiusvector rH "(R, z #z !2Z ) .
(76)
The corresponding van der Waals image and interference plane positions Z and Z , respective ly, are de"ned below. In terms of these coordinates the three components of the van der Waals interaction can be written in the form
1! C P (cos ) , ; (r)"!2(1#2) 1# 45 1#2 r
(77)
C ; (r)" ; (rH ) , 45 45 C
(78)
C ; (r)"2 (4!3 cos !3 cos H ) 45 (rrH ) ! (1!)[5#9 cos cos H !6(cos #cos H ) #9 sin sin H cos cos H ] .
(79)
Here " / is the ratio of transverse (perpendicular) and longitudinal (parallel) adsorbate , dynamic polarizabilities relative to the molecular axis, the three constants C determining the strengths of the interactions are de"ned below, P (cos ) denotes the second-order Legendre polynomial, and the geometrical meaning of r, rH , rH , , H , H , and the positions of the relevant VdW reference planes Z and Z are sketched in Fig. 7. Hence, it is seen that the radius
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vectors rH and rH connect the position of the projectile with the position of the adsorbate re#ected across the VdW plane located at Z and Z in front of the equivalent jellium background edge, respectively. The VdW constants C's have been calculated in Ref. [232] and are given by
C "
du (iu) (iu) , 2
(80)
C "
du (iu) (iu)R (iu) , 2
(81)
C "
du (iu) (iu)R (iu) , 2
(82)
where () is the projectile dynamic polarizability, denotes the longitudinal adsorbate polariza bility in the adsorbed phase (i.e. ()" ()), and other symbols are the same as in Eq. (46). The XX positions of relevant VdW planes have been calculated in Ref. [233] and read as
du (iu) (iu)R (iu)d (iu) , Z " '. C 2
(83)
du Z " (iu) (iu)R (iu)d (iu) , C '. 2
(84)
where d () was de"ned earlier in connection with Eq. (47). The modi"cation of the free adsorbate '. polarizabilities P as a result of adsorption has been discussed in Refs. [187,234]. The calculations of the total cross section for He atom elastic scattering from adsorbed CO molecules using the VdW potentials (77)}(79) in combination with repulsive He}CO potentials typical of the gas phase yielded a good quantitative agreement [234] with the experimental data reviewed in Ref. [14]. 3.2.1.4. Wavefunctions of the laterally averaged projectile}surface potential. In the case of projectile interaction with a statically #at surface one has ;(r)";(z) and the surface averaged potential ;G 0 (z) appearing in expansion (19) then coincides with ;(z). The unperturbed projectile motion is described by the wavefunction given by the G"0 term in the expansion on the RHS of Eq. (20). The latter is a solution of the three-dimensional SchroK dinger equation for projectile motion in ;(z):
! #;(z) k (r)"Ek k (r) . 2M
(85)
Also, in the case of surfaces with very weak static corrugation, manifesting itself through ; (z) EO ¹4(0) , d e\ & O
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coupling) give, under the scattering conditions investigated, much smaller contribution to the inelastic re#ection coe$cient than a sequential exchange of the same number of phonons but arising from single-phonon vertices originating from the linear coupling of the form (96). Fig. 1 in Ref. [294] illustrates a few lower-order Feynman diagrams arising from the expansion of the ¹4-matrix and comprising one- and two-phonon vertices that arise from linear and nonlinear projectile}phonon coupling. Irrespective of the simplicity of the scattering model used, this theoretical "nding justi"es the introduction of an enormous simpli"cation in the structure of the scattering potential, viz., that under standard experimental conditions in HAS the form of the potential given by the sum of the "rst two terms on the RHS of Eq. (93) can be used with con"dence because all higher-order terms would produce only negligible corrections. 3.4. The search for a unixed approach As has been pointed out in preceding sections, various theoretical approaches have been developed to interpret the results of atom}surface scattering experiments. The applicability of these approaches depends to a large extent on the scattering regime (classical, semiclassical or quantum) in which the experiments have been carried out. At higher incident energies and for larger projectile mass the semiclassical nonperturbative treatments (e.g. the eikonal and TA approaches discussed earlier in this section) proved useful in describing the inelastic scattering probabilities in the dominantly multiphonon scattering regime. However, the majority of HAS experiments that have been carried out so far require a quantum interpretation of the results of measurements. This is particularly important in the single-phonon scattering regime in which the dispersion curves of surface-localized vibrations of di!erent polarizations have been investigated. Here, the utility of HAS is fully exploited by a careful analysis of the exchange of projectile parallel momentum and energy with surface vibrations, and hence the interpretation of the results depends crucially on the ability to accurately describe the threedimensional character of the interactions. Whereas there exist detailed formalisms (e.g. the standard ¹-matrix approach) and recipes on how to ful"ll this requirement in the single-phonon scattering regime, the passage to the multiphonon scattering regime has usually been made at the expense of a full account of the complex character of interactions governing the collision. Namely, in a general situation it is only possible to evaluate higher-order terms in perturbation expansion of the ¹-matrix in terms of the number of excited phonons by invoking extremely simplifying assumptions, and to achieve the unitarity conditions for the scattering amplitudes is usually a very problematic task. On the other hand, in the alternative semiclassical approaches which allow full treatment of the multiphonon excitations and preserve the unitarity of the excitation spectrum, like the TA, the rigor is lost because of the neglect of a quantum character of propagation of the projectile. This, for instance, gives rise to erroneous predictions on the total energy transfer in the quantum multiphonon scattering regime [297], which represents a serious de"ciency in the theoretical studies of gas}surface collisions. Hence, from all that has been said so far about the various theoretical formalisms developed and applied to analyze inelastic He atom scattering spectra, it turns out that a more uni"ed approach that would cover several di!erent scattering regimes is needed for the interpretation of TOF experiments. Such an approach should combine the merits of formerly developed formalisms best applicable in particular scattering regimes and also should enable smooth transitions in the
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parameter space between the neighboring regimes. Thus, the requirements on a uni"ed approach for treating inelastic HAS could be summarized in the following few points: (i) Quantum formalism: The scattering conditions typical of HAS and the interpretation of measurements clearly require a uni"ed description in which the projectile motion and the surface vibrations are both treated fully quantum mechanically. Also, it should be possible, at least in principle, to establish a formal equivalence between this approach and the one based on the ¹-matrix formalism. (ii) Unitarity: The uni"ed approach should exhibit the property of unitarity in that the sum of all scattering probabilities, elastic and inelastic, should be equal to unity for unit incident current of the projectile particles. In other words, the scattering amplitudes should satisfy the optical theorem. In the jargon of surface scattering this statement may be reformulated in terms of the existence of a Debye}Waller factor which provides normalization proper for the spectrum of all scattering intensities. (iii) Full account of the surface vibrational dynamics: It should also be possible to incorporate a complete description of surface vibrational dynamics into the uni"ed scattering formalism, either by using the dynamical matrix approach, Green's function or a similar method for the description of lattice dynamics. (iv) Correct limit in the one-phonon scattering regime: In the limit of weak coupling the formalism should reproduce the quantum results obtained in the one-phonon scattering regime with full account of the three-dimensional character of projectile}phonon interactions and phonon polarizations. (v) Quantum treatment of multiphonon scattering in three dimensions: In the limit of enhanced projectile}phonon coupling and the increase of projectile incident energy the formalism should smoothly interpolate between the quantum one-phonon scattering regime, described by the DWBA, and the semiclassical scattering regime in which the trajectory approximation for projectile motion yields reasonable results. However, it should be possible to establish such a transition without resorting to simplifying forms of the one-dimensional interactions and oversimpli"ed phonon densities of states. In this limit all the limits discussed in connection with the application of the TA should be retrieved. The above-listed requirements, in fact, imply the search for a possibility of `resummationa of the perturbation series for the S-matrix in powers of the coupling constant so as to generate a new series characterized by renormalized small parameters which would guarantee its faster convergence and preservation of unitarity. This is a rather demanding goal but, as will be demonstrated in the subsequent sections, it is possible to achieve it within the scattering spectrum formalism devised for treating atom}surface collisions.
4. Scattering spectrum approach in the theoretical description of inelastic inert atom scattering from surfaces 4.1. Formulation of the scattering spectrum expression and its relation to the TOF spectra The beginnings of a uni"ed quantum treatment of inelastic particle}surface scattering, which would ful"ll the requirements listed and discussed in Section 3.4, can be traced back to the 1970s
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when MuK ller-Hartmann et al. [250] and Brenig [298] proposed the study of the scattering probability distribution as a function of energy transfer between the projectile particle and the dynamical degrees of freedom of the solid in order to gain information on the collision. This approach is based on a quantum treatment of temporal evolution of the system described by the total Hamiltonian H"H #H#H ,
(160)
where H "p/2M and H describe the noninteracting subsystems of the projectile particle and the crystal degrees of freedom, respectively, and H is the projectile}crystal interaction. If not stated otherwise, H will denote in the following the Hamiltonian of quantized harmonic vibrations of a perfect crystal lattice, i.e. free bulk and surface phonons (cf. Eq. (41)). Generalizations of H to other degrees of freedom typical of the crystal heatbath are straightforward [42}44]. H describes the interaction giving rise to scattering of the projectile from the initial unperturbed eigenstate of H into all possible scattering channels, elastic and inelastic. To simplify the procedure one usually resorts to the description in which H is "rst separated into the strong static and much weaker dynamic projectile}crystal interactions ; and k . G This state is an eigenstate of H "H #; which satis"es the scattering boundary conditions and hence can be obtained from the limiting procedure described by Eq. (123). Such renormalized motion is then perturbed by the dynamic interaction < which induces inelastic transitions of the projectile from one distorted wave state to another. In this process the energy and parallel momentum transfer between the projectile and the crystal takes place whereby the latter is excited from the initial state to a "nal state . The "nal state of the entire scattering system is obtained from the limiting procedure of collision theory in which the long time limit of the evolution operator of the system or the S-operator, S"U(tPR, tP!R), is allowed to act on the initial unperturbed state of the system. In doing so we assume that at time tP!R the projectile is prepared far away from the surface in a well-de"ned sharp and coherent initial state characterized by k (or, equivalently, in terms of the projectile parallel momentum K and energy E ), which is G G G feasible with the present-day beam-scattering apparatuses (cf. Section 2). We also assume that the crystal is in contact with a heat reservoir and the probability that at temperature ¹ the crystal is in a state with energy is given by p . Then the state of the system at tPR, which has evolved ? ? I 4)> I 4 is on-thefrom the state > k , "> k , is given by S> k , "(1!2i¹ k , where ¹ G G G G energy shell component of ¹4 de"ned by Eq. (136).
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We shall be interested in the probability density NkG (E) that an amount of energy 2 E"E !E has been transferred during the scattering event from the projectile to the crystal D G degrees of freedom. This quantity, hereafter referred to as the scattering spectrum, is obtained by projecting out from the "nal state wavefunction of the system all the components satisfying the condition of a given amount of energy transfer E, and then summing up the absolute squares of their amplitudes. Taking for the required projection operator the operator function (E! #H), the desired scattering spectrum can be set up in the form [24,25, ? 28,250,251,254,298] NkG
2
(E)" p > , SR (E! #H)S> k , . G ? kG ? ?
(162)
This can be calculated once the free crystal Hamiltonian H is speci"ed and the outgoing distorted waves > k and the scattering operator S are known. Note also in passing that since we G have de"ned expression (162) in terms of distorted waves, the present scattering operator S corresponds to S4 of Section 3.3.3. Inserting the complete set of incoming scattering states \ k , , on D which H is diagonal, between the S-operator and the -function in the expression on the RHS of (162), we obtain NkG
2
(E)" p \ k , S> k , (E! # ) . D G ? @ ? k D ?@
(163)
A comparison of this expression with expressions (124) and (126) clearly demonstrates why quantity (162) can be termed the scattering spectrum. Thus, the problem of calculating the scattering spectrum (162) is equivalent to calculating the matrix elements of the scattering operator in expression (163). This establishes the desired formal link between the scattering spectrum formalism and the standard S-matrix (or the ¹-matrix) formalism that was required in Section 3.4. Here it should also be observed that on-the-energy-shell character of the scattering event is controlled by the scattering operator S and not by the -function on the RHS of (163) which only acts to select the probability amplitudes satisfying the given energy transfer condition. Expressions (162) and (163) for the scattering spectrum satisfy the unitarity condition
\
N kG
2
(E) d(E)"1 ,
(164)
by virtue of the unitarity properties of the operator S. Although the authors of Refs. [250,298], after having introduced expressions of the form (162), resorted to the recoilless trajectory approximation to describe the projectile motion already at the early stage of their evaluation of the S-operator in expression (162), and thereby obtained a formula equivalent to Eq. (110), the starting expression (162) enables a uni"ed treatment along the lines discussed in Section 3.4. Moreover, since the TOF experiments provide information on all three good quantum numbers specifying the projectile translational motion, i.e. in the present case on the change of the projectile two-dimensional parallel momentum K and energy E, the approach of Eq. (162) can be conveniently generalized [28] to de"ne the parallel momentum and energy
B. Gumhalter / Physics Reports 351 (2001) 1}159
71
resolved scattering spectrum which could be directly related to the measured TOF spectra. Denoting by PK the parallel momentum operator of the crystal and assuming "rst a simpler case in which is an eigenstate of this operator with the corresponding eigenvalue P , the energy and ? parallel momentum resolved scattering spectrum is de"ned by [24,25,28,251] (E, K)" p > , SR (E! #H) ( K!P #PK )S> (165) k , . G ? kG ? ? ? In a more general case in which the state is not a momentum eigenstate but is represented by a Bloch wave composed of the eigenstates Q #G of PK , viz., ? NkG
2
" C (G)Q #G , ? ? G
(166)
the generalization of expression (165) in terms of Bloch waves is straightforward [28]: NkG
2
(E, K)
" p C (G) ?G ? ? ; > k , Q #GSR (E! #H) [ K! (Q #G)#PK ]S> k , Q #G . (167) G G ? ? ? ? Upon integrating expression (167) over d(K) one retrieves the energy-resolved scattering spectrum (162). Hence, the energy and parallel momentum scattering spectrum given by expressions (165) and (167) satis"es the unitarity condition in (E, K) space:
N kG
2
(E, K) d(E) d(K)"1 .
(168)
In an analogous fashion as for spectrum (162) we can represent the energy and parallel momentum resolved scattering spectrum in terms of the S-matrix. Resorting for illustrative purposes to simpler expression (165) only, and making use of the completeness of the incoming distorted waves, we "nd (E, K)" p \ (169) k , S> k , (E! # ) ( K!P #P ) . D G ? ? @ ? @ k D ?@ In a TOF experiment only a limited subset of "nal projectile states described by the quantum numbers (K , E ) is measured. These states are selected by the TOF detection procedure in which D D the apparatus imposes an `instrumental windowa on the values of K and E detectable in a particular experimental arrangement. In the following, we shall for simplicity assume an ideal TOF apparatus that can produce incoming beams in the form of plane waves k "K , E and G G G detect the scattered particles only in the sagittal plane de"ned by K and the surface normal. We G shall model this role of the apparatus by a quantity called `instrumental functiona, FkG 1" (E, K), F where " # is the "xed total scattering angle. 1" G D NkG
2
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To illustrate the relation between the quantum-mechanically calculated scattering spectrum (165) and the TOF spectrum NkG (E, , ), which is expressed as a function of E and the 2 2-$ 2-$ "nal scattering angles and , it is convenient to introduce the coordinates in terms of 2-$ 2-$ which the constraint of in-sagittal-plane scattering can be most conveniently implemented. To this end we shall "rst orient the coordinate system so that the components of K point in the directions parallel and perpendicular to the sagittal plane. The condition that the TOF apparatus detects the scattered particles whose motion is restricted to the sagittal plane is then formulated such that only the particles with unchanged direction of in-surface-plane velocity enter the detector, viz., the particles for which v " K /MP0. The second condition is that for given E the change , , of parallel wavevector in the sagittal plane, K , is such that the projectile "nal velocity points in the direction . Third, we must also include a condition that takes account of a "nite detector 2-$ aperture of area A normal to the direction . Here we shall assume that the area of the surface D D illuminated by the incident beam is much larger than A . Then, the number of particles scattered D from the illuminated surface and hitting the area A is proportional to A /cos since the latter is D D D the e!ective area of the surface seen by the detector [34]. These conditions can be modelled by introducing the `instrumental functiona in the form A Const. D f (K )f ( ! ) , FkG 1" (E, K)K , F D 2-$ F v(E #E) cos D G
(170)
where for an idealized TOF apparatus: f (x)P (x) .
(171)
The prefactor given by the inverse of the "nal velocity of the particle, v "v(E #E), on the RHS D G of Eq. (170) arises from the proportionality of the number of ionized scattered beam particles counted by the detector with the time J1/v(E #E) that the particles need to traverse the G length of the detector ionization chamber. Next, we make use of the relation for the change of the in-sagittal-plane component of the parallel wavevector: M K " v(E #E) sin !k sin , G D G G
(172)
which enables us to write R(K ) dv d(K)"d(K ) d(K )"d(K ) DF , , Rv DF M ), " v(E #E) cos d d(K G D D ,
(173)
where the identity dv "v(E #E) d has been used. Expression (173) enables passing from DF G D integration of the scattering spectrum over d(E) d(K) to integration over the variables appear-
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73
ing on the RHS of expression (170). Combining expressions (170) and (173) enables us to establish the desired connection between the measured TOF and calculated scattering spectrum in the form NkG
2
(E, , sag)K NkG (E, K)FkG 1" (E, K) d(K) 2 F 2-$
"Const. NkG (E, K) 2
" Const.
MA D (K ) ( ! ) d d(K ) , D 2-$ D ,
MA D Nk (E, , sag) . G 2 2-$
(174)
Here sag in the argument of the spectra serves as a reminder that the initial and "nal state projectile wavevectors, k and k , respectively, are constrained to the sagittal plane. G D Expression (174) demonstrates that for in-sagittal-plane scattering the measured TOF and calculated scattering spectra are directly proportional and that the proportionality factor is a constant for a particular TOF spectrum (i.e. for "xed and ). This enables a direct 1" G comparison of the relative intensities of spectral features occurring at the same E in the measured and calculated spectra that correspond to the same scattering conditions. In other words, for in-sagittal-plane scattering the experimental and theoretical spectral intensities scale by a common factor. Finally, it should be noted that in the case of out-of-sagittal-plane scattering the simple proportionality (174) does not hold because for an arbitrary scattering direction, , the 2-$ instrumental function acquires a di!erent form. Also, whenever the role of the instrumental function can be more conveniently expressed through the projectile "nal state variables, it may prove more convenient to carry out the integration over the "nal projectile parallel wave vector K instead of over K: D d(K)PdK "[k(E #E)] cos sin d d "k k d . D G D D D D D DX D
(175)
Due to these additional complications arising in the description of out-of-sagittal-plane TOF spectra we shall avoid applications of the scattering spectrum formalism to such situations. 4.2. Development of the scattering spectrum formalism (SSF) In this subsection we shall develop a formalism for calculating the energy and parallel momentum resolved scattering spectrum starting from de"nition (165). A generalization in which the point of departure would be de"nition (167) is straightforward but as in this latter case the notation becomes cumbersome we shall refrain from pursuing it for the bene"t of clearer and more concise presentation. In the end, however, we shall restore the result which encompasses both forms of the initial state averages. We start from a standard trick to represent the energy and parallel momentum projecting -functions in expression (165) in terms of their Fourier transforms in - and R-space, respectively.
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This yields NkG
d dR (E, K)" p exp!(i/ )[(E! )!( K!P )R] 2 ? 2 (2 ) ? ? ?
K R)]S> ; > k , SR exp[!(i/ )(H!P k , . G G
(176)
Here it should be noted that the auxiliary integration variables (, R) used to represent the -functions in the form of Fourier integrals do not describe spatio-temporal evolution of the scattering system and that this information is contained solely in the scattering operator S. We proceed by expressing the total Hamiltonian of the scattering system as H"H #g< ,
(177)
where H is given by Eq. (122) and g< describes the dynamical interaction of the projectile with the phonons of the target. Here we have introduced for later convenience the coupling constant g which will eventually be set equal to unity. Expression (177) leads to the representation of the scattering operator in the interaction picture, S , according to ' S" lim e\ & R\R U (t, t )"S S , ' ' RR \
(178)
exp[!iH (t!t )/ ]. This enables us to transform the initial state where S "lim RR \ averages in (176) as NkG
?2
(, R)" > K R)]S> k , SR exp[!(i/ )(H!P k , G G " > K R)]S > , k , SR exp[!(i/ )(H!P G ' ' kG " > K R)]> k , exp[!(i/ )(H!P k , , G G ' '
(179)
where H and PK are de"ned by the canonical transformations: ' ' H"SRHS , ' ' '
(180)
PK "SRPK S . ' ' '
(181)
Now, according to a general theorem [299] the operator U (t, t ), and thereby also S , can be ' ' represented in exponential form:
U (t, t )"e\ %RR "exp !i gLG (t, t ) , ' L L
(182)
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75
where G (t, t )'s are hermitian operators obtained from nested commutator expansions in powers L of the coupling constant g:
g R gG (t, t )" dt < (t ) ,
' R
(183)
R i(g/ ) R gG (t, t )"! dt dt [< (t ), < (t )] , ' ' 2 R R (g/ ) gG (t, t )" 4
R
R
dt
R
R
dt
R
R
(184)
dt [< (t ), [< (t ), < (t )]] ' ' '
1 R R R # dt dt dt [[< (t ), < (t )], < (t )] , ' ' ' 3 R R R
(185)
(186)
g (2) , gG (t, t )"
etc., where all other higher-order terms in the coupling constant comprise higher-order commutators [., [.., [2,.]]] of the dynamic interaction < (t )"e & RH <e\ & RH . ' H
(187)
Thus, the S-matrix in the interaction picture can be written in a general form S " lim e\ %RR "e\ %\"e\ % , ' RR \
(188)
which can be conveniently used to carry out the canonical transformation in expression (179) and to evaluate the diagonal matrix elements of the thus-obtained exponential operator. This procedure can be presented in a compact form by introducing a uni"ed vector notation for the variables and exponentiated operators: (, R)"(, X, Y)P( , , )" ,
E ,!K ,!K P( , , )" , V W
( , P )P , ? ? ?
H PK PK ,! V ,! W P(H , H , H )"H ,
(189)
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where due to the property that H and PK commute, also the components of the vector operator H commute with each other, i.e. [H , H ]"0 . J JY
(190)
Using the notation of expressions (189) we can write E !(K)R" " , J J
J PK PK H ! V X! W Y" H "H , J J
J H PK "(L , L , L )"L . SRHS " ' ,! ' ' '
(191)
(192)
(193)
This enables us to re-express Eq. (176) in a more compact form as NkG
2
(E, K)"NkG
2
d ()" p exp[!i(! )] > k , exp(!iL)> k , . G G ? (2 ) ? ? (194)
In the next step we make use of the identity iK SRAS "e %Ae\ %"A# GK[A] , (195) ' ' m! K where GK[A] denotes mth-order repeated commutator of G with an arbitrary operator A. We apply this to expression (193) to obtain the exponent in expression (194) in the form L"H#W
(196)
where H is given by Eq. (192) and iK W"(W , W , W )" GK[H] . & V W m! K This enables us to write > k , exp(!iL)> k , " > k , exp[!i(H#W)]> k , G G G G
(197)
(198)
and this form is amenable to standard treatments of evolution operator in quantum mechanics. On noticing that H" ?
(199)
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and introducing an auxiliary parameter , eventually to be set equal to unity, we can write > " > , k , exp[!i(H#W) ]> k , k , exp(!iH )U (, )> k , G G G G H ' H
(200)
where U (, ) satis"es a di!erential equation: ' R U (, )"!iW (, )U (, ) ' ' R '
(201)
W (, )"exp(iH )W exp(!iH ) , '
(202)
with
which is calculated in the `interaction representationa in the -space. Upon integrating Eq. (201) we obtain
U (, )"¹ exp !i ' H
H
d W (, ) , '
(203)
where ¹ is the ordering operator associated with the parameter . Hence, U (, ) acquires the H ' form of an evolution operator generated by the perturbing operator W in the -space. This "nally yields NkG
2
(E, K)"NkG
2
()"
d exp(!i)[> k U (, "1)> k \, G G ' (2 )
(204)
where ... includes the initial state thermal average. Now, since the phonon quantum numbers in expression (204) are associated only with thermal averaging, it is clear that the same result is also obtained if one starts from expression (167) and repeats the above procedure. Hence, expression (204) is a general result for the scattering spectra (165) and (167) and yields information on the relative change of the quantum state of the heatbath with reference to its initial temperature only. Expression (204) satis"es the unitarity condition (168) as can be easily veri"ed by carrying out integration over d and making use of Eqs. (200)}(203). The explicit calculation of expression (204) may become very tedious because it requires taking the averages of exponentiated operators. Namely, expanding the operator U (, "1) from Eq. ' (203) in a power series and averaging term by term would be exceedingly impractical unless the in"nite series is truncated. In the latter case, however, the procedure would lead to nonunitary results, and that would be in con#ict with the previously imposed requirements on the properties of the scattering spectrum. To circumvent this di$culty we shall resort to the generalized cumulant expansion to calculate the averages of exponential operators [300]. Following this procedure and reverting back to (, R)-variables we can write [> k U (, )> k \"exp[C(, R, )] G G '
(205)
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and hence NkG
2
(E, K)"
d dR exp!(i/ )[(E)! (K)R]#C(, R) , 2 (2 )
(206)
where C(, R)"C(, R, "1) .
(207)
The property C(, R, "0)"0
(208)
which immediately follows from Eqs. (203) and (205) guarantees that the unitarity of spectrum (206) is obeyed. This holds irrespective of any particular form of C( ) as long as condition (208) is satis"ed. The cumulant expansion [300] gives C(, R, )" C (, R, ) , L L in which the nth-order cumulants are de"ned by
C (, R, )"(!i)L L
H
d 2
HL\
(209)
W (, R, )2W (, R, )> \ d [> ' ' L kG L kG
H H (!i)L ¹ W (, R, )2W (, R, )> \ , (210) " d 2 d [> H ' ' L kG L kG n! and the subscript c denotes the cumulant average of the operators that is constructed from the `ordinarya averages > k 2> k . The procedure for carrying out cumulant averages is detailed G G in Ref. [300] and here we only write down for illustration the explicit forms of the "rst few cumulants of arbitrary operators A, B, C2 in an obvious notation:
A " A , AB " AB! A B , ABC " ABC![ A BC# B CA# C AB]#2 A B C ,
(211)
etc. In a simpler case in which A"A( ), B"A( ), C"A( ),2 etc., and A( )" A( )" A( )" A2 etc., we have A( ) " A , A( )A( ) " [A( )! A][A( )! A] , A( )A( )A( ) " [A( )! A][A( )! A][A( )! A] , (212) etc.
B. Gumhalter / Physics Reports 351 (2001) 1}159
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Eqs. (206)}(210) provide a framework for a systematic calculation of the energy and parallel momentum resolved scattering spectrum (176). These equations are exact as no approximations have been made in their derivation. In this respect they are equivalent to the ¹-matrix formalism for calculating the scattering intensities which was discussed in Section 3.3.3. Yet, the main advantage of the present approach over the standard ¹-matrix approach is that it o!ers a uni"ed and from the calculational point of view a much easier approximate and nonperturbative treatment of the single- and multiphonon scattering spectra that satisfy the requirements discussed in Section 3.4. At this stage it is important to observe that expression (209) is not an expansion in powers of the coupling constant g, i.e. C 's do not exhibit simple scaling behavior JgL. Rather, each operator L W comprises all powers of g, whereby the operator W ( ) de"ned through Eq. (202) comprises all ' powers of the original interaction \ d [> ' kG kG
(213)
already represents an in"nite ascending series in powers of g (i.e. k W> k \ G G
(214)
and this is, according to Eqs. (183)}(186) and (197), an in"nite ascending series in powers of g. It should also be noted that contributions to C(, R, ) which are linearly proportional to , and thereby to , arise also in higher-order C 's. For later convenience we shall denote the sum of all L such terms by !i and call it `relaxation shifta following the analogy with a similar contribution occurring in the propagators of localized states describing photoemission from atomic core levels [175]. Since expression (205) has the appearance of a vacuum #uctuation amplitude in the variable , we can make use of Eq. (201) to write in an obvious compact notation: [> R k W (, )U (, )> k \ G G ' ' , C( )" C(, R, )"!i [> U (,
)> \ R
k k G G '
(215)
where > k (, )"U (, )> k can be thought of as the -evolved wavefunction > k in the G G G '
-space. From this we can conclude R C( )"C(R)"const. lim R
H
(216)
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This limit enables us to anticipate the expression for C( ) in a general form [301]:
C( )"C ( )!
\
d ()[1!e\ HJ!i ] ,
(217)
where the integral term on the RHS describes all `dynamica contributions generated by `interactionsa W (, ) in which plays the role of a free, yet unspeci"ed parameter. To "nd the explicit ' form of the `weighted density of excitationsa ()50 we take the derivative of expression (217) with respect to and observe that in the limit PR the oscillating term () exp(!i ) gives zero contribution to the integral on the RHS of (217). Thus, we obtain
C( )!C(R)"!i
\
()e\ HJ d"!i
\
()e\ HJ d"!i , H
(218)
where ()"(). Taking the inverse Fourier transform of (218) we "nd
i e HJ[C( )!C(R)] d
()" 2 \
(219)
and this expression can, in principle, be calculated by making use of Eq. (215). Its substitution back into Eq. (217) enables, again in principle, the calculation of C(, R, ). However, the signi"cance of expression (217) is not in its practical applicability but rather in its general structure. Thus, quite generally, C(, R, ) comprises the terms which are constant (independent of ), the terms linear in (described by the term !i and deducible from (217)), and oscillatory and bounded functions of . A glance at expression (200) reveals that the same applies to the dependence of C(, R, ) on the projecting variables (, R) comprised in . The appearance of the `relaxation shifta represents a subtle issue whose resolution requires careful inspection of the origin and properties of . It is intuitively clear that for the scattering boundary conditions one should have "0, and here we shall present only a general argument in support of that. With the help of de"nitions from Eqs. (179)}(193) we can write lim exp(i ) > k , SR exp(!iH )S> k , G G ? H " lim > k , exp(iH )SR exp(!iH )S> k , G G H " lim > k , SR( )S(0)> k , G G H " > k , S> k , , G G
(220)
where in the last step the factorization property of the correlation function SR( )S(0) in the limit
PR has been invoked following the general arguments of Levi and Suhl [276]. This property arises from the fact that the variables (, R) introduced in Eq. (176) are associated with vibrational
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#uctuations of the unperturbed crystal, and these are uncorrelated in the (PR, RPR)-limit that is equivalent to the limit PR. Now, the last line of Eq. (220) implies that the large- limit of expression (205), which is by de"nition equal to the expression on the LHS of (220), is also
-independent. Making use of Eqs. (217) and (220) we can write
lim exp !i !i ()[1!e\ HJ] d "[> k S> k \"const. G G \ H
(221)
and this can be satis"ed only if
!i "C ( )!i
\
() d"0 .
(222)
This implies "0 or, in other words, that there is no `relaxation shifta in the scattering spectrum (206) and hence neither in (176). We shall also see in later applications of the cumulant expansion in concrete calculations of the scattering spectra how the property "0 directly arises from the structure of W and the (, R)-dependence of cumulants (210). ' Hence, the only term surviving in the limit of Eq. (221) is
exp !
\
d () "exp(!2=) ,
(223)
which through comparison of Eq. (221) with Eq. (145) provides another justi"cation for the expression of the Debye}Waller factor because the distorted wave eigenstates > k and \ k can G G di!er only by an irrelevant phase factor. The above-discussed general properties of C( ) can be summarized by writing the latter in the most general form:
C( )"!
\
()(1!e\ HJ) d
(224)
and this has the following important implications on the scattering spectrum de"ned by Eq. (176): (i) Existence of a sharp line arising from the (, R)-independent component of C(, R, "1). This line, hereafter referred to as elastic, is given by N k (E, K)"exp(!2=) (E) (K) , G 2
(225)
and its weight is described by the Debye}Waller factor, Eq. (223), which is obtained from exponentiation of the sum of constant factors present in C(, R, "1). This property enables identixcation of the thus-derived elastic line in spectrum (176) with the true no-loss line describing specular elastic scattering. (ii) Existence of inelastic side bands arising from the oscillating (dynamic) component in the integrand on the RHS of Eq. (224). The total spectral weight of the inelastic structure follows from the unitarity of the spectrum and is given by 1!exp(!2=).
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Apart from the above-discussed features, expression (176) exhibits additional interesting properties, one of which is the shape of the scattering spectrum in the limit of strong coupling, viz., 2= > > I I H \ \ \ \ I I H K QG K QG IX IX H IX IX H
"g > > I I H K QG IX IX H IX G K (#)cRK Q G X cK X aRQ , #g VHI K XQ \ \ \ \ I I H K QG IX IX H
(233)
in which on-the-energy-shell matrix elements V for one-phonon emission, (#), and absorption, (!), are given by IX ($)"2GH
"(Q ,!(Q#G))"(Q ,!Q !G ,!Q !G ) . H H V V W W
(252)
Here one should note the minus sign in front of the second term in the square brackets on the RHS of Eq. (251). Taking higher-order commutators the next terms in the series (250) can be calculated. We "rst write down the expression for W . In order to avoid cumbersome notation we shall again suppress GO0 contributions from the summation over parallel wavevectors because these can be easily restored in "nal expressions. Thus, restricting the phonon wave vectors to the "rst SBZ and using the combination rule (238) we "nd g W "! [G , [G , H]] 2 g R Q cK Q aQ aQ R [(Q #Q )VHI IXK (!)VHI K KX IXQ (#)cK K XQ H YHY > > Y > IX > YIX H HY 2 K QQ IX IX IX YHHY
"
R Q cK Q aQ R a X IX (#)VHYI !(Q #Q )VHI K KX IQX (!)cK H HY K\QY) \ \ YIX \ IX HY QH X IX (!)VHYI a a !(Q !Q )VHI K Q cK Q K KX IQX (!)cR H HY K>QK \ Y > IX \ YIX QH QHY X IX (#)VHYI #(Q !Q )VHI aR aR ] . K Q cK Q K KX IQX (#)cR H HY K\QK > Y \ IX > YIX QH QHY
(253)
The "rst two terms in the square brackets on the RHS of expression (253) describe an interplay between phonon creation and annihilation processes and as such comprise both diagonal and nondiagonal operator components. On the other hand, the last two terms describe the two-phonon creation or annihilation processes and comprise only the nondiagonal components. The next term in series (250) which is also quadratic in the coupling constant g derives from G . In the present approximation for G (Eq. (248)) it is given by W "ig[G , H]"0
(254)
and vanishes because it is obtained by taking a commutator of two diagonal operators. Higherorder terms W K can be obtained in a similar fashion but writing down their explicit forms becomes increasingly tedious due to the presence of m-fold products of the on-shell matrix elements V($). Once the various terms W K in series (250) have been calculated one can readily determine their (, R)-dependence in the `interaction representationa de"ned by Eqs. (202), (229) and (230). This is
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91
accomplished by replacing the phonon operators in on-the-parallel-momentum and energy shell expressions for WK by their interaction representation forms given by aQ ()"exp(iH)aQ exp(!iH)"aQ exp!i[Q !(Q#G)R] H H H H
(255)
and analogously so for aRQ () which is obtained by hermitian conjugation of expression (255). H Substituting WK, obtained from W K by applying (255), into Eq. (210) one can systematically ' calculate the contributions to nth-order cumulants in expansion (209). The contributions to "rst-order cumulant C can come only from W K that are diagonal in both the projectile and phonon number operators. In the present approximation leading to (250) the "rst such term is derived from W given by Eq. (253). Recovering the G-dependence in this expression and constructing W() thereof we "nd using Eqs. (213) and (214): ' C (, R)"!i [Q !(Q#G)R] H QG H ;VHI IX (#)[n(Q )#1]!VHI IX (!)n(Q ) . K XQ K XQ \ \GK H > >GK H
(256)
According to the discussion in Section 4.2 this expression represents the `relaxation shifta term because it is linear in the (, R)-variables. The second-order cumulant C (, R) in expansion (209) is obtained by substituting expression (250) into (210) and carrying out the double ` -ordereda integrals. Here W and W give rise to separate contributions describing emission and subsequent reabsorption of one and two phonons originating from the same vertex in the -space, respectively. Following the arguments for small contributions of simultaneous two-phonon processes to the phonon matrix elements (cf. discussion following Eqs. (246) and (247)), and to remain consistent with the neglect of nondiagonal components of W , we shall consider only the lowest-order e!ect arising from the diagonal terms of W , and these have already been accounted for in C (Eq. (256)). Thus, by substituting (251) into (210) we "nd Q GR Q C (, R)"! VHI K X I QXG G K (#)[n(Q )#1][1!e\ S H O\ > ] G\ \ G H QG HIX Q GR Q #VHI QXG G K (!)n(Q )[1!e S H O\ > ]!C (, R) . K X I G> > G H
(257)
The term !C (, R) on the RHS of (257) is the `relaxation shifta of the second-order cumulant originating from W , which is exactly cancelled out by the `relaxation shifta occurring in and equal to the "rst-order cumulant (256) but originating from W . Thus, the absence of `relaxation shiftsa from C(, R) which was established in Section 4.2 on a general level is here explicitly demonstrated on the example of "rst- and second-order cumulants. The same procedure as described above can be followed to obtain higher-order cumulants (i.e. for n'2). In this process special attention must be paid to consistently take into account the e!ects of relaxation shifts produced by higher-order W N's in the lower-order cumulants and by lowerorder W O's in the higher-order cumulants, so that they cancel out. Their cancellation also serves as a test of consistency of the various approximations invoked at the various stages of calculation
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of C 's. However, the explicit calculation of C for n'2 becomes increasingly tedious and will not L L be pursued here. Estimates of the e!ects of such higher-order correction terms have been discussed by Dunn [302] for a simpler problem of polaron propagation, and there they have been found to be very small.
5. Scattering from 6at surfaces in the SSF approach 5.1. Single-phonon scattering regime and distorted wave Born approximation (DWBA) The results of Section 4.3 can be most directly implemented in the calculation of inelastic scattering spectrum in the distorted wave Born approximation (DWBA). To illustrate this we "rst approximate the full C(, R) by the sum of expressions (256) and (257), viz., C(, R)KC# (, R)"C (, R)#C (, R) ,
(258)
where the meaning of superscript EBA will become clear in the next subsection. Substituting this on the RHS of Eq. (206), expanding the so-obtained exp[C# (, R)] into a power series and retaining only the "rst two terms, we "nd N"5 (E, K) k G 2 "e\5#ki Ts (E) (K) #e\5k#G2
VKHIG X IQXG G KG (#)[n(Q )#1] (E# Q ) ( K# Q# G) \ \ H H QG HIX
X XG # VHI Q G K (!)n(Q ) (E! Q ) ( K! Q! G) , K I G> > G H H QG HIX where we have introduced the notation
(259)
X XG (260) 2=#kG " VKHIG X IQXG G KG (#)[n(Q )#1]#VHI Q G K (!)n(Q ) . K I G> > G 2 \ \ H H QG HIX Expression (260) plays the role of the Debye}Waller exponent (DWE) for the Debye}Waller factor (DWF) pertinent to the scattering spectrum obtained by substituting approximate expression (258) into Eq. (206). Its features will be discussed in more detail in the next subsection in connection with multiphonon scattering events but here we only observe that 2=#kG Jg. However, it is evident 2 already at this point that the quantity exp[!2=#kG ] gives the weight of the elastic specular 2 component of the scattering spectrum represented by the (E) (K)-term on the RHS of expression (259). In future discussions we shall also refer to this factor as the intensity of the no-loss line. The remaining two terms on the RHS of Eq. (259) represent the DWBA expression for the component of the scattering spectrum describing one-phonon emission and absorption processes, respectively. They are multiplied by the same DWF as for the no-loss line. Apart from the Bose
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factors the intensities of the phonon emission and absorption events are determined by the corresponding on-the-energy-shell factors V(#) and V(!), respectively, which according to de"nition (234), are seemingly highly singular because of the presence of energy-conserving -functions. However, in the case of scattering processes proper (i.e. k Pk which excludes G D transitions into bound states of the atom}surface potential ;(r)), the square of this -function should be converted into a Kronecker symbol [24,25,251] following Eq. (91), so that X IX X IX ($)"2 > the exchange of one-phonon quantum are strongly correlated by the on-shell con"nement and X XG hence the di!erence between VKHIG X IQXG G KG (#) and VHI K I Q G K (!) for the same initial quantum G> > G \ \ numbers can be signi"cant in the quantum scattering regime. In this situation it may prove advantageous to re-express the EBA scattering spectrum (263) in a form in which the relative magnitudes of V($) for one-phonon emission and absorption, respectively, could be exploited so that the e!ect of recoil can be traced more easily. Introducing a compact notation N#kG (, R)"exp[!2=# (0, 0)#2=# (, R)] 2
(266)
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and employing trigonometric identities to separate real and imaginary parts from the products of exp$i[Q !(Q#G)R] and the Bose occupation distributions in the scattering function H 2=# (, R), we can transform (266) to the form [297]
N#kG (, R)"e\5k#G2 exp 2
(4[n(Q )#1]n(Q ) VHI K XKIXG Q G (#)VHI K XKIXG Q G (!) G G\ \ G G> > H H QG IX H
;cos i ln
Q G [n(Q )#1]VHI K XKIXG \ \ (#) G H #[Q !(Q#G)R] H n(Q )VHI K XKIXG Q G (!) G G> > H
. (267)
The expression under the "rst square root in the exponential on the RHS of Eq. (267) is invariant under the permutation V(#)V(!) of the quantities describing one-phonon emission and absorption processes. By contrast, the argument of the cosine function in the large round brackets on the RHS of (267) is clearly sensitive to this permutation. This demonstrates the overall sensitivity of the EBA scattering spectrum to the recoil e!ects associated with the one-phonon interaction vertices. On the other hand, it is also clear from expression (267) that the EBA spectrum does not keep memory of the cumulative recoil e!ects arising from sequential phonon exchange processes. This intervertex correlation, which has been neglected by adopting approximation (258), can be approximately restored by con"ning the calculated EBA spectrum (263) or (267) to the energy and parallel momentum shell, as has already been stated earlier. This approximate restoration of correlation among a manifold of phonon exchange processes will work increasingly better with the diminution of the relative magnitude of recoil. Hence, the validity of the EBA is expected to be enhanced in the scattering regime in which the cumulative e!ect of recoil, and hence of the intervertex correlation, on the projectile propagation diminishes. This statement is, in fact, a generalization of criterion (240) which served as a guideline in the derivation of expressions leading to the EBA scattering spectrum (263) and the above discussion of its validity. 5.2.2. Debye}Waller factor in atom}surface scattering The quantity termed Debye}Waller factor (DWF) in atom}surface scattering theory, and in particular the general expression introduced in Eq. (223), or the more specialized on-the-energyshell EBA expression (260), have neither the same form nor the same physical meaning as the more popular counterpart derived, for instance, in the theory of X-ray or neutron scattering from crystals. In this section we shall clarify the di!erences between these two types of the DWF but also point out some of their common features. To this end we "rst brie#y review the properties of the DWF encountered in neutron scattering theory and pinpoint some of its characteristics relevant to our further discussion. The standard notion of the Debye}Waller factor introduced in the theory of neutron scattering from crystal lattices [243,244] describes the attenuation of scattered beam intensities due to quantized thermal vibrations of the lattice atoms. In this theory the DWF e!ect is calculated in the "rst-order Born approximation which holds in the case of weak scattering of neutron beams by the short-range potential of crystal ion cores. However, the dependence of the projectile}ion core potential on the displacement u of the lth ion in the crystal is not assumed weak in this case. Hence, J
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an expansion of the potential in powers of displacements, in the form (93), and its truncation beyond the linear term does not represent a good approximation to the inelastic neutron scattering problem. Rather, the potential is left in the form of a pairwise sum of the projectile}ion pair pseudopotentials, l v(r!rl !ul ), so that its matrix element taken between the incoming and outgoing plane waves, r k "exp(ikr )/(¸ and r k "exp(ikr )/(¸, respectively, where G G D D ¸ is the volume of the quantization box, comes out as
k v(r!rl !ul ) k "v(k !k ) exp[!i(k !k )(rl #ul )] , D l G D G l D G
(268)
where
1 dr e\ krv(r) . v(k)" ¸
(269)
The displacements in the argument of the exponential function on the RHS of Eq. (268) can be expanded in the normal modes of the bulk crystal:
eq e\ qrl (aq #aR!q ) , ul " H H H 2NM q q J H H
(270)
where q, j and eq are the wavevector, index and polarization vector of a quantized normal mode, H respectively, and M is the ion mass assumed equal for all ions in the lattice. Expansion (270) then J enables manipulation with exponentiated displacements in the products of matrix elements (268) by the application of boson algebra to phonon mode creation and annihilation operators aRq and H aq , respectively. This leads to the di!erential neutron scattering cross section in the "rst Born H approximation (BA) in the form [307]
d k (M¸) " D v(k) d e\ #O e\ krl \rp Sl p (k, ) d(E) d k (2) lp D \ G
(271)
where E"E !E and k"k !k denote the energy and momentum transfers to the target, D G D G respectively, d denotes the di!erential of the "nal scattering angle, M is the projectile mass, and D Sl p (k,) is the exponentiated displacement}displacement correlation function de"ned as Sl p (k, )"[e\ kul e kup O\
eq k H [2n(q )#1] "exp ! H q 2NM q J H H
eq k H exp ! ([n(q )#1]e\ SqH O\qrl !rp #n(q )e SqH O\qrl !rp ) . H H 2NM q J qH H (272)
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Here the "rst factor on the RHS of (272), viz., the exponential
eq k H [2n(q )#1] e\U k"exp ! H q 2NM q J H H
(273)
is recognized as a standard Debye}Waller factor multiplying the square of the transition matrix element in the "rst-order BA expression for neutron scattering cross section. A rough estimate of the magnitude of the Debye}Waller attenuation in atom}surface scattering was given long ago by Weare [308] by using a slightly modi"ed neutron scattering formalism extended also beyond the "rst-order BA. The resulting approximate quasiclassical expression for the DW exponent as a function of the substrate temperature ¹ , rederived also later by Levi and Suhl [276], can be cast in the form
3( k ) ¹ ME cos ¹ X G G "24 lim 2=(¹ )" M k Mk J " J " " " 2 "
.
(274)
Here is the surface Debye temperature of the substrate, k is the change of the projectile " X momentum normal to the surface, E and are the incoming energy and angle of scattering of the G G projectile, respectively, and k is the Boltzmann constant. E in this expression is sometimes also G corrected for the surface potential well depth D (Beeby's correction [66]) in which case ( k ) is X replaced by [( k )#8MD]. However, the form of the DW exponent (274) can be justi"ed only in X the regime of impulsive scattering [245,276] and therefore its validity is of limited range. In particular, for incident energies typical of the thermal energy He atom scattering from surface phonons and soft projectile}surface interactions the approximation of impulsive scattering has been shown to become unreliable [251] for making quantitative comparisons with the experimental data, as is illustrated in Fig. 12. Hence, in the majority of the studied systems the full EBA calculation of the DWF is required. To pinpoint the features that distinguish the neutron scattering DWF, Eq. (273), from the atom}surface scattering DWF calculated in the EBA, we shall rewrite the latter as an exponential function of expression (260) and then recall the properties of the re#ection coe$cient calculated in the DWBA (Eqs. (151) and (152)). This gives
X XG HIX IQXG G K (!)n(Q ) e\5 k#G2 "exp ! VHI Q G K (#)[n(Q )#1]#VK K I G\ \ G G> > G H H QG HIX
"exp ! R"5 . DG D$G
(275)
This relation clearly demonstrates the relation between the EBA DWF in atom}surface scattering and the corresponding inelastic on-the-energy-shell re#ection coe$cients calculated in the DWBA. Now, the comparison of expressions (273) and (275) reveals several important di!erences between the two types of DWF. The "rst essential di!erence shows up in the on-the-energy-shell vs. o!-the-energy shell character of the matrix elements appearing in the exponent of the two DWFs.
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Fig. 10. Schematic of diagrammatic representation of the cumulant expansion of N# (, 0) [energy resolved-only version of expression (266)] expressed in terms of propagators exp[!i ( ! )] of nondispersive phonons which are K L represented by wavy lines ( " ). The light full line denotes positive direction (propagation) along the -axis. KL Intermediate and are integrated over the interval (04 4 4). The on-the-energy-shell matrix elements V($) associated with phonon creation (#) and phonon annihilation (!) vertices, respectively, are represented by crosses. Double cross denotes absolute square of the same factor. The "rst and second terms in the exponent in the second line of the "gure represent the -only contributions C () and C () corresponding to expressions (256) and (257), respectively.
In atom}surface scattering the assumed linear projectile}phonon coupling (93) leads to the exponent of the EBA DWF in terms of the on-the-energy-shell interaction matrix elements V($) which determine the amplitudes of emission (#) or absorption (!) of a single real phonon. The exponential form of this DWF is due to the sequential or repeated uncorrelated emission or absorption of in"nitely many single real phonons in the course of scattering [245], and its appearance is connected with the conservation of the norm of the projectile wavefunction, in the sense used in the derivation of the optical theorem. This may be illustrated by the diagrammatic expansion of expression for N#kG (, R) de"ned by Eq. (266). Assuming for simplicity zero substrate 2 temperature, suppressing the momentum projecting variable R and retaining only the energy projecting variable , the diagrammatic representation of this expansion is sketched in Fig. 10. On the other hand, as has also been demonstrated in Ref. [245], the exponential form of the DWF typical of neutron scattering cross section given by Eq. (273) arises from the coupling of the projectile to all orders of crystal ion displacements (because the corresponding matrix elements (268) contain displacements in the exponent). Among other things, this type of nonlinear coupling gives rise to instantaneous multiple virtual phonon exchange represented in diagrammatic expansion of the correlation function (272) by closed phonon loops attached to the two endpoint vertices associated with the matrix elements v(k) and vH(k) in expression (271) for the scattering cross section (see Fig. 11). These closed loops describe o!-shell virtual phonon exchange or quantum #uctuations, and sum up to an exponential that is given by the expression on the RHS of Eq. (273). The described formal di!erence between the two types of the DWF arises in connection with their di!erent physical origin. The physical interpretation of the EBA DWF corresponding to the scattering spectrum (263) can be visualized by calculating the mean number of phonons n (k , ¹ ) that have been exchanged between the projectile and the target in the course of collision. G
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Fig. 11. Schematic of diagrammatic representation of van Hove}Glauber exponentiated displacement correlation function, Eq. (272), for nondispersive phonons (q " ) expressed in terms of phonon propagators exp[!i ( ! )] K L which are represented by wavy lines ( ", 0). Light full line denotes positive direction along the -axis. The number of KL dots overlapping in a vertex is equal to the number of endpoints of phonon lines terminating in the same vertex. O!-the-energy-shell factors $i(k)u are associated with the vertices at and 0, respectively.
Thus, we have n # (k , ¹ )" p > , SR aRQ aQ S> , , G ? kG H H kG Q H ? (276) "2=#kG , 2 where the average has been calculated by resorting to the EBA formalism outlined in the previous subsections. Hence, it is seen that the DWF of the EBA atom}surface scattering spectrum is an exponential function of the mean number of real phonons exchanged during the collision, exp[!n # (k , ¹ )]. It is seen that there is a complete analogy with semiclassical expression (114), G with a supplement that the EBA expression (276) includes the e!ects of quantum recoil of the projectile. On the other hand, in the case of neutron scattering from crystals the Debye}Waller factor appearing in the "rst-order Born approximation cross-section formula (271) is given by an exponential function of the mean number of virtual or #uctuating phonons associated with the
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vibrating crystal. The properties of the latter DWF, which arise from the projectile}phonon coupling to all orders in crystal displacements, can be generalized also beyond the "rst-order Born approximation result [276]. The second di!erence manifests itself in the dependence of the two types of Debye}Waller exponents on the initial and "nal quantum numbers of the scattered projectile. In the case of atom}surface scattering the Debye}Waller exponent in (275) is summed over all "nal scattering vectors and hence depends only on the initial quantum number k . For this reason it appears as G a normalization factor proper for the scattering spectrum (263). On the other hand, the standard neutron scattering Debye}Waller exponent in (273) is k-resolved and varies with the "nal scattering angle. Thereby it plays the role of a renormalization factor for the strength of the scattering matrix element v(k) (in the form of Holstein renormalization [239], cf. Eq. (106)), which appears due to quantum #uctuations of the scattering centers rather than representing a normalization factor for the total scattering cross section. In the quantum scattering regime the di!erence between the magnitudes of the two types of DWF (273) and (275) can be signi"cant because of the di!erent matrix elements and the absence or presence of the on-shell constraints in the respective exponents. However, in some scattering regimes these constraints may become less restrictive and thereby less important. In this case the di!erence between the two distinct Debye}Waller exponents may become increasingly small. To illustrate this we consider a simple example of a one-dimensional motion of the projectile in the quasiclassical regime of fast scattering from the surface [245]. In the quasiclassical scattering regime the matrix element of the force operator in the coupling interaction in expression (93) can be replaced by its classical analog F (t), which is equivalent to treating the projectile motion in the trajectory approximation (see Section 3.3.2). Then, on account of the slow motion of lattice atoms during the fast scattering event, we can carry out the integration over the time t in G (Eq. (183)) by taking the average value u of the displacement u(t) out of the time integral. This is e!ectively equivalent to decoupling the particle from the lattice dynamics and yields
u 1 \ dt F (t )u (t ) i P dt F (t )"!(k)u , f G i" f ' '
\
(277)
instead of giving rise to expressions V($) that contain the energy-conserving -function on the RHS of Eq. (234). Hence, in this limit the di!erences between the matrix elements and on- and o!-the-energy-shell contributions are washed out and the exponents in the expressions on the RHS of Eqs. (273) and (275) become equivalent. In other words, in the classical limit in which the particle motion and the lattice dynamics are decoupled, the #uctuations of the phonon "eld yield the same form of the two DWFs [245]. A similar conclusion, but starting from di!erent, semiclassical representation of the collision dynamics and the ¹-matrix, was arrived at by Levi and Suhl [276]. 5.2.3. Scattering from Einstein modes The multiphonon e!ects in atom}surface scattering should manifest themselves most clearly in the case of excitation of low-energy vibrational modes which exhibit little or no dispersion. With incoming He atom energies exceeding several times the energies of such modes and for not too low substrate temperatures, one can clearly observe spectral peaks corresponding to multiple excitation
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of dispersionless modes both on the energy loss and gain sides in the TOF spectra of a number of systems [105,309}313]. Depending on the speci"cities of the systems studied, these modes can be associated with either a parallel or a perpendicular motion of surface atoms. The problem of multiple excitation of dispersionless modes of various provenance has been extensively studied in the past because of its relevance to the thermodynamical properties of a large variety of systems. The peculiarity of multiple excitation of dispersionless quanta by external probes lies in the possibility of strong interference among the processes with di!erent exchange of total momentum but the same total energy balance, viz., loss or gain or even zero energy transfer, which cannot occur in the case of dispersive modes. Depending on the level on which the exchange of momentum between the probe and nondispersive excitations is treated, di!erent results are obtained for the excitation probabilities. In the case when the probe is a scattering projectile this depends on the approximation in which the projectile recoil and correlation in multiquantum emission or absorption events is taken into account. In the regime of fully uncorrelated events of emission and absorption of nondispersive bosons the closed-form solutions have been discussed by Vineyard in Ref. [314] for neutron scattering from nondispersive modes of harmonic oscillators, by Ibach and Mills in Ref. [113] for electron scattering from optical phonons in EELS, by Manson in Ref. [315] for He atom scattering from Einstein modes, and by Mahan in Ref. [316] for photon absorption by Einstein oscillators. In all these results the interference e!ects give rise to a modi"cation of the fundamental excitation probabilities which, in one way or another, are expressed through the modi"ed Bessel functions with total momentum transfer in the argument. In the following, we shall demonstrate that the EBA scattering spectrum approach o!ers a solution for atom scattering from Einstein phonons which goes beyond these results in that it also takes into account the recoil e!ects during each phonon emission or absorption event. A closed-form solution for the EBA scattering spectrum describing inelastic atom scattering from Einstein-like modes is obtained by separating the nondispersive branch j of frequency out of the scattering function in the exponent in expression (266). This gives (, R)N (, R) , N#kG (, R)"N# k k G 2 G 2 2
(278)
where the superscripts Ein and dis denote components comprising Einstein and all other dispersive modes, respectively. In the remainder of this subsection we shall for the sake of simplicity disregard (, R), which is retrieved from expression (267), as its e!ect on the total scattering spectrum N k G 2 can be easily restored by a simple convolution procedure [311,312]. Denoting the single frequency of a set of Einstein oscillators by , the corresponding component of the scattering function (264) can be written in the form 2=# (, R)"V(R,#)[n( )#1]e\ S O#V(R,!)n( )e S O ,
(279)
where V(R,$) are obtained by carrying out the (K, Q, G, k ) summations over the separated j th X component of expression (264): V(R,$)" VHI K X I QXG G K ($) exp[$i(Q#G)R] . G8 8 G QG HIX
(280)
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Now, observing that [n( )#1]/n( )"exp( /k ¹ ) and introducing the notation V(R,#)#V(R,!)"M(R) ,
(281)
i[V(R,#)!V(R,!)]"N(R) ,
(282)
()"( !i /2k ¹ )
(283)
i V(R,#) , (R)"arctan[N(R)/M(R)]" ln 2 V(R,!)
(284)
and
we may write (, R)"e\5 exp(n( )[n( )#1][M(R)#N(R)] cos[()#(R)] N# k G 2 "e\5 exp(4n( )[n( )#1]V(R,#)V(R,!) cos[()#(R)] , (285) where 2="2=# ("0, R"0)"[n( )#1]V(0,#)#n( )V(0,!) . (286) Making use of the generating function expansion for the modi"ed Bessel function of the "rst kind, exp(z cos )" I (z) exp(il), we "nd J\ J (, R)"e\5 P (R)e\ JS O , (287) N# k G 2 J J\ where
[n( )#1]V(R,#) J I ((4n( )[n( )#1]V(R,#)V(R,!)) . (288) J n( )V(R,!) This gives for the separated Einstein phonon component of the scattering spectrum: P (R)" J
N# (E, K)"e\5 N (K) (E#l ) , k G 2 J J\ where
dR N (K)" e\ KRP (R) . J J (2)
(289)
(290)
Expression (289) encompasses the recoil e!ects to the same level of accuracy as does expression (267) describing scattering from dispersive phonons. The sensitivity of the scattering spectrum (289) to recoil e!ects, i.e. to the permutation V(#)V(!), manifests itself through the "rst term in the large round brackets on the RHS of Eq. (288). In this respect the present expression (289) describes the projectile recoil to higher-order e!ects than the corresponding formulae derived in Refs. [113,314,315] and [316], but at the expense of the simplicity of expression for N (K). J
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The angular integrated scattering spectrum corresponding to expression (289) is obtained upon replacing N (K) on the RHS of Eq. (289) by P (R"0). Accordingly, the DWF of such lateral J J momentum integrated spectrum [317] is given by e\5P (R"0) where P (R"0)O1 and both factors contain the contributions from all inelastic scattering channels. Expressions (288) and (289) exhibit interesting structure. Expression (288) contains complex quantities V(R,#) and V(R,!) whose di!erence !iN(R) measures the recoil of the projectile in the one-phonon creation and annihilation events. Namely, in the limit of classical recoilless trajectory approximation one has V(R,#)"V(R,!) and consequently N(R)"0 (E) is given by a generalized and (R)"0. In this limit the angular integrated spectrum N# k G 2 Poisson distribution (cf. Eq. (86) in Ref. [251]) typical of the forced oscillator model applied to Einstein phonons. However, as is seen from expressions (288) and (289), the lateral momentum resolution and recoil e!ects destroy such a simple structure. The deviations of N# (E) from the k G 2 generalized Poisson distribution grow larger as N(R) increases, i.e. as the quantum recoil e!ects become more important. (E"0, KO0)O0, meaning that due to coupling to the For "nite ¹ one generally has N# k G 2 phonon heatbath a "nite momentum transfer may occur also in nondi!ractive elastic collisions. The spectral intensity of such o!-specular elastic transitions in the K direction is given by e\5N (K). In the limit of specular elastic transitions (KP0) this tends to e\5P (RPR) (K). The quantity e\5P (RPR) may be identi"ed with the DWF corre sponding to the elastic specular peak. However, from de"nitions of P (R), Eq. (288), and V(R,$) we can deduce that P (RPR)P1 because of the destructive interference e!ects in the argument of P (R). Hence, the DWF corresponding to the elastic specular peak is again given by e\5 .
5.2.4. Extreme multiphonon approximation and semiclassical trajectory approximation (TA) as the limiting cases of the EBA As has already been pointed out in Section 4.2 on the general level, the extreme multiphonon limit of the scattering spectrum (263) is reached in the regime in which the Debye}Waller exponent 2=, Eq. (223), of the scattering spectrum is large, i.e. when the mean number of exchanged phonons is large [25,28,238,251]. Focusing the discussion on the EBA spectrum, we observe that in this case the major contribution to the Fourier transform in Eq. (263) comes from small values of the exponent [2=# (,R)!2=# ] upon expanding it into a power series in and R and retaining only the linear and quadratic terms. Collecting the leading contributions to this series we obtain
[E! (¹ )] ( K ) ( K ) ! V ! W exp ! 2 (¹ ) 2(¹ ) 2(¹ ) S V W lim NkG (E, K)" 2 (2) (¹ ) (¹ ) (¹ ) # < S V W 5 Here (¹ ) is given by
.
(291)
(¹ )" Q VKHIG X IQXG G KG (#)#[VKHIG X IQXG G KG (#)!VKHIG X IQXG G KG (!)]n(Q ) H \ \ \ \ > > H IX QGH (292)
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and has the meaning of the mean total energy transfer or the "rst moment of the energy resolved scattering spectrum N#kG (E) obtained by integrating expression (263) over d(K). Note here the 2 minus sign in the square brackets on the RHS of Eq. (292) which gives rise to a recoil-induced temperature-dependent contribution to (¹ ). The temperature-dependent spectral widths (¹ ) ? are given by (¹ )" VHI QXG G K (#)[n(Q )#1]#VHI QXG G K (!)n(Q ) , K X I K X I G \ \ Y G G> > G H H ? GQ IX H
(293)
where stands for either Q , (K!K ) , or (K!K ) . H GV GW Expression (291) should represent a good approximation to the starting EBA expression (263) in the multiphonon limit in which E and K, which are con"ned to the scan curve, do not exceed much the corresponding spectral widths (292) and (293), respectively. Due to the on-the-scan-curve condition imposed on E and K, the maximum of the spectrum (291) does not generally coincide with , viz., it may occur on either the positive or negative energy transfers, depending on the scattering conditions. The intensity prefactor [(2) ]\ exhibits the temperature dependS V W ence which approaches ¹\ behavior in the high-¹ limit. The detailed structure of the interaction matrix elements is of no importance for this behavior (e.g. pairwise interactions vs. some di!erent expressions) as long as the general structure of the scattering function (264) persists in that form. Note also that although expression (291) represents a limiting case of the momentum and energy resolved scattering spectrum, the value of the energy transfer is here given by (¹ ), Eq. (292), which is integrated over all "nal momenta and not by the corresponding K-resolved quantity as one might expect. This is due to the uncertainty in energy and momentum #uctuation involved in the short- and small-R component of the response of the phonon system which is only relevant in deriving expression (291). The temperature dependence of (¹ ) and (¹ ) explicitly a!ects the ? position of the spectral maximum in the extreme multiphonon regime. In (¹ ) this dependence is ? strong and analogous to that of the Debye}Waller exponent. On the other hand, in (¹ ) it is weaker as it only appears in the recoil correction in the square brackets on the RHS of Eq. (292). As for relatively high incoming projectile energies the di!erence between the gain and loss values of the interaction matrix elements, V(#) and V(!), respectively, becomes very small, the temperature variation of (¹ ) becomes weak and altogether vanishes in the recoilless scattering approxima tions like the TA (cf. Eq. (116)). However, the position of the spectral maximum of expression (291) may, nevertheless, be strongly temperature dependent as it also depends on , and which are S V W all strongly ¹ -dependent. The recoilless TA limit of the EBA scattering spectrum (263) is obtained upon replacing the DWBA scattering probability amplitude by its quasiclassical limit [24,255,319]: IX ($)"2 > G G H H QG HIX
(306)
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(k , ¹ ) vanishes in the recoilless trajectory approximation (TA) for the projectile motion [259], G yielding the result given by expression (116). However, the TA may fail even for heavier atoms (cf. Section 6.4 and Ref. [263]), and in the quantum scattering regime of gas}surface collisions at thermal energies the recoil is large which makes (k , ¹ ) strongly ¹ -dependent. This means that G term (306), which is generally negative, can become equal or exceed in magnitude the positive term (305) as ¹ is varied and the other parameters are kept constant. The surface temperature at which the two terms cancel each other de"nes the recovery temperature ¹ . 7.2. Quantum vs. classical results for energy transfer in benchmark systems The EBA calculations for the total energy transfer in HAS have been carried out for the same three prototype systems [297,313] whose scattering spectra have been calculated and compared with experiment in Section 6. Application of expression (302) to the HePXe(1 1 1) TOF spectra described in Section 6.1 yields the values of the angular resolved energy transfer denoted by full squares in Fig. 41 as a function of the angle of incidence for "xed experimental beam energy E "10.4 meV. The same type of calculation was then repeated with the theoretical EBA scattering G spectra and these results are denoted by a full line in Fig. 41. As seen from the two plots, the theoretical results for the angular resolved energy transfer are consistent with the values deduced from the experimental spectra, inasmuch as the calculated EBA values of the scattering spectra are (cf. inset in Fig. 41 and Section 6.1). Thereby the consistency required for the EBA-based calculation of the total energy transfer is satis"ed for this system. In a completely analogous fashion one proceeds with the system HePCu(0 0 1). The values of the angular resolved energy transfer deduced from the experimental and theoretical scattering
Fig. 41. Comparison of the theoretical angular resolved energy transfer in HePXe(1 1 1) collisions (full line) calculated from Eq. (302) with the values deduced from the available experimental HAS-TOF spectra (squares), given as a function of " ! /2 and for the scattering conditions as denoted. Inset: comparison of experimental (open squares) and G 1" calculated (full line) scattering spectrum for corresponding to the experimental point denoted by arrow.
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Fig. 42. Comparison of the theoretical angular resolved energy transfer in HePCu(0 0 1) collisions (full line) calculated from expression (302) with the values deduced from the available experimental HAS-TOF spectra (open circles), given as a function of " ! /2 and for the scattering conditions as denoted. Inset: comparison of the experimental G 1" multiphonon HAS-TOF (open circles) and theoretical (full curve) scattering spectra for corresponding to the experimental point denoted by arrow.
spectra as a function of the angle of incidence are compared in Fig. 42, in which they are denoted by the open circles and full line, respectively. Again, the theoretical results are consistent with the ones deduced from experiment. Finally, Fig. 43 shows a comparison between the experimental and theoretical values of the angular resolved energy transfer as a function of the surface temperature for the collision system HePXe/ Cu(1 1 1). The data are presented for several experimental incident He atom beam energies E and G "xed incident beam angle. Given all the uncertainties connected with numerical integration of the experimental TOF spectra and computation of their "rst moments in expression (302), the agreement between the two sets of results is very satisfactory, as much as it has been found for comparisons between the experimental and theoretical scattering spectra (cf. inset in Fig. 43 and Section 6.3). A good agreement between the experimental and theoretical angular resolved energy transfer established above for HAS from the three prototype surfaces enables a reliable estimate of the total energy transfer in these experiments. Fig. 44 illustrates the behavior of total energy transfer in the discussed systems as a function of the projectile incident energy over the interval which is relevant to HAS experiments as well as to the conditions of wind tunnel investigations of gas}surface collisions in space. It is noticeable that the magnitude of energy transfer is surface speci"c in the studied scattering regime, in contrast to the results from standard classical accommodation theories [360,361]. In particular, the energy transfer is larger for surfaces with enhanced surfaceprojected phonon density of states with perpendicular polarization. Here it is largest for the Xe/Cu(1 1 1) surface because the corresponding S-phonon exhibits perpendicular polarization and Xe adlayer-localization over almost the entire SBZ. This points to the dominant role which the surface modes with perpendicular polarization, that couple most strongly to He atoms, play in the energy transfer in gas interactions with atomically smooth surfaces.
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Fig. 43. Comparison of the temperature dependence of the angular resolved energy transfer (k , ¹ , ) calculated from G D HePXe/Cu(1 1 1) TOF spectra for four experimental E and "xed scattering geometry (open symbols), and from the G EBA formalism (full lines). The inset shows a comparison of the measured and calculated multiphonon scattering spectrum for E "45.11 meV, "503, ¹ "58.2 K. G G Fig. 44. Total energy transfer characteristic of HePXe(1 1 1), HePCu(0 0 1) and HePXe/Cu(1 1 1) collisions calculated from expression (304) and plotted as functions of the incoming He atom energy for "xed incident angle and substrate temperature. The starred line denotes the energy transfer obtained from the classical Baule formula (307) applied to HePXe binary collisions.
A quick semiquantitative estimate of the energy transfer in binary collisions is often made by using the classical Baule formula [364] applicable to collinear scattering geometry: 4M M E . " \ (M #M ) G
(307)
Here M and E are the mass and incident energy, respectively, of the incoming particle which G collides with particle of mass M initially at rest. This formula has also been frequently applied to gas}surface collisions although its applicability is limited to impulsive scattering. In the scattering regime spanned by the collision parameters and energy interval of Fig. 44 the Baule formula applies semiquantitatively only to the systems HePXe/Cu(1 1 1) and HePXe(1 1 1). In these two cases the frequencies of surface vibrations of heavy Xe atoms (with the corresponding mode energies below 4 meV, see Section 6) are much smaller than the inverse collision time and hence the impulsive scattering limit is reached. This is also in accord with independent testings of the applicability of the Baule formula to HAS from condensed multilayers of noble gas atoms on Si(1 0 0) surface [365]. On the other hand, the surface Debye frequency of copper is much higher ( !&24 meV [338]) and of the order of the inverse collision time characteristic of the " HePCu(0 0 1) system in the studied scattering regime. As a consequence, the impulsive scattering limit is not yet reached in the studied regime which makes the Baule formula inapplicable. The best agreement between the experimental and theoretical scattering spectra and angular resolved energy transfer that has been achieved for the prototype system HePXe/Cu(1 1 1) make this system the best candidate for a reliable calculation of the recovery temperature. To this end the
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Fig. 45. Temperature dependence of the total energy transfer in HePXe/Cu(1 1 1) collisions normalized to E and XG plotted for "503 and E "80 meV (full curve), 60 meV (long-dashed curve), 20 meV (short-dashed curve) and G G 2.4 meV (dash}dotted curve). The corresponding normalized values of at ¹ "0 obtained from the Baule formula corrected for the well depth are denoted by inverted triangle, circle, diamond and triangle, respectively. The scattered beam is heated on the average if (0. Inset: relative contributions of the phonon modes to the recovery temperature ¹ (see text). Fig. 46. Recovery factor ¹ /¹ for a prototype heatbath sustaining phonon modes typical of Xe/Cu(1 1 1) system plotted as a function of the incident energy E or stagnation temperature ¹ of the gas for three representative incident angles . G G The dashed}dotted line denotes the classical result ¹ /¹ "1.25.
¹ -dependence of the total energy transfer (304) must be computed over a wide interval and Fig. 45 shows the corresponding values normalized to vertical component of the projectile incident energy, E "E cos . The ¹ -dependence of (k , ¹ ) hinders energy transfer to phonons and causes XG G G G negative slopes in the plots. This arises from the larger phase space for projectile cPc transitions into "nal states with E 'E . There it may give rise to negative (k , ¹ ) (e.g. for E "2.4 meV and X XG G G "503 at ¹ '62 K) and hence to a heating of the scattered beam. In the classical theory this G e!ect is independent of the accommodation coe$cient and hence of [363]. Here, the universal G behavior of (k , ¹ ) for higher E , as exempli"ed by the near coincidence of the two highest energy G G curves in Fig. 45 and con"rmed by additional calculations at high E , manifests itself only for xxed G because the -dependence of three-dimensional scattering matrix elements is not contained G G solely in the factorizable scaling factor E [251,310]. Extension of the classical Baule expression in XG the cube model (K"0) to the present system is demonstrated in comparison with the quantum results for ¹ "0. Since ;(z) with the well depth of 6.6 meV supports three He atom bound states the inset shows peculiar low-E dependence of the recovery temperature (for which (k , ¹ )"0) G G calculated for the present phonon heatbath for "xed , for He coupling either only to S- or to S-, LG and SH-phonon modes. The small di!erence indicates that the major contribution is from strong He atom coupling to the vertically polarized S-modes [310]. Rapid variations in ¹ are caused by the kinematic focusing in S-phonon assisted cPb transitions for large parallel momentum transfer. The present quantum theory enables essential progress beyond the classical results by allowing the parallel momentum exchange with the phonons, multiphonon interference and quantum recoil of the projectile. Their interplay gives the recovery factor as a function of E (or ¹ ) and which G G for the prototype heatbath Xe/Cu(1 1 1) is shown in Fig. 46. Quite generally, ¹ is largest for normal
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incidence and only at higher E (i.e. higher ¹ ) quantum results may approach the classical limit G ¹ /¹ "1.25 so far observed only for rough technical surfaces [361]. Large deviations from the classical limit at low E are due to the quantum regime which allows larger K and transitions G a!ected by the bound states of the He}surface potential. The results for the recovery factor pertaining to the collision system HePXe/Cu(1 1 1) are qualitatively not system speci"c as they have been derived from the theory which is quite general and can be readily extended to the calculations of heat transfer in collisions of He [313] or heavier rare gas atoms [357,358] with clean surfaces in a wide range of the scattering conditions. All these results signify that in inelastic gas}surface scattering under the conditions of free molecular #ow the combination of quantum and temperature e!ects gives rise to a violation of the universality of the energy transfer and recovery factor predicted by the classical accommodation theory. Determination of the recovery temperature for atomically clean but strongly corrugated surfaces presents a more complicated problem. The recovery temperature of a clean LiF(1 1 1) surface has been measured in wind tunnel experiments and interpreted within a model adopted from the theory of neutron scattering from crystals [362]. The calculated results encompassing elastic di!ractive and inelastic scattering e!ects and heuristically normalized so as to satisfy the unitarity condition, yield a recovery factor which is consistent with the general trends discussed above for the case of #at surfaces.
8. Concluding remarks and protocol for the use of EBA-formalism in interpretations of atom}surface scattering experiments The scattering spectrum approach formulated and developed in Section 4 proves exceedingly convenient in establishing a direct correspondence between the theoretical scattering intensities and the measured intensities of the He or other inert atom beams inelastically scattered from surfaces. The formalism lends itself to the various approximate treatments among which the exponentiated Born approximation (EBA) spans a broad interval of applicability ranging from the extreme quantum, one-phonon He atom scattering regime to the quasiclassical regime of multiphonon scattering of heavier particles from surfaces. This property of the EBA formalism has been demonstrated for He atom scattering from several prototype surfaces characterized by the very di!erent projectile}surface interactions and phonon dynamics which may be regarded as benchmark systems for testing the validity of the developed theory. A very good agreement between the values of the measured and calculated scattering intensities and the Debye}Waller factors in the passage from one-, to two-, to multiphonon scattering regime (cf. Section 6) illustrates the general validity and interpretational potentiality of the EBA formalism in applications to atom}surface scattering. The development of the EBA formalism is rather demanding and derivations of the expressions for the scattering spectrum relevant to the various experimental regimes require rigorous and sometimes also lengthy procedures. However, the resulting formulae for the scattering from statically #at surfaces are rather simple on their own and also intuitive enough so as that they can be easily implemented to inelastic noble gas scattering from surfaces. To facilitate their use and make a compact summary of the derived results that would be free from the excessive material, we conclude this review with a brief protocol for the use of the EBA formalism in interpretations of inelastic HAS-TOF spectra from #at surfaces.
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The energy and parallel momentum scattering spectrum which can be directly compared with a HAS-TOF spectrum measured in the sagittal plane is calculated from (cf. Eq. (263))
d dR e\ #O\ KR exp[2=# (, R)!2=# (0, 0)] , N#kG (E, K)" 2 (2 )
(308)
where k "(K , k )"(k , , ). The symbols E"E !E and K"K !K denote the energy G G XG G G G D G D G and parallel momentum exchanged between the projectile atom and the surface at a temperature ¹ . The quantity 2=# (, R)" VKHIG X IQXG G KG (#)[n(Q )#1]e\ S/H O\Q>GR
\ \ H QG HIX
Q GR / X XG # VHI (309) K I Q G K (!)n(Q )e S H O\ > , G> > G H QG HIX denotes the EBA scattering function in which G is the surface reciprocal lattice vector, n(Q ) H stands for the Bose distribution of phonons of wavevector Q, branch index j and energy Q at the H temperature ¹ , and the on-shell inelastic scattering matrix elements V($) are expressed in terms of the inelastic scattering matrix elements \ interval y for any relativistic heavy-ion experiment. Similar analyses can be performed for any two kinds of distinguishable particles. The net positive charge from the protons in the colliding nuclei is much smaller than the total charge produced in an ultrarelativistic heavy-ion collision. For example, N exceeds N by > \ only &15% at in Pb#Pb collisions at SPS energies. The #uctuations in the number of positive and negative (or neutral) pions are also very similar, > K \ . Charged particle #uctuations , , have been estimated in thermal as well as participant nucleon models [24] including e!ects of resonances, acceptance, and impact parameter #uctuations. By varying the acceptance and centrality, the degree of thermalization can actually be determined empirically. Detailed analysis indicates
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that the #uctuations in central Pb#Pb collisions at the SPS are thermal whereas peripheral collisions are a superposition of pp #uctuations [64]. The #uctuations in the total (N "N #N ) and net (Q"N !N ) charge are de"ned as > \ > \ [54] N (N $N )!N $N N > \ > \ " > # \ $C , > , N N ,\ N #N > \ where the correlation is given by
(23)
N N !N N > \ . (24) C" > \ N /2 Fluctuations in positive, negative, total and net charge can be combined to yield both the intrinsic #uctuations in the numbers of N and the correlations in their production as well as a consistency ! check. These quantities can change as a consequence of thermalization and a possible phase transition. In practice, > + \ , so that the #uctuation in total charge simpli"es to , , N !N " #C (25) , ,> , N and that for the net charge becomes Q!Q , " > !C . (26) / , N The #uctuation in net charge can be related to the #uctuation in the ratio of positive to negative particles KN /N N \ > /4 , (27) / > \ , , plus volume (or impact parameter) #uctuations [51,53]. The virtue of this expression is that volume #uctuations can in principle be extracted empirically. Alternatively one can vary the centrality bin size or the acceptance. Furthermore, the volume #uctuations for net and total charge are proportional to the net (N !N ) and total (N #N ) charge, respectively, with the same > \ > \ prefactor. In the following we shall assume that such `triviala volume #uctuations have been removed. The analysis has so far been general and Eqs. (23)}(26) apply to any kind of distinguishable particles, e.g. positive and negative particles, pions, kaons, baryons, etc. } irrespective of what phase the system may be in, or whether it is thermal or not. In the following, we shall consider thermal equilibrium, which seems to apply to central collisions between relativistic nuclei, in order to reveal possible e!ects on #uctuations of the presence of a quark}gluon plasma. 4.2. Charge yuctuations in a thermal hadron gas In a thermal hadron gas (HG) as created in relativistic in nuclear collisions, pions can be produced either directly or through the decay of heavier resonances, , ,2 . The resulting
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#uctuation in the measured number of pions is > " \ "f #f #f #2 , (28) , , L L M M S S where f is the fraction of measured pions produced from the decay of resonance r, and f "1. P P P These mechanisms are assumed to be independent, which is valid in a thermal system. The heavier resonances such as , ,2 decay into pairs of >\ and thus lead to a correlation 1 C&%" f #f #2 . S 3 M
(29)
Resonances reduce the #uctuations in net charge in a HG in chemical equilibrium at temperature ¹"170 MeV and baryon chemical potential "270 MeV and strangeness chemical potential "74 MeV to "0.70 [51,17]. In [52] the value "0.70 is found. / / In addition, overall charge conservation reduces #uctuations in net charge when the acceptance is large and thus increases correlations as will be discussed below. 4.3. Charge yuctuations in a quark}gluon plasma A phase transition to the QGP can alter both #uctuations and correlations in the production of charged pions. To the extent that these e!ects are not eliminated by subsequent thermalization of the HG, they may remain as observable remnants of the QGP phase. As shown in Refs. [52,53], net charge #uctuations in a plasma of u, d quarks and gluons are reduced partly due to the intrinsically smaller quark charge and partly due to correlations from gluons 1 , N q , (30) " D / N $ N SB2 where N is the number of quark #avors, q their charges, and N the number of quarks. The total D number of charged particles (but not the net charge) can be altered by the ultimate hadronization of the QGP. Assuming a pion gas as the "nal state, this e!ect can be estimated by equating the entropy of all pions to the entropy of the quarks and gluons. Since 2/3 of all pions are charged and since the entropy per fermion is 7/6 times the entropy per boson in a two-#avor QGP N K(N #N ) , (31) where the number of gluons is N "(16/9N )N . Inserting this result in (30), we see that the resulting #uctuations are "0.18 in a two-#avor QGP (and "0.12 for three #avors). As / / pointed out in [53], lattice results give K0.25. However, according to [55] a substantial / fraction of the pions are decay products from the HG, and the entropy of the HG exceeds that of a pion gas by a factor 1.75}1.8. As described in [52] the net charge #uctuations should be increased by this factor in the QGP, i.e. K0.33 in a two-#avor QGP, whereas it remains similar in the / HG, K0.6. /
It is amusing to note that this number gives a very poor (i.e., negative) estimate for N /N in Eq. (31).
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The above models are all grand canonical ones, i.e. no net charge conservation, as opposed to microcanonical models that now will be discussed. If the high-density phase is initially dominated by gluons with quarks produced only at a later stage of the expansion by gluon fusion, the production of positively and negatively charged quarks will be strongly correlated on su$ciently small rapidity scales. An increased entropy density in the collisions volume will lead to enhanced multiplicity as compared to a standard hadronic scenario if total entropy is conserved. The associated particle production must conserve net charge on large rapidity scales (y91) due to causality because "elds cannot communicate over large distances and therefore must conserve charge within the `event horizona. Therefore the net charge, N , is approximately conserved whereas the total charge, Q, increase by an amount proportional to the additional entropy produced. If the entropy density increases from s to s going from a HG to &% /%. QGP without additional net charge production, #uctuations in net charge will be reduced correspondingly, s /%.K &% &% . / / s /%.
(32)
The resulting #uctuation in net charge is necessarily smaller than that from thermal quark production as given by Eq. (30). A similar phenomenon occurs in string models where particle production by string breaking and qq pair production results in #avor and charge correlations on a small rapidity scale [42]. If droplets or density #uctuations appear, they are expected not to produce additional net charge. Consequently, the net charge #uctuations should still vanish K0 whereas / . K2 >&2 /%. , The strangeness #uctuation in kaons K! might seem less interesting at "rst sight since strangeness is not suppressed in the QGP: The strangeness per kaon is unity, and the total number of kaons is equal to the number of strange quarks. However, if strange quarks are produced at a late stage in the expansion of a #uid initially dominated by gluons, the net strangeness will again be greatly reduced on su$ciently small rapidity scale. Consequently, #uctuations in net/total strangeness would be reduced/enhanced. The baryon number #uctuations have been estimated in a thermal model [52] in a grand canonical model. It is, however, not known how possible variations in baryon stopping event-byevent and subsequent di!usion and annihilation of the baryons and antibaryons in the hadronic phase a!ect these results. If only charged particles are detected, but not K, KM , neutrons and antineutrons, the #uctuations have smaller correlations as compared to the total and net strangeness or baryon number. 4.4. Total charge conservation Total charge conservation is important when the acceptance y is a non-negligible fraction of the total rapidity. It reduces the #uctuations in the net charge as calculated within the canonical ensemble, Eqs. (28)}(31). If the total positive charge (which is exactly equal to the total negative charge plus the incoming nuclear charges) is randomly distributed, the resulting #uctuations are
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smaller than the intrinsic ones by a factor (1!f f
"(N)\
dN dy dy
181
), where (33)
W is the acceptance fraction or the probability that a charged particle falls into the acceptance y assuming full p coverage. Since charged particle rapidity distributions are peaked near R mid-rapidity, charge conservation e!ectively kills #uctuations in the net charge even when y is substantially smaller than the laboratory rapidity, y K6 (11) at SPS (RHIC) energies. Total charge conservation also has the e!ect of increasing towards 2 > according to , Eq. (25). Similar e!ects can be seen in photon #uctuations when photons are produced in pairs through P2. In the WA98 experiment, K2 is found after the elimination of volume A #uctuations [49]. On the other hand, if the acceptance y is too small, particles can di!use in and out of the acceptance during hadronization and freezeout [52]. Furthermore, correlations due to resonance production will disappear when the average separation in rapidity between decay products exceeds the acceptance. Each of these e!ects tends to increase all #uctuations towards Poisson statistics when y: y, where y denotes the rapidity interval that particles di!use during hadronization, freezeout and decay. We "nd approximately
y 2 y # (1!f ) , K / / y#2 y y#2 y
(34)
where is the canonical thermal #uctuation of Eq. (26), with (29), and Eq. (30) and is the / / #uctuation corrected for both y and total charge conservation. The resulting #uctuations in total and net charge are shown in Fig. 9 assuming > " K1.1 , L and y"0.5. As mentioned above, f and y are related by the measured charge particle rapidity distributions [11]. The total charge #uctuations in a HG (C"0.4) from Eq. (31) agree well with
Fig. 9. Acceptance dependence of thermal #uctuations in net charge ( of Eq. (34), lower curves) and total ( , upper / , curves). Correlations increase from a hadron gas (CK0.4), to a QGP (CK0.8) and a gluon plasma (CK1.0) (see text). The HG result with a rapidity di!usion of y"0.8 is also shown for comparison to the other curves which use y"0.5. The large error bar on the NA49 data point is not statistical but re#ects the uncertainty in the subtraction of impact parameter #uctuations from #uctuations in charged particles [11,24]. The corresponding net charge #uctuation predicted by UrQMD [58] is shown by the open circle. From [54].
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NA49 data [11] after subtraction of residual impact parameter #uctuations. Data on charge particle ratios, which do not contain impact parameter #uctuations, will be able to test the net charge #uctuations of Eq. (34) to higher accuracy. Predictions from UrQMD are also shown for comparison [58]. The sensitivity to di!usion is small as seen in Fig. 9 where for the #uctuations are also shown for y"0.8 as recently used in [59]. The curves in Fig. 9 apply to RHIC energies as well after scaling y with y.
5. Fluctuations in particle ratios By taking ratios of particles, e.g. K/, >/\, /!,2, one conveniently removes volume and impact parameter #uctuations to "rst approximation. Simply increasing/decreasing the volume or centrality, the average number of particles of both species scales up/down by the same amount and thus cancel in the ratio. 5.1. >/\ ratio and entropy production Most particles produced in relativistic nuclear collisions are pions and they therefore constitute most of the number of positive and negatively charged particles. The #uctuations in the >/\ ratio and thus the ratio of positive and negative particles are intimately related to the #uctuations in net charge [51,53] 4 N /N # , (35) \ >" , , N > \ / where is the impact parameter or volume #uctuations and are the net charge #uctuations / as given by Eq. (27). The >/\ ratio has been studied in detail in [51]. Resonances such as , ,2 decaying into two or three pions correlate the > and \ production as for positively and negatively charged particles discussed above. Consequently, the #uctuation in the >/\ ratio is reduced by &30% in agreement with NA49 data [11]. 5.2. K/ ratio and strangeness enhancement To second order in the #uctuations of the numbers of K and , we have [24,51]
K!K K 1# L ! . K/" K
(36)
The corresponding #uctuation in K/ is given by K!K . D, )L " ) # L !2 K K/ K
(37)
The #uctuation in the kaon-to-pion ratio is dominated by the #uctuations in the number of kaons alone. The third term in Eq. (37) includes correlations between the number of pions and kaons.
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Fig. 10. Event-by-event #uctuations in the K/ ratio measured by NA49 in central Pb#Pb collisions at the SPS [11].
It contains a negative part from volume #uctuations, which removes the volume #uctuations in and since such #uctuations cancel in any ratio. In the NA49 data [11] shown in Fig. 10 the ) L average ratio of charged kaons to charged pions is K/"0.18 and K200. Excluding volume #uctuations, we take K K1.2}1.3 as discussed above. The "rst two terms in Eq. (37) ) L then yield DK0.20}0.21 in good agreement with preliminary measurements D"0.23 [11]. Thus at this stage the data give no evidence for correlated production of K and , as described by the "nal term in Eq. (37), besides volume #uctuations. The similar #uctuations in mixed event analyzes D "0.208 [11] con"rm this conclusion.
Strangeness enhancement has been observed in relativistic nuclear collisions at the SPS. For example, the number of kaons and therefore also K/ is increased by a factor of 2}3 in central Pb#Pb collisions. It would be interesting to study the #uctuations in strangeness as well. By varying the acceptance one might be able to gauge the degree of thermalization as discussed above. The #uctuations in the K/ ratio as function of centrality would in that case reveal whether strangeness enhancement is associated with thermalization or other mechanisms lie behind. In a plasma of decon"ned quarks strangeness is increased rapidly by ggPss and qq Pss processes and lead to enhancement of total strangeness s#s whereas the net strangeness s!s remains zero. The #uctuations in net and total strangeness will qualitatively behave like net and total charge, however, with unit strangeness quantum numbers as compared to the fractional charges. 5.3. /! ratio and chiral symmetry restoration Fluctuations in neutral relative to charged pions would be a characteristic signal of chiral symmetry restoration in heavy-ion collisions. If, during expansion and cooling, domains of chiral condensates gets `disorienteda (DCC) [36], anomalous #uctuations in /! ratios could result if DCC domains are large. For a single DCC domain the probability distribution of ratios d"/(#>#\) is P(d)"1/(2d with mean d"1/3 and #uctuation "4/15, i.e. much B
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larger than ordinary #uctuations in such ratios (see Eq. (37)) which decrease inversely with the number of pions. Neutral pions are much harder to measure than charged pions but with respect to #uctuations, it su$ces to measure the charged pions only. The anomalous #uctuations in due to a DCC are anti-correlated to !, i.e. they are of same magnitude but opposite sign. A DCC can equally well be searched for in total charge #uctuations as in the /! ratio, except for the troublesome impact parameter #uctuations. 5.4. J/ multiplicity correlations and absorption mechanisms J/ suppression has been found in relativistic nuclear collisions [60] and it is yet unclear how much is due to absorption on participant nucleons and produced particles (comovers). Whether `anomalous suppressiona is present in the data is one of the most discussed signals from a hot and dense phase at early times [60]. It was originally suggested that the formation of a quark}gluon plasma would destroy the cc bound states [61]. In relativistic heavy-ion collisions very few J/'s are produced in each collision. Of these only 6.9% branch into dimuons that can be detected and so the chance to detect two dimuon pairs in the same event is very small. Therefore, it will be correspondingly di$cult to measure #uctuations and other higher moments of the number of J/. Another more promising observable is the correlation between the multiplicities in, e.g., a rapidity interval y of a charmonium state "J/, ,2 (N ) and all particles (N) [54]. The correlator R NN !NN also enters in the ratio /N (see Eq. (36)). The correlator has as good statistics R R as the total number of and it may contain some very interesting anti-correlations, namely that absorption grows with multiplicity N. The physics behind can be comover absorption, which grows with comover density, or formation of quark}gluon plasma, which may lead to both anomalous suppression of and large multiplicity in y. Contrarily, direct Glauber absorption should not depend on the multiplicity of produced particles N since it is caused by collisions with participating nucleons. To quantify this anti-correlation we model the absorption/destruction of 's by simple Glauber theory N R "e\6NAR M J,e\A,6,7 , (38) N R where N is the number of J/'s before comover or anomalous absorption sets in but after direct R Glauber absorption on participant nucleons. In Glauber theory the exponent is the absorption cross section times the absorber density and path length traversed in matter. The density and therefore also the exponent is proportional to the multiplicity N with coe$cient d log N R . (39) "! d log N ,6,7 In a simple comover absorption model for a system with longitudinal Bjorken scaling, it can be calculated approximately [62] dN v A AR AR log R/ , K dy 4R A
(40)
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where dN /dy, , v and are the comover rapidity density, absorption cross section, relative A AR AR velocity and formation time, respectively. On average comover or anomalous absorption is responsible for a suppression factor e\A. It is di$cult to determine because only the total suppression including direct Glauber absorption on participants is measured. The anti-correlation is straightforward to calculate when the #uctuations in the exponent are small (i.e. ( /N;1). It is , NN !NN "! N . R R , R
(41)
It is negative and proportional to the amount of comover and anomalous absorption and obviously vanishes when the absorption is independent of multiplicity ("0). The anti-correlation can be accurately determined as the current accuracy in determining N is a few percent (NA50 R minimum bias [60]) in each E bin. 2 The anti-correlations in Eq. (41) may seem independent of the rapidity interval. However, if it is less than the typical relative rapidities between comovers and the , the correlations disappear. Preferably, the rapidity interval should be of the order of the typical rapidity #uctuations due to density #uctuations. The anticorrelations of Eq. (41) quantify the amount of comover or anomalous absorption and can therefore be exploited to distinguish between these and direct Glauber absorption mechanisms. In that respect it is similar to the elliptic #ow parameter for [62] for the comover absorption part but di!ers for the anomalous absorption. 5.5. Photon yuctuations: thermal emission vs. P2 WA98 have measured photon and charged particle multiplicities and their #uctuations vs. centrality and E binning size. As mentioned above impact parameter #uctuations are propor2 tional to the E binning size; the WA98 analysis nicely con"rms this, and can subsequently remove 2 impact parameter #uctuations. The resulting charged particle multiplicity #uctuations with impact parameter #uctuations subtracted, !n K1.1}1.2 were shown in Fig. 7. , , The #uctuations in photon multiplicities were found to be almost twice as large as for charged particles !n K2.0. This has the simple explanation that photons mainly are produced in A , P2 decays. The #uctuations are then the double of the #uctuations in to "rst approximation as seen from Eq. (18). If the photons were directly produced from a `shininga thermal "reball one would expect that they would exhibit Bose}Einstein #uctuations, " #"1.37 for massless particles. In addition A , the 's in the hadronic background will produce photons with " #"2.0. The measured A , #uctuation in the number of photons will therefore lie between these two numbers and can be exploited to quantify the amount of thermal photon emission vs. P2 decay from a hadronic gas 2.0! N A . A " 2.0!1.37 N #NL A A
(42)
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The impact parameter #uctuations must be subtracted from the measured photon #uctuations by, e.g., taking the ratio of photons to some other particle with known behavior. A 6. Transverse momentum 6uctuations Fluctuations in average transverse momentum were among the "rst event-by-event analyses studied. In a series of papers MroH wczynH ski et al. have studied transverse momentum #uctuations in heavy-ion collisions with the purpose of studying thermalization and other e!ects. Fluctuations in temperature and thus average transverse momentum event-by-event were studied by a number of people [15}18] in connection with critical phenomena relevant if the transition is close to a critical point. Experimental analyses by NA49 [11,14] reveal that a careful evaluation of systematic e!ects are required before substantial equilibration can be claimed in central heavy-ion collisions from transverse momentum #uctuations. They also have found strong correlations between multiplicity and transverse momentum. The total transverse momentum per event , P " p (43) G G is very similar to the transverse energy, for which #uctuations have been studied extensively [10,8]. The mean transverse momentum and inverse slopes of distributions generally increase with centrality or multiplicity. Assuming that ,d log(p )/d log N is small, as is the case for pions , [63], the average transverse momentum per particle for given multiplicity N is to leading order p "p (1#(N!N)/N) , (44) , where p is the average over all events of the single-particle transverse momentum. With this parametrization, the average total transverse momentum per particle in an event obeys P /N"p . When the transverse momentum is approximately exponentially distributed with inverse slope ¹ in a given event, p "2¹, and (p )"2¹"p /2. G G The total transverse momentum and also the transverse energy contains both #uctuations in multiplicity and #uctuations in the individual particle transverse momenta and energy (see Appendix C). An interesting quantity is therefore the total transverse momentum per particle, P /N, where the multiplicity #uctuations are removed to "rst order although important correla tions remain. The total transverse momentum per particle in an event has #uctuations
1 (p p !p ) . (45) N(P /N)"(p )#p # G H G , N G$H The three terms on the right are, respectively: (i) The individual #uctuations (p )"p !p , the main term. In the NA49 data, G G p "377 MeV and N"270. From Eq. (45) we thus obtain (P /N)/p K1/(2N" 4.3%, which accounts for most of the experimentally measured #uctuation 4.65% [11]. The data contains no indication of intrinsic temperature #uctuations in the collisions.
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(ii) E!ects of correlations between p and N, which are suppressed with respect to the "rst term by a factor &. In NA49 the multiplicity of charged particles is mainly that of pions for which ¹Kp /2 increases little compared with pp collisions, and K0.05}0.1. Thus, these correlations are small for the NA49 data. However, for kaons and protons, can be an order of magnitude larger as their distributions are strongly a!ected by the #ow observed in central collisions [63]. (iii) Correlations between transverse momenta of di!erent particles in the same event. In the WNM the momenta of particles originating from the same participant are correlated. In Lund string fragmentation, for example, a quark-antiquark pair is produced with the same p but in opposite direction. The average number of pairs of hadrons from the same participant is n(n!1), where n is the number of particles emitted from the same participant nucleon, and therefore the latter term in Eq. (45) becomes (n(n!1)/n)(p p !p ). To a good G H$G approximation, n is Poisson distributed, i.e., n(n!1)/n"n, equal to 0.77 for the NA49 acceptance, so that this latter term becomes K(p p !p ). The momentum correlation G H$G between two particles from the same participant is expected to be a small fraction of (p ). G To quantify the e!ect of rescatterings, the di!erence between N(P /N) and (p ) has been studied in detail [12] via the quantity (p )K(N(P /N)!((p ) . G
(46)
As we see from Eq. (45), in the applicable limit that the second and third terms are small, 1 (p #(p p !p )) . (p )K , G H$G ((p ) G
(47)
In the Fritiof model, based on the WNM with no rescatterings between secondaries, one "nds (p )K4.5 MeV. In the thermal limit the correlations in Eq. (46) should vanish for classical particles but the interference of identical particles (HBT correlations) contributes to these correlations &6.5 MeV [13]; they are again slightly reduced by resonances. The NA49 experimental value, (p )"5 MeV (corrected for two-track resolution) seems to favor the thermal limit [11]. Note however that with K0.05}0.1, the second term on the right side of Eq. (47) alone leads to K1}4 MeV, i.e., the same order of magnitude. If (p p !p ) is not positive, then one G H$G cannot a priori rule out that the smallness of (p ) does not arise from a cancellation of this term with p , rather than from thermalization. , A comparison of the transverse momentum #uctuations of charged particles to those in mixed events, where correlations thus are removed, showed a small enhancement of only 0.002$0.002 [11]. It was estimated that Bose e!ects should enhance this ratio by 1}2% but that total energy conservation introduces an anticorrelation that partially cancels the Bose enhancement [17,18]. Experimental problems with two-track resolution have also been estimated to lead to a ratio that is 1}2% lower. Consequently, the numbers seem to be compatible. The covariance matrix between multiplicity and transverse momentum has been analyzed by NA49 [11]. Strong but trivial correlations are found due to the fact that higher multiplicity gives larger total transverse momentum event-by-event. This correlation is removed in the quantity P /N and its covariance matrix with multiplicity appears diagonal.
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7. Event-by-event 6uctuations at RHIC The theoretical analysis above leads to a qualitative understanding of event-by-event #uctuations and speculations on how phase transitions may show up. It gives a quantitative description of AGS and SPS data without the need to invoke new physics. We shall here look towards RHIC experiments and attempt to describe how #uctuations may be searched for. General correlators between all particle species should be measured event-by-event, e.g., the ratios [24] N N !N N N /N G H , G H K1# ,H ! G H N N N N /N H G H G H
(48)
where N are the multiplicities in acceptances i and j of any particle. Volume #uctuations are GH automatically removed in such ratios, their #uctuations and correlations. If the energy deposition, transverse energy or momentum are measured, these latter will have additional #uctuation due to the multiplicity #uctuations as explained in Appendix C. More generally we de"ne the multiplicity correlations between any two bins N N !N N G H " G H GH (N N G H
(49)
also referred to as the covariance. When i, j refer to two rapidity bins the covariance is also proportional to the rapidity (auto-)correlation function C(y !y ). G H It is instructive to consider "rst completely random (uncorrelated or statistical) particle emission. For a "xed total multiplicity N , the probability for a particle to end up in bin i is 2 p "N /N KE /E . The distribution is a simple multinomial distribution for which G G 2 G 2
1!p , i"j G . " GH !(p p , iOj G H
(50)
The i"j result is the well known one for a binomial distribution. The iOj result is negative because particles in di!erent bins are anti-correlated: more (less) particles in one bin leads to less (more) in other bins on average due to a "xed total number of particles. As shown above there are nonstatistical #uctuations due to various sources: Bose}Einstein #uctuations, resonances, etc., and } in particular } density #uctuations. As in Eq. (21) we assume that the multiplicity consist of particles from a HM and a QM phase. The covariances in Eq. (50) are derived analogously to Eq. (22) N G/+ , " #( ! ) GH GH&+ GH/+ GH&+ N G
(51)
when N "N ; when di!erent the general formula is a little more complicated. Now, the G H hadronic #uctuations is of order unity for i"j, smaller for adjacent bins and vanishes or GH&+ even becomes slightly negative according to (50) for bins very di!erent in pseudorapidity or azimuthal angle . The QM #uctuations can be much larger: &N (see the discussion G/+ G/+
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after Eq. (20)). To discriminate the QM #uctuations from the hadronic ones, Eq. (51) requires (52) N 9(( ! )N . G/+ GH GH&+ G The charged particle multiplicity in central Au#Au collisions at RHIC is dN /dK500}600 per unit pseudorapidity [23]. To see a clear increase in #uctuations, say , ! &1, GH GH&+ a density #uctuation of only N 9(N K25 particles are required per unit rapidity correG/+ G sponding to a few percent of the average. By analyzing many events (of the same total multiplicity) the accuracy by which #uctuations are measured experimentally is greatly improved. Generally, , and so #uctuations can in principle be determined with immense accuracy. &1/(N It may be advantageous to correlate bins with the same pseudorapidity but di!erent azimuthal angles since the hadronic correlations between these are small whereas QM #uctuations remain. No experimental determination of the purely statistical uncertainties associated with any one-body distribution } such as multiplicity as a function of rapidity } can be performed without measuring and diagonalizing the correlation matrix C "N N !N N . While it is convenGH G H G H tional to assign uncertainties according to the diagonal elements M , the correlations in the GG covariance matrix are required for a correct error analysis and can also reveal physical important results.
8. Summary In a phase transition in high-energy nuclear collisions, whether it is "rst order or a soft cross-over, density #uctuations may be expected that show up in rapidity and multiplicity #uctuations event-by-event. The #uctuations can be enhanced signi"cantly in case of droplet formation as compared to that from an ordinary hadronic scenario. A combined analysis of, e.g., positive, negative, total and net charge, allows one to extract the various #uctuations and correlations uniquely. Likewise a number of other observables as charged and neutral pions, kaons, photons, J/, etc., and their ratios can show anomalous correlations and enhancement or suppression of #uctuations. This clearly demonstrates the importance of event-by-event #uctuations accompanying phase transitions, and illustrates how monitoring such #uctuations vs. centrality becomes a promising signal, in the upcoming RHIC experiments, for the onset of a transition. The potential for enhanced or suppressed #uctuations (orders of magnitude) from a transition makes it worth looking for at RHIC considering the relative simplicity and accuracy of multiplicity #uctuation measurements. An analysis of #uctuations in central Pb#Pb collisions as currently measured in NA49 does, however, not show any sign of anomalous #uctuations. Fluctuations in multiplicity, transverse momentum, K/ and other ratios can be explained by standard statistical #uctuation and additional impact parameter #uctuations, acceptance cuts, resonances, thermal #uctuations, etc. This understanding by `standarda physics should be taken as a baseline for future studies at RHIC and LHC and searches for anomalous #uctuations and correlations from phase transitions that may show up in a number of observables. By varying the centrality one should be able to determine quantitatively the amount of thermalization in relativistic heavy-ion collisions as de"ned in Eq. (19). For peripheral collisions,
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where only few rescatterings occur, we expect the participant model (WNM) to be approximately valid and the degree of thermalization to be small. For central collisions, where many rescatterings occur among produced particles, we expect to approach the thermal limit and the degree of thermalization should be close to 100%. At RHIC and LHC energies the #uctuations in the number of charged particles consequently decrease drastically with centrality whereas at SPS energies the two limits are accidentally very close. Event-by-event physics is an important tool to study thermalization and phase transitions through anomalous #uctuations and correlations } as in rain. Acknowledgements Thanks are due to G. Baym and A.D. Jackson for inspiration and collaboration on some of the work described in this report. Discussion with S. Voloshin and G. Roland (NA49), J.J. Ga rdh+je and collaborators in NA44 and BRAHMS, T. Nayak (WA98), J. Bondorf, S. Jeon, V. Koch, and many suggestions for improvement from an anonymous referee are gratefully acknowledged. Appendix A. Damping of initial density 6uctuations Hydrodynamic #ow with Bjorken scaling is stable according to a stability analysis carried out in [35]. By linearing the hydrodynamic equations in small perturbations in entropy density s and rapidity y around the Bjorken scaling solution and looking for solutions in the form of harmonic perturbations, e IE, the hydrodynamic equations could be written in matrix form (Eq. (A.13) in [35])
0 !ik
s/s "
y !ikc !(1!c) Q Q The eigenvalues of the above matrix
s/s
y
1 1 "! (1!c)$ (1!c)!ck , ! Q Q Q 2 4
.
(A.1)
(A.2)
always have real negative part for c kO0 and #uctuations are therefore damped. For long wave Q length #uctuations in rapidity and not too soft equations of state, c k'1!c, the solution is Q Q a damped oscillator. Note that the long wave length solution k"0 reproduces the Bjorken scaling. The exact solution for the entropy density #uctuation
s "c eH> OO #c eH\ OO , > \ s
(A.3)
is sensitive to the equation of state through c , the initial conditions for the rapidity density Q #uctuations (the constants c ), and their wave length k\. ! At large times the eigenvalue with the largest real part dominates and
s 0 H! . J s
(A.4)
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Here the oscillating factor has been ignored, leaving the power law fall-o! of #uctuations with exponent
1 Min Re[ ] " (1!c)!Re Q ! 2
1 (1!c)!ck . Q Q 4
(A.5)
One notes that density yuctuations are undamped for soft equation of states (c "0). They are also Q undamped if their wave length is long (kK0). To estimate the resulting damping we take a typical rapidity #uctuation for a droplet
y&(¹/m &1 discussed above, which corresponds to a wavenumber kK1. For an ideal equation of state with sound speed c "1/(3 the last term in Eq. (A.2) is then either imaginary or Q small and real, and the real part of the eigenvalue is dominated by the "rst term of Eq. (A.2), Re[ ]K!1/3. If we take a typical formation time K1 fm/c and a freezeout time K8 fm/c ! as extracted from HBT studies [1], the resulting suppression of a density #uctuation during expansion is a factor &8\"0.5 according to Eq. (A.4).
Appendix B. Fluctuations in source models As #uctuations for a source model appears again and again (see Eqs. (10), (11), (18), (22)) we shall derive this simple equation in detail. We de"ne the #uctuations for any stochastic variable x as x!x . " V x
(B.1)
It is usually of order unity and therefore more convenient than variances. For a Poisson distribution, P "e\?,/N!, the #uctuation is "1. For a binomial distribution with tossing probabil, , ity p the #uctuation is "1!p, independent of the number of tosses. In heavy-ion collisions , several processes add to #uctuations so that typically &1}2. Correlations can in some cases , double the #uctuations as, for example, P2 doubles the #uctuations in photon multiplicity and net charge conservation doubles the #uctuation in total charge. Impact parameter #uctuations further increases the total charge #uctuations to "3}5 in peripheral nuclear , collisions [64]. Generally, when the multiplicity (N) arise from independent sources (N ) such as participants, resonances, droplets or whatever, G, (B.2) N" n , G G where n is the number of particles produced in source i. In the absence of correlations between G N and n, the average multiplicity is N"N n. Here, .. refer to averaging over each individual (independent) source as well as the number of sources. The number of sources vary from event to event and average is performed over typically N &100,000 events as in NA49 or N &10 in WA98.
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Squaring Eq. (B.2) assuming that the sources emit particles independently, i.e. n n "n n G H G H for iOj, the square consists of the diagonal and o!-diagonal elements: N"N n#N (N !1)n . G G With (B.1) we obtain the multiplicity #uctuations
(B.3)
N!N " " #n , , L , N as in Eq. (10).
Appendix C. Fluctuations in the energy deposited Many experiments do not measure individual particle tracks or multiplicities but instead the energy deposited in arrays of detector segments, E , in a given event. One could also project the G energy transversely by weighting with the sine of the scattering angle to study #uctuations in transverse energy [7}10]. Since particles mostly have relativistic speeds in relativistic heavy-ion collisions, the transverse energy is almost the same as the total transverse momentum in an event. The total energy deposited in the event is " E " E (C.1) 2 G G and can be used as a measure of the centrality of the collision. The energy deposited in each element (or group of elements) is the sum over the number of particle tracks (N ) hitting detector i of the G individual ionization energy of each particle ( ) G ,G E " . (C.2) G L L The average is: E "N . The energy will approximately be gaussian distributed, G G d/dE Jexp(!(E !E )/2 G E ), with #uctuations (see Appendix B) G G G # G E!E G " # . G, G (C.3) # C ,G E G Here, the #uctuation in ionization energy per particle C" !1 ,
(C.4)
depends on the typical particle energies in the detector and the corresponding ionization energies for the detector type and thickness. For the BRAHMS detectors we estimate / K0.3 [65]. C This number will, however, depend on rapidity since the longitudinal velocity enters the ionization power. As these are `triviala detector parameter, we shall exclude the #uctuations in most C analyses and concentrate on the second term in Eq. (C.3) which is the #uctuations in the number of particles as examined in detail above.
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References [1] R. Hanbury-Brown, R.Q. Twiss, Philos. Mag. 45 (1954) 633. S. Pratt, Phys. Rev. Lett. 53 (1984) 1219. T. CsoK rgo , B. LoK rstad, Phys. Rev. C 54 (1996) 1390. U. Heinz, B.V. Jacak, Annu. Rev. Nucl. Part. Sci. 49 (1999), and references therein. [2] S. Trentalange, S.U. Pandey, J. Acoust. Soc. Am. 99 (1996) 2439; C. Slotta, U. Heinz, Phys. Rev. E 58 (1998) 526. [3] M.R. Andrews, C.G. Townsend, H.-J. Miesner, D.S. Durfee, D.M. Kurn, W. Ketterle, Science 275 (1997) 637; Y. Castin, J. Dalibard, Phys. Rev. A 55 (1997) 4330. [4] See, e.g., J. Phillips et al., astro-ph/0001089. [5] M. Toscano et al., Mon. Not. R. Astron. Soc., astro-ph/9811398. [6] P. Braun-Munziger, J. Stachel, Nucl. Phys. A 638 (1998) 3c, and refs. herein. [7] G. Baym, G. Friedman, I. Sarcevic, Phys. Lett. B 219 (1989) 205. [8] H. Heiselberg, G.A. Baym, B. BlaK ttel, L.L. Frankfurt, M. Strikman, Phys. Rev. Lett. 67 (1991) 2946; B. BlaK ttel, G.A. Baym, L.L. Frankfurt, H. Heiselberg, M. Strikman, Nucl. Phys. A 544 (1992) 479c. [9] G. Baym, B. BlaK ttel, L.L. Frankfurt, H. Heiselberg, M. Strikman, Phys. Rev. C 52 (1995) 1604. [10] T. As kesson et al. (Helios collaboration), Z. Phys. C 38 (1988) 383. [11] G. Roland et al. (NA49 collaboration), Nucl. Phys. A 638, 91c (1998); H. AppelhaK user et al. (NA49 collaboration), Phys. Lett. B 459 (1999) 679; J.G. Reid (NA49 collaboration), Nucl. Phys. A 661 (1999) 407c; K. Perl, NA49 note 244. [12] M. GazH dzicki, S. MroH wczynH ski, Z. Phys. C 54 (1992) 27. [13] S. MroH wczynH ski, Phys. Rev. C 57 (1998) 1518; Phys. Lett. B 430, 9; ibid. B 439 (1998) 6; ibid. B 465 (1999) 8; Acta Phys. Polon. B 31 (2000) 2065. [14] T.A. Trainor, hep-ph/0001148; T.A. Trainor, J.G. Reid, hep-ph/0004258. [15] L. Stodolsky, Phys. Rev. Lett. 75 (1995) 1044. [16] E.V. Shuryak, Phys. Lett. B 430 (1998) 9. [17] M. Stephanov, K. Rajagopal, E. Shuryak, Phys. Rev. Lett. 81 (1998) 4816; Phys. Rev. D 60 (1999) 114028; K. Rajagopal, Proceedings of the Minnesota Conference on Continuous Advances in QCD, 1998 (hep-th/9808348). [18] B. Berdnikov, K. Rajagopal, Phys. Rev. D 61 (2000) 105017; K. Rajagopal, Proceedings of International Conference on Quark Nuclear Physics, Adelaide, Australia, February 2000 (hep-ph/0005101). [19] S. Gavin, C. Pruneau, nucl-th/9907040; S. Gavin, nucl-th/9908070. [20] S.A. Voloshin, V. Koch, H.G. Ritter, nucl-th/9903060. [21] A. Bialas, R. Peschanski, Nucl. Phys. B 273 (1986) 703. [22] M.A. Bloomer et al. (WA80), Nucl. Phys. A 544 (1992) 543c. [23] B.B. Back et al. PHOBOS Collaboration, Phys. Rev. Lett. 85 (2000) 3100. [24] G. Baym, H. Heiselberg, Phys. Lett. B 469 (1999) 7. [25] L. Van Hove, Phys. Lett. B 118 (1982) 138. [26] C. Bernard et al., Phys. Rev. D 55 (1997) 6861; Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, T. Yoshi, Phys. Rev. D 54 (1996) 7010. [27] G. Boyd et al., Phys. Rev. Lett. 75 (1995) 4169. E. Laermann, Proceedings of Quark Matter '96, Nucl. Phys. A 610 (1996) 1c. [28] JLQCD Collaboration, Nucl. Phys. Proc. 73 (Suppl.) (1999) 459. [29] K. Eskola, hep-ph/9911350. [30] L.D. Landau, E.M. Lifshitz, Statistical Physics, Part 1, Pergamon, New York, 1980. [31] L. Van Hove, Z. Phys. C 21 (1984) 93; J.I. Kapusta, A.P. Vischer, Phys. Rev. C 52 (1995) 2725; E.E. Zabrodin, L.P. Csernai, J.I. Kapusta, G. Kluge, Nucl. Phys. A 566 (1994) 407c. [32] H. Heiselberg, C.J. Pethick, E.F. Staubo, Phys. Rev. Lett. 70 (1993) 1355. [33] M.A. Halasz, A.D. Jackson, R.E. Shrock, M.A. Stephanov, J.J.M. Verbaarschot, Phys. Rev. D 58 (1998) 96007. [34] H. Heiselberg, A.D. Jackson, Proceedings of Advances in QCD, Minnesota, May 1998, nucl-th/9809013. [35] G. Baym, B.L. Friman, J.-P. Blaizot, M. Soyeur, W. Czyz, Nucl. Phys. A 407 (1983) 541. [36] J. Bjorken, Phys. Rev. D 27 (1983) 140; M. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B 422 (1998) 247; Nucl. Phys. B 558 (1999) 219. [37] A. Bialas, M. Bleszynski, W. Czyz, Nucl. Phys. B 111 (1976) 461.
194 [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65]
H. Heiselberg / Physics Reports 351 (2001) 161}194 K. Werner, private communication. W. Thome et al., Nucl. Phys. B 129 (1977) 365. J. Whitmore, Phys. Rep. 27 (1976) 187. UA5 Collaboration, G.J. Alner et al., Z. Phys. C 33 (1986) 1; Phys. Rep. 154 (1987) 247. H. B+ggild, T. Ferbel, Annu. Rev. Nucl. Sci. 24 (1974) 451. E735 Collaboration, C.S. Lindsey et al., Nucl. Phys. A 544 (1992) 343c. Z. Koba, H.B. Nielsen, P. Olesen, Nucl. Phys. B 40 (1972) 317. E877 coll., J. Barrette et al., Phys. Rev. C 56 (1997) 3254; E895 coll., H. Liu et al., A 638 (1998) 451c. M. Gazdzicki, O. Hansen, Nucl. Phys. A 528 (1991) 754; W. Wroblewski, Acta Phys. Pol. B 4 (1973) 857. J. Schukraft et al. (NA34 collaboration), Nucl. Phys. A 498 (1989) 79c. H. Heiselberg, A. Levy, Phys. Rev. C 59 (1999) 2716. T.K. Nayak (WA98 collaboration), private communication. G. Bertsch, Phys. Rev. Lett. 72 (1994) 2349. S. Jeon, V. Koch, Phys. Rev. Lett. 83 (1999) 5435. M. Asakawa, U. Heinz, B. MuK ller, Phys. Rev. Lett. 85 (2000) 2072. S. Jeon, V. Koch, Phys. Rev. Lett. 85 (2000) 2076. H. Heiselberg, A.D. Jackson, nucl-th/0006021. J. Sollfrank, P. Koch, U. Heinz, Z. Phys. C 52 (1991) 593; J. Sollfrank, U. Heinz, Phys. Lett. B 289 (1992) 132; G.E. Brown, J. Stachel, G.M. Welke, Phys. Lett. B 253 (1991) 19. J.P. Sullivan et al., Phys. Rev. Lett. 70 (1993) 3000. H. Heiselberg, Phys. Lett. B 379 (1996) 27. U.A. Wiedemann, U. Heinz, Phys. Rev. C 56 (1997) 3265. M. Bleicher, S. Jeon, V. Koch, Phys. Rev. C 62 (2000) 061902. M. Stephanov, E. Shuryak, hep-ph/0010100. NA50 Collaboration, M.C. Abreu et al., Phys. Lett. B 410, 327 (1997); ibid 337; CERN-EP/99-13, Phys. Lett. B, to appear. T. Matsui, H. Satz, Phys. Lett. B 178 (1991) 416. H. Heiselberg, R. Mattiello, Phys. Rev. C 60 (1999) 44902. I.G. Bearden et al. (NA44 collaboration), Phys. Rev. Lett. 78 (1997) 2080. S. Voloshin (NA49), private communication. J.J. Gaardh+je, BRAHMS collaboration, private communication.
SUPERFLUID ANALOGIES OF COSMOLOGICAL PHENOMENA
G.E. VOLOVIK Low Temperature Laboratory, Helsinki University of Technology Box 2200, FIN-02015 HUT, Finland L.D. Landau Institute for Theoretical Physics, 117334 Moscow, Russia
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 351 (2001) 195}348
Super#uid analogies of cosmological phenomena G.E. Volovik Low Temperature Laboratory, Helsinki University of Technology, Box 2200, FIN-02015 HUT, Finland L.D. Landau Institute for Theoretical Physics, 117334 Moscow, Russia Received December 2000; editor: C.W.J. Beenakker Contents 1. Introduction. Physical vacuum as condensed matter 2. Landau}Khalatnikov two-#uid hydrodynamics as e!ective theory of gravity 2.1. Super#uid vacuum and quasiparticles 2.2. Dynamics of super#uid vacuum 2.3. Normal component } `mattera 2.4. Quasiparticle spectrum and e!ective metric 2.5. E!ective metric for bosonic collective modes in other systems 2.6. E!ective quantum "eld and e!ective action 2.7. Vacuum energy and cosmological constant. Nulli"cation of vacuum energy 2.8. Einstein action and higher derivative terms 3. `Relativistica energy}momentum tensor for `mattera moving in `gravitationala super#uid background in two #uid hydrodynamics 3.1. Kinetic equation for quasiparticles (matter) 3.2. Momentum exchange between super#uid vacuum and quasiparticles 3.3. Covariance vs. conservation 3.4. Energy}momentum tensor for `mattera 3.5. Local thermodynamic equilibrium 3.6. Global thermodynamic equilibrium. Tolman temperature. Pressure of `mattera and `vacuuma pressure 4. Universality classes of fermionic vacua 4.1. Fermi surface as topological object
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4.2. Fully gapped systems: `Dirac particlesa in superconductors and in super#uid He-B 4.3. Systems with Fermi points 4.4. Gapped systems with nontrivial topology in 2#1 dimensions 5. Fermi points: He-A vs. Standard Model 5.1. Super#uid He-A 5.2. Standard Model and its momentumspace topology 6. E!ective relativistic quantum "eld theory emerging in a system with Fermi point 6.1. Collective modes of fermionic vacuum } electromagnetic and gravitational "elds 6.2. Physical laws in vicinity of Fermi point: Lorentz invariance, gauge invariance, general covariance, conformal invariance 6.3. E!ective electrodynamics 6.4. E!ective S;(N) gauge "elds from degeneracy of Fermi point 7. Chiral anomaly in condensed matter systems and Standard Model 7.1. Adler}Bell}Jackiw equation 7.2. Anomalous nonconservation of baryonic charge 7.3. Analog of baryogenesis in He-A: momentum exchange between super#uid vacuum and quasiparticle matter 7.4. Axial anomaly and force on He-A vortices
E-mail address:
[email protected]." (G.E. Volovik). 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 3 9 - 3
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G.E. Volovik / Physics Reports 351 (2001) 195}348 7.5. Experimental veri"cation of Adler} Bell}Jackiw equation in rotating He-A Macroscopic parity violating e!ects 8.1. Helicity in parity violating systems 8.2. Chern}Simons energy term 8.3. Helical instability and `magnetogenesisa by chiral fermions 8.4. Mixed axial-gravitational Chern}Simons term Fermion zero modes and spectral #ow in the vortex core 9.1. Fermion zero modes on vortices 9.2. Spectral #ow in singular vortices: Callan}Harvey mechanism of anomaly cancellation Interface between two di!erent vacua and vacuum pressure in super#uid He 10.1. Interface between vacua of di!erent universality classes and Andreev re#ection 10.2. Force acting on moving mirror from thermal relativistic fermions 10.3. Vacuum pressure and vacuum energy in He Vierbein defects 11.1. Vierbein domain wall 11.2. Conical spaces
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11.3. Vortex vs. spinning cosmic string 11.4. Gravitational Aharonov}Bohm e!ect and Iordanskii force 12. Horizons, ergoregions, degenerate metric, vacuum instability and all that 12.1. Event horizons in vierbein wall and Hawking radiation 12.2. Landau critical velocity and ergoregion 12.3. PainleveH }Gullstrand metric in e!ective gravity in super#uids. Vacuum resistance to formation of horizon 12.4. Stable event horizon and its momentumspace topology 12.5. Hawking radiation 12.6. Extremal black hole 12.7. Thermal states in the presence of horizons. Modi"ed Tolman's law 12.8. PainleveH }Gullstrand vs. Schwarzschild metric in e!ective gravity. Incompleteness of space}time in e!ective gravity 12.9. Vacuum under rotation 13. How to improve helium-3 13.1. Gradient expansion 13.2. E!ective action in inert vacuum 14. Discussion References
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Abstract In a modern viewpoint relativistic quantum "eld theory is an emergent phenomenon arising in the low-energy corner of the physical fermionic vacuum } the medium, whose nature remains unknown. The same phenomenon occurs in condensed matter systems: In the extreme limit of low-energy condensed matter systems of special universality class acquire all the symmetries, which we know today in high-energy physics: Lorentz invariance, gauge invariance, general covariance, etc. The chiral fermions as well as gauge bosons and gravity "eld arise as fermionic and bosonic collective modes of the system. Inhomogeneous states of the condensed matter ground state } vacuum } induce nontrivial e!ective metrics of the space, where the free quasiparticles move along geodesics. This conceptual similarity between condensed matter and the quantum vacuum allows us to simulate many phenomena in high-energy physics and cosmology, including the axial anomaly, baryoproduction and magnetogenesis, event horizon and Hawking radiation, cosmological constant and rotating vacuum, etc., probing these phenomena in ultra-low-temperature super#uid helium, atomic Bose condensates and superconductors. Some of the experiments have been already conducted. 2001 Elsevier Science B.V. All rights reserved. PACS: 67.57.!z; 12.10.!g; 04.50.#h Keywords: Quantum liquid; E!ective quantum "eld theory; E!ective gravity; Quantum vacuum; Cosmological constant; Chiral anomaly
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1. Introduction. Physical vacuum as condensed matter The traditional Grand Uni"cation view is that the low-energy symmetry of our world is the remnant of a larger symmetry, which exists at high energy, and is broken when the energy is reduced. According to this philosophy the higher the energy the higher is the symmetry: ;(1);S;(3)P;(1);S;(2);S;(3)PSO(10)Psupersymmetry, etc. The less traditional view is quite opposite: it is argued that starting from some energy scale one probably "nds that the higher the energy the poorer are the symmetries of the physical laws, and "nally even the Lorentz invariance and gauge invariance will be smoothly violated [1,2]. From this point of view relativistic quantum "eld theory is an e!ective theory [3,4]. It is an emergent phenomenon arising as a "xed point in the low-energy corner of the physical vacuum whose nature is inaccessible from the e!ective theory. In the vicinity of the "xed point the system acquires new symmetries which it did not have at higher energy. It is quite possible that even such symmetries as Lorentz symmetry and gauge invariance are not fundamental, but gradually appear when the "xed point is approached. From this viewpoint it is also possible that grand uni"cation schemes make no sense if the uni"cation occurs at energies where the e!ective theories are no longer valid. Both scenaria occur in condensed matter systems. In particular, super#uid He-A provides an instructive example. At high temperature the He gas and at lower temperature the He liquid have all the symmetries that ordinary condensed matter can have: translational invariance, global ;(1) group and global SO(3) symmetries of spin and orbital rotations. When the temperature decreases further the liquid He reaches the super#uid transition temperature ¹ , below which it spontan eously looses all its symmetries except for the translational one } it is still liquid. This breaking of symmetry at low temperature, and thus at low energy, reproduces the Grand Uni"cation scheme, where the symmetry breaking is the most important element. However, this is not the whole story. When the temperature is reduced further, the opposite `anti-grand-uni"cationa scheme starts to work: in the limit ¹P0 the super#uid He-A gradually acquires from nothing almost all the symmetries, which we know today in high-energy physics: (an analog of) Lorentz invariance, local gauge invariance, elements of general covariance, etc. It appears that such an enhancement of symmetry in the limit of low energy happens because He-A belongs to a special universality class of Fermi systems [5]. For the condensed matter of such class, the chiral fermions and gauge bosons arise as fermionic and bosonic collective modes together with the chirality itself and with corresponding symmetries. The inhomogeneous deformations of the condensed matter ground state } quantum vacuum } induce nontrivial e!ective metrics of the space, where the free quasiparticles move along geodesics, thus simulating the gravity "eld. This conceptual similarity between condensed matter and quantum vacuum gives some hint on the origin of symmetries and also allows us to simulate many phenomena in high-energy physics and cosmology. The quantum "eld theory, which we have now, is incomplete due to ultraviolet divergences at small scales. The crucial example is provided by the quantum theory of gravity, which after 70 yr of research is still far from realization in spite of numerous beautiful achievements [6]. This is a strong indication that gravity, both classical and quantum, is not fundamental: It is e!ective "eld theory which is not applicable at small scales where the `microscopica physics of vacuum becomes important, and, according to the `anti-granduni"cationa scenario, some or all of the known symmetries in Nature are violated. The analogy between the quantum vacuum and condensed
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matter could give an insight into this trans-Planckian physics since it provides examples of the physically imposed deviations from Lorentz and other invariances at higher energy. This is important in many di!erent areas of high-energy physics and cosmology, including possible CPT violation and black holes, where the in"nite red shift at the horizon opens a route to the trans-Planckian physics. Condensed matter teaches us that the low-energy properties of di!erent condensed matter vacua (magnets, super#uids, crystals, superconductors, etc.) are robust, i.e. they do not depend much on the details of microscopic (atomic) structure of these substances. The principal role is played by symmetry and topology of the condensed matter: they determine the soft (low-energy) hydrodynamic variables, the e!ective Lagrangian describing the low-energy dynamics, topological defects and quantization of physical parameters. The microscopic details provide us only with the `fundamental constantsa, which enter the e!ective phenomenological Lagrangian, such as speed of `lighta (say, the speed of sound), super#uid density, modulus of elasticity, magnetic susceptibility, etc. Apart from these `fundamental constantsa, which can be rescaled, the systems behave similarly in the infrared limit if they belong to the same universality and symmetry classes, irrespective of their microscopic origin. The detailed information on the system is lost in such acoustic or hydrodynamic limit [7]. From the properties of the low-energy collective modes of the system } acoustic waves in case of crystals } one cannot reconstruct the atomic structure of the crystal since all the crystals have similar acoustic waves described by the same equations of the same e!ective theory, in crystals it is the classical theory of elasticity. The classical "elds of collective modes can be quantized to obtain quanta of acoustic waves } phonons. This quantum "eld remains the e!ective "eld which is applicable only in the long-wave-length limit, and does not give a detailed information on the real quantum structure of the underlying crystal (except for its symmetry class). In other words one cannot construct the full quantum theory of real crystal using the quantum theory of elasticity. Such theory would always contain divergences on atomic scale, which cannot be regularized in unique way. The same occurs in other e!ective theories of condensed matter. In particular the naive approach to calculate the ground state (vacuum) energy of super#uid liquid He using the zero point energy of phonons gives even the wrong sign of the vacuum energy, as we shall see in Section 2.7. It is quite probable that in the same way the quantization of classical gravity, which is one of the infrared collective modes of the quantum vacuum, will not add more to our understanding of the `microscopica structure of the vacuum [7}9]. Indeed, according to this `anti-granduni"cationa analogy, such properties of our world, as gravitation, gauge "elds, elementary chiral fermions, etc., all arise in the low-energy corner as a low-energy soft modes of the underlying `condensed mattera. At high energy (of the Planck scale) these modes merge with the continuum of the all high-energy degrees of freedom of the `Planck condensed mattera and thus cannot be separated anymore from each other. Since the gravity is not fundamental, but appears as an e!ective "eld in the infrared limit, the only output of its quantization would be the quanta of the low-energy gravitational waves } gravitons. The more deep quantization of gravity makes no sense in this philosophy. In particular, the e!ective theory cannot give any prediction for the vacuum energy and thus for the cosmological constant. The main advantage of the condensed matter analogy is that in principle we know the condensed matter structure at any relevant scale, including the interatomic distance, which plays the part of
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one of the Planck length scales in the hierarchy of scales. Thus the condensed matter can suggest possible routes from our present low-energy corner of `phenomenologya to the `microscopica physics at Planckian and trans-Planckian energies. It can also show the limitation of the e!ective theories: what quantities can be calculated within the e!ective "eld theory using, say, renormalization group approach, and what quantities depend essentially on the details of the transPlanckian physics. The condensed matter analogy is in some respect similar to the string theory, where the gauge invariance and general covariance are not imposed, and fermions, gravitons and gauge quanta are the emergent low-energy properties of an underlying physical object } the string. At the moment the string theory is viewed as the most successful attempt to quantize gravity so far. However, as distinct from the string theory which requires higher dimensions, the `relativistica fermions and bosons arise in the underlying nonrelativistic condensed matter in an ordinary 3#1 space}time, provided that the condensed matter belongs to the proper universality class. In the main part of the review we consider super#uid He in its A-phase which belongs to that special universality class of Fermi liquids, where the e!ective gravity, gauge "elds and chiral fermions appear in the low-energy corner together with Lorentz and gauge invariance [5,10], and discuss the correspondence between the phenomena in super#uid He-A and that in relativistic particle physics. However, some useful analogies can be provided even by Bose liquid } super#uid He, where a sort of the e!ective gravitational "eld appears in the low-energy corner. That is why it is instructive to start with the simplest e!ective "eld theory of Bose super#uid which has a very restricted number of e!ective "elds.
2. Landau}Khalatnikov two-6uid hydrodynamics as e4ective theory of gravity 2.1. Superyuid vacuum and quasiparticles According to Landau and Khalatnikov [11] a weakly excited state of the collection of interacting He atoms can be considered as a small number of elementary excitations } quasiparticles (phonons and rotons). In addition, the state without excitation } the ground state, or vacuum } can experience the collective motion. The super#uid vacuum can move without friction, and inhomogeneity of the #ow serves as the gravitational and/or other e!ective "elds. The matter propagating in the presence of this background is represented by fermionic (in Fermi super#uids) or bosonic (in Bose super#uids) quasiparticles, which form the so called normal component of the liquid. Such two-#uid hydrodynamics introduced by Landau and Khalatnikov [11] is the example of the e!ective "eld theory which incorporates both the collective motion of the super#uid background (gravitational "eld) and the quasiparticle excitations (matter). This is the counterpart of the Einstein equations, which incorporate both gravity and matter. One must distinguish between the bare particles and quasiparticles in super#uids. The particles are the elementary objects of the system on a microscopic `trans-Planckiana level, these are the atoms of the underlying liquid (He or He atoms). The many-body system of the interacting atoms form the quantum vacuum } the ground state. The nondissipative collective motion of the super#uid vacuum with zero entropy is determined by the conservation laws experienced by the atoms and by their quantum coherence in the super#uid state. The quasiparticles are the
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particle-like excitations above this vacuum state. The bosonic excitations in super#uid He and fermionic and bosonic excitations in super#uid He represent the matter in our analogy and correspond to elementary particles. In super#uids they form the viscous normal component responsible for the thermal and kinetic low-energy properties of super#uids. 2.2. Dynamics of superyuid vacuum In the simplest super#uid, the coherent motion of the super#uid vacuum is characterized by two soft collective (hydrodynamic) variables: the mean particle number density n(r, t) of atoms comprising the liquid, and the super#uid velocity * (r, t) of their coherent motion. In super#uid He the vacuum is the coherent state described by the macroscopic phase, and the super#uid velocity is the gradient of the phase: * "( /m), where m is the bare mass of particle } the mass of He atom. The #ow of such vacuum is curl-free: ;* "0. This is not however a rule: as we shall see in Section 5.1.3, in He-A the macroscopic coherence is more complicated and the #ow of super#uid vacuum can have a continuous vorticity, ;* O0. 2.2.1. Galilean transformation for quasiparticles The liquids considered here are nonrelativistic and obeying the Galilean transformation law. Under the Galilean transformation to the coordinate system moving with the velocity u the super#uid velocity transforms as * P* #u. The transformational properties of bare particles (atoms) and that of quasiparticles are essentially di!erent. The transformation law for the momentum and energy of atoms contains the mass m of the atom: pPp#mu and EPE#p ) u#(1/2)mu. But such characteristic of the microscopic world as the bare mass m cannot enter the transformation law for quasiparticles: quasiparticles in e!ective low-energy theory have no information on the trans-Planckian world of the bare atoms comprising the vacuum state. This implies that the momentum of quasiparticle is invariant under the Galilean transformation: pPp, while the quasiparticle energy is Doppler shifted: EI PEI #p ) u. The Galilean invariance is the symmetry of the underlying microscopic physics. When the low-energy corner is approached and the e!ective "eld theory emerges, the Galilean transformations are gradually extended towards the general coordinate transformations of the e!ective `relativistica theory. This is another example of how the memory on the microscopic physics is erased in the low-energy corner. 2.2.2. Current and continuity equation The particle number conservation provides one of the equations of the e!ective theory of super#uids } the continuity equation: Rn # ) J"0 . Rt
(1)
In a strict microscopic theory of monoatomic liquid, n and the particle current density J are given by the particle distribution function n( p, r)"q e q raRp a : \q p>q 1 n(r)" n( p, r), J(r)" pn( p, r) . (2) m p p
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In the Galilean system the momentum of particles equals the mass current, and thus the particle current and the momentum density are related by the second in Eq. (2). The particle distribution function n( p, r) is typically a rather complicated function of momentum even at ¹"0 because of the strong interaction between the bare atoms in a real liquid. n( p) can be determined only in a fully microscopic theory and thus never enters the e!ective theory, in which the details of macroscopic physics are lost. The latter instead is determined by quasiparticle distribution function f ( p, r), which is simple because at low ¹ the number of quasiparticles is small and their interaction can be neglected. That is why in equilibrium f ( p) is given by the thermal Bose distribution (or by the Fermi distribution for fermionic quasiparticles) and in nonequilibrium it can be found from the conventional kinetic equation for quasiparticles. In the e!ective theory the particle current has two contributions J"n* #J ,
1 J " P, P" pf ( p) . m p
(3)
The "rst term n* is the current transferred coherently by the collective motion of super#uid vacuum with the super#uid velocity * . In equilibrium at ¹"0 this is the only current, but if quasiparticles are excited above the ground state, their momentum P gives an additional contribution to the particle current providing the second term in Eq. (3). Since the momentum of quasiparticle is invariant under the Galilean transformation, the particle current in Eq. (3) transforms in a proper way: JPJ#nu. 2.2.3. London equation and energy density The second equation for the collective variables is the London equation for the super#uid velocity (see Section 2.2.4), which is curl-free in super#uid He (;* "0): E R* (4) m Q # "0 . n Rt Together with the kinetic equation for the quasiparticle distribution function f ( p), Eqs. (4) and (1) for collective "elds * and n give the complete e!ective theory for the kinetics of quasiparticles (matter) and coherent motion of vacuum (gravitational "eld) once the energy functional E is known. In the limit of low temperature, where the density of thermal quasiparticles is small, the interaction between quasiparticles can be neglected. Then the simplest Ansatz satisfying the Galilean invariance is
E" dr
m n*#(n)!n# EI ( p, r) f ( p, r) . 2 p
(5)
Here (n) (or (n)"(n)!n) is the vacuum energy density as a function of the particle density; is the overall constant chemical potential, which is the Lagrange multiplier responsible for the conservation of the total number N"dx n of the He atoms; EI ( p, r)"E( p, n(r))#p ) * (r) (6) is the Doppler shifted quasiparticle energy in the laboratory frame with E( p, n(r)) being the quasiparticle energy measured in the frame comoving with the super#uid vacuum.
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2.2.4. Absence of canonical Lagrangian formalism in ewective theories Eqs. (1) and (4) can be obtained from the Hamiltonian formalism, R * " H, * , R n" H, n , R R using the energy in Eq. (5) as Hamiltonian, H"E, and the following Poisson brackets 1 * (r ), n(r ) " (r !r ), n(r ), n(r ) " * (r ), * (r ) "0 . m
(7)
The Poisson brackets between components of super#uid velocity are zero only for curl-free super#uidity. In a more general case they are [11,12] (for further generalization see [13]): 1 v (r ), v (r ) "! e (;* ) (r !r ) . I G H mn GHI
(8)
In this case even at ¹"0, when the quasiparticles are absent, the Hamiltonian description of the hydrodynamics is only possible: There is no Lagrangian, which can be expressed in terms of the hydrodynamic variables * and n (Lagrangian can be introduced in terms of the nonlocal Clebsch variables). The absence of the Lagrangian for the soft collective variables in many condensed matter systems [12,14] is one of the consequences of the reduction of the degrees of freedom in e!ective "eld theory, as compared with the fully microscopic description where the Lagrangian exists on the fundamental level [15]. When the high-energy microscopic degrees are integrated out, the nonlocality of the remaining action is a typical phenomenon, which shows up in many faces. In ferromagnets, for example, the number of the hydrodynamic variables is odd: three components of the magnetization vector M. They thus cannot form the canonical pairs of conjugated variables. One can (i) use the Hamiltonian description in terms of Poisson brackets, M (r ), M (r ) " G H !e M (r )(r !r ); (ii) introduce the nonlocal variables, such as spherical coordinates of GHI I unit vector m( "M/M; or (iii) introduce the multi-valued action [12]. Such action can be written as the Novikov}Wess}Zumino term, which contains an extra coordinate . For ferromagnets this term is [15]:
S " dx dt d Mm( ) (R m( ;R m( ) . R O ,58
(9)
The integral here is over the 5D space, whose boundary is an ordinary 3#1 space}time. Though the action is written in "ctitious 5D space, its variation is the total derivative and thus is determined in the physical space}time. According to the condensed matter analogy, the presence of the non-local Wess}Zumino term in a relativistic quantum "eld theory would indicate that such theory is e!ective. Probably the same happens in gravity: the absence of the covariant energy}momentum tensor simply re#ects existence of underlying `microscopica degrees of freedom, which are responsible for nonlocality of the energy}momentum for the `collectivea gravitational "eld (see also Section 3.3). 2.3. Normal component } `mattera In a local thermal equilibrium the distribution of quasiparticles is characterized by local temperature ¹ and by local velocity of the quasiparticle gas * , which is called the normal
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component velocity:
fT ( p)" exp
\ EI ( p)!p* $1 , ¹
(10)
where the sign # is for the fermionic quasiparticles in Fermi super#uids and the sign ! is for the bosonic quasiparticles in Bose super#uids. Since EI ( p)"E( p)#p ) * , the equilibrium distribution is determined by the Galilean invariant quantity * !* ,w, which is the normal component velocity measured in the frame comoving with super#uid vacuum. It is called the counter#ow velocity. In the limit when the counter#ow velocity * !* is small, the quasiparticle (`mattera) contribution to the liquid momentum and thus to the particle current in Eq. (3) is proportional to the counter#ow velocity: p p RfT , J "n (v !v ), n "! G I G GI I I GI m RE p
(11)
where the tensor n is the so called density of the normal component. In this linear regime the GI total current in Eq. (3) can be represented as the sum of the currents carried by the normal and super#uid components J "n v #n v , (12) G GI I GI I where tensor n "n !n is the so called density of super#uid component. In the isotropic GI GI GI super#uids, He and He-B, where the quasiparticle spectrum in super#uid-comoving frame in Eq. (6) is isotropic, E( p)"E( p ), the normal component density is an isotropic tensor, n "n . GI GI In super#uid He-A the normal component density is a uniaxial tensor which re#ects an uniaxial anisotropy of quasiparticle spectrum [16]. At ¹"0 the quasiparticles are frozen out and one has n "0 and n "n in all monoatomic super#uids. GI GI GI 2.4. Quasiparticle spectrum and ewective metric The structure of the quasiparticle spectrum in super#uid He becomes more and more universal the lower the energy. In the low-energy corner the spectrum of these quasiparticles, phonons, can be obtained in the framework of an e!ective theory. The e!ective theory is unable to describe the high-energy part of the spectrum } rotons. These can be determined in a fully microscopic theory only. On the contrary, the spectrum of phonons is linear, E( p, n)Pc(n) p , and only the `fundamental constanta } the speed of `lighta c(n) } depends on the physics of the higher-energy hierarchy rank. Phonons represent the quanta of the collective modes of the super#uid vacuum, sound waves, with the speed of sound obeying c(n)"(n/m)(d/dn). All other information on the microscopic atomic nature of the liquid is lost. The Lagrangian for sound waves propagating above the smoothly varying background is obtained from Eqs. (1) and (4) at ¹"0 by decomposition of the super#uid velocity and density into the smooth and #uctuating parts: * "* # [17,18]. The quadratic part of the Lagrangian for the scalar "eld is [19]:
1 1 m L" n ( )! ( #(* ) ) ) , (!ggIJR R . I J c 2 2
(13)
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The quadratic Lagrangian for sound waves has necessarily the Lorentzian form, where the e!ective Riemann metric experienced by the sound wave, the so called acoustic metric, is simulated by the smooth parts of the hydrodynamic "elds: vG cGH!vG vH 1 , , gG"! , gGH" g"! mnc mnc mnc
(14)
mn mnv mn mn G , g " , (!g" g "! (c!*), g "! . G GH c c c GH c
(15)
Here and further * , n and c mean the smooth parts of the velocity, density and `speed of lighta. Sound waves in super#uids and crystals provide a typical example of how an enhanced symmetry and e!ective Lorentzian metric appear in condensed matter in the low-energy corner. The energy spectrum of sound wave quanta, phonons, which represent the (scalar) `gravitonsa in this e!ective gravity, is determined by gIJp p "0 or (EI !p ) * )"cp, i.e. E"cp . I J
(16)
2.5. Ewective metric for bosonic collective modes in other systems The e!ective action in Eq. (13) is typical of the low-energy collective modes in ordered systems. A more general case is provided by the Lagrangian for the Goldstone bosons in antiferromagnets } the spin waves. The spin wave dynamics in x}y antiferromagnets and in He-A is governed by the Lagrangian for the Goldstone variable , which is the angle of the antiferromagnetic vector: (17) L"GH !( #(* ) ) ),(!ggIJR R . G H I J Here the matrix GH is the spin rigidity, which is tensor both in crystalline antiferromganet and in anisotropic He-A; is the spin susceptibility; and * is the local velocity of crystal in antiferromagnets and the super#uid velocity, *"* , in He-A. In antiferromagnets these 10 coe$cients give rise to all ten components of the e!ective Riemann metric:
g"!(), gG"!()vG, gGH"
(GH!vGvH), \"det(GH) ,
(18)
g "!()\(1! vGvH), g "! vH, g " , (!g" . GH G GH GH GH (19)
The e!ective interval of the space}time, where quasiparticles (magnons) are propagating along the geodesics, is
1 ds"! (dxG!vG dt)(dxH!vH dt) . dt# GH ()
(20)
This form of the interval corresponds to the Arnowitt}Deser}Misner decomposition of the space}time metric, where the function 1 N" , ()
(21)
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is known as lapse function; g "(/) gives the three-metric describing the geometry of the GH GH e!ective space; and the velocity vector * plays the part of the so-called shift function (see, e.g. the book [20]). 2.6. Ewective quantum xeld and ewective action The e!ective action for the bosonic quasiparticles, Eq. (13) for phonons and Eq. (17) for spin waves (magnons), is formally general covariant. In addition, in the classical limit of Eq. (16) corresponding to geometrical optics (in our case this is geometrical acoustics) the propagation of phonons is invariant under the conformal transformation of metric, gIJPgIJ. The latter symmetry is lost at the quantum level: Eq. (13) is not invariant under general conformal transformations; however the reduced symmetry is still there: Eq. (13) is invariant under global scale transformation, gIJPgIJ and P with "Const. As we shall see further, in super#uid He-A other e!ective "elds and new symmetries appear in the low-energy corner, including also e!ective S;(2) gauge "elds and gauge invariance. The symmetry of Lagrangian for fermionic quasiparticles induces, after integration over the fermionic degrees of freedom, the corresponding symmetry of the e!ective action for the gauge "elds. Moreover, in addition to super#uid velocity "eld there are other collective degrees of freedom which simulate the gravity with the spin-2 gravitons. However, as distinct from the e!ective gauge "elds, whose e!ective action is very similar to that in particle physics, the e!ective gravity cannot reproduce in a full scale the Einstein theory: the e!ective action for the metric is contaminated by the noncovariant terms, which come from the `trans-Planckiana physics [5]. The origin of the noncovariant terms in the e!ective action for the `gravitya in condensed matter is actually the same as the source of the problems related to quantum gravity and cosmological constant: the e!ective quantum "eld theory for gravity contains nonrenormalizable ultraviolet in"nities. The quantum quasiparticles interact with the classical collective "elds * and n, and with each other. In Fermi super#uid He the fermionic quasiparticles interact with many collective "elds describing the multicomponent order parameter and with their quanta. That is why one obtains the interacting Fermi and Bose quantum "elds, which are in many respect similar to that in particle physics. However, this e!ective "eld theory can be applied to the lowest order of the perturbation theory only. The higher-order diagrams are divergent and nonrenormalizable, which simply means that the e!ective theory is valid when only the low-energy/momentum quasiparticles are involved even in their virtual states. This means that only those terms in the e!ective action can be derived by integration over the quasiparticle degrees of freedom, whose integrals are concentrated solely in the low-energy region. For the other processes one must go beyond the e!ective "eld theory and consider the higher levels of description, such as Fermi liquid theory, or further the microscopic level of the underlying liquid with atoms and their interactions. In short, all the terms in e!ective action come from the microscopic `Plancka physics, but only some fraction of them can be derived in a self-consistent way within the e!ective "eld theory itself. In Bose super#uids the fermionic degrees of freedom are absent, that is why the quantum "eld theory there is too restrictive to serve as a model for relativistic particle physics, but nevertheless it is useful to consider it since it provides the simplest example of the e!ective theory. On the other hand the Landau}Khalatnikov scheme is rather universal and is easily extended to super#uids with more complicated order parameter and with fermionic degrees of freedom (see the book [16]).
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2.7. Vacuum energy and cosmological constant. Nullixcation of vacuum energy 2.7.1. Nullixcation of vacuum energy in quantum liquids The vacuum energy density (n), or the other relevant potential of the vacuum (n)"(n)!n, as well as the parameters which characterize the quasiparticle energy spectrum cannot be determined by the e!ective theory: they are provided solely by the higher (microscopic) level of description. The equilibrium state of the vacuum is obtained by minimization of the energy (n) at given number N of bare atoms, or which is the same by minimization of the potential (n)"(n)!n where the chemical potential is the Lagrange multiplier. We consider here only the states with spatially homogeneous n since the liquid helium (or other underlying liquid) must be homogeneous in equilibrium. (This in particular means that in equilibrium the e!ective metric g must be spatially homogeneous, i.e. the e!ective space viewed by IJ quasiparticles must be #at in equilibrium. The #atness thus arises naturally in the e!ective theory as the property of the underlying system, without having to invoke the in#ationary concept.) The equilibrium state of the vacuum is characterized by the equilibrium value of the particle number density n (), which is obtained from equation d /dn"0. The equilibrium value of the potential (n) is related to the pressure in the liquid created by external sources provided by the environment. From the de"nition of the pressure, P"!d( P Q * 0 u (3) # 0 * u (3) 0 # 0 d (3) ! 0 ! * d (3) 0 ! ! ! 0 # 0 !1 ! 0 * !1 0 0 0 # 0 e ! 0 !1 ! !1 * !1 !1 !1 e 0 ! 0
(115)
In the above G(224) model, 16 fermions of one generation can be represented as the product Cw of four bosons and four fermions [65]. This scheme is similar to the slave-boson approach in condensed matter, where the particle is considered as a product of the spinon and holon. Spinons are fermions which carry spin, while holons are `slavea-bosons which carry electric charge [66]. In the Terazawa scheme [65] the `holonsa C form the S;(4) quartet of spin-0 S;(2)-singlet particles ! which carry baryonic and leptonic charges, their B!¸ charges of the S;(4) group are ! (, , ,!1). The `spinonsa are spin- particles w, which are S;(4) singlets and S;(2)-isodoublets; !
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they carry spin and isospin.
u
d u d C * 0 0 u d u d C * * 0 0 " ;(w> w\ w> w\) . (116) * * 0 0 u d u d C * * 0 0 e e C * * 0 0 \ Here $1/2 means the charge = for the left spinons and = for the right spinons, which * 0 coincides with the electric charge of spinons: Q"(B!¸)#= #= "= #= . In * 0 * 0 Terazawa notations w "(w>, w>) forms the doublet of spinons with Q"#1/2 and * 0 w "(w\, w\) } with Q"!1/2. These four spinons, two left and two right, transform under * 0 S;(2) ;S;(2) symmetry group. * 0 *
5.2.2. Momentum-space topological invariants In the case of one chiral fermion the massless (gapless) character of its energy spectrum in Eq. (67) is protected by the momentum-space topological invariant. However, in case of the equal number of left and right fermions the total topological charge N in Eq. (65) is zero for the Fermi point at p"0, if the trace is over all the fermionic species. Thus the topological mechanism of mass protection does not work and in principle an arbitrary small interaction between the fermions can provide the Dirac masses for all eight pairs of fermions. This indicates that the Standard Model is marginal in the same way as the planar state of super#uid He in Eq. (94). However, in systems with marginal Fermi points, the mass (gap) would not appear for some or all fermions, if the interaction has some symmetry elements. This situation occurs both in the planar phase of He and in the Standard Model. In both cases the weighted momentum-space topological invariants can be introduced which provide the mass protection. These invariants are robust to such perturbations, which conserve given symmetry, and they are the functions of parameters of this symmetry group. In the Standard Model the relevant symmetries are the electroweak symmetries ;(1) and S;(2) generated by the hypercharge and by the weak charge 7 * correspondingly. 5.2.3. Generating function for topological invariants constrained by symmetry Let us introduce the matrix N whose trace gives the invariant N in Eq. (65): 1 dSA GR I G\GR J G\GR H G\ , e (117) N" N N N 24 IJHA N where as before the integral is about the Fermi point in the 4D momentum}energy space. Let us consider the expression
(N, Y)"tr[NY] ,
(118)
where Y is the generator of the ;(1) group, the hypercharge matrix. It is clear that the Eq. (118) is 7 robust to any perturbation of Green's function, which does not violate the ;(1) symmetry, since in 7 this case the hypercharge matrix Y commutes with Green's function G. The same occurs with any power of Y, i.e. (N, YL) is also invariant under symmetric deformations. That is why one can introduce the generating function for all the topological invariants containing powers of the
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hypercharge (e F7 Y, N)"tr[e F7 YN] .
(119)
All the powers (N, YL), which are topological invariants, can be obtained by di!erentiating of Eq. (119) over the group parameter . Since the above parameter-dependent invariant is robust to 7 interactions between the fermions, it can be calculated for the noninteracting particles. In the latter case the matrix N is diagonal with the eigenvalues C "#1 and C "!1 for right and left ? ? fermions, respectively. The trace of this matrix N over given irreducible fermionic representation introduced by Froggatt and Nielsen in of the gauge group is (with minus sign) the symbol N W? '5 Ref. [67]. In their notations y/2(">), a, and I denote hypercharge, color representation and the 5 weak isospin correspondingly. For the Standard Model with hypercharges for 16 fermions given in Eq. (115) one has the generating function:
(120) (e F7 Y, N)" C e F7 7? "2 cos 7 !1 (3e F7 #e\ F7 ) . ? 2 ? The factorized form of the generating function re#ects the factorization in Eq. (116) and directly follows from this equation: The generating function for the momentum space topological invariants for `holonsa is (e F * B\L, N )"(3e F * #e\ F * ), which must be multiplied by the `spinona factor 2(cos( /2)!cos( /2)). 0 * In addition to the hypercharge the weak charge is also conserved in the Standard Model above the electroweak transition. The generating function for the topological invariants which contain the powers of both the hypercharge > and the weak charge = also has the factorized form: (121) (e F5 W* e F7 Y, N)"2 cos 7 !cos 5 (3e F7 #e\ F7 ) . 2 2
The generators of the S;(3) color group, which is left}right symmetric, do not change the form of ! the generating function in Eq. (121). The nonzero values of Eq. (121) show that Green's function is singular at p"0 and p "0, which means that some fermions must be massless. 5.2.4. Discrete symmetry and massless fermions Choosing the parameters "0 and "2 one obtains the maximum possible value of the 7 5 generating function: (e W* , N)"16 .
(122)
which means that all 16 fermions of one generation are massless above the electroweak scale 200 GeV. This also shows that in many cases only the discrete symmetry group, such as the Z group e W* , is enough for the mass protection. In the planar state of He in Eq. (94) each of the two Fermi points at p"$p lK has zero $ topological charge, N "0. Nevertheless the gapless fermions in the planar state are supported by the topological invariant (P, N) containing the discrete Z symmetry (PP"1) of the planar state vacuum. This symmetry is the combination of discrete gauge transformation and spin rotation by around z axis: e CX A "A (on discrete symmetries of super#uid phases of He see Ref. [16]). IG IG
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Applying to the Bogoliubov}Nambu Hamiltonian (94), for which the generator of the ;(1) gauge rotation is \ , this symmetry operation has the form P"e O\ e NX "!\ : PH"HP. The X nonzero topological invariants, which support the mass (gap) protection for the Fermi points at p"$p lK , are correspondingly (P, N)"$2. Thus at each Fermi point there are gapless $ fermions. They acquire the relativistic energy spectrum in the low-energy corner. In the relativistic limit the discrete symmetry P, which is responsible for the mass protection, is equivalent to the -symmetry for Dirac fermions. If the symmetry is obeyed, i.e. it commutes with the Dirac Hamiltonian, H"H, then the Dirac fermion has no mass. This is consistent with the nonzero value of the topological invariant: it is easy to check that for the massless Dirac fermion one has (, N)"2. This connection between topology and mass protection looks trivial in the relativistic case, where the absence of mass due to symmetry can be directly obtained from the Dirac Hamiltonian. However the equations in terms of the topological charge, such as Eq. (122), appears to be more general, since they remain valid even if the Lorentz symmetry is violated at higher energy and the Dirac equation is not applicable any more. In the nonrelativistic case even the chirality is not a good quantum number at high energy (this in particular means that transitions between the fermions with di!erent chirality are possible at high energy, see Section 10.1.2 for an example). The topological constraints, such as in Eq. (122), protect nevertheless the gapless fermionic spectrum in non-Lorentz-invariant Fermi systems. 5.2.5. Nullixcation of topological invariants below electroweak transition and massive fermions When the electroweak symmetry ;(1) ;S;(2) is violated to ;(1) , the only remaining charge 7 * / } the electric charge Q">#= } produces zero value for the whole generating function according to Eq. (121): (e F/ Q, N)"(e F/ Ye F/ W* , N)"0 .
(123)
The zero value of the topological invariants implies that even if the singularity in Green's function exists it can be washed out by interaction. Thus each elementary fermion in our world must have a mass after such a symmetry breaking. What is the reason for such a symmetry breaking pattern, and, in particular, for such choice of electric charge Q? Why the nature had not chosen the more natural symmetry breaking, such as ;(1) ;S;(2) P;(1) , ;(1) ;S;(2) PS;(2) or ;(1) ;S;(2) P;(1) ;;(1) ? The pos7 * 7 7 * * 7 * 7 5 sible reason is provided by Eq. (121), according to which the nulli"cation of all the momentumspace topological invariants occurs only if the symmetry breaking scheme ;(1) ;S;(2) P;(1) 7 * / takes place with the charge Q"$>$= . Only in such cases the topological mechanism for the mass protection disappears. This can shed light on the origin of the electroweak transition. It is possible that the elimination of the mass protection is the only goal of the transition. This is similar to the Peierls transition in condensed matter: the formation of mass (gap) is not the consequence but the cause of the transition. It is energetically favorable to have masses of quasiparticles, since this leads to decrease of the energy of the fermionic vacuum. Formation of the condensate of top quarks, which generates the heavy mass of the top quark, could be a relevant scenario for that (see review [68]). In the G(224) model the electric charge Q"(B!¸)#= #= is left}right symmetric. * 0 That is why, if only the electric charge is conserved in the "nal broken symmetry state, the only Q relevant topological invariant (e F/ , N) is always zero, there is no mass protection and the Weyl
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fermions must be paired into Dirac fermions. This fact does not depend on the de"nition of the hypercharge, which appears at the intermediate stage where the symmetry is G(213). It also does not depend much on the de"nition of the electric charge Q itself: the only condition for the nulli"cation of the topological invariant is the symmetry (or antisymmetry) of Q with respect to the parity transformation. 5.2.6. Relation to axial anomaly The momentum-space topological invariants determine the axial anomaly in fermionic systems. In particular, the charges related to the gauge "elds cannot be created from vacuum; the condition for that is the nulli"cation of some invariants (see Section 7): (Y, N)"(Y, N)"((W* )Y, N)"(YW* , N)"2"0 . (124) Nulli"cation of all these invariants is provided by the form of the generating function in Eq. (121), though in this equation it is not assumed that the groups ;(1) and S;(2) are local. 7 * 6. E4ective relativistic quantum 5eld theory emerging in a system with Fermi point 6.1. Collective modes of fermionic vacuum } electromagnetic and gravitational xelds Let us consider the collective modes in the system with Fermi points. The e!ective "elds acting on a given particle due to interactions with other moving particles cannot destroy the Fermi point. That is why, under the inhomogeneous perturbation of the fermionic vacuum the general form of Eqs. (66) and (67) is preserved. However the perturbations lead to a local shift in the position of the Fermi point p in momentum space and to a local change of the vierbein eI (which in particular I @ includes slopes of the energy spectrum. This means that the low-frequency collective modes in such Fermi liquids are the propagating collective oscillations of the positions of the Fermi point and of the slopes at the Fermi point (Fig. 7). The former is felt by the right- or the left-handed quasiparticles as the dynamical gauge (electromagnetic) "eld, because the main e!ect of the electromagnetic "eld A "(A , A) is just the dynamical change in the position of zero in the energy I spectrum: in the simplest case (E!eA )"c( p!eA). The collective modes related to a local change of the vierbein eI correspond to the dynamical @ gravitational "eld. The quasiparticles feel the inverse tensor g as the metric of the e!ective space IJ in which they move along the geodesic curves ds"g dxI dxJ . (125) IJ Therefore, the collective modes related to the slopes play the part of the gravity "eld. Thus near the Fermi point the quasiparticle is the chiral massless fermion moving in the e!ective dynamical electromagnetic and gravitational "elds. 6.2. Physical laws in vicinity of Fermi point: Lorentz invariance, gauge invariance, general covariance, conformal invariance In the low-energy corner the fermionic propagator in Eq. (66) is gauge invariant and even obeys the general covariance near the Fermi point. For example, the local phase transformation of the
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wave function of the fermion, Pe C?rR can be compensated by the shift of the `electromagnetica "eld A PA #R . These attributes of the electromagnetic (A ) and gravitational (gIJ) "elds arise I I I I spontaneously as the low-energy phenomena. Now let us discuss the dynamics of the bosonic sector } collective modes of A and gIJ. Since I these are the e!ective "elds their motion equations do not necessarily obey gauge invariance and general covariance. However, in some special cases such symmetries can arise in the low-energy corner. The particular model with the massless chiral fermions has been considered by Chadha and Nielsen [2], who found that the Lorentz invariance becomes an infrared "xed point of the renormalization group equations. What are the general conditions for such symmetry of the bosonic "elds in the low-energy corner? The e!ective Lagrangian for the collective modes is obtained by integrating over the vacuum #uctuations of the fermionic "eld. This principle was used by Sakharov and Zeldovich to obtain an e!ective gravity [24] and e!ective electrodynamics [69], both arising from #uctuations of the fermionic vacuum. If the main contribution to the e!ective action comes from the vacuum fermions whose momenta p are concentrated near the Fermi point, i.e. where the fermionic spectrum is linear and thus obeys the `Lorentz invariancea and gauge invariance of Eq. (66), the result of the integration is necessarily invariant under gauge transformation, A PA #R , and has I I I a covariant form. The obtained e!ective Lagrangian then gives the Maxwell equations for A and I the Einstein equations for g , so that the propagating bosonic collective modes do represent the IJ gauge bosons and gravitons. Thus two requirements must be ful"lled } (i) the fermionic system has a Fermi point and (ii) the main physics is concentrated near this Fermi point. In this case the system acquires at low energy all the properties of the modern quantum "eld theory: chiral fermions, quantum gauge "elds, and gravity. All these ingredients are actually low-energy (infra-red) phenomena. In this extreme case when the vacuum fermions are dominatingly relativistic, the bosonic "elds acquire also another symmetry obeyed by massless relativistic Weyl fermions, the conformal invariance } the invariance under transformation g P(r, t)g . The gravity with the conforIJ IJ mally invariant e!ective action, the so-called Weyl gravity, is still a viable rival to Einstein gravity in modern cosmology [70,71]: The Weyl gravity (i) can explain the galactic rotation curves without dark matter; (ii) it reproduces the Schwarzschild solution at small distances; (iii) it can solve the cosmological constant problem, since the cosmological constant is forbidden if the conformal invariance is strongly obeyed; etc. (see [72]). 6.3. Ewective electrodynamics 6.3.1. Ewective action for `electromagnetica xeld Let us consider what happens in a practical realization of systems with Fermi points in condensed matter } in He-A. From Eqs. (108) and (111) it follows that the "elds, which act on the `relativistica quasiparticles as electromagnetic and gravitational "elds, have a nontrivial behavior. For example, the same texture of the lK -vector is felt by quasiparticles as the e!ective magnetic "eld B"p o ;lK according to Eq. (111) and simultaneously it enters the metric according to Eq. (108). $ Such "eld certainly cannot be described by the Maxwell and Einstein equations together. Actually the gravitational and electromagnetic variables coincide in He-A only when we consider the vacuum manifold: Outside of this manifold they split. He-A, as any other fermionic system with
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Fermi point, has enough number of collective modes to provide the analogs for the independent gravitational and electromagnetic "elds. But some of these modes are massive in He-A. For example the gravitational waves correspond to the modes, which are di!erent from the oscillations of the lK -vector. As distinct from the photons (orbital waves } propagating oscillations of the lK -vector) the gravitons are massive (Section 10.3). All these troubles occur because in He-A the main contribution to the e!ective action for the most of the bosonic "elds come from the integration over vacuum fermions at the `Plancka energy scale, E& . These fermions are far from the Fermi points and their spectrum is nonlinear. That is why in general the e!ective action for the bosonic "elds is not symmetric. There are, however, situations when the He-A behave as `perfecta condensed matter, i.e. when there is an exact correspondence between some terms in action for relativistic quantum "eld theory and for He-A. We discuss them below. Considering this correspondence one must keep in mind that the two systems are described by di!erent sets of variables. For example, the vector "eld lK is an observable variable in He-A, but at low energy it plays the role of vector potential of gauge "eld A, which is not observable in relativistic quantum "eld theory. To transform from one set of variables to the other, the free energy or Lagrangian of the two systems should be expressed in a covariant and gauge invariant form and should not contain any material parameters, such as `speed of lighta [48]. Then it can be equally applied to both systems, Standard Model and He-A. Note that the e!ective quantum "eld theory, if it does not contain the high energy cuto!, should not contain the speed of light c explicitly: it is hidden within the metric tensor. 6.3.2. Running coupling constant: zero charge ewect The "rst example, where the exact correspondence occurs, is the action for the lK -"eld, which contains the term with the logarithmically divergent factor ln(/) (see Ref. [48] and Section 13). It comes from the zero charge e!ect, the logarithmic screening of the `electric chargea by the massless fermions, for whom the lK -"eld acts as electromagnetic "eld. Due to its logarithmic divergence this term is dominating at low frequency : the lower the frequency the larger is the contribution of the vacuum fermions from the vicinity of the Fermi point and thus the more symmetric is the Lagrangian for the lK -"eld. This happens, for example, in the physically important case discussed in Section 8.3, where the Lagrangian for the lK -texture is completely equivalent to the conventional Maxwell Lagrangian for the (hyper-) magnetic and electric "elds. In this particular case the equilibrium state is characterized by the homogeneous direction of the lK vector, which is "xed by the counter#ow: lK * !* . The e!ective electromagnetic "eld is simulated by the small deviations of the lK vector from its equilibrium direction, A"p lK . Since lK is a unit vector, its variation lK NlK . This corresponds to the $ gauge choice A"0, if z axis is chosen along the background orientation, z( "lK . In the considered case only the dependence on z and t is relevant. As a result in the low-energy limit the e!ective Lagrangian for the A becomes gauge invariant, so that in this regime the A "eld does obey the I I Maxwell equations coming from the Lagrangian: (!g gIJg?@F F . ¸" I? J@ 4
(126)
Here is a running coupling constant. To apply this to He-A one must express the metric and gauge "eld in terms of the He-A observables. The e!ective metric gIJ is given by Eq. (108), where in
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the linear approximation one can use the homogeneous background "eld lK , and the gauge "eld is A"p lK . Substituting these into Eq. (126) one reproduces the Lagrangian for the lK "eld, which was $ earlier obtained in a microscopic theory [73]:
p v 1 $ $ (R lK )! (R lK #(* ) )lK ) . (127) X ¹ 24
v R $ The running coupling constant is logarithmically divergent because of the polarization of the vacuum of gapless fermions: ln
1 \" ln . 12 ¹
(128)
This is in a complete analogy with the logarithmic divergence of the "ne structure constant e/4 c in quantum electrodynamics, which is provided by polarization of the fermionic vacuum with two species of Weyl fermions (or with one Dirac fermion if its mass M is small compared to ¹, otherwise ¹ is substituted by M). The gap amplitude , constituting the ultraviolet cuto! of the logarithmi cally divergent coupling, plays the part of the Planck energy scale, while the infrared cuto! is provided by temperature. To extend the Eq. (128) to the moving super#uid it must be written in covariant form introducing the four-temperature and the cuto! four vector " u where the four-temperature J and I I four-velocity u are determined in Section 3: J 1 ln(I ) . (129) \" I 6 At ¹"0 the infrared cuto! is provided by the magnetic "eld itself:
(gIJ ) 1 I J ln . (130) \" gIJg?@F F 12 I? J@ Note that has a parallel with the Planck energy in some other situations, too (see Section 8.3). Another example is the analog of the cosmological constant, which arises in the e!ective gravity of He-A and has the value /6 (see Section 10.3 and also Ref. [74]). This parameter also determines the gravitational constant G&\ (see Sections 11.2 and 8.2.4 and also Section 13, where it was found that G\"(2/9) ). 6.4. Ewective S;(N) gauge xelds from degeneracy of Fermi point 6.4.1. Why all fermions and bosons have the same speed of light in low-energy limit In He-A the Fermi point (say, at the north pole p"#p lK ) is doubly degenerate owing to the $ ordinary spin of the He atom (Section 5.1.5). This means that in equilibrium the two zeros, each with the topological invariant N "!1, are at the same point in momentum space. These two species of fermions living in the vicinity of the Fermi point form the spinor representation of the global S;(2) group of spin rotations. Due to the S;(2) group connecting two fermions at the 1 1 same Fermi point, the two fermions have the same energy spectrum, which means that they have the same `speeds of lighta. The Lagrangian for the collective gauge "eld in Eq. (126), which is
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obtained after integration over these fermions, contains the same metric tensor gIJ as the fermionic spectrum in Eq. (108). This means that the `photonsa (orbital waves) have the same `speed of lighta as fermions. As a result in a `perfecta system of the Fermi-point universality class, all fermions and bosons acquire the same speed of light in the low-energy corner. 6.4.2. Local symmetry from global continuous and discrete groups There is another important consequence of the double degeneracy of the Fermi point. The global S;(2) group which makes the Fermi point degenerate is applicable only for the vacuum or 1 thermal equilibrium state. In nonequilibrium the collective motion of the vacuum splits the Fermi points: positions of the two points oscillate separately. This does not violate the momentum space topology: the total topological charge of the two Fermi points, N "!2, is conserved in such oscillations. The collective dynamical degrees of freedom of the vacuum, which are responsible for the separate motion of the Fermi points, are viewed by the fermions as local S;(2) gauge "eld. The propagator describing the two fermions, each being the spinor in the Bogoliubov}Nambu space, is the 4;4 matrix. If one neglects the degrees of freedom related to the vierbein then the collective variables which describe the dynamics of the doubly degenerate Fermi point enter the fermionic propagator as G\"\ @eI(p !e A !e =? ) . (131) @ I > I > ? I The cross-term coupling the fermions of di!erent spin projections, =? contains the collective ? I variable =? . This new e!ective "eld acts on chiral quasiparticles as a `weaka S;(2) gauge "eld. I Thus in this e!ective "eld theory the ordinary spin of the He atoms plays the part of the weak isospin [58,48]. The global S;(2) symmetry of the underlying liquid produces the local S;(2) 1 symmetry in e!ective low-energy theory. The `weaka "eld =? is also dynamical and in the leading logarithmic order obeys the Maxwell I (actually Yang}Mills) equations. It is worthwhile to mention that the `weaka charge is also logarithmically screened by the fermionic vacuum with the same coupling constant as in Eq. (128). There is a zero charge e!ect for the S;(2) gauge "eld in He-A [75] instead of the asymptotic freedom in the Standard Model, where the antiscreening is produced by the bosonic degrees of freedom of the vacuum. Since the S;(2) gauge bosons appear in He-A only in the low-energy limit, their contribution to the vacuum polarization is small compared with the fermionic contribution, and thus the antiscreening e!ect can be neglected in spite of the prevailing number of bosons. The same can in principle occur in the Standard Model above, say, GUT scale. Appearance of the local S;(2) symmetry in the low-energy physics of He-A implies that the higher local symmetry groups of our vacuum can, in principle, arise as a consequence of the Fermi point degeneracy. For example, in the Terazava decomposition of 16 fermions into 4 spinons and 4 holons (Section 5.2.1) the four-fold degeneracy can produce both the S;(4) and S;(2) ;S;(2) ! * 0 gauge groups. In particle physics the collective modes related to the shift of the four-momentum are also discussed in terms of the `generalized covariant derivativea [76,77]. In this theory the gauge "elds, the Higgs "elds, and Yukawa interactions, all are realized as shifts of positions of the degenerate Fermi point, with degeneracy corresponding to di!erent quarks and leptons. In principle the S;(N) gauge "elds can appear as a result of discrete symmetry, which being combined with the momentum space topology can play a decisive role in the degeneracy of the massless fermions. In Section 5.2.4 we considered the planar phase of super#uid He where
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momentum-space topology is nontrivial only due to the discrete Z symmetry. The resulting momentum-space invariant is responsible for the degenerate Fermi points, which in turn give rise to chiral fermions and to the e!ective S;(2) gauge "eld in the low-energy corner. In principle, the discrete symmetry can be at the origin of the degeneracy of the Fermi point in the Standard Model too, and it thus can be responsible for the fact that all fermions and bosons have the same speed of light. On the importance and possible decisive role of the discrete symmetries in relativistic quantum "elds see Refs. [78}80]. 6.4.3. Diwerent metrics for diwerent fermions In Eq. (131) we did not take into account that dynamically each of the two elementary Fermi points can have its own vierbein "eld: though in equilibrium their vierbeins coincide due to internal symmetry, they can oscillate separately. As a result the number of the collective modes could increase even more. This is an interesting problem which must be investigated in detail. If the degenerate Fermi point mechanism has really some connection to the dynamical origin of the non-Abelian gauge "elds, we must connect the degeneracy of the Fermi point (number of the fermionic species) with the symmetry group of the gauge "elds. Naive approach leads to extremely high symmetry group. That is why there should be some factors which can restrict the number of the gauge and other massless bosons. For example the extra massless bosons can be killed by some special discrete symmetry between the fermions of the degenerate point. Another source of the reduction of the number of the e!ective "elds has been found by Chadha and Nielsen [2]. They considered the massless electrodynamics with di!erent metric (vierbein) for the left-handed and right-handed fermions. In this model the Lorentz invariance is violated. They found that the two metrics converge to a single one as the energy is lowered. Thus in the low-energy corner the Lorentz invariance becomes better and better, and at the same time the number of independent massless bosonic modes decreases. There is however an open question in the Chadha and Nielsen approach: If the correct covariant terms in action are provided only by the logarithmic selection, then the logarithm is too slow function to account for the high accuracy with which symmetries are observed in nature [81]. As the He-A analogy indicates, the noncovariant terms in e!ective action appear due to integration over fermions far from the Fermi point, where the `Lorentza invariance is not obeyed. Thus to obtain the S;(N) gauge "eld (and Einstein gravity) with high precision the Lorentz invariance in the large range of the trans-Planckian region is needed. In this sense the Lorentz invariance appears to be more fundamental, since it established the local gauge invariance and general covariance of the e!ective theory. 6.4.4. Mass of =-bosons, yat directions and supersymmetry In He-A the S;(2) gauge "eld acquires mass due to the nonrenormalizable terms, which come from the `Planckiana physics. However, in the BCS model the mass of the `=-bosona is exactly zero due to the hidden symmetry of the BCS action, and becomes nonzero only due to the non-BCS corrections: m & /v p . It is interesting that in the BCS theory applied to the He-A state the 5 $ $ hidden symmetry is extended up to the S;(4) group. The reason for such enhancement of symmetry is still unclear. Probably this can be related with the #at directions in the Ginzburg}Landau potential for super#uid He-A obtained within the BCS scheme (some discussion of that can be found in [82,12] and in Section 5.15 of the book [48]) or with the supersymmetry in the BCS
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systems discussed by Nambu [83]. In any case the natural appearance of the groups S;(2) and S;(4) in condensed matter e!ective quantum "eld theory reinforces the G(224) group, discussed in Section 5.2.1, as the candidate for uni"cation of electroweak and strong interactions.
7. Chiral anomaly in condensed matter systems and Standard Model Massless chiral fermions give rise to a number of anomalies in the e!ective action. The advantage of He-A is that this system is complete: not only the `relativistica infrared regime is known, but also the behavior in the ultraviolet `nonrelativistica (or `trans-Planckiana) range is calculable, at least in principle, within the BCS scheme. Since there is no need for a cuto!, all subtle issues of the anomaly can be resolved on physical grounds. The measured quantities related to the anomalies depend on the correct order of imposing limits, i.e. on what parameters of the system tend to zero faster: temperature ¹; external frequency ; inverse quasiparticle lifetime due to collisions with thermal fermions 1/; inverse volume; the distance between the energy levels of fermions, etc. All this is very important for the ¹P0 limit, where is formally in"nite. An example of the crucial di!erence between the results obtained using di!erent limiting procedures is the so called `angular momentum paradoxa in He-A, which is also related to the anomaly: The orbital momentum of the #uid at ¹"0 di!ers by several orders of magnitude, depending on whether the limit is taken while keeping P0 or PR. The `angular momentum paradoxa in He-A has possibly a common origin with the anomaly in the spin structure of hadrons [84]. 7.1. Adler}Bell}Jackiw equation The chiral anomaly is the phenomenon which allows the nucleation of the fermionic charge from the vacuum [85,86]. Such nucleation results from the spectral #ow of the fermionic charge through the Fermi point to high energy. Since the #ux in the momentum space is conserved, it can be equally calculated in the infrared or in the ultraviolet limits. In He-A it is much easier to use the infrared regime, where the fermions obey all the `relativistica symmetries. As a result one obtains the same anomaly equation, which has been derived by Adler and by Bell and Jackiw for the relativistic systems. In relativistic theories the rate of production of the fermionic charge q from the vacuum by applied electric and magnetic "elds is (see Fig. 8) 1 C q eFIJFH . (132) q "R JI" ? ? ? IJ I 8 ? Here q is the charge carried by the ath fermion which is nucleated together with the fermion; e is ? ? the charge of the ath fermion with respect to the gauge "eld FIJ; C "$1 is the chirality of the ? fermion; and FH is the dual "eld strength. IJ In a more general case when the chirality is not readily de"ned the above equation can be presented in terms of the momentum-space topological invariant 1 (QE, N)FIJFH , q " IJ 8 where Q is the matrix of the charges q and E is the matrix of the `electrica charges e . ? ?
(133)
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Fig. 8. Spectrum of massless right-handed and left-handed particles with electric charges e and e correspondingly in 0 * a magnetic "eld B along z; the thick lines show the occupied negative-energy states. Motion of the particles in the plane perpendicular to B is quantized into the Landau levels shown. The free motion is thus e!ectively reduced to onedimensional motion along B with momentum p . Because of the chirality of the particles the lowest (n"0) Landau level, X for which E"cp if the particle is right handed or E"!cp if the particle is left handed, is asymmetric: it crosses zero X X only in one direction. If we now apply an electric "eld E along z, the spectral #ow of levels occurs: the right-handed particles are pushed together with the energy levels from negative to positive energies according to the equation of motion p "e E. The whole Dirac sea of the right-handed particles moves up, creating particles and the fermionic charge X 0 q from the vacuum into the positive energy continuum of matter. The same electric "eld pushes the Dirac sea of the 0 left-handed particles down, annihilating the fermionic charge q . There is a net production of the fermionic charge from * the vacuum, if the left}right symmetry is not exact, i.e. if the charges of left and right particles are di!erent. The rate of particle production is proportional to the density of states at the Landau level, which is Je B , so that the rate of production of fermionic charge q from the vacuum is q "(1/4)(q e !q e )E ) B. 0 0 * *
The Adler}Bell}Jackiw equation (132) is fully covariant and thus can be applied to He-A after expressing the gauge "eld and fermionic charge in terms of the He-A observables. The e!ective `magnetica and `electrica "elds in He-A are simulated by the space and time dependent lK -texture: B"p o ;lK and E"p R lK . The Adler}Bell}Jackiw equation has been veri"ed in He-A experi$ $ R ments (see Section 7.5). In particle physics the only evidence of axial anomaly is related to the decay of the neutral pion P2, although the anomalous nonconservation of the baryonic charge has been used in di!erent cosmological scenaria explaining an excess of matter over antimatter in the Universe (see review [87]).
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7.2. Anomalous nonconservation of baryonic charge In the standard electroweak model there is an additional accidental global symmetry ;(1) whose classically conserved charge is the baryon number B. Each of the quarks is assigned B"1/3 while the leptons (neutrino and electron) have B"0. The baryonic number is not fundamental quantity, since it is not conserved in uni"ed theories, such as G(224) or SO(10), where leptons and quarks are combined in the same multiplet. At low energy the matrix elements for transformation of quarks to leptons are extremely small and the baryonic charge can be considered as a good quantum number with high precision. However, it can be produced due to the axial anomaly, in which it is generated from the vacuum due to spectral #ow. In the Standard Model there are two gauge "elds whose `electrica and `magnetica "elds become a source for baryoproduction: The hypercharge "eld ;(1) and the weak "eld S;(2) . Let us "rst consider the e!ect of the hypercharge 7 * "eld. The production rate of baryonic charge in the presence of hyperelectric and hypermagnetic "elds is N 1 (YB, N)B ) E " $ (> #> !> !> )B ) E , 7 7 4 B0 S0 B* S* 7 7 4
(134)
where N is the number of families, > , > , > and > are hypercharges of right and left u and $ B0 S0 B* S* d quarks. Since the hypercharges of left and right fermions are di!erent (see Eq. (115)), one obtains the nonzero value of (YB, N)"1/2, and thus a nonzero production of baryons by the hypercharge "eld N $ B )E . 8 7 7
(135)
The weak "eld also contributes to the production of the baryonic charge: 1 N (W* W* B, N)B@ ) E "! $ B@ ) E . 5 @5 4 8 5 @5
(136)
Thus the total rate of baryon production in the Standard Model takes the form 1 BQ " [(YB, N)B ) E #((W* )B, N)B@ ) E ] 7 7 5 @5 4 N " $ (B ) E !B@ ) E ) . 5 @5 8 7 7
(137) (138)
The same equation describes the production of the leptonic charge ¸: one has ¸Q "BQ since B!¸ is conserved due to anomaly cancellation. This means that production of one lepton is followed by production of three baryons. The second term in Eq. (137), which comes from non-Abelian S;(2) "eld, shows that the * nucleation of baryons occurs when the topological charge of the vacuum changes, say, by sphaleron or due to de-linking of linked loops of the cosmic strings [88}90]. This term is another example of interplay of the momentum-space and real-space topologies discussed in Section 4.4. It is the density of the topological charge in real space multiplied by the factor ((W* )B, N), which is
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the invariant in the momentum space. The "rst nontopological term in Eq. (137) describes the exchange of the baryonic (and leptonic) charge between the hypermagnetic "eld and the fermionic degrees of freedom. It is important that Eq. (137) is completely determined by the invariants of the Fermi point and is valid even in the nonrelativistic systems. That is why the same equation can be applied to He-A after being adjusted to the He-A symmetry. 7.3. Analog of baryogenesis in He-A: momentum exchange between superyuid vacuum and quasiparticle matter In He-A the relevant fermionic charge, which is important for the dynamics of super#uid liquid, is the linear momentum. The super#uid background moving with velocity * and the normal component moving with velocity * can exchange momentum. This exchange is mediated by the texture of the lK "eld, which carries continuous vorticity (see Eq. (103)). The momentum of the #owing vacuum is transferred to the momentum carried by texture, and then from texture to the system of quasiparticles. The force between the super#uid and normal components arising due to this momentum exchange is usually called the mutual friction, though the term friction is not very good since some or essential part of this force is reversible and thus nondissipative. In super#uids and superconductors with curl-free super#uid velocity * , the mutual friction is produced by the dynamics of quantized vortices which serve as mediator. Here we are interested in the process of the momentum transfer from the texture to quasiparticles. It can be described in terms of the chiral anomaly, since as we know the lK texture plays the part of the ;(1) e!ective gauge "eld acting on relativistic quasiparticles, and these quasiparticles are chiral. Thus we have all the conditions to apply the axial anomaly equation (132) to this process of transformation of the fermionic charge carried by magnetic "eld (texture) to the fermionic charge carried by chiral particles (normal component) (Fig. 9). When a chiral quasiparticle crosses zero energy in its spectral #ow it carries with it its linear momentum P"$p lK . That is why this P is the $ proper fermionic charge q which enters the Eq. (132), and the rate of the momentum production from the texture is 1 1 (PE, N)B ) E" B ) E P C e . (139) PQ " ? ? ? 4 4 ? Here B"(p / );lK and E"(p / )R lK are e!ective `magnetica and `electrica "elds; E is the matrix $ $ R of corresponding `electrica charges in Eq. (98): e "!C (the `electrica charge is opposite to the ? ? chirality of the He-A quasiparticle, see Eq. (99)); and P "!C p lK is the momentum (fermionic ? ? $ charge) carried by the ath fermionic quasiparticle. Using this translation to the He-A language one obtains that the momentum production from the texture per unit time per unit volume is p PQ "! $ lK (R lK ) (;lK )) . 2 R
(140)
It is interesting to follow the history of this term in He-A. First the nonconservation of the momentum of the super#uid vacuum at ¹"0 has been found from the general consideration of the super#uid hydrodynamics of the vacuum [91]. Later it was found that the quasiparticles must
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Fig. 9. Production of the fermionic charge in He-A (linear momentum) and in Standard Model (baryonic number) are described by the same Adler}Bell}Jackiw equation. Integration of the anomalous momentum production over the cross section of the moving continuous vortex gives the loss of linear momentum and thus the additional force per unit length acting on the vortex due to spectral #ow.
be nucleated whose momentum production rate is described by the same Eq. (140), but with the opposite sign [92]. Thus the total momentum of the system has been proved to conserve. In the same paper Ref. [92] it was "rst found that the quasiparticle states in He-A in the presence of twisted texture of lK (i.e. the texture with ;lK O0) has a strong analogy with the eigenstates of a massless charged particle in a magnetic "eld. Then it became clear [93] that the momentum production is described by the same equation as the axial anomaly in relativistic quantum "eld theory. Now we know why it happens: The spectral #ow from the texture to the `mattera occurs through the Fermi point and thus it can be described by the physics in the vicinity of the Fermi point, where the `relativistica quantum "eld theory with chiral fermions necessarily arises and thus the anomalous #ow of momentum can be described in terms of the Adler}Bell}Jackiw equation. 7.4. Axial anomaly and force on He-A vortices 7.4.1. Continuous vortex texture From the underlying microscopic theory we know that the total linear momentum of the liquid is conserved. The Eq. (140) thus implies that in the presence of a time-dependent texture the momentum is transferred from the texture (the distorted super#uid vacuum or magnetic "eld) to the heat bath of quasiparticles (analogue of matter). The rate of the momentum transfer gives an extra force acting on a moving lK -texture. This force in#uences the dynamics of the continuous texture, which represents the vortex in He-A (it is an analog of stringy texture in the Standard Model [94]), and this force has been measured in experiments on the rotating He-A [45] (see Section 7.5). The continuous vortex texture, "rst discussed by Chechetkin [95] and Anderson and Toulouse [96] (ATC vortex, Fig. 10), has in its simplest axisymmetric form the following distribution of the lK -"eld (z( , ( and K are unit vectors of the cylindrical coordinate system) lK (, )"z( cos ()#r( sin () ,
(141)
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Fig. 10. Top: An n "2 continuous vortex in He-A. The arrows indicate the local direction of the order parameter vector lK . Under experimental conditions the direction of the lK -vector in the bulk liquid far from the soft core is kept in the plane perpendicular to applied magnetic "eld far from the core. This does not change the topology of the Anderson} Toulouse}Chechetkin vortex: the lK -vector covers the whole 4 sphere within the soft core. As a result there is 4 winding of the phase of the order parameter around the soft core, which corresponds to n "2 quanta of anticlockwise circulation. Bottom: The NMR absorption in the characteristic vortex satellite originates from the soft core where the lK orientation deviates from the homogeneous alignment in the bulk. Each soft core contributes equally to the intensity of the satellite peak and gives a practical tool for measuring the number of vortices.
where () changes from (0)"0 to (R)". Such skyrmionic lK -texture forms the so called soft core of the vortex, since the region of texture contains nonzero vorticity of super#uid velocity in Eq. (102):
[1!cos ()]K , ;* " sin R z( . * (, )" 2m M 2m
(142)
In comparison to a more familiar singular vortex, the continuous vortex has a regular super#uid velocity "eld * , with no singularity on the vortex axis. However, the circulation of the super#uid velocity about the soft core is still quantized: Z dx ) * " with "2 /m. This is twice the conventional circulation quantum number in the pair-correlated system, "2 /2m, where 2m is the mass of the Cooper pair, i.e. the winding number of this vortex is n "2. Quantization of circulation in continuous vortex is related to the topology of the lK -"eld: according to Mermin-Ho relation (103) the lK -vector covers the whole 4 sphere when the soft core is swept:
dx dy lK )
RlK RlK ; "4 . Rx Ry
(143)
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This also re#ects the interplay of the real-space and momentum-space topologies, since the lK -vector shows the position of the Fermi point in the momentum space. The general rule is: if the Fermi point a with the momentum-space topological charge N sweeps the 4n solid angle in ? ? the soft core of the vortex, the real-space topological charge of the vortex (the winding number) will be [31] 1 (144) n " n N . ? ? 2 ? For the Anderson}Toulouse}Chechetkin texture in Eq. (141) the Fermi point with N "2 sweeps 4 angle, while the Fermi point with N "!2 sweeps !4 solid angle, as a result the winding number of the vortex is n "2 (see also Section 9.1.3). 7.4.2. Spectral-yow force acting on a continuous vortex The stationary vortex has nonzero e!ective `magnetica "eld, B"(p );lK . If the vortex moves $ with a constant velocity * with respect to the heat bath the moving texture acquires the time * dependence, lK (r!* t), and this leads to the e!ective `electrica "eld * p 1 (145) E" R A"! $ (* ) )lK .
*
R Since B ) EO0, the motion of the vortex leads to the production of the quasiparticle momenta by the spectral #ow. This means the transfer of the momentum from the vortex texture to the heat bath of quasiparticles, if the vortex moves with respect to the heat bath (normal component of the liquid or `mattera). In other words, if * O* there is a force acting between the normal component and * the vortex. This force (per unit length) is obtained by integration of the anomalous momentum transfer in Eq. (140) over the cross-section of the soft core of the moving ATC vortex:
p p F " d $ lK (R lK ) (;lK ))" d $ lK (((* !* ) ) )lK ) (;lK )) * 2 R 2 "!2 C z( ;(* !* ) , *
(146)
where p C " $ . 3
(147)
The spectral-#ow force in Eq. (146) is transverse to the relative motion of the vortex with respect to the heat bath and thus is nondissipative (reversible). In this derivation it was assumed that the quasiparticles and their momenta, created by the spectral #ow from the inhomogeneous vacuum, are "nally absorbed by the normal component. The retardation in the process of absorption and also the viscosity of the normal component lead to a dissipative (friction) force between the vortex and the normal component: F "!(* !* ). There is no momentum exchange between * the vortex texture and the normal component if they move with the same velocity; according to Section 3.6 the condition that * "0 in the texture frame is one of the conditions of the global thermodynamic equilibrium, when the dissipation is absent.
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7.4.3. Topological stability of spectral yow force. Spectral yow force from Novikov}Wess}Zumino action The same result for the force in Eq. (146) was obtained in a microscopic theory by Kopnin [97]. He used the quasiclassical approach, which is valid at energies well above the "rst Planck scale, E with di!erent excitation methods of the Nd> for the di!erent sites [159].
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Fig. 13. PL spectra of GaN implanted with Er (60 keV, RT) with co-implanted oxygen (bottom) and without (top). (From Ref. [96])
4. Electrical properties The objective of ion implantation, in the context of semiconductors, is to modify the electrical properties of the semiconducting material. The goal can be either to increase or decrease the conductivity of the semiconductor, which is accomplished by dopant implantation or implantation isolation, respectively. Dopant implantation introduces an electrically active n- or p-type dopant for the purpose of increasing the free carrier concentration. The particular dopant introduced depends on the semiconductor and the desired "nal electrical properties of the semiconductor. Implantation isolation involves the implantation of various elements to produce a high resistivity layer through the introduction of mid-gap levels, that trap both electrons and holes. These levels can be either created through the implantation process itself by defects or through the chemical nature of the implanted species. 4.1. Implantation isolation Kahn et al. were the "rst to report the use of ion implantation to modify the electrical properties of GaN [191]. They implanted Be> and N> to compensate the background n-type behavior. The GaN showed a signi"cantly reduced free carrier concentration after implantation of
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1.2;10 cm\ of either Be> or N> and annealing at 10003C. The fact that Be> and N> had the same e!ect implies that the compensation is due to the induced defects and not to the chemical nature of the implanted impurity. As discussed above, the annealing temperature of 10003C is insu$cient to remove all defects within reasonable time. The implantation of the light elements of H [74,151,192] and He [74,151,193], and the isovalent elements of N [192,194}196] and P [193], have been investigated in GaN for the purpose of creating damage during implantation, which facilitates the isolation. The resistivity obtained after implantation scaled with the mass of the implanted ion, i.e. the larger the ion, the larger the concentration of defects, and the higher the resistivity [151,192,193]. Similarly, the higher the implanted dose the higher the observed resistivity [151,192]. However, after He implantation doses exceeding 4;10 cm\, the resistivity actually began to decrease [192]. This is due to an increase in hopping or defect conduction where the trap density is so high that carriers can hop between the defects. Levels generated by the implantation of H [197,198], He [198}200], and N [201] in GaN have been investigated using deep-level transient spectroscopy (DLTS). Three defect levels located at 0.13, 0.16, and 0.20 eV were observed after H implantation [197,198]. The level at 0.20 eV was also present after He implantation, along with two deep levels located at 0.78 and 0.95 eV [198}200]. These defect levels are electron traps. The implantation of N produced a deep level located at 0.67 eV [201], which was attributed to nitrogen interstitials. The concentration of this level increased with increasing implantation dose and decreased after annealing at 9003C. An activation energy of 0.76$0.02 eV was also determined by temperaturedependent resistivity measurements for He- and N-implanted GaN, while the activation energy for H-implanted GaN was 0.29$0.04 eV [192]. This `activationa energy is a measure of the main level controlling the electrical properties and is in good agreement with the observations made by DLTS. Cao et al. implanted Ti, O, Fe, and Cr into GaN to investigate the possibility of chemically induced isolation [202]. The activation energies of the various implanted ions were all similar, indicating that the implantation defects dominate the electrical properties as opposed to the chemical nature of the implanted ion. However, all the implantations reported in this section have resulted in device quality high-resistivity GaN. To understand the in#uence of the implantation-induced damage on the electrical properties of implanted GaN, we implanted Ar into GaN as a control dopant, since it should not introduce any energy levels due to the chemical nature of the dopant. Hall e!ect and C}< electrical measurements showed that all the Ar-implanted GaN samples exhibited a high resistivity in the as-implanted state and thus could not be measured. This is in good agreement with the results of other groups reported above. Recovery of the electrical properties of GaN was observed after annealing the Ar-implanted GaN samples at 11003C for 1 h in a tube furnace. The sheet carrier concentration after annealing was in the range of 2}5;10 cm\. Taking the thickness of 2 m of the GaN "lm into account, the sheet carrier concentration can be converted to a volumetric carrier concentration in the range of 1}2.5;10 cm\, which was the background carrier concentration of the GaN "lms before implantation. This indicates that the implantation-induced electronic traps have been partially removed allowing the bulk properties of the GaN "lm to dominate the Hall e!ect measurement. This is also con"rmed by the high mobility observed in the implanted "lms. Further annealing at 12003C did not change the electrical properties signi"cantly.
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Table 3 Electrical properties of GaN implanted with Si and annealed in a tube furnace or RTA at various temperatures and times. Ion energy: 200 keV Implantation conditions
Sheet resistivity (/䊐)
Sheet carrier concentration (cm\)
Mobility (cm/V s)
Percent activation (%)
2.47;10 1.67;10 1.77;10 '10
6.88;10 1.44;10 1.52;10 N/A
367.47 260.87 232.51 N/A
0.69 0.14 0.15 N/A
12003C, 15 s RTA anneal 1;10 cm\, RT 1;10 cm\, RT 1;10 cm\, 6503C 1;10 cm\, 6503C
3.61;10 '10 2.41;10 5.81;10
5.4;10 N/A 6.5;10 3.99;10
321 N/A 398.56 269.8
0.054 N/A 0.0065 0.004
11003C, 1 h furnace anneal 1;10 cm\, RT 1;10 cm\, 6503C 1;10 cm\, 6503C
1.67;10 1.77;10 2.27;10
5.33;10 1.12;10 4.88;10
70.24 31.51 56.42
53.3 112 48.8
12003C, 1 h furnace anneal 1;10 cm\, RT 1;10 cm\, RT 1;10 cm\, 6503C 1;10 cm\, 6503C
N/A 2.90;10 4.87;10 1.32;10
N/A 1.04;10 1.13;10 9.97;10
N/A 20.74 113.57 47.45
N/A 104 113 99.7
11003C, 15 s RTA anneal 1;10 cm\, RT 1;10 cm\, RT 1;10 cm\, 6503C 1;10 cm\, 6503C
4.2. Donor doping Electrical measurements showed that GaN is also too resistive to measure directly after Si implantation. Annealing at 9003C for 60 s in a RTA-system did not signi"cantly change the resistivity of the samples and, therefore, electrical measurements could not be performed, but they were possible after annealing at 11003C for 15 s. However, the sheet carrier concentrations were low resulting in a very low Si activation percentages (see Table 3). The values are much lower than the 8% reported by Zolper et al. for a similar annealing procedure [71,72,194}196]. However, further work by Zolper et al. has also resulted in similar low activation percentages in the range of &0.05% [17,88,89]. The reason for this discrepancy is unclear but may be due to di!erences in the quality of the host GaN material. The low sheet carrier concentration is similar to that observed after Ar implantation and furnace annealing for our samples. This implies that the carriers observed in the Si-implanted GaN may in fact be due to the native n-type background carriers present in the GaN substrate. This is supported by the high mobility observed in the Si implanted GaN samples, which one would not expect in an implantation damaged crystal. Annealing at 12003C for 15 s in an RTA did not improve the sheet carrier concentration. Thus, either the
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Fig. 14. Carrier concentration versus reciprocal temperature as determined from variable temperature Hall-e!ect measurements for GaN implanted with Si at 6503C with an energy of 200 keV and a dose of (a) 1;10 cm\ and (b) 1;10 cm\. Included are the data from GaN samples doped with Si during growth with similar doping levels. The solid lines are "ts of the experimental data, and the resulting activation energy (E ) for each "t is indicated.
annealing time or temperature are still too low for electrical activation of the implanted Si. Increasing the annealing time should improve this situation. Table 3 also shows the electrical measurements of GaN implanted with Si and annealed in a tube furnace for 1 h at 11003C. The long annealing time provided a much higher percentage of Si-activated compared to RTA annealing with &100% activation for the 1;10 cm\ dose and &50% activation for the 1;10 cm\ dose. These results were con"rmed by C}< measurements. The observed mobilities are low compared to the bulk GaN due to the remaining high concentration of defects (compare with PL-results) that act as scattering centers for the free carriers. These results indicate that the Si-implanted region in the GaN is dominating the electrical properties of the "lm and that the implanted Si ions have become electrically active in the GaN substrate. Fig. 14 shows temperature-dependent Hall measurements for GaN implanted at 6503C with (a) 1;10 cm\ and (b) 1;10 cm\ Si ions along with data from a GaN doped with similar levels of Si during the growth. All samples show very little temperature dependence at low temperatures. This behavior was also observed by other groups [203}206] and can be attributed to impurity band conduction in the GaN [207]. Thus, only data above 200 K were used for the determination
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of the activation energy. The GaN sample doped with Si during growth showed a donor activation energy, E , of 17.3 meV. This value is in good agreement with results given in Refs. [205,206,208,209]. Analysis of the 1;10 cm\ Si-implanted GaN sample indicated that two independent donor levels with activation energies of 18.1 and 273.9 meV described the temperature-dependent behavior. The activation energy of the "rst donor level is very close to that observed in the Si-doped GaN sample and, therefore, is due to the implanted Si donors. This con"rms that the implanted Si has become electrically active in the GaN. The second donor level is most likely due to deep levels related to implantation-induced defects. This is in agreement with the results reported in Refs. [197}200], where defect levels with similar activation energies were detected by DLTS. However, Zolper et al. [71,195,196] found only one activation energy level at 62 meV, which is most likely due to a combination of the implanted Si donor level with deep levels and should not be taken as a Si donor energy level. The temperature dependence of the high-dose Si-implanted GaN shows an activation energy of 5.5 meV, which is almost identical to the results determined in the sample doped with Si at a similar level during growth, indicating the electrical activation of the implanted Si ions. The results are in agreement with Ref. [17]. The low activation energy indicates a doping level above the degeneracy limit that is consistent with the high doping level. The electron mobilities in both Si-implanted GaN samples were low and did not change signi"cantly with temperature. This is due to the large amount of remaining defects that act as scattering centers. Higher temperature anneals at 12003C for 1 h in a tube furnace further increased the activation of the implanted Si and resulted in the highest mobility observed (see Table 3). However, this mobility is still lower then that observed in GaN doped during growth. This demonstrates that the optimal annealing temperature for GaN is a bit higher than 12003C, as predicted by the two-thirds rule. High activation percents have also been observed in Si implanted GaN after annealing at 13003C [90,91] and 14003C [73] in a RTA for 10 s. Electrical conductivity was observed in the Ge-implanted GaN after a 1-h 12003C tube furnace anneal. The measured sheet carrier concentrations and mobilities were lower compared to Si, resulting in activation percentages between 15 and 45%. This could be due to the higher remaining amount of implantation induced defects by the heavier ion or due to the deeper donor level of the Ge, as seen by the PL measurements above. However, the implanted Ge dominates the electrical properties of the sample and is electrically active. Other donors, including O [18,69,70], Te [10,75,85], and S [10,75,85], have been implanted and electrically investigated. Oxygen is a shallow donor in GaN [120,210] and may cause background n-type behavior [211]. However, studies on O implanted in GaN indicate a low implantation activation [18,69,70]. This was also found for S and Te implanted into GaN even after hightemperature annealing at 14003C [10,75,85]. Both S and Te have high activation energies of 48$10 and 50$20 meV, respectively, which likely results in the low implantation activation e$ciency. 4.3. Acceptor doping The challenge in ion implantation activation is to reach p-type behavior after implantation of acceptors into a material with n-type background carriers. In the case of GaN, this challenge is even more di$cult due to the absence of shallow acceptors. Table 4 summarizes the electrical
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Table 4 Electrical properties of GaN implanted with acceptors annealed in a tube furnace at various temperatures and times. The ion implantation was performed at room temperature Ion species and implantation conditions
Sheet resistivity (/䊐)
Sheet carrier concentration (cm\)
Mobility (cm/V s)
Percent activation
11003C, 1 h furnace annealing Mg, 160 keV, 1;10 cm\ Mg, 160 keV, 1;10 cm\ Ca, 300 keV, 1;10 cm\ Ca, 300 keV, 1;10 cm\
1.02;10 8.46;10 4.96;10 8.14;10
1.90;10 1.10;10 2.96;10 1.10;10
330.4 67.59 425.1 69.46
N/A N/A N/A N/A
1.66;10 '10 6.31;10 '10
3.40;10 N/A 8.22;10 N/A
109.5 N/A 120.4 N/A
N/A (n-type) N/A N/A (n-type) N/A
12503C, 30 min furnace annealing Mg, 60 keV, 1;10 cm\ Mg, 60 keV, 1;10 cm\ Mg, 60 keV, 1;10 cm\ Mg, 120 keV, 1;10 cm\ Mg, 120 keV, 1;10 cm\
1.4;10 2;10 1.57;10 9.5;10 1.63;10
5.3;10 3;10 5.5;10 6.7;10 1.3;10
80 100 71 97 300
N/A (n-type) n/p-type N/A (n-type) N/A (n-type) N/A (n-type)
13003C, 10 min furnace annealing Mg, 60 keV, 1;10 cm\ Mg, 60 keV, 1;10 cm\ Mg, 120 keV, 1;10 cm\
2.06;10 4.0;10 6.3;10
8;10 2.8;10 n"1.1;10
38 2.2 63
n/p-type n/p-type N/A (n-type)
12003C, 1 h furnace annealing Mg, 160 keV, 1;10 cm\ Mg, 160 keV, 1;10 cm\ Ca, 300 keV, 1;10 cm\ Ca, 300 keV, 1;10 cm\
(n-type) (n-type) (n-type) (n-type)
properties after implantation of Mg and Ca followed by subsequent annealing at various temperatures. Annealing at high temperatures and di!erent conditions enabled electrical measurements to be performed. However, the sign of the Hall coe$cient was, in the most cases, negative indicating n-type behavior in the implanted samples as opposed to the expected p-type behavior. The results of most of other studies [17,18,45,71,72,194}196,212] are in agreement with this result and showed GaN remaining n-type after Mg implantation and annealing at high temperatures. Cao et al. [84,85] were able to achieve p-type GaN after Mg implantation and annealing at very high temperatures (1100}14003C). The large deviations in the reported electrical properties of Mgimplanted GaN is due to the large activation energy of &200 meV, which means that at room temperature only &1% of the Mg ions are ionized. Thus, for a Mg implantation dose of 1;10 cm\ and an implantation activation of 100% the sheet hole concentration is expected to be 1;10 cm\, which is approximately the order of the n-type background carriers. In the case of Ca, the same argument must be applied due to a similar high activation energy of &160 meV. Only the low mobility observed in the acceptor implanted samples compared to Ar-implanted samples is an indication of electrical activation, which is compensated by the native n-type carriers.
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Other acceptors have been predicted to be shallow, but reports of C [10,19,75,85] and Be [40] implantation have not resulted in p-type GaN. Zolper et al. reported that GaN implanted with C had increasing n-type behavior with increasing annealing temperature. 4.4. Device applications The "rst use of ion implantation for device processing was by Kahn et al., who used N> and Be> implants to increase the performance of Schottky barriers on GaN [191]. The implantation lowered the native free carrier concentration at the surface such that deposited Schottky contacts had good rectifying properties. Implantation has been also used to improve ohmic contacts by increasing the free carrier concentration at the surface of the "lm under the contact [213,214]. In both cases, Si implantations with the associated activation anneal lowered the speci"c contact resistivity of the metal contact. Simple p}n junctions have been accomplished either by implantation of Si into Mg-doped GaN "lms [45,83,215] or by implantation of Mg into Si-doped GaN [76]. The junctions yielded rectifying behavior after annealing, but they had relatively high turn-on voltages, low breakdown voltages, low forward currents and high ideality factors due to the remaining implantation defects. Zolper et al. [70,72,216] produced a JFET using Si implantation to produce the n-type channel along with the n-type source and the drain regions. Implanted Ca was used as the p-type gate in the JFET. Following an 11503C activation anneal the JFET showed good channel modulation with a measured transconductance of 7 mS/mm and a saturation current of 33 mA/mm. The transconductance value is low compared to other devices produced in the III-nitrides and is attributed to a high access resistance or a low mobility in the channel. Similarly to the implanted p}n junctions there is still signi"cant damage remaining after annealing at this temperature that is expected to degrade the performance of the device.
5. Summary, conclusions, and future work The crystal structure of GaN is very resistant to ion bombardment due to the high ionicity of the Ga}N bond. Very high doses are required for amorphization; i.e. the amorphization threshold is much higher than in other semiconductors. The damage caused during implantation consists of vacancies, interstitials, anti-site defects, and extended defects including dislocations and stacking faults. Amorphization in GaN occurs in small local regions. These regions are subsequently enlarged with increasing implantation dose until the crystal structure collapses to create an amorphous layer. Structural defects can be readily created with medium implantation doses (10}10 cm\) and lead to lattice expansion. Computer simulations usually overestimate the damage compared to that experimentally observed. The discrepancy is due to signi"cant dynamic annealing during implantation and due to the strong bonding. Implantation at high-temperatures does not necessarily result in lower damage. The implanted impurities occupy de"ned lattice sites: the alkali elements mainly occupy interstitial sites; whereas, all other elements have been found on substitutional sites directly after ion implantation. The optical properties are greatly a!ected by implantation resulting in a complete loss of the luminescence even for low doses. The largest e!ect is the introduction of
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Fig. 15. Di!usion, recovery, and activation processes of ion-implanted impurities in GaN as function of annealing temperature.
nonradiative recombination centers due to the damage. The introduced defects have mainly deep levels within the band gap; therefore, implanted GaN is electrically highly resistive. The damage must be annealed out to achieve optical and electrical activation of the implanted impurities. Several sophisticated annealing procedures to achieve high annealing temperatures without decomposition of the GaN surface were described herein. Fig. 15 summarizes schematically the fundamental di!usion, recovery, and activation processes that occur in ion implanted GaN as a function of annealing temperature. Implanted light elements undergo measurable di!usion via interstitial mechanisms at low temperatures. Lithium becomes mobile at 4503C until it recombines with a vacancy resulting in substitutional Li that is stable to higher temperatures. Hydrogen should be also mobile on interstitial sites at even lower temperatures, as complete out-di!usion occurs at 5003C and 8003C for p- and n-type GaN, respectively. With the exception of Zn and Mg at &12003C, no other implanted element (even Be) showed signi"cant di!usion in the investigated temperature ranges. The high thermal stability of impurities is due to the strong bonding con"gurations, i.e. the high Debye temperature of GaN. The onset of di!usion of implantation defects starts around 6003C. Each interstitial atom di!uses on interstitial sites until it "nds a vacancy for recombination. This e!ect occurs only when amorphization is not reached; it results in a reduction and elimination of the lattice expansion that is visible in a change of the RBS and XRD spectra. However, anti-site defects are likely to be created during this recovery of the structural defects. The recovery of point defects, visible by PL and PAC, starts around 8003C and is mainly associated with the onset of vacancy di!usion. Small microscopic areas in the range of several tens of nanometers are completely recovered resulting in optical activation of some implanted impurities. These recovered regions grow with increasing annealing temperature to the micrometer
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range at 12003C, which is the size of excitons in GaN. Thus, most of the implanted impurities become optically activated, but defect complexes are still present. The latter require higher temperatures to break up due to the additional formation or/and coulomb energy. These defects cause an intense yellow band in the PL spectra and are visible by PAC in this temperature range. The concentration of defect complexes signi"cantly decreases when annealed above 13003C leading to macroscopic electrical activation that can be measured by Hall or C}uxes since they appear in the currents of the respective balance laws for momentum and director [77,27]. In hydrodynamics the viscous forces are assumed to be small, and they are written as linear functions of all the >uxes: sym T A = : (2.28) N h The matrix must be compatible with the uniaxial symmetry of the nematic phase, and it must be invariant when n is changed into −n. Furthermore, it has to obey Onsager’s theorem [52], which demands a symmetric matrix for zero magnetic 0eld. Ful0lling all these requirements results in Eqs. (2.22) and (2.26). One additional, important Onsager relation is due to Parodi [170]: !2 + !3 = !6 − !5 :
(2.29)
It reduces the number of independent viscosities in a nematic liquid crystal to 0ve. The Leslie coeNcients of the compound 5CB are [39] !1 = −0:111 P; !2 = −0:939 P; !3 = −0:129 P ; !4 = 0:748 P; !5 = 0:906 P; !6 = −0:162 P :
(2.30)
At the end, we explain two typical situations that help to clarify the meaning of the possible viscous processes in a nematic and how they are determined by the Leslie coeNcients. In the 0rst situation we perform typical shear experiments as illustrated in Fig. 3. The director 0eld between the plates is spatially uniform, and the upper plate is moved with a velocity v0 relative to the lower one. There will be a constant velocity gradient along the vertical z direction. Three simple geometries exist with a symmetric orientation of the director; it is either parallel to the velocity 0eld C, or perpendicular to C and its gradient, or perpendicular to C and parallel to its gradient. The director can be 0rmly aligned in one direction by applying a magnetic 0eld strong enough to largely exceed the viscous torques. For all three cases, the shear forces T per unit area are calculated from the stress tensor T of Eq. (2.22), yielding T = )i C0 =d, where d is the separation between the plates. The viscosities as a function of the Leslie coeNcients for all three cases are given in Fig. 3. They are known as Mie8sowicz viscosities after the scientist who 0rst measured them [150,151]. If one chooses a non-symmetric orientation for the director, the viscosity !1 is accessible in shear experiments too [84].
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Fig. 3. De0nition of the three MiZesowicz viscosities in shear experiments.
Fig. 4. Permeation of a >uid through a helix formed by the nematic director.
The second situation describes a Gedanken experiment illustrated in Fig. 4 [102]. Suppose the nematic director forms a helical structure with wave number q0 inside a capillary. Such a con0guration is found in cholesteric liquid crystals that form when the molecules are chiral. Strictly speaking, the hydrodynamics of a cholesteric is more complicated than the one of nematics [139]. However, for what follows we can use the theory formulated above. We assume that the velocity 0eld in the capillary is spatially uniform and that the helix is not distorted by the >uid >ow. Do we need a pressure gradient to press the >uid through the capillary, although there is no shear >ow unlike a Poiseuille experiment? The answer is yes since the molecules of the >uid, when >owing through the capillary, have to rotate constantly to follow the director in the helix, which determines the average direction of the molecules. The dissipated energy follows from the second term of the entropy production rate in Eq. (2.27). The rate of change, N = C0 · grad n, is non-zero due to the convective time derivative. The energy dissipated per unit time and unit volume has to be matched by the work per unit time performed by the pressure gradient p . One 0nally arrives at p = %1 q02 v0 :
(2.31)
Obviously, the Gedanken experiment is determined by the rotational viscosity %1 . It was suggested by Helfrich [102] who calls the motion through a 0xed orientational pattern permeation. This motion is always dissipative because of the rotational viscosity of the molecules which have to follow the local director. Of course, the Gedanken experiment is not suitable for measuring %1 . A more appropriate method is dynamic light scattering from director >uctuations [99,29]. Together with the shear experiments it is in principle possible to measure all 0ve independent viscosities of a nematic liquid crystal.
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2.4. Topological defects Topological defects [111,146,226,27], which are a necessary consequence of broken continuous symmetry, exist in systems as disparate as super>uid helium 3 [230] and 4 [235], crystalline solids [224,81,160], liquid crystals [30,121,127], and quantum-Hall >uids [204]. They play an important if not determining role in such phenomena as response to external stresses [81,160], the nature of phase transitions [27,164,222], or the approach to equilibrium after a quench into an ordered phase [21]; and they are the primary ingredient in such phases of matter as the Abrikosov >ux-lattice phase of superconductors [1,13] or the twist-grain-boundary phase of liquid crystals [193,94,95]. They even arise in certain cosmological models [34]. Topological defects are points, lines or walls in three-dimensional space where the order parameter of the system under consideration is not de0ned. The theory of homotopy groups [111,146,226,27] provides a powerful tool to classify them. To identify, e.g., line defects, homotopy theory considers closed loops in real space which are mapped into closed paths in the order parameter space. If a loop can be shrunk continuously to a single point, it does not enclose a defect. All other loops are divided into classes of paths which can be continuously transformed into each other. Then, each class stands for one type of line defect. All classes together, including the shrinkable loops, form the 4rst homotopy or fundamental group. The group product describes the combination of defects. In the case of point singularities, the loops are replaced by closed surfaces, and the defects are classi0ed via the second homotopy group. In the next two subsections we deal with line and point defects in nematic liquid crystals whose order parameter space is the projective plane P 2 = S 2 =Z2 , i.e., the unit sphere S 2 with opposite points identi0ed. They play a determining role for the behavior of colloidal dispersions in a nematic environment. There exist several good reviews on defects in liquid crystals [111,146,226,27,30,121,127]. We will therefore concentrate on facts which are necessary for the understanding of colloidal dispersions. Furthermore, rather than being very formal, we choose a descriptive path for our presentation. 2.4.1. Line defects = disclinations Line defects in nematic liquid crystals are also called disclinations. Homotopy theory tells us that the fundamental group ,1 (P 2 ) of the projective plane P 2 is the two-element group Z2 . Thus, there is only one class of stable disclinations. Fig. 5 presents two typical examples. The defect line with the core is perpendicular to the drawing plane. The disclinations carry a winding number of strength + or −1=2, indicating a respective rotation of the director by + ◦ or −360 =2 when the disclination is encircled in the anticlockwise direction (see left part of Fig. 5). Note that the sign of the winding number is not 0xed by the homotopy group. Both types of disclinations are topologically equivalent since there exists a continuous distortion of the director 0eld which transforms one type into the other. Just start from the left disclination in Fig. 5 and rotate the director about the vertical axis through an angle when going outward from the core in any radial direction. You will end up with the right picture. The line defects in Fig. 5 are called wedge disclinations. In a Volterra process [111,27] a cut is performed so that its limit, the disclination line, is perpendicular to the spatially uniform director 0eld. Then the surfaces of the cut are rotated with respect to each other by an angle of 2S about the disclination line, and material is either 0lled in (S = +1=2) or removed (S = −1=2). In twist
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Fig. 5. Disclinations of winding number ±1=2. For further explanations see text.
disclinations the surfaces are rotated by an angle of about an axis perpendicular to the defect line. Disclinations of strength ±1=2 do not exist in a system with an vector order parameter since it lacks the inversion symmetry of the nematic phase with respect to the director. In addition, one 0nds ,1 (S 2 ) = 0, i.e., every disclination line of integral strength in a ferromagnet is unstable; “it escapes into the third dimension”. The same applies to nematic liquid crystal as demonstrated by Cladis et al. [35,236] and Bob Meyer [149] for S = 1. The director 0eld around a disclination follows from the minimization of the Frank free energy (2.6) [111,29,51]. In the one-constant approximation the line energy Fd of the disclination can be calculated as , R 1 Fd = K : (2.32) + ln 4 2 rc The surface term in Eq. (2.6) is neglected. The second term on the right-hand side of Eq. (2.32) stands for the elastic free energy per unit length around the line defect where R is the radius of a circular cross-section of the disclination (see Fig. 5). Since the energy diverges logarithmically, one has to introduce a lower cut-o9 radius rc , i.e., the radius of the disclination core. Its line energy, given by the 0rst term, is derived in the following way [111]. One assumes that the core of the disclination contains the liquid in the isotropic state with a free energy density 'c necessary to melt the nematic order locally. Splitting the line energy of the disclination as in Eq. (2.32) into the sum of a core and elastic part, Fd = ,rc2 'c + K, ln(R=rc )=4, and minimizing it with respect to rc , results in 'c =
K 1 ; 8 rc2
(2.33)
so that we immediately arrive at Eq. (2.32). The right-hand side of Eq. (2.33) is equivalent to the Frank free energy density of the director 0eld at a distance rc from the center of the disclination. Thus rc is given by the reasonable demand that the nematic state starts to melt when this energy density equals 'c . With an estimate 'c = 10−7 erg=cm3 , which follows from a description of the nematic-isotropic phase transition by the Landau–de Gennes theory [126], and K = 10−6 dyn, we obtain a core radius rc of the order of 10 nm. In the general case (K1 = K2 = K3 ), an analytical expression for the elastic free energy does not exist. However,
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Fig. 6. Radial and hyperbolic hedgehog of point charge Q = 1.
a rough approximation for the core energy per unit length, Fc , can be found by averaging over the Frank constants: , K1 + K2 + K3 Fc = : (2.34) 8 3 In Section 4.3 we will make use of this form for Fc . A more re0ned model of the disclination core is derived from Landau–de Gennes theory [48,96,212], which employs a traceless second-rank tensor Q as an order parameter (see Section 8.1). The tensor also describes biaxial liquid crystalline order. Investigations show that the core of a disclination should indeed be biaxial [141,144,207], with a core radius of the order of the biaxial correlation length b , i.e., the length on which deviations from the uniaxial order exponentially decay to zero. Outside of the disclination core, the nematic order is essentially uniaxial. Therefore, the line energy of a disclination is still given by Eq. (2.32) with rc ∼ b , and with a core energy now determined by the energy di9erence between the biaxial and uniaxial state rather than the energy di9erence between the isotropic and nematic state. 2.4.2. Point defects Fig. 6 presents typical point defects in a nematic liquid crystal known as radial and hyperbolic hedgehogs. Both director 0elds are rotationally symmetric about the vertical axis. The second homotopy group ,2 (P 2 ) of the projective plane P 2 is the set Z of all integer numbers. They label every point defect by a topological charge Q. The result is the same as for the vector order parameter space S 2 since close to the point singularity the director 0eld constitutes a unique vector 0eld. For true vectors it is possible to distinguish between a radial hedgehog of positive and negative charge depending on their vector 0eld that can either represent a source or a sink. In a nematic liquid crystal this distinction is not possible because n and −n describe the same state. Note, e.g., that the directors close to a point defect are reversed if the defect is moved around a ± 1=2 disclination line. Therefore, the sign of the charge Q has no meaning in nematics, and by convention it is chosen positive. The charge Q is determined by the number of times the unit sphere is wrapped by all the directors on a surface enclosing the defect core. An analytical expression for Q is 1 Q= dSi 'ijk n · (∇j n × ∇k n) ; (2.35) 8 where the integral is over any surface enclosing the defect core. Both the hedgehogs in Fig. 6 carry a topological charge Q=1. They are topologically equivalent since they can be transformed
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Fig. 7. The hyperbolic hedgehog at the center is transformed into a radial point defect by a continuous distortion of the director 0eld. Nails indicate directors tilted relative to the drawing plane.
Fig. 8. A radial (left) and a hyperbolic (right) hedgehog combine to a con0guration with total charge 0 = |1 − 1|.
into each other by a continuous distortion of the director 0eld. Just start from the hyperbolic hedgehog and rotate the director about the vertical axis through an angle when going outward from the core in any radial direction. By this procedure, which is illustrated in Fig. 7 with the help of a nail picture, we end up with a radial hedgehog. The length of the nail is proportional to the projection of the director on the drawing plane, and the head of the nail is below the plane. Such a transition was observed by Lavrentovich and Terentjev in nematic drops with homeotropic, i.e., perpendicular anchoring of the director at the outer surface [126]. In systems with vector symmetry, the combined topological charge of two hedgehogs with respective charges Q1 and Q2 is simply the sum Q1 + Q2 . In nematics, where the sign of the topological charge has no meaning, the combined topological charge of two hedgehogs is either |Q1 + Q2 | or |Q1 − Q2 |. It is impossible to tell with certainty which of these possible charges is the correct one by looking only at surfaces enclosing the individual hedgehogs. For example, the combined charge of two hedgehogs in the presence of a line defect depends on which path around the disclination the point defects are combined [226]. In Fig. 8 we illustrate how a radial and a hyperbolic hedgehog combine to a con0guration with total charge 0 = |1 − 1|. Since the distance
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Fig. 9. A hyperbolic hedgehog can be opened up to a −1=2 disclination ring.
d of the defects is the only length scale in the system, dimensional arguments predict an interaction energy proportional to Kd [169]. It grows linear in d reminiscent to the interaction energy of quarks if one tries to separate them beyond distances larger than the diameter of a nucleus. The energies of the hedgehog con0gurations, shown in Fig. 6, are easily calculated from the Frank free energy Fel + F24 [see Eqs. (2.2) and (2.3)]. The director 0elds of these con0gurations are n = (x; y; z)=r for the radial and n = (−x; −y; z)=r for the hyperbolic hedgehog, where r = (x; y; z) and r = |r|. In a spherical region of radius R with free boundary conditions at the outer surface, we obtain for their respective energies: Fradial = 4(2K1 − K24 )R → 4(2K − K24 )R ; 4 4 Fhyper = (6K1 + 4K3 + 5K24 )R → (2K + K24 )R ; (2.36) 15 3 where the 0nal expressions apply to the case of equal Frank constants. When K24 = 0, these energies reduce to those calculated in Ref. [126]. Note that the Frank free energy of point defects does not diverge in contrast to the distortion energy of disclinations in the preceding subsection. The hyperbolic hedgehog has lower energy than the radial hedgehog provided K3 ¡ 6K1 − 5K24 or K ¿ K24 for the one-constant approximation. Thus, if we concentrate on the bulk energies, i.e., K24 = 0, the hyperbolic hedgehog is always energetically preferred, since K1 is always of the same order as K3 . This seems to explain the observation of Lavrentovich and Terentjev, already mentioned [126], who found the con0guration illustrated in Fig. 7 in a nematic drop with radial boundary conditions at the outer surface. However, a detailed explanation has to take into account the Frank free energy of the strongly twisted transition region between hyperbolic and radial hedgehog [126]. In Section 7.4 we will present a linear stability analysis for the radial hedgehog against twisting. In terms of the Frank constants, it provides a criterion for the twist transition to take place, and its shows that the twisting starts close to the defect core. If, in addition to the one-constant approximation, K24 also ful0ls the Cauchy relation (2.5), i.e., K = K24 , the energies of the two hedgehog con0gurations in Eqs. (2.36) are equal, as one could have predicted from Eq. (2.6), which is then invariant with respect to rigid rotations of even a spatially varying n. The twisting of a hedgehog in a nematic drop takes place at a length scale of several microns [26,126,182]. However, point defect also possess a 0ne structure at smaller length scales, which has attracted a lot of attention. Fig. 9 illustrates how a hyperbolic hedgehog opens up to a −1=2
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disclination ring by 0lling in vertical lines of the director 0eld. The far0eld of the disclination ring is still given by the hedgehog so that the ring can be assigned the same topological point charge Q = 1. Similarly, a radial point defect is topologically equivalent to a 1=2 disclination ring. Mineev pointed out that the characterization of a ring defect requires two parameters; the index of the line and the charge of the point defect [152]. The classi0cation of ring defects within homotopy theory was developed by Garel [86] and Nakanishi et al. [161]. It can be asked whether it is energetically favorable for a hedgehog to open up to a disclination ring [155,225]. One can obtain a crude estimate of the radius R0 of the disclination ring with the help of Eqs. (2.32) and (2.36) for disclination and hedgehog energies. When R0 rc , the director con0guration of a charge 1 disclination ring is essentially that of a simple disclination line, discussed in the previous subsection. It extends up to distances of order R0 from the ring center. Beyond this radius, the director con0guration is approximately that of a hedgehog (radial or hyperbolic). Thus, we can estimate the energy of a disclination ring of radius R0 centered in a spherical region of radius R to be 1 R0 Fring ≈ 2R0 K + 8!K(R − R0 ) ; (2.37) + ln 4 2 rc where ! = 1 for a radial hedgehog and ! = 1=3 for a hyperbolic hedgehog. We also set K24 = 0. Minimizing over R0 , we 0nd 16 3 !− : (2.38) R0 = rc exp 32 Though admittedly crude, this approximation yields a result that has the same form as that calculated in Refs. [155,225] using a more sophisticated ansatz for the director 0eld. It has the virtue that it applies to both radial and hyperbolic far-0eld con0gurations. It predicts that the core M of a radial hedgehog should be a ring with radius R0 ≈ rc e3:6 , or R0 ≈ 360 nm for rc ≈ 100 A. The core of the hyperbolic hedgehog, on the other hand, will be a point rather than a ring because R0 ≈ rc e−0:2 ≈ rc . As in the case of disclinations, more re0ned models of the core of a point defect use the Landau–de Gennes free energy, which employs the second-rank tensor Q as an order parameter. Schopohl and Sluckin [208] chose a uniaxial Q but allowed the degree of orientational order, described by the Maier–Saupe parameter S [29,51], to continuously approach zero at the center of the defect. A stability analysis of the Landau–de Gennes free energy demonstrates that the radial hedgehog is either metastable or unstable against biaxial perturbations in the order parameter depending on the choice of the temperature and elastic constants [195,88]. Penzenstadler and Trebin modeled a biaxial defect core [171]. They found that the core radius is of the order of the biaxial correlation length b , which for the compound MBBA gives approximately 25 nm. This is an order of magnitude smaller than the estimate above. The reason might be that the ansatz function used by Penzenstadler and Trebin does not include a biaxial disclination ring. Such a ring encircles a region of uniaxial order, as illustrated in the right part of Fig. 9, and it possesses a biaxial disclination core. Numerical studies indicate the existence of such a ring [218,87] but a detailed analysis of the competition between a biaxial core and a biaxial disclination ring is still missing. We expect that Eq. (2.37) for a disclination ring and therefore Eq. (2.38) for its radius can be justi0ed within the Landau–de Gennes theory for R0 rc ∼ b .
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However, the Frank elastic constant and the core energy will be replaced by combinations of the Landau parameters. Since the ring radius R0 varies exponentially with the elastic constants and the core energy, and since rc is only roughly de0ned, it is very diNcult to predict with certainty even the order of magnitude of R0 . Further investigations are needed. 3. Nematic colloidal dispersions In this section we 0rst give a historic account of the topic relating it to recent developments in the liquid crystal 0eld and reviewing the work performed on colloidal dispersions in nematic liquid crystals. Then, with nematic emulsions, we introduce one particular model system for such colloidal dispersions. 3.1. Historic account Liquid crystal emulsions, in which surfactant-coated drops, containing a liquid crystalline material, are dispersed in water, have been a particularly fruitful medium for studying topological defects for thirty years [147,61,26,126,121,60]. The liquid crystalline drops typically range from 10 to 50 m in diameter and are visible under a microscope. Changes in the Frank director n are easily studied under crossed polarizers. The isolated drops in these emulsions provide an idealized spherical con0ning geometry for the nematic phase. With the introduction of polymer-dispersed liquid crystals as electrically controllable light shutters [58,60], an extensive study of liquid crystals con0ned to complex geometries, like distorted drops in a polymer matrix or a random porous network in silica aerogel, was initiated [60,44]. Here, we are interested in the inverse problem that is posed by particles suspended in a nematic solvent. Already in 1970, Brochard and de Gennes studied a suspension of magnetic grains in a nematic phase and determined the director 0eld far away from a particle [22]. The idea was to homogeneously orient liquid crystals with a small magnetic anisotropy by a reasonable magnetic 0eld strength through the coupling between the liquid-crystal molecules and the grains. The idea was realized experimentally by two groups [31,75]. However, even in the highly dilute regime the grains cluster. Extending Brochard’s and de Gennes’ work, Burylov and Raikher studied the orientation of an elongated particle in a nematic phase [24]. Chaining of bubbles or microcrystallites was used to visualize the director 0eld close to the surface of liquid crystals [191,36]. A bistable liquid crystal display was introduced based on a dispersion of agglomerations of silica spheres in a nematic host [62,118,117,91]. The system was called 4lled nematics. Chains and clusters were observed in the dispersion of latex particles in a lyotropic liquid crystal [181,188,189]. The radii of the particles were 60 and 120 nm. Therefore, details of the director 0eld could not be resolved under the polarizing microscope. Terentjev et al. [225,119,201,213] started to investigate the director 0eld around a sphere by both analytical and numerical methods, 0rst concentrating on the Saturn-ring and surface-ring con0guration. Experiments of Philippe Poulin and coworkers on inverted nematic emulsions, which we describe in the following subsection, clearly demonstrated the existence of a dipolar structure formed by a water droplet and a companion hyperbolic hedgehog [182,179,183,184]. A similar observation at a nematic-isotropic interface was made by Bob Meyer in 1972 [148].
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Lately, Poulin et al. [180] were able to identify the dipolar structure in suspensions of latex particles and they could observe an equatorial ring con0guration in the weak-anchoring limit of nematic emulsions [153]. In a very recent paper, Gu and Abbott reported Saturn-ring con0gurations around solid microspheres [97]. Particles in contact with the glass plates of a cell were studied in Ref. [98]. Lubensky, Stark, and coworkers presented a thorough analytical and numerical analysis of the director 0eld around a spherical particle [182,140,219]. It is discussed in Section 4. Ramaswamy et al. [190] and Ruhwandl and Terentjev [200] determined the long-range quadrupolar interaction of particles surrounded by a ring disclination, whereas Lubensky et al. addressed both dipolar and quadrupolar forces [140] (see Section 5). Recently, Lev and Tomchuk studied aggregates of particles under the assumption of weak anchoring [130]. Work on the Stokes drag of a spherical object immersed into a uniformly aligned nematic was performed by Diogo [57], Roman and Terentjev [194], and Heuer et al. [112,105]. The calculations were extended to the Saturn-ring con0guration by Ruhwandl and Terentjev [199] and to the dipolar structure by Ventzki and Stark [228], whose work is explored in detail in Section 6. The Stokes drag was also determined through molecular dynamics simulations by Billeter and Pelcovits [10]. Stark and Stelzer [220] numerically investigated multiple nematic emulsions [182] by means of 0nite elements. We discuss the results in Section 7. It is interesting to note that dipolar con0gurations also appear in two-dimensional systems including (1) free standing smectic 0lms [132,175], where a circular region with an extra layer plays the role of the spherical particle, and (2) Langmuir 0lms [74], in which a liquid-expanded inclusion in a tilted liquid-condensed region acts similarly. Pettey et al. [175] studied the dipolar structure in two dimensions theoretically. In cholesteric liquid crystals particle-stabilized defect gels were found [239], and people started to investigate dispersions of particles in a smectic phase [80,108,12]. Sequeira and Hill were the 0rst to measure the viscoelastic response of concentrated suspensions of zeolite particles in nematic MBBA [209]. Meeker et al. [143] reported a gel-like structure in nematic colloidal dispersions with a signi0cant shear modulus. Perfectly ordered chains of oil droplets in a nematic were produced from phase separation by Loudet et al. [137]. Very recent studies of the nematic order around spherical particles are based on the minimization of the Landau–de Gennes free energy using an adaptive grid scheme [83], or they employ molecular dynamics simulations of Gay–Berne particles [10,6]. The 0ndings are consistent with the presentation in Section 4. With two excellent publications [210,211], Ping Sheng initiated the interest in partially ordered nematic 0lms above the nematic-isotropic phase transition temperature Tc using the Landau–de Gennes approach. In 1981, Horn et al. [106] performed 0rst measurements of liquid crystal-mediated forces between curved mica sheets. Motivated by both works, Poniewierski \ and Sluckin re0ned Sheng’s study [177]. Bor\stnik and Zumer explicitly considered two parallel plates immersed into a liquid crystal slightly above Tc [18], and thoroughly investigated short-range interactions due to the surface-induced nematic order. An analog work was presented by de Gennes, however, assuming a surface-induced smectic order [50]. The e9ect of such a presmectic 0lm was measured by Moreau et al. [154]. Recent studies address short-range forces of spherical objects using either analytical methods [15], which we report in Section 8, or numerical calculations [85]. In Section 8 we also demonstrate that such forces can induce >occulation of colloidal particles above the nematic-isotropic phase transition [16,17]. In a more general context, they were also suggested by LUowen [135,136]. Mu\sevi\c et al. probe these
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interactions with the help of an atomic force microscope [158,159,113], whereas BUottger et al. [19] and Poulin et al. [178] are able to suspend solid particles in a liquid crystal above Tc . Even Casimir forces arising from >uctuations in the liquid-crystalline order parameter were investigated both in the nematic [3,2,223] and isotropic phase [240] of a liquid crystal. 3.2. Nematic emulsions In 1996, Philippe Poulin succeeded in producing inverted and multiple nematic emulsions [182,183]. The notion “inverted” refers to water droplets dispersed in a nematic solvent, in contrast to direct liquid-crystal-in-water emulsions. If the solvent itself forms drops surrounded by the water phase, one has multiple emulsions. We introduce them here since they initiated the theoretical work we report in the following sections. Philippe Poulin dispersed water droplets of 1 to 5 m in diameter in a nematic liquid crystal host, pentylcyanobiphenyl (5CB), which formed larger drops (∼ 50 m diameter) surrounded by a continuous water phase. This isolated a controlled number of colloidal droplets in the nematic host which allowed to observe their structure readily. As a surfactant, a small amount of sodium dodecyl sulfate was used. It is normally ine9ective at stabilizing water droplets in oil. Nevertheless, the colloidal water droplets remained stable for several weeks, which suggested that the origin of this stability is the surrounding liquid crystal—a hypothesis that was con0rmed by the observation that droplets became unstable and coalesced in less than one hour after the liquid crystal was heated to the isotropic phase. The surfactant also guaranteed a homeotropic, i.e., normal boundary condition of the director at all the surfaces. The multiple nematic emulsions were studied by observing them between crossed polarizers in a microscope. Under such conditions, an isotropic >uid will appear black, whereas regions in which there is the birefringent nematic will be colored. Thus the large nematic drops in a multiple emulsion appear predominately red in Fig. 10a, 1 whereas the continuous water phase surrounding them is black. Dispersed within virtually all of the nematic drops are smaller colloidal water droplets, which also appear dark in the photo; the number of water droplets tends to increase with the size of the nematic drops. Remarkably, in all cases, the water droplets are constrained at or very near the center of the nematic drops. Moreover, their Brownian motion, usually observed in colloidal dispersions, has completely ceased. However, when the sample is heated to change the nematic into an isotropic >uid, the Brownian motion of the colloidal droplets is clearly visible in the microscope. Perhaps the most striking observation in Fig. 10a is the behavior of the colloidal droplets when more than one of them cohabit the same nematic drop: the colloidal droplets invariably form linear chains. This behavior is driven by the nematic liquid crystal: the chains break, and the colloidal droplets disperse immediately upon warming the sample to the isotropic phase. However, although the anisotropic liquid crystal must induce an attractive interaction to cause the chaining, it also induces a shorter range repulsive interaction. A section of a chain of droplets under higher magni0cation (see Fig. 10b) shows that the droplets are prevented from approaching each other too closely, with the separation between droplets being a signi0cant 1
Reprinted with permission from P. Poulin, H. Stark, T.C. Lubensky, D.A. Weitz, Novel colloidal interactions in anisotropic >uids, Science 275 (1997) 1770. Copyright 1997 American Association for the Advancement of Science.
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Fig. 10. (a) Microscope image of a nematic multiple emulsion taken under crossed polarizers, (b) a chain of water droplets under high magni0cation, (c) a nematic drop containing a single water droplet.
fraction of their diameter. A careful inspection of Fig. 10b even reveals black dots between the droplets which we soon will identify as topological point defects. The distance between droplets and these host->uid defects increases with the droplet radius. To qualitatively understand the observation, we start with one water droplet placed at the center of a large nematic drop. The homeotropic boundary condition enforces a radial director 0eld between both spherical surfaces. It exhibits a distinctive four-armed star of alternating bright and dark regions under crossed polarizers that extend throughout the whole nematic drop as illustrated in Fig. 10c. Evidently, following the explanations in Section 2:4:2 about point defects, the big nematic drop carries a topological point charge Q = 1 that is matched by the small water droplet which acts like a radial hedgehog. Each water droplet beyond the 0rst added to the interior of the nematic drop must create orientational structure out of the nematic itself to satisfy the global constraint Q = 1. The simplest (though not the only [140]) way to satisfy this constraint is for each extra water droplet to create a hyperbolic hedgehog in the nematic host. Note that from Fig. 8 we already know that a radial hedgehog (represented by the water droplet) and a hyperbolic point defect carry a total charge zero. Hence, N water droplets in a large nematic drop have to be accompanied by N − 1 hyperbolic hedgehogs. Fig. 11 presents a qualitative picture of the director 0eld lines for a string of three droplets. It is rotationally symmetric about the horizontal axis. Between the droplets, hyperbolic hedgehogs appear. They prevent the water droplets from approaching each other and from 0nally coalescing since this would involve a strong distortion of the director 0eld. The defects therefore mediate a short-range repulsion between the droplets.
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Fig. 11. The director 0eld lines of a nematic drop containing a string of three spherical particles.
In the following sections, we demonstrate the physical ideas which evolved from the experiments on multiple nematic emulsions. We will explain the chaining of droplets by introducing the topological dipole formed by one spherical particle and its companion hyperbolic defect. This leads us to the next chapter where we investigate the simplest situation, i.e., one particle placed in a nematic solvent which is uniformly aligned at in0nity. 4. The paradigm---one particle The multiple nematic emulsions that we introduced in Section 3.2 are already a complicated system. In this section we investigate thoroughly by both analytical and numerical means what I regard as the paradigm for the understanding of inverted nematic emulsions. We ask which director 0eld con0gurations do occur when one spherical particle that prefers a radial anchoring of the director at its surface is placed into a nematic solvent uniformly aligned at in0nity. This constitutes the simplest problem one can think of, and it is a guide to the understanding of more complex situations. 4.1. The three possible con4gurations If the directors are rigidly anchored at the surface, the particle carries a topological charge Q = 1. Because of the boundary conditions at in0nity, the total charge of the whole system is zero; therefore, the particle must nucleate a further defect in its nematic environment. One possibility is a dipolar structure where the particle and a hyperbolic hedgehog form a tightly bound object which we call dipole for short (see Fig. 12). As already explained in Fig. 8, the topological charges +1 of a radial hedgehog, represented by the particle, and of a hyperbolic point defect “add up” to a total charge of zero. In the Saturn-ring con0guration, a −1=2 disclination ring encircles the spherical particle at its equator (see Fig. 12). Of course, the disclination ring can be moved upward or downward, and by shrinking it to the topologically
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Fig. 12. A spherical particle with a preferred homeotropic anchoring at its surface that is placed into a uniformly aligned nematic liquid crystal exhibits three possible structures: the dipole con0guration where the particle is accompanied by a hyperbolic hedgehog, the Saturn-ring con0guration where the particle is surrounded by a −1=2 disclination ring at the equator, and the surface-ring con0guration.
equivalent hyperbolic hedgehog, the Saturn ring is continuously transformed into the dipole con0guration. However, our calculations show that a non-symmetric position of the defect ring is never stable. When the surface anchoring strength W is lowered (see Fig. 12), the core of the disclination ring prefers to sit directly at the surface of the particle. For suNciently low W , the director 0eld becomes smooth everywhere, and a ring of tangentially oriented directors is located at the equator of the sphere. In the case of tangential boundary conditions, there exists only one structure. It possesses two surface defects, called boojums, at the north and south pole of the particle [145,26,120,231]. We will not investigate it further. It is instructive to 0rst consider the director 0eld far away from the particle, which crucially depends on the global symmetry of the system [22,140]. With its knowledge, ansatz functions for the director con0gurations around a particle can be checked. Furthermore, the far 0eld determines the long-range two-particle interaction. Let the director n0 at in0nity point along the z axis. Then, in the far 0eld, the director is approximated by n(r) ≈ (nx ; ny ; 1) with nx ; ny 1. In leading order, the normalization of the director can be neglected, and the Euler–Lagrange equations for nx and ny arising from a minimization of the Frank free energy in the one-constant
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approximation are simply Laplace equations: ∇2 n3 = 0 :
(4.1)
The solutions are the well-known multipole expansions of electrostatics that include monopole, dipole, and quadrupole terms. They are all present if the suspended particle has a general shape or if, e.g., the dipole in Fig. 12 is tilted against n0 . In the dipole con0guration with its axial symmetry about n0 , the monopole is forbidden, and we obtain x zx y zy nx = p + 2c 5 and ny = p + 2c 5 ; (4.2) r3 r r3 r where r = (x2 + y2 + z 2 )1=2 . We use the expansion coeNcients p and c to assign both a dipole (p) and quadrupole (c ) moment to the con0guration: p = p n0
and
c = c(n0 ⊗ n0 − 1=3) :
(4.3)
The symbol ⊗ means tensor product, and 1 is the unit tensor of second rank. We adopt the convention that the dipole moment p points from the companion defect to the particle. Hence, if p ¿ 0, the far 0eld of Eqs. (4:2) belongs to a dipole con0guration with the defect sitting below the particle (see Fig. 12). Note, that by dimensional analysis, p ∼ a2 and c ∼ a3 , where a is the radius of the spherical particle. Saturn-ring and surface-ring con0gurations possess a mirror plane perpendicular to the rotational axis. Therefore, the dipole term in Eqs. (4:2) is forbidden, i.e., p = 0. We will show in Section 6 that the multipole moments p and c determine the long-range two-particle interaction. We will derive it on the basis of a phenomenological theory. In the present section we investigate the dipole by both analytical and numerical means. First, we identify a twist transition which transforms it into a chiral object. Then, we study the transition from the dipole to the Saturn ring con0guration, which is induced either by decreasing the particle radius or by applying a magnetic 0eld. The role of metastability is discussed. Finally, we consider the surface-ring con0guration and point out the importance of the saddle-splay free energy F24 . Lower bounds for the surface-anchoring strength W are given. 4.2. An analytical investigation of the dipole Even in the one-constant approximation and for 0xed homeotropic boundary conditions, analytical solutions of the Euler–Lagrange equations, arising from the minimization of the Frank free energy, cannot be found. The Euler–Lagrange equations are highly non-linear due to the normalization of the director. In this subsection we investigate the dipole con0guration with the help of ansatz functions that obey all boundary conditions and possess the correct far-0eld behavior. The free parameters in these ansatz functions are determined by minimizing the Frank free energy. We will see that this procedure already provides a good insight into our system. We arrive at appropriate ansatz functions by looking at the electrostatic analog of our problem [182,140], i.e., a conducting sphere of radius a and with a reduced charge q which is exposed to an electric 0eld of unit strength along the z axis. The electric 0eld is E (r ) = ez + qa2
r
r3
− a3
r 2 ez − 3z r : r5
(4.4)
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Fig. 13. Frank free energy (in units of ,Ka) for the topological dipole as a function of the reduced distance rd =a from the particle center to the companion hedgehog.
In order to enforce the boundary condition that E be normal to the surface of the sphere, an electric image dipole has to be placed at the center of the sphere. The ansatz function for the director 0eld follows from a normalization: n(r) = E (r)= |E (r)|. An inspection of its far 0eld gives x xz y yz n(r ) ≈ qa2 3 + 3a3 5 ; qa2 3 + 3a3 5 ; (4.5) r r r r in agreement with Eqs. (4:2). The electrostatic analog assigns a dipole moment qa2 and a quadrupole moment 3a3 =2 to the topological dipole. The zero of the electric 0eld determines the location −rd ez of the hyperbolic hedgehog on the z axis. Thus, q or the distance rd from the center of the particle are the variational parameters of our ansatz functions. Note that for q = 3, the hedgehog just touches the sphere, and that for q ¡ 3, a singular ring appears at the surface of the sphere. In Fig. 13 we plot the Frank free energy in the one-constant approximation and in units of ,Ka as a function of the reduced distance rd =a. The saddle-splay term is not included, since for rigid anchoring it just provides a constant energy shift. There is a pronounced minimum at rd0 = 1:17a corresponding to a dipole moment p = qa2 = 3:08a2 . The minimum shows that the hyperbolic hedgehog sits close to the spherical particle. To check the magnitude of the thermal >uctuations of its radial position, we determine the curvature of the energy curve at rd0 ; its approximate value amounts to 33,K=a. According to the equipartition theorem, the average thermal displacement rd0 follows from the expression
rd0 kB T ≈ ≈ 2 × 10−3 ; (4.6) a 33Ka where the 0nal estimate employs kB T ≈ 4 × 10−14 erg; K ≈ 10−6 dyn, and a=1 m. These >uctuations in the length of the topological dipole are unobservably small. For angular >uctuations of the dipole, we 0nd 5 ≈ 10−2 , i.e., still diNcult to observe [140]. We conclude that the spherical particle and its companion hyperbolic hedgehog form a tightly bound object. Interestingly, we note that angular >uctuations in the 2D version of this problem diverge logarithmically with the sample size [175]. They are therefore much larger and have indeed been observed in free standing smectic 0lms [132].
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Fig. 14. (Top) Image of a single droplet with its companion defect as observed under crossed polarizers obtained by Poulin [183]. (Bottom) Simulated image of the same con0guration using the Jones matrix formalism [140]. The two pictures are very similar. From Ref. [140].
The droplet-defect dipole was observed by Philippe Poulin in inverted nematic emulsions [183]. In the top part of Fig. 14 we present how it looks like in a microscope under crossed polarizers, with one polarizer parallel to the dipole axis. In the bottom part of Fig. 14 we show a calculated image using the Jones matrix formalism [60] based on the director 0eld of the electrostatic analog. Any refraction at the droplet boundary is neglected. The similarity of the two images is obvious and clearly con0rms the occurrence of the dipole con0guration. The electric 0eld ansatz is generalized by no longer insisting that it originates in a true electric 0eld. This allows us to introduce additional variational parameters [140]. The Frank free energy at rd0 is lowered, and the equilibrium separation amounts to rd0 = 1:26a. The
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respective dipole and quadrupole moments turn out to be p = 2:20a2 and c = −1:09a2 . We are also able to construct ansatz functions for the dipole–Saturn ring transition utilizing the method of images for the related 2D problem and correcting the far 0eld [140]. The results agree with the numerical study presented in the next subsection. 4.3. Results and discussion of the numerical study Before we present the results of our numerical study, we summarize the numerical method. Details can be found in [219]. 4.3.1. Summary of numerical details The numerical investigation is performed on a grid which is de0ned by modi0ed spherical coordinates. Since the region outside the spherical particle is in0nitely extended, we employ a radial coordinate 6 = 1=r 2 . The exponent 2 is motivated by the far 0eld of the dipole con0guration. Such a transformation has two advantages. The exterior of the particle is mapped into a 0nite region, i.e., the interior of the unit sphere (6 6 1). Furthermore, equally spaced grid points along the coordinate 6 result in a mesh size in real space which is small close to the surface of the particle. In this area the director 0eld is strongly varying, and hence a good resolution for the numerical calculation is needed. On the other hand, the mesh size is large far away from the sphere where the director 0eld is nearly homogeneous. Since our system is axially symmetric, the director 0eld only depends on 6 and the polar angle 5. The director is expressed in the local coordinate basis (er ; e5 ; e7 ) of the standard spherical coordinate system, and the director components are parametrized by a tilt [(6; 5)] and a twist [(6; 5)] angle: nr = cos ; n5 = sin cos , and n7 = sin sin . The total free energy Fn of Eq. (2.1) is expressed in the modi0ed spherical coordinates. Then, the Euler–Lagrange equations in the bulk and at the surface are formulated with the help of the chain rules of Eqs. (2.14) and (2.15) and by utilizing the algebraic program Maple. A starting con0guration of the director 0eld is chosen and relaxed into a local minimum via the Newton–Gauss–Seidel method [187] which was implemented in a Fortran program. So far we have described the conventional procedure of a numerical investigation. Now, we address the problem of how to describe disclination rings numerically. Fig. 15 presents such a ring whose general position is determined by a radial (rd ) and an angular (5d ) coordinate. The free energy Fn of the director 0eld follows from a numerical integration. This assigns some energy to the disclination ring which certainly is not correct since the numerical integration does not realize the large director gradients close to the defect core. To obtain a more accurate value for the total free energy F, we use the expression F = Fn − Fn |torus + Fc=d × 2rd sin 5d ;
(4.7)
where Fc and Fd are the line energies of a disclination introduced in Eqs. (2.32) and (2.34). The quantity Fn |torus denotes the numerically calculated free energy of a toroidal region of cross section R2 around the disclination ring. Its volume is R2 × 2rd sin 5d . The value Fn |torus is replaced by the last term on the right-hand side of Eq. (4.7), which provides the correct free energy with the help of the line energies Fc or Fd . We checked that the cross section R2 of the cut torus has to be equal or larger than 3V6 V5=2, where V6 and V5 are the lattice
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Fig. 15. Coordinates (rd ; 5d ) for a −1=2 disclination ring with a general position around the spherical particle. From Ref. [219].
Fig. 16. The reduced distance rd =a of the hyperbolic hedgehog from the center of the sphere as a function of the reduced splay (K1 =K3 ) and twist (K2 =K3 ) constants.
constants of our grid. For larger cross sections, the changes in the free energy F for 0xed core radius rc were less than 1%, i.e., F became independent of R2 . What is the result of this procedure? All lengths in the free energy Fn can be rescaled by the particle radius a. This would suggest that the director con0guration does not depend on the particle size. However, with the illustrated procedure a second length scale, i.e., the core radius rc of a disclination, enters. All our results on disclination rings therefore depend on the ratio a=rc . In discussing them, we assume rc ≈ 10 nm [111] which then determines the radius a for a given a=rc . 4.3.2. Twist transition of the dipole con4guration In this subsection we present our numerical study of the topological dipole. We always assume that the directors are rigidly anchored at the surface (W → ∞) and choose a zero magnetic 0eld. In Fig. 16 we plot the reduced distance rd =a of the hedgehog from the center of the sphere as a function of the reduced splay (K1 =K3 ) and twist (K2 =K3 ) constants. In the
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Fig. 17. (a) Nail picture of a closeup of the twisted dipole con0guration. Around the hyperbolic hedgehog the directors are tilted relative to the drawing plane. From Ref. [219]. (b) Phase diagram of the twist transition as a function of the reduced splay (K1 =K3 ) and twist (K2 =K3 ) constants. A full explanation is given in the text.
one-constant approximation, we 0nd rd =1:26 ± 0:02, where the mesh size of the grid determines the uncertainty in rd . Our result is in excellent agreement with the generalized electric-0eld ansatz we introduced in the last subsection [140]. However, Ruhwandl and Terentjev using a Monte-Carlo minimization report a somewhat smaller value for rd [200]. In front of the thick line rd is basically constant. Beyond the line, rd starts to grow which indicates a structural change in the director 0eld illustrated in the nail picture of Fig. 17a. Around the hyperbolic hedgehog the directors develop a non-zero azimuthal component n7 , i.e., they are tilted relative to the drawing plane. This introduces a twist into the dipole. It should be visible under a polarizing microscope when the dipole is viewed along its symmetry axis. In Fig. 17b we draw a phase diagram of the twist transition. As expected, it occurs when K1 =K3 increases or when K2 =K3 decreases, i.e., when a twist deformation costs less energy than a splay distortion. The open circles are numerical results for the transition line which can well be 0tted by the straight line K2 =K3 ≈ K1 =K3 − 0:04. Interestingly, the small o9set 0.04 means that K3 does not play an important role. Typical calamatic liquid crystals like MBBA, 5CB, and PAA should show the twisted dipole con0guration. Since the twist transition breaks the mirror symmetry of the dipole, which then becomes a chiral object, we describe it by a Landau expansion of the free energy: 2 max 4 F = F0 + a(K1 =K3 ; K2 =K3 )[nmax 7 ] + c[n7 ] :
(4.8)
we have introduced a simple order parameter. With the maximum azimuthal component nmax 7 Since the untwisted dipole possesses a mirror symmetry, only even powers of nmax are allowed. 7 The phase transition line is determined by a(K1 =K3 ; K2 =K3 ) = 0. According to Eq. (4.8), we expect a power-law dependence of the order parameter with the exponent 1=2 in the twist region close to the phase transition. To test this idea, we choose a constant K2 =K3 ratio and determine nmax for varying K1 . As the log–log plot in Fig. 18 illustrates, when approaching the 7 phase transition, the order parameter obeys the expected power law: nmax ∼ (K1 =K3 − 0:4372)1=2 7
with K2 =K3 = 0:4 :
(4.9)
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Fig. 18. Log–log plot of the order parameter nmax versus K1 =K3 close to the twist transition (K2 =K3 = 0:4); 7 ◦ : : : numerical values, – : : : 0t by a straight line. Fig. 19. The free energy F in units of K3 a as a function of the angular coordinate 5d . The parameter of the curves is the particle size a. Further parameters are indicated in the inset.
4.3.3. Dipole versus Saturn ring There are two possibilities to induce a transition from the dipole to the Saturn-ring con0guration; either by reducing the particle size or by applying, e.g., a magnetic 0eld. We always assume rigid anchoring in this subsection, set K24 = 0, and start with the 0rst point. 4.3.3.1. E<ect of particle size. In Fig. 19 we plot the free energy F in units of K3 a as a function of the angular coordinate 5d of the disclination ring. For constant 5d , the free energy F was chosen as the minimum over the radial coordinate rd . The particle radius a is the parameter of the curves, and the one-constant approximation is employed. Recall that 5d = =2 and 5d = correspond, respectively, to the Saturn-ring or the dipole con0guration. Clearly, for small particle sizes (a = 180 nm) the Saturn ring is the absolutely stable con0guration, and the dipole enjoys some metastability. However, thermal >uctuations cannot induce a transition to the dipole since the potential barriers are much higher than the thermal energy kB T . E.g., a barrier of 0:1K3 a corresponds to 1000kB T (T = 300 K; a = 1 m). At a ≈ 270 nm, the dipole assumes the global minimum of the free energy, and 0nally the Saturn ring becomes absolutely unstable at a ≈ 720 nm. The scenario agrees with the results of Ref. [140] where an ansatz function for the director 0eld was used. Furthermore, we stress that the particle sizes were calculated with the choice of 10 nm as the real core size of a line defect, and that our results depend on the line energy (2.32) of the disclination. The reduced radial coordinate rd =a of the disclination ring as a function of 5d is presented in Fig. 20. It was obtained by minimizing the free energy for 0xed 5d . As long as the ring is open, rd does not depend on 5d within an error of ±0:01. Only in the region where it closes to the hyperbolic hedgehog, does rd increase sharply. The 0gure also illustrates that the ring sits closer to larger particles. The radial position of rd =a = 1:10 for 720 nm particles agrees very well with analytical results obtained by using an ansatz function (see Refs. [140]) and with numerical calculations based on a Monte-Carlo minimization [200]. Recent observations of
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Fig. 20. The reduced radial coordinate rd =a of a disclination ring as a function of 5d for two particle sizes. Further parameters are indicated in the inset.
the Saturn-ring con0guration around glass spheres of 40, 60, or 100 m in diameter [97] seem to contradict our theoretical 0ndings. However, we explain them by the strong con0nement in 120-m-thick liquid crystal cells which is equivalent to a strong magnetic 0eld. 4.3.3.2. E<ect of a magnetic 4eld. A magnetic 0eld applied along the symmetry axis of the dipole can induce a transition to the Saturn-ring con0guration. This can be understood from a simple back-of-the-envelope calculation. Let us consider high magnetic 0elds, i.e., magnetic coherence lengths much smaller than the particle size a. The magnetic coherence length H was introduced in Eq. (2.9) as the ratio of elastic and magnetic torques on the director. For H a, the directors are basically aligned along the magnetic 0eld. In the dipole con0guration, the director 0eld close to the hyperbolic hedgehog cannot change its topology. The 0eld lines are “compressed” along the z direction, and high densities of the elastic and magnetic free energies occur in a region of thickness H . Since the 0eld lines have to bend around the sphere, the cross section of the region is of the order of a2 , and its volume is proportional to a2 H . The Frank free energy density is of the order of K=2H , where K is a typical Frank constant, and therefore the elastic free energy scales with Ka2 =H . The same holds for the magnetic free energy. In the case of the Saturn-ring con0guration, high free energy densities occur in a toroidal region of cross section ˙2H around the disclination ring. Hence, the volume scales with a2H , and the total free energy is of the order of Ka, i.e., a factor a=H smaller than for the dipole. Fig. 21 presents a calculation for a particle size of a=0:5 m and the liquid crystal compound 5CB. We plot the free energy in units of K3 a as a function of 5d for di9erent magnetic 0eld strengths which we indicate by the reduced inverse coherence length a=H . Without a 0eld (a=H = 0), the dipole is the energetically preferred con0guration. The Saturn ring shows metastability. A thermally induced transition between both states cannot happen because of the high potential barrier. At a 0eld strength a=H = 0:33, the Saturn ring becomes the stable con0guration. However, there will be no transition until the dipole loses its metastability at a 0eld strength a=H = 3:3, which is only indicated by an arrow in Fig. 21. Once the system has changed to the Saturn ring, it will stay there even for zero magnetic 0eld. Fig. 22a schematically illustrates how a dipole can be transformed into a Saturn ring with the help of a magnetic 0eld.
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Fig. 21. The free energy F in units of ,K3 a as a function of the angular coordinate 5d . The parameter of the curves is the reduced inverse magnetic coherence length a=H . Further parameters are indicated in the inset. Fig. 22. (a) The Saturn ring is metastable at H =0. The dipole can be transformed into the Saturn ring by increasing the magnetic 0eld H beyond Ht2 where the dipole loses its metastability. Turning o9 the 0eld the Saturn ring remains. (b) The Saturn ring is unstable at H = 0. When the magnetic 0eld is decreased from values above Ht2 , the Saturn ring shrinks back to the dipole at Ht1 where the Saturn ring loses its metastability. A hysteresis occurs. From Ref. [219].
If the Saturn ring is unstable at zero 0eld, a hysteresis occurs (see Fig. 22b). Starting from high magnetic 0elds, the Saturn ring loses its metastability at Ht1 , and a transition back to the dipole takes place. In Fig. 19 we showed that the second situation is realized for particles larger than 720 nm. We also performed calculations for a particle size of 1 m and the liquid crystal compound 5CB and still found the Saturn ring to be metastable at zero 0eld in contrast to the result of the one-constant approximation. To be more concrete, according to Eq. (2.9), a=H = 1 corresponds to a 0eld strength of 4:6 T when 0:5 m particles and the material parameters of 5CB (K3 = 0:53 × 10−6 dyn; V = 10−7 ) are used. Hence, the transition to the Saturn ring in Fig. 21 occurs at a rather high 0eld of 15 T. Assuming that there is no dramatic change in a=H = 3:3 for larger particles, this 0eld decreases with increasing particle radius. Alternatively, the transition to the Saturn ring is also induced by an electric 0eld with the advantage that strong 0elds are much easier to apply. However, the large dielectric anisotropy V' = ' − '⊥ complicates a detailed analysis because of the di9erence between applied and local electric 0elds. Therefore, the electric coherence length E = [4K3 =(V' E 2 )]1=2 , which replaces H , only serves as a rough estimate for the applied 0eld E necessary to induce a transition to the Saturn ring. 4.3.4. In>uence of 4nite surface anchoring In the last subsection we investigate the e9ect of 0nite anchoring on the director 0eld around the spherical particle. The saddle-splay term with its elastic constant K24 is important now. We always choose a zero magnetic 0eld. In Fig. 23 we employ the one-constant approximation and plot the free energy versus the reduced surface extrapolation length S =a for di9erent reduced
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Fig. 23. The minimum free energy F in units of ,K3 a as a function of the reduced surface extrapolation length S =a for di9erent K24 =K3 . A 0rst-order phase transition from the dipole to the surface ring occurs. Further parameters are indicated in the inset.
Fig. 24. The saddle-splay free energy F24 in units of ,K3 a as a function of S =a for the same curves as in Fig. 23. Inset: F24 in units of ,K24 a versus the angular width of the surface ring calculated from the ansatz functions in Eqs. (4.10). Fig. 25. Phase diagram of the dipole-surface ring transition as a function of S =a and K24 =K3 . Further parameters are indicated in the inset.
saddle-splay constants K24 =K3 . Recall that S is inversely proportional to the surface constant W [see Eq. (2.11)]. The straight lines belong to the dipole. Then, for decreasing surface anchoring, there is a 0rst-order transition to the surface-ring structure. We never 0nd the Saturn ring to be the stable con0guration although it enjoys some metastability. For K24 =K3 =0, the transition takes place at S =a ≈ 0:085. This value is somewhat smaller than the result obtained by Ruhwandl and Terentjev [200]. One could wonder why the surface ring already occurs at such a strong anchoring like S =a ≈ 0:085 where any deviation from the homeotropic anchoring costs a lot of energy. However, if V5 is the angular width of the surface ring where the director deviates from the homeotropic alignment (see inset of Fig. 24) then a simple energetical estimate allows V5 to be of the order of S =a. It is interesting to see that the transition point shifts to higher anchoring
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strengths, i.e., decreasing S =a when K24 =K3 is increased. Obviously, the saddle-splay term favors the surface-ring con0guration. To check this conclusion, we plot the reduced saddle-splay free energy F24 versus S =a in Fig. 24. The horizontal lines belong to the dipole. They correspond to the saddle-splay energy 4K24 a which one expects for a rigid homeotropic anchoring at the surface of the sphere. In contrast, for the surface-ring con0guration the saddle-splay energy drops sharply. The surface ring at the equator of the sphere introduces a “saddle” in the director 0eld as illustrated in the inset of Fig. 24. Such structures are known to be favored by the saddle-splay term. We modeled the surface ring with an angular width V5 by the following radial and polar director components: 5 − =2 5 − =2 −1 nr = −tanh and n5 = − cosh ; (4.10) V5 V5 where V5=2 to ensure that nr = 1 at 5 = 0; , and calculated the saddle-splay energy versus V5 by numerical integration. The result is shown in the inset of Fig. 24. It 0ts very well to the full numerical calculations and con0rms again that a narrow “saddle” around the equator can considerably reduce the saddle-splay energy. For the liquid crystal compound 5CB we determined the stable con0guration as a function of K24 =K3 and S =a. The phase diagram is presented in Fig. 25. With its help, we can derive a lower bound for the surface constant W at the interface of water and 5CB when the surfactant sodium dodecyl sulfate is involved. As the experiments by Poulin et al. clearly demonstrate, water droplets dispersed in 5CB do assume the dipole con0guration. From the phase diagram we conclude S =a ¡ 0:09 as a necessary condition for the existence of the dipole. With a ≈ 1 m; K3 = 0:53 × 10−6 dyn, and de0nition (2.11) for S we arrive at W ¿ 0:06 erg=cm2 :
(4.11)
If we assume the validity of the Cauchy relation (2.5), which for 5CB gives K24 =K3 = 0:61, we conclude that W ¿ 0:15 erg=cm2 . Recently, Mondain-Monval et al. were able to observe an equatorial ring structure by changing the composition of a surfactant mixture containing sodium dodecyl sulfate (SDS) and a copolymer of ethylene and propylene oxide (Pluronic F 68) [153]. We conclude from our numerical investigation that they observed the surface-ring con0guration. 4.4. Conclusions In this section we presented a detailed study of the three director 0eld con0gurations around a spherical particle by both analytical and numerical means. We clearly 0nd that for large particles and suNciently strong surface anchoring, the dipole is the preferred con0guration. For conventional calamitic liquid crystals, where K2 ¡ K1 , the dipole should always exhibit a twist around the hyperbolic hedgehog. It should not occur in discotic liquid crystals where K2 ¿ K1 . According to our calculations, the bend constant K3 plays only a minor role in the twist transition. The Saturn ring appears for suNciently small particles provided that one can realize a suNciently strong surface anchoring. According to our investigation, for 200 nm particles the surface constant has to be larger than W = 0:3 erg=cm2 . However, the dipole can be transformed into the Saturn ring by means of a magnetic 0eld if the Saturn ring is metastable at H = 0. Otherwise a hysteresis is visible. For the liquid crystal compound 5CB, we 0nd the Saturn ring
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to be metastable at a particle size a = 1 m. Increasing the radius a, this metastability will vanish in analogy with our calculations within the one-constant approximation (see Fig. 19). Lowering the surface-anchoring strength W , the surface-ring con0guration with a quadrupolar symmetry becomes absolutely stable. We never 0nd a stable structure with dipolar symmetry where the surface ring possesses a general angular position 5d or is even shrunk to a point at 5d = 0; . The surface ring is clearly favored by a large saddle-splay constant K24 . The dispersion of spherical particles in a nematic liquid crystal is always a challenge to experimentalists. The clearest results are achieved in inverted nematic emulsions [182,179,183,153,184]. However, alternative experiments with silica or latex spheres do also exist [181,188,189,153, 180,98] and produce impressive results [97]. We hope that the summary of our research stimulates further experiments which probe di9erent liquid crystals as a host >uid [180], manipulate the anchoring strength [153,97,98], and investigate the e9ect of external 0elds [97,98]. 5. Two-particle interactions To understand the properties of, e.g., multi-droplet emulsions, we need to determine the nature of particle–particle interactions. These interactions are mediated by the nematic liquid crystal in which they are embedded and are in general quite complicated. Since interactions are determined by distortions of the director 0eld, there are multi-body as well as two-body interactions. We will content ourselves with calculations of some properties of the e9ective two-particle interaction. To determine the position-dependent interaction potential between two particles, we should solve the Euler–Lagrange equations, as a function of particle separation, subject to the boundary condition that the director be normal to each spherical object. Solving completely these non-linear equations in the presence of two particles is even more complicated than solving them with one particle, and again we must resort to approximations. Fortunately, interactions at large separations are determined entirely by the far-0eld distortions and the multipole moments of an individual topological dipole or Saturn ring, which we studied in Section 4.1. The interactions can be derived from a phenomenological free energy. We will present such an approach in this section [190,182,140]. 5.1. Formulating a phenomenological theory In Section 4, we established that each spherical particle creates a hyperbolic hedgehog to which it binds tightly to create a stable topological dipole. The original spherical inclusion is described by three translational degrees of freedom. Out of the nematic it draws a hedgehog, which itself has three translational degrees of freedom. The two combine to produce a dipole with six degrees of freedom, which can be parametrized by three variables specifying the position of the particle, two angles specifying the orientation of the dipole, and one variable specifying the magnitude of the dipole. As we have seen, the magnitude of the dipole does not >uctuate much and can be regarded as a constant. The direction of the dipole is also fairly strongly constrained. It can, however, deviate from the direction of locally preferred orientation (parallel to a local director to be de0ned in more detail below) when many particles are present. The particle–defect pair is in addition characterized by its higher multipole
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moments. The direction of the principal axes of these moments is speci0ed by the direction of the dipole as long as director con0gurations around the dipole remain uniaxial. The magnitudes of all the uniaxial moments like the strengths p and c of the dipole and quadrupole moment (see Section 4.1) are energetically 0xed, as we have shown in Section 4.2. When director con0gurations are not uniaxial, the multipole tensors will develop additional components, which we will not consider here. We can thus parametrize topological dipoles by their position and orientation and a set of multipole moments, which we regard as 0xed. Let e! be the unit vector specifying the direction of the dipole moment associated with droplet !. Its dipole and quadrupole moments are then p! = pe! and c ! = c(e! ⊗ e! − 1=3), where p and c are the respective magnitudes of the dipole and quadrupole moments calculated, e.g., by analytical means in Section 4.2. The symbol ⊗ means tensor product, and 1 is the second-rank unit tensor. Note, that this approach also applies to the Saturn-ring and surface-ring con0guration but with a vanishing dipole moment p = 0. It even applies to particles with tangential boundary conditions where two surfaces defects, called boojums [145,26,120], are located at opposite points of the sphere and where the director 0eld possesses a uniaxial symmetry, too. We now introduce dipole- and quadrupole-moment densities, P (r) and C (r), in the usual way. Let r! denote the position of droplet !, then P (r ) = p! (r − r ! ) and C (r ) = c ! (r − r ! ) : (5.1) !
!
In the following, we construct an e9ective free energy for director and particles valid at length scales large compared to the particle radius. At these length scales, we can regard the spheres as point objects (as implied by the de0nitions of the densities given above). At each point in space, there is a local director n(r) along which the topological dipoles or, e.g., the Saturn rings wish to align. In the more microscopic picture, of course, the direction of this local director corresponds to the far-0eld director n0 . The e9ective free energy is constructed from rotationally invariant combinations of P , C , n, and the gradient operator ∇ that are also even under n → −n. It can be expressed as a sum of terms F = Fel + Fp + FC + Falign ;
(5.2)
where Fel is the Frank free energy, Fp describes interactions between P and n, FC describes interactions between C and n involving gradient operators, and Falign = −D d 3 r Cij (r)ni (r)nj (r) = −DQ {[e! ·n(r ! )]2 − 1=3} (5.3) !
e!
forces the alignment of the axes along the local director n(r! ). The leading contribution to Fp is identical to the treatment of the >exoelectric e9ect in a nematic [147,51] Fp = 4K d 3 r[ − P ·n(∇·n) + :P · (n × ∇ × n)] ; (5.4) where : is a material-dependent unitless parameter. The leading contribution to FC is FC = 4K d 3 r[(∇·n)n ·∇(ni Cij nj ) + ∇(ni Cij nj ) · (n × ∇ × n)] :
(5.5)
There should also be terms in FC like Cij ∇k ni ∇k nj . These terms can be shown to add contributions to the e9ective two-particle interaction that are higher order in separation than those arising
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from Eq. (5.5). One coeNcient in Fp and all coeNcients in FC are 0xed by the requirement that the phenomenological theory yields the far 0eld of one particle given by Eq. (4.2) (see next subsection). Eq. (5.5) is identical to that introduced in Ref. [190] to discuss interactions between Saturn rings, provided ni Cij nj is replaced by a scalar density 6(r) = ! (r − r! ). The two energies are absolutely equivalent to leading order in the components n3 of n perpendicular to n0 provided all e! are restricted to be parallel to n0 . Since P prefers to align along the local director n, the dipole-bend coupling term in Eq. (5.4) can be neglected to leading order in deviations of the director from uniformity. The −P ·n(∇·n) term in Eq. (5.4) shows that dipoles aligned along n create local splay as is evident from the dipole con0guration depicted in Fig. 12. In addition, this term says that dipoles can lower their energy by migrating to regions of maximum splay while remaining aligned with the local director. Experiments on multiple nematic emulsions [182,183] support this conclusion. Indeed, the coupling of the dipole moment to a strong splay distortion explains the chaining of water droplets in a large nematic drop whose observation we reported in Section 3.2. We return to this observation in Section 7. 5.2. E<ective pair interactions In the following we assume that the far-0eld director n0 and all the multipole moments of the particles point along the z axis, i.e., e! = ez = n0 . Hence, we are able to write the dipole and quadrupole densities as P (r ) = P(r )n0
and
C (r ) = 32 C(r )(n0 ⊗ n0 − 1=3) ;
(5.6)
where P(r) and C(r) can be both positive and negative. We are interested in small deviations from n0 ; n = (nx ; ny ; 1), and formulate the e9ective energy of Eq. (5.2) up to harmonic order in n3 : (5.7) F = K d 3 r[ 12 (∇n3 )2 − 4P 93 n3 + 4(9z C)93 n3 ] : The dipole-bend coupling term of Eq. (5.4) does not contribute because P is aligned along the far-0eld director. The Euler–Lagrange equations for the director components are ∇2 n3 = 493 [P(r ) − 9z C(r )] ;
which possess the solution 1 9 [P(r ) − 9z C(r )] : n 3 (r ) = − d 3 r |r − r | 3
(5.8)
(5.9)
For a single droplet at the origin, P(r) = p(r) and C(r) = 23 c(r), and the above equation yields exactly the far 0eld of Eq. (4.2). This demonstrates the validity of our phenomenological approach. Particles create far-0eld distortions of the director, which to leading order at large distances are determined by Eq. (5.8). These distortions interact with the director 0elds of other particles which leads to an e9ective particle–particle interaction that can be expressed to leading order
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Fig. 26. A chain of three topological dipoles formed due to their dipolar interaction.
as pairwise interactions between dipole and quadrupole densities. Using Eq. (5.9) in Eq. (5.7), we obtain F 1 d 3 r d 3 r [P(r)VPP (r − r )P(r ) + C(r)VCC (r − r )C(r ) = 4K 2 + VPC (r − r )][C(r)P(r ) − P(r)C(r )] ; (5.10) with 1 1 VPP (r) = 93 93 = 3 (1 − 3 cos3 5) r r 1 1 VCC (r) = −92z 93 93 = 5 (9 − 90 cos2 5 + 105 cos4 5) r r 1 cos 5 VPC (r) = 9z 93 93 = 4 (15 cos2 5 − 9) ; r r
(5.11)
where 5 is the angle enclosed by the separation vector r and n0 . The interaction energy between droplets at positions r and r with respective dipole and quadrupole moments p; p ; c; and c is thus 4 2 U (R) = 4K pp VPP (R) + cc VCC (R) + (cp − cp)VPC (R) ; (5.12) 9 3 where R = r − r . The leading term in the potential U (R) is the dipole–dipole interaction which is identical to the analogous problem in electrostatics. Minimizing it over the angle 5, one 0nds that the dipoles prefer to form chains along their axes, i.e., pp ¿ 0; 5 = 0; . Such a chain of dipoles is illustrated in Fig. 26. It is similar to con0gurations seen in other dipolar systems such as magnetorheological >uids and in magnetic emulsions under the in>uence of an external 0eld [90,133]. The chaining was observed by Poulin et al. in inverted emulsions [179,183] or in a suspension of micron-size latex particles in a lyotropic discotic nematic [180]. Both systems were placed in a thin rectangular cell of approximate dimensions 20 m × 1 cm × 1 cm. The upper and lower plates were treated to produce tangential boundary conditions. Thus the total topological charge in the cell was zero. The dipolar forces were measured recently by a method introduced by Poulin et al. [179]. When small droplets are 0lled with a magnetorheological >uid instead of pure water, a small magnetic 0eld of about 100 G, applied perpendicular to the
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chain axis, induces parallel magnetic dipoles. Since they repel each other, the droplets in the chain are forced apart. When the magnetic 0eld is switched o9, the droplets move towards each other to reach the equilibrium distance. In a chain of two moving droplets, the dipolar force on one droplet has to be balanced by the Stokes drag, pp 24K 4 = 6)e9 av ; (5.13) R where v is the velocity of one particle, and )e9 is an e9ective viscosity, which we will address in Section 6. Inertial e9ects can be neglected since the movement is overdamped. By measuring the velocity as a function of R, Poulin et al. could show that the origin of the attractive force is indeed of dipolar nature down to a separation of approximately 4a. Furthermore, they found that the prefactor of the dipolar force scales as a4 , as expected since both the dipole moments p and p scale as a2 (see Section 4.1). In Section 6 we will calculate the Stokes drag of a spherical particle. If p; p = 0, the quadrupolar interaction is dominant. A minimization over 5 predicts that ◦ the quadrupoles should chain under an angle of 5 = 49 [190]. In experiments with tangential boundary conditions at the droplet surface, where a quadrupolar structure with two opposite ◦ surface defects (boojums) forms, the chaining occurred under an angle of 5 = 30 , probably due to short-range e9ects [183]. A similar observation was made in a suspension of 50 nm latex particles in a lyotropic discotic nematic [153], where one expects a surface-ring con0guration because of the homeotropic surface anchoring (see Section 4.3.4). Finally, we discuss the coupling between dipoles and quadrupoles in Eq. (5.12). Their moments scale, respectively, as a2 or a3 . The coupling is only present when the particles have di9erent radii. Furthermore, for 0xed angle 5, the sign of the interaction depends on whether the small particle is on the right or left side of the large one. With this rather subtle e9ect, which is not yet measured, we close the section about two-particle interactions. 6. The Stokes drag of spherical particles In Section 2.3 we introduced the Ericksen–Leslie equations that govern the hydrodynamics of a nematic liquid crystal. Due to the director as a second hydrodynamic variable besides the >uid velocity, interesting new dynamical phenomena arise. With the MiZesowicz viscosities and Helfrich’s permeation, we presented two of them in Section 2.3. Here we deal with the >ow of a nematic around a spherical particle in order to calculate the Stokes drag, which is a well-known quantity for an isotropic liquid [217,202]. 2 Via the celebrated Stokes–Einstein relation [63– 65], it determines the di9usion constant of a Brownian particle, and it is, therefore, crucial for a 0rst understanding of the dynamics of colloidal suspensions [202]. In Section 6.1 the existing work on the Stokes drag, which has a long-standing tradition in liquid crystals, is reviewed. Starting from the Ericksen–Leslie equations, we introduce the theoretical concepts for its derivation in Section 6.2. We calculate the Stokes drag for three director con0gurations; a uniform director 0eld, the topological dipole, and the Saturn-ring 2
We cite here on purpose the excellent course of Sommerfeld on continuum mechanics. An English edition of his lectures on theoretical physics is available.
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structure. Since a full analytical treatment is not possible, we have performed a numerical investigation. A summary of its details is presented in Section 6.3. Finally, we discuss the results and open problems in Section 6.4. 6.1. Motivation Due to the complexity of the Ericksen–Leslie equations, only few examples with an analytical solution exist, e.g., the >ow between two parallel plates, which de0nes the di9erent MiZesowicz viscosities [47], the Couette >ow [8,46], the Poiseuille >ow [7], which was 0rst measured by Cladis et al. [234], or the back >ow [176]. Besides the exploration of new e9ects, resulting from the coupling between the velocity and director 0eld, solutions to the Ericksen–Leslie equations are also of technological interest. They are necessary to determine the switching times of liquid crystal displays. A common way to measure viscosities of liquids is the falling-ball method, where the velocity of the falling particle is determined by a balance of the gravitational, the buoyancy, and Stokes’s friction force. Early experiments in nematic liquid crystals measured the temperature and pressure dependence of the e9ective viscosity )e9 in the Stokes drag [234,122]. Cladis et al. [234] argued that )e9 is close to the MiZesowicz shear viscosity )b , i.e., to the case where the >uid is >owing parallel to the director (see Fig. 3 in Section 2.3). Nearly twenty years later, Poulin et al. used the Stokes drag to verify the dipolar force between two topological dipoles in inverted nematic emulsions [179]. BUottger et al. [19] observed the Brownian motion of particles above the nematic-isotropic phase transition. Measuring the di9usion constant with the help of dynamic light scattering, they could show that close to the phase transition the e9ective viscosity in the Stokes drag increases due to surface-induced nematic order close to the particle. It is obvious that the hydrodynamic solution for the >ow of a nematic liquid crystal around a particle at rest, which is equivalent to the problem of a moving particle, presents a challenge to theorists. Diogo [57] assumed the velocity 0eld to be the same as the one for an isotropic >uid and calculated the drag force for simple director con0gurations. He was interested in the case where the viscous forces largely exceed the elastic forces of director distortions, i.e., Ericksen numbers much larger than one, as we shall explain in the next subsection. Roman and Terentjev, concentrating on the opposite case, obtained an analytical solution for the >ow velocity in a spatially uniform director 0eld, by an expansion in the anisotropy of the viscosity [194]. Heuer et al. presented analytical and numerical solutions for both the velocity 0eld and the Stokes drag again assuming a uniform director 0eld [112,105]. They were 0rst investigating a cylinder of in0nite length [104]. Ruhwandl and Terentjev allowed for a non-uniform but 0xed director con0guration, and they numerically calculated the velocity 0eld and the Stokes drag of a cylinder [198] or a spherical particle [199]. The particle was surrounded by the Saturn-ring con0guration (see Fig. 12 of Section 4.1), and the cylinder was accompanied by two disclination lines. Billeter and Pelcovits used molecular dynamics simulations to determine the Stokes drag of very small particles [10]. They observed that the Saturn ring is strongly deformed due to the motion of the particles. The experiments on inverted nematic emulsions [182,179] motivated us to perform analogous calculations for the topological dipole [228], which we present in the next subsections. Recently, Chono and Tsuji performed a numerical solution of the Ericksen–Leslie equations around a cylinder determining both the velocity and director 0eld [32]. They could show that
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the director 0eld strongly depends on the Ericksen number. However, for homeotropic anchoring their director 0elds do not show any topological defects required by the boundary conditions. The Stokes drag of a particle surrounded by a disclination ring strongly depends on the presence of line defects. There exist a few studies, which determine both experimentally [37] and theoretically [107,49,203] the drag force of a moving disclination. In the multi-domain cell, a novel liquid crystal display, the occurrence of twist disclinations is forced by boundary conditions [206,205,192]. It is expected that the motion of these line defects strongly determines the switching time of the display. 6.2. Theoretical concepts We 0rst review the Stokes drag in an isotropic liquid and then introduce our approach for the nematic environment. 6.2.1. The Stokes drag in an isotropic >uid The Stokes drag in an isotropic >uid follows from a solution of the Navier–Stokes equations. Instead of considering a moving sphere, one solves the equivalent problem of the >ow around a sphere at rest [217]. An incompressible >uid (div C =0) and a stationary velocity 0eld (9C= 9t =0) are assumed, so that the 0nal set of equations reads div C = 0 and − ∇p + div T = 0 : (6.1) In an isotropic >uid the viscous stress tensor T is proportional to the symmetrized velocity gradient A, T = 2)A, where ) denotes the usual shear viscosity. We have subdivided the pressure p = p0 + p in a static (p0 ) and a hydrodynamic (p ) part. The static pressure only depends on the constant mass density % and, therefore, does not appear in the momentum-balance equation of the set (6.1). The hydrodynamic contribution p is a function of the velocity. It can be chosen zero at in0nity. Furthermore, under the assumption of creeping >ow, we have neglected the non-linear velocity term in the momentum-balance equation resulting from the convective part of the total time derivative d C=dt. That means, the ratio of inertial (%v2 =a) and viscous ()v=a2 ) forces, which de0nes the Reynolds number Re = %va=), is much smaller than one. To estimate the forces, all gradients are assumed to be of the order of the inverse particle radius a−1 , the characteristic length scale of our problem. Eqs. (6.1) are solved analytically for the non-slip condition at the surface of the particle [C(r = a) = 0], and for a uniform velocity C∞ at in0nity. Once the velocity and pressure 0elds are known, the drag force FS follows from an integration of the total stress tensor −p1 + T over the particle surface. An alternative method demands that the dissipated energy per unit time, (T ·A) d 3 r, which we introduced in Eq. (2.27) of Section 2.3, should be FS v∞ [11]. The 0nal result is the famous Stokes formula for the drag force: FS = %v∞ with % = 6)a : (6.2) The symbol % is called the friction coeNcient. The Einstein–Stokes relation relates it to the di9usion constant D of a Brownian particle [63– 65]: kB T D= ; (6.3) 6)a where kB is the Boltzmann constant and T is temperature.
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We can also calculate the Stokes drag for a 0nite spherical region of radius r = a=' with the particle at its center [228]. The result is FS = %' v'
with %' = 6)a
1 − 3'=2 + '3 − '5 =2 ; (1 − 3'=2 + '3 =2)2
(6.4)
where v' denotes the uniform velocity at r = a='. The correction term is a monotonically increasing function in ' on the interesting interval [0; 1]. Hence, the Stokes drag increases when the particle is con0ned to a 0nite volume. For ' = 1=32 the correction is about 5%. 6.2.2. The Stokes drag in a nematic environment To calculate the Stokes drag in a nematic environment, we have to deal with the Ericksen– Leslie equations, which couple the >ow of the >uid to the director motion. We do not attempt to solve these equations in general. Analogous to the Reynolds number, we de0ne the Ericksen number [49] as the ratio of viscous ()v∞ =a2 ) and elastic (K=a3 ) forces in the momentum balance of Eq. (2.18): )v∞ a Er = : (6.5) K The elastic forces are due to distortions in the director 0eld, where K stands for an average Frank constant. In the following, we assume Er 1, i.e., the viscous forces are too weak to distort the director 0eld, and we will always use the static director 0eld for C = 0 in our calculations. The condition Er 1 constrains the velocity v∞ . Using typical values of our parameters, i.e., K = 10−6 dyn; ) = 0:1 P, and a = 10 m, we 0nd m v∞ 100 : (6.6) s Before we proceed, let us check for three cases if this constraint is ful0lled. First, in the measurements of the dipolar force by Poulin et al., the velocities of the topological dipole are always smaller than 10 m=s [179]. Secondly, in a falling-ball experiment the velocity v of the falling particle is determined by a balance of the gravitational, the buoyancy, and Stokes’s friction force, i.e., 6)e9 av = (4=3)a3 (% − %> )g, and we obtain v=
m 2 (% − %> )a2 g → 10 : 9 )e9 s
(6.7)
To arrive at the estimate, we choose )e9 =0:1 P and a=10 m. We take %=1 g=cm3 as the mass density of the particle and % − %> = 0:01 g=cm3 as its di9erence to the surrounding >uid [202]. Thirdly, we consider the Brownian motion of a suspended particle. With the time t = a2 =6D that the particle needs to di9use a distance equal to the particle radius a [202], we de0ne an averaged velocity v=
a 6D m → 10−3 = : t a s
(6.8)
The estimate was calculated using the Stokes–Einstein relation of Eq. (6.3) with thermal energy kB T = 4 × 10−14 erg at room temperature and the same viscosity and particle radius as above.
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After we have shown that Er 1 is a reasonable assumption, we proceed as follows. We 0rst calculate the static director 0eld around a sphere from the balance of the elastic torques, n × h0 = 0 [see Eqs. (2.19) and (2.25)]. It corresponds to a minimization of the free energy. For C = 0, the static director 0eld de0nes a static pressure p0 via the momentum balance, −∇p0 + div T 0 = 0, where the elastic stress tensor T 0 depends on the gradient of n [see Eqs. (2.18) and (2.11)]. If we again divide the total pressure into its static and hydrodynamic part, p=p0 +p , the velocity 0eld is determined from the same set of equations as in (6.1), provided that we employ the viscous stress tensor T of a nematic liquid crystal [see Eq. (2.22)]. In the case of an inhomogeneous director 0eld, both the di9erent shear viscosities and the rotational viscosity %1 , discussed in Section 2.3, contribute to the Stokes drag. In general, the friction force FS does not point along C∞ , and the friction coeNcient is now a tensor . In the following, all our con0gurations are rotationally symmetric about the z axis, and the Stokes drag assumes the form FS = C∞
with = %⊥ 1 + (% − %⊥ )ez ⊗ ez :
(6.9)
There only exist two independent components % and %⊥ for a respective >ow parallel or perpendicular to the symmetry axis. In these two cases, the Stokes drag is parallel to C∞ . Otherwise, a component perpendicular to C∞ , called lift force, appears. In analogy with the isotropic >uid, we introduce e9ective viscosities )e9 and )⊥ e9 via
% = 6)e9 a
and
%⊥ = 6)⊥ e9 a :
(6.10)
It is suNcient to determine the velocity and pressure 0elds for two particular geometries with C∞ either parallel or perpendicular to the z axis. Then, the friction coeNcients are calculated with the help of the dissipated energy per unit time [see Eq. (2.27)] [11,57]: =⊥ FS v∞ = (T ·A + h ·N ) d 3 r : (6.11) It turns out that the alternative method via an integration of the stress tensor at the surface of the particle is numerically less reliable. Note that the velocity and pressure 0elds for an arbitrary angle between C∞ and ez follow from superpositions of the solutions for the two selected geometries. This is due to the linearity of our equations. It is clear that the Brownian motion in an environment with an overall rotational symmetry is governed again by two independent di9usion constants. The generalized Stokes–Einstein formula of the di9usion tensor D takes the form kB T D = D⊥ 1 + (D − D⊥ )ez ⊗ ez with D=⊥ = : (6.12) %=⊥ At the end, we add some critical remarks about our approach which employs the static director 0eld. From the balance equation of the elastic and viscous torques [see Eqs. (2.19), (2.25) and (2.26)], we derive that the change n of the director due to the velocity C is of the order of the Ericksen number: n ∼ Er. This adds a correction T 0 to the elastic stress tensor T 0 in the momentum balance equation. In the case of a spatially uniform director 0eld, the correction T 0 is by a factor Er smaller than the viscous forces, and it can be neglected. However, for a non-uniform director 0eld, it is of the same order as the viscous term, and, strictly speaking,
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should be taken into account. Since our problem is already very complex, even when the directors are 0xed, we keep this approximation for a 0rst approach to the Stokes drag. How the friction force changes when the director 0eld is allowed to relax, must be investigated by even more elaborate calculations. Two remarks support the validity of our approach. First, far away from the sphere, n has to decay at least linearly in 1=r, and T0 is negligible against the viscous forces. Secondly, the non-linear term in the Navier–Stokes equations usually is omitted for Re1. However, whereas the friction and the pressure force for the Stokes problem decay as 1=r 3 , the non-linear term is proportional to 1=r 2 , exceeding the 0rst two terms in the far0eld. Nevertheless, performing extensive calculations, Oseen could prove that the correction of the non-linear term to the Stokes drag is of the order of Re [217]. One might speculate that the full relaxation of the director 0eld introduces a correction of the order of Er to the Stokes drag. 6.3. Summary of numerical details In this subsection we only review the main ideas of our numerical method. A detailed account will be given in Ref. [228]. The numerical investigation is performed on a grid which is de0ned by modi0ed spherical coordinates. Since the region outside the spherical particle is in0nitely extended, we employ a reduced radial coordinate = a=r. The velocity and director 0elds are expressed in the local spherical coordinate basis. With this choice of coordinates, the momentum balance of Eqs. (6.1) with the viscous stress tensor of a nematic becomes very complex. We, therefore, used the algebraic program Maple to formulate it. The two equations in (6.1) are treated by di9erent numerical techniques. Given an initial velocity 0eld, the momentum balance including the inertial term 9C= 9t can be viewed as a relaxation equation towards the stationary velocity 0eld, which we aim to determine. The Newton– Gauss–Seidel method, introduced in Section 2.2, provides an e9ective tool to implement this relaxation. Employing the discretized version of the momentum balance equation, the velocity at the grid point r relaxes according to vinew (r) = viold (r) −
[ − ∇p + div T ]i : [9(−∇p + div T )]i = 9vi (r)
(6.13)
Note that the denominator can be viewed as the inverse of a variable time step for the 0ctitious temporal dynamics of C. A relaxation equation for the pressure involving div C = 0 is motivated by the method of arti0cial compressibility [33]. Let us consider the complete mass-balance equation. For small variations of the density, we obtain 9p % 9p = − 2 div C with c = : (6.14) 9t c 9% The quantity c denotes the sound velocity for constant temperature, and c2 =% is the isothermal compressibility. In discretized form we have % pnew = pold − 2 Vt div C : (6.15) c
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Note that the reduced 0ctitious time step % Vt=c2 cannot be chosen according to the Newton– Gauss–Seidel method since div C does not contain the pressure p. Instead, it should be as large as possible to speed up the calculations. In Ref. [187] upper bounds are given beyond which the numerical scheme becomes unstable. To obtain the friction coeNcient % , an e9ective two-dimensional problem has to be solved due to the rotational symmetry of the director con0gurations about the z axis. In the case of %⊥ (C∞ ⊥ez ), the velocity 0eld possesses at least two mirror planes which are perpendicular to each other and whose line of intersection is the z axis. As a result, the necessary three-dimensional calculations can be reduced to one quadrant of the real space. A description of all the boundary conditions will be presented in Ref. [228]. The director 0elds for the topological dipole and the Saturn ring are provided by the respective ansatz functions of Eqs. (22) and (33) in Ref. [140]. The parameters of minimum free energy are chosen. In Section 4 we showed that these ansatz functions give basically the same results as the numerical investigation. We checked our programs in the isotropic case. It turned out that the three-dimensional version is not completely stable for an in0nitely extended integration area. We therefore solved Eqs. (6.1) in a 0nite region of reduced radius r=a = 1=' = 32. For ' = 1=32, our programs reproduced the isotropic Stokes drag, calculated from Eq. (6.4), with an error of 1%. 6.4. Results, discussion, and open problems We begin with an investigation of the stream line patterns, discuss the e9ective viscosities, and formulate some open problems at the end. 6.4.1. Stream line patterns In Fig. 27 we compare the stream line patterns around a spherical particle for an isotropic liquid and a spatially uniform director 0eld parallel to C∞ . A uniform n can be achieved by weak surface anchoring and application of a magnetic 0eld with a magnetic coherence length smaller than the particle radius. In the isotropic >uid the bent stream lines occupy more space around the particle, whereas for a uniform director con0guration they seem to follow the vertical director 0eld lines as much as possible. This can be understood from a minimum principle. In Section 2.3 we explained that a shear >ow along the director possesses the smallest shear viscosity, called )b . Hence, in such a geometry the smallest amount of energy is dissipated. Indeed, for a uniform director 0eld, one can derive the momentum balance from a minimization of the dissipation function stated in Eq. (2.27) [104]. A term −2p div C has to be added because of the incompressibility of the >uid. It turns out that the Lagrange multiplier −2p is determined by the pressure p. In the case of the topological dipole parallel to C∞ , we observe a clear asymmetry in the stream lines as illustrated in Fig. 28. The dot indicates the position of the point defect. It breaks the mirror symmetry of the stream line pattern, which exists, e.g., in an isotropic liquid relative to a plane perpendicular to the vertical axis. In the far0eld of the velocity, the splay deformation in the dipolar director con0guration is clearly recognizable. Since we use the linearized momentum balance in C, the velocity 0eld is the same no matter if the >uid >ows upward or downward. The stream line pattern of the Saturn ring [see Fig. 29 (right)] exhibits the mirror symmetry,
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Fig. 27. Stream line pattern around a spherical particle for an isotropic liquid (right) and a uniform director 0eld parallel to C∞ (left). Fig. 28. Stream line pattern around a spherical particle for an isotropic liquid (right) and the topological dipole parallel to C∞ (left).
and the position of the ring disclination is visible by a dip in the stream line close to the equator of the sphere. If C∞ is perpendicular to the dipole axis, the missing mirror plane of the dipole con0guration is even more pronounced in the stream line pattern. It is illustrated in Fig. 30, where the point defect is indicated by a dip in the stream line. Although the pattern resembles the one of the Magnus e9ect [217], symmetry dictates that FS⊥ C∞ . A lift force perpendicular to C∞ does not exist. We 0nd a non-zero viscous torque acting on the particle whose direction for a >uid >ow from left to right is indicated in Fig. 30. Symmetry allows such a torque M since the cross product of the dipole moment p and C∞ gives an axial or pseudovector M ˙ p × C∞ . In the Saturn-ring con0guration a non-zero dipole moment and, therefore, a non-zero torque cannot occur. 6.4.2. E<ective viscosities In Table 1 we summarize the e9ective viscosities of the Stokes drag, de0ned in Eq. (6.10), for a uniform director 0eld, the dipole and the Saturn-ring con0guration. The values are calculated for the two compounds MBBA and 5CB. For a reference, we include the three MiZesowicz viscosities. In the case of C∞ parallel to the symmetry axis of the three con0gurations, we might expect that )e9 is close to )b as argued by Cladis et al. [234]. For a uniform director 0eld, )e9 exceeds )b by 30% or 60%, respectively. The increase originates in the stream lines bending around the particle. The e9ective viscosity )e9 of the dipole and the Saturn ring are larger than )b by an approximate factor of two. In addition to the bent stream lines, there exist
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Fig. 29. Stream line pattern around a spherical particle for the Saturn ring (right) and the topological dipole (left) with their respective symmetry axis parallel to C∞ . Fig. 30. Stream line pattern around a spherical particle for the topological dipole perpendicular to C∞ .
Table 1 E9ective viscosities of the Stokes drag for the two compounds MBBA and 5CB and for three di9erent director con0gurations. As a reference, the three MiZesowicz viscosities are included
)e9 (P) )⊥ e9 (P) )⊥ e9 =)e9
MBBA: )a = 0:416 P; )b = 0:283 P; )c = 1:035 P
5CB: )a = 0:374 P; )b = 0:229 P; )c = 1:296 P
Uniform n
Dipole
Saturn ring
Uniform n
Dipole
Saturn ring
0.380 0.684 1.80
0.517 0.767 1.48
0.493 0.747 1.51
0.381 0.754 1.98
0.532 0.869 1.63
0.501 0.848 1.69
strong director distortions close to the particle which the >uid has to >ow through constantly changing the local direction of the moving molecules. Recalling our discussion of the permeation in Section 2.3, a contribution from the rotational viscosity %1 arises which does not exist in a uniform director 0eld. In all three cases, we 0nd )e9 either close to or larger than )a , so that )b is not the only determining quantity of )e9 , as argued by Cladis et al. [234]. For C∞ perpendicular to the symmetry axis, )e9 ⊥ assumes a value between )a and )c , which is understandable since the >ow velocity is mainly perpendicular to the director 0eld. The ratio )⊥ e9 =)e9 for the uniform director 0eld is the largest since the extreme cases of a respective >ow parallel or perpendicular to the director 0eld is realized the best in this con0guration. Furthermore, both the dipole and the Saturn ring exhibit nearly the same anisotropy, and we conclude that they cannot be distinguished from each other in a falling-ball experiment.
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The ratios )⊥ e9 =)e9 that we determine for the Saturn ring and the uniform director 0eld in the case of the compound MBBA agree well with the results of Ruhwandl and Terentjev who 0nd ⊥ )⊥ e9 =)e9 |uniform = 1:69 and )e9 =)e9 |Saturn = 1:5 [199]. However, they di9er from the 0ndings of Billeter and Pelcovits in their molecular dynamics simulations [10]. In the ansatz function of the dipolar con0guration, we vary the separation rd between the hedgehog and the center of the particle. Both the e9ective viscosities increase with rd since the non-uniform director 0eld with its strong distortions occupies more space. However, the ratio )⊥ e9 =)e9 basically remains the same. For the Saturn ring, )e9 increases stronger with the radius rd than does )⊥ e9 . This seems to be reasonable since a >ow perpendicular to the plane of the Saturn ring experiences more resistance than a >ow parallel to the plane. As a result, )⊥ e9 =)e9 decreases when the ring radius rd is enlarged. 6.4.3. Open problems One should try to perform a complete solution of the Ericksen–Leslie equations including a relaxation of the static director 0eld for C = 0. In the case of Er 1, a linearization in the small deviation n from the static director 0eld would suNce. Such a procedure helps to gain insight into several open problems. First, it veri0es or falsi0es the hypothesis that the correction to the Stokes drag is of the order of Er. Secondly, the Stokes drag of the topological dipole is the same whether the >ow is parallel or anti-parallel to the dipole moment. This is also true for an object with a dipolar shape in an isotropic >uid. If such an object is slightly turned away from its orientation parallel to C∞ , it will experience a viscous torque and either relax back or reverse its direction to 0nd its absolute stable orientation. The topological dipole will not turn around since it experiences an elastic torque towards its initial direction, as explained in Section 5.1. Nevertheless, a full solution of the Ericksen–Leslie equations would show whether and how much the dipole deviates from its preferred direction under the in>uence of a velocity 0eld. It would also clarify its orientation when C∞ is perpendicular to the dipolar axis. Furthermore, we speculate that the non-zero viscous torque, discussed in Section 6.4.1, is cancelled by elastic torques. Preliminary results [228] for the two-dimensional problem with the relaxation of the director 0eld included show that the Stokes drag of the dipolar con0guration varies indeed linearly in Er for Er ¡ 1. Furthermore, it is highly non-linear depending on C∞ being either parallel or anti-parallel to the topological dipole. The Stokes drag of particles in a nematic environment still presents a challenging problem to theorists. On the other hand, clear measurements of, e.g., the anisotropy in Stokes’s friction force are missing. 7. Colloidal dispersions in complex geometries In this section we present a numerical investigation of water droplets in a spherically con0ned nematic solvent. It is motivated by experiments on multiple nematic emulsions which we reported in Section 3.2. However, it also applies to solid spherical particles. Our main purpose is to demonstrate that the topological dipole provides a key unit for the understanding of multiple emulsions. In Sections 7.1–7.3 we 0rst state the questions and main results of our investigation.
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Fig. 31. Scenario to explain the chaining of water droplets in a large nematic drop. The right water droplet and its companion hyperbolic hedgehog form a dipole, which is attracted by the strong splay deformation around the droplet in the center (left picture). The dipole moves towards the center until at short distances the repulsion mediated by the point defect sets in (middle picture). A third droplet moves to the region of maximum splay to form a linear chain with the two other droplets.
Then we de0ne the geometry of our problem and summarize numerical details. In particular, we employ the numerical method of 0nite elements [227] which is most suitable for non-trivial geometries. Finally we present our results in detail and discuss them. The last subsection contains an analytical treatment of the twist transition of a radial director 0eld enclosed between two concentric spheres. It usually occurs when the inner sphere is not present. We perform a linear stability analysis and thereby explain the observation that a small water droplet at the center of a large nematic drop suppresses the twisting. 7.1. Questions and main results In our numerical investigation we demonstrate that the dipolar con0guration formed by one spherical particle and its companion hyperbolic point defect also exists in more complex geometries, e.g., nematic drops. This provides an explanation for the chaining reported in Section 3.2 and in Refs. [182,183]. One water droplet 0ts perfectly into the center of a large nematic drop, which has a total topological charge +1. Any additional water droplet has to be accompanied by a hyperbolic hedgehog in order not to change the total charge. If the dipole forms (see Fig. 31, left), it is attracted by the strong splay deformation in the center, as predicted by the phenomenological theory of Section 5.1 and in Refs. [182,140], until the short-range repulsion mediated by the defect sets in (see Fig. 31, middle). Any additional droplet seeks the region of maximum splay and forms a linear chain with the two other droplets. In the following we present a detailed study of the dipole formation in spherical geometries. For example, when the two water droplets in the middle picture of Fig. 31 are moved apart symmetrically about the center of the large drop, the dipole forms via a second-order phase transition. We also identify the dipole in a bipolar con0guration which occurs for planar boundary conditions at the outer surface of the nematic drop. Two boojums, i.e., surface defects appear [145,26,120], and the dipole is attracted by the strong splay deformation in the vicinity of one of them [182,183,140]. Besides the dipole we 0nd another stable con0guration in this geometry, where the hyperbolic hedgehog sits close to one of the boojums, which leads to a hysteresis in the formation of the dipole. In the experiment it was found that the distance d of the point defect from the surface of a water droplet scales with the radius r of the droplet like d ≈ 0:3r [182,183]. In the following we will call this relation the scaling law. By our numerical investigations, we con0rm this scaling
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Fig. 32. (a) Geometry parameters for two water droplets with respective radii r1 and r2 in a large nematic drop with radius r3 . The system is axially symmetric about the z axis, and cylindrical coordinates 6; z are used. The coordinates z1 ; z2 , and zd are the respective positions of the two droplets and the hyperbolic hedgehog. The two distances of the hedgehog from the surfaces of the droplets are d1 and d2 . From Ref. [220]. (b) Triangulation of the integration area (lattice constant: b = 0:495). Between the small spheres a re0ned net of triangles is chosen. From Ref. [220].
law within an accuracy of ca. 15%, and we discuss the in>uence of the outer boundary of the large drop. Finally, we show that water droplets can repel each other without a hyperbolic defect placed between them. 7.2. Geometry and numerical details We numerically investigate two particular geometries of axial symmetry. The 0rst problem is de0ned in Fig. 32a. We consider two spherical water droplets with respective radii r1 and r2 in a large nematic drop with radius r3 . The whole system possesses axial symmetry, so that the water droplets and the hyperbolic hedgehog, indicated by a cross, are located always on the z axis. We employ a cylindrical coordinate system. The coordinates z1 ; z2 , and zd denote, respectively, the positions of the centers of the droplets and of the hyperbolic hedgehog on the z axis. The distances of the hedgehog from the surfaces of the two water droplets are, respectively, d1 and d2 . Then, the quantity d1 + d2 means the distance of the two small spheres, and the point defect is situated in the middle between them if d1 = d2 . We, furthermore, restrict the nematic director to the (6; z) plane, which means that we do not allow for twist deformations. 3 3
In nematic droplets with homeotropic anchoring a twist in the director 0eld is usually observed (see [26] and Section 7.4). In Section 4.3.2 we demonstrated that it even appears in the dipole con0guration close to the hyperbolic hedgehog. However, for the Frank elastic constants of 5CB, the distance of the defect from the surface of the water droplet di9ers only by 10% if the director 0eld is not allowed to twist. We do not expect a di9erent behavior in the geometry under consideration in this section. Here, we want to concentrate, as a 0rst step, on the principal features of the system. Therefore, we neglect twist deformations to simplify the numerics. The same simpli0cation to catch the main behavior of nematic drops in a magnetic 0eld was used by other authors, see, e.g., [115,114].
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The director is expressed in the local coordinate basis of the cylindrical coordinate system, n(6; z)=sin (6; z)e6 +cos (6; z)ez , where we introduced the tilt angle . It is always restricted to the range [−=2; =2] to ensure the n → −n symmetry of the nematic phase. At all the boundaries we assume a rigid homeotropic anchoring of the director, which allows us to omit any surface term in the free energy. In Ref. [140] it was shown that rigid anchoring is justi0ed in our system and that any deformation of the water droplets can be neglected. In the second problem we have only one water droplet insider a large nematic drop. We use the same coordinates and lengths as described in Fig. 32a, but omit the second droplet. The anchoring of the director at the outer surface of the large nematic sphere is rigid planar. At the surface of the small sphere we again choose a homeotropic boundary condition. Because of the non-trivial geometry of our problem, we decided to employ the method of 0nite elements [227], where the integration area is covered with triangles. We construct a net of triangles by covering our integration area with a hexagonal lattice with lattice constant b. Vertices of triangles that only partially belong to the integration area are moved onto the boundary along the radial direction of the appropriate sphere. As a result, extremely obtuse triangles occur close to the boundary. We use a relaxation mechanism to smooth out these irregularities. The 0nal triangulation is shown in Fig. 32b. In the area between the small spheres, where the hyperbolic hedgehog is situated, the grid is further subdivided to account for the strong director deformations close to the point defect. The local re0nement helps us to locate the minimum position of the defect between the spheres within a maximum error of 15% by keeping the computing time to a reasonable value [220]. In the following, we express the Frank free energy, introduced in Section 2.1, in units of K3 a ^ The quantity a is the characteristic length scale of our system, and denote it by the symbol F. typically several microns. The saddle-splay term, a pure surface term, is not taken into account. The Frank free energy is discretized on the triangular net. For details, we refer the reader to Ref. [220]. To 0nd a minimum of the free energy, we start with a con0guration that already possesses the hyperbolic point defect at a 0xed position zd and let it relax via the standard Newton–Gauss–Seidel method [187], which we illustrate in Eq. (2.16) of Section 2.2. Integrating the free energy density over one triangle yields a line energy, i.e., an energy per unit length. As a rough estimate for its upper limit we introduce the line tension Fl =(K1 +K3 )=2 of the isotropic core of a disclination [51]. Whenever the numerically calculated local line energy is larger than Fl , we replace it by Fl . Note that Fl di9ers from Eq. (2.34). However, its main purpose is to stabilize the hyperbolic point defect against opening up to a disclination ring whose radius would be unphysical, i.e., larger than the values discussed in Section 2.4.2. All our calculations are performed for the nematic liquid crystal pentylcyanobiphenyl (5CB), for which the experiments were done [182,183]. Its respective bend and splay elastic constants are K3 = 0:53 × 10−6 dyn and K1 = 0:42 × 10−6 dyn. The experimental ratio r3 =r1=2 of the radii of the large and small drops is in the range 10 –50 [182,183]. The diNculty is that we want to investigate details of the director 0eld close to the small spheres which requires a 0ne triangulation on the length scale given by r1=2 . To keep the computing time to a reasonable value we choose the following lengths: r3 = 7; r1=2 = 0:5–2, and b = 0:195 for the lattice constant of the grid. In addition, we normally use one step of grid re0nement between the small spheres (geometry 1) or between the small sphere and the south pole of the large nematic drop (geometry 2). With such parameters we obtain a lattice with 2200 –2500 vertices.
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Fig. 33. The free energy F^ as a function of the distance d1 + d2 between the small spheres which are placed symmetrically about z = 0 (r1 = r2 = 1). Curve 1: zd = 0, curve 2: position zd of the defect can relax along the z axis. From Ref. [220]. Fig. 34. The free energy F^ − F^min as a function of d1 =r1 = d2 =r2 . The small spheres are placed symmetrically about z = 0. Curve 1: r1 = r2 = 0:5, curve 2: r1 = r2 = 1, and curve 3: r1 = r2 = 2. From Ref. [220].
7.3. Results and discussion of the numerical study In this subsection we discuss the results from our numerical investigation. First, we con0rm the scaling law d1=2 ≈ 0:3r1=2 , which was observed in experiment, by varying the di9erent lengths in our geometry. Secondly, we demonstrate that the topological dipole is also meaningful in complex geometries. Finally, we show that the hyperbolic hedgehog is not necessary to mediate a repulsion between the water droplets. 7.3.1. Scaling law In Fig. 33 we plot the reduced free energy F^ as a function of the distance d1 + d2 between the surfaces of the small spheres, which are placed symmetrically about the center, i.e., z2 = −z1 . Their radii are r1 = r2 = 1. Curve 1 shows a clear minimum at d1 + d2 ≈ 0:7, the defect stays in the middle between the two spheres at zd = 0. In curve 2 we move the defect along the z axis and plot the minimum free energy for each 0xed distance d1 + d2 . It is obvious that beyond d1 + d2 = 2 the defect moves to one of the small spheres. We will investigate this result in more detail in the following subsection. In Fig. 34 we take three di9erent radii for the small spheres, r1 = r2 = 0:5; 1; 2, and plot the free energy versus d1 =r1 close to the minimum. Recall that d1 is the distance of the hedgehog from the surface of sphere 1. Since for such small distances d1 + d2 the defect always stays at zd = 0, i.e., in the middle between the two spheres, we have d1 =r1 = d2 =r2 . The quantity F^min refers to the minimum free energy of each curve. For each of the three radii we obtain an energetically preferred distance d1 =r1 in the range of [0:3; 0:35], which agrees well with the experimental value of 0.3. Why does a scaling law of the form d1=2 = (0:325 ± 0:025)r1=2 occur? When the small spheres are far away from the surface of the large nematic drop, its 0nite radius r3 should hardly in>uence the distances d1 and d2 . Then, the only length scale in the system is r1 = r2 , and we expect d1=2 ˙ r1=2 . However, in Fig. 34 the in>uence from the boundary of the
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large sphere is already visible. Let us take curve 2 for spheres with radii r1=2 = 1 as a reference. It is approximately symmetric about d1 =r1 = 0:35. The slope of the right part of curve 3, which corresponds to larger spheres of radii r1=2 = 2, is steeper than in curve 2. Also, the location of the minimum clearly tends to values smaller than 0.3. We conclude that the small spheres are already so large that they are strongly repelled by the boundary of the nematic drop. On the other hand, the slope of the right part of curve 1, which was calculated for spheres of radii r1=2 = 0:5, if less steep than in curve 2. This leads to the conclusion that the boundary of the nematic drop has only a minor in>uence on such small spheres. When we move the two spheres with radii r1=2 = 1 together in the same direction along the z axis, the defect always stays in the middle between the droplets and obeys the scaling law. We have tested its validity within the range [0; 3] for the defect position zd . Of course, the absolute minimum of the free energy occurs in the symmetric position of the two droplets, z2 = −z1 . We further check the scaling law for r1 = r2 . We investigate two cases. When we choose r1 = 2 and r2 = 0:6, we obtain d1=2 ≈ 0:3r1=2 . In the second case, r1 = 2 and r2 = 1, we 0nd d1 ≈ 0:37r1 and d2 ≈ 0:3r2 . As observed in the experiment, the defect sits always closer to the smaller sphere. There is no strong deviation from the scaling law d1=2 = (0:325 ± 0:025)r1=2 , although we would allow for it, since r1 = r2 . 7.3.2. Identi4cation of the dipole In this subsection we demonstrate that the topological dipole is meaningful in our geometry. We place sphere 2 with radius r2 = 1 in the center of the nematic drop at z2 = 0. Then, we determine the energetically preferred position of the point defect for di9erent locations z1 of sphere 1 (r1 = 1). The position of the hedgehog is indicated by > = (d2 − d1 )=(d1 + d2 ). If the defect is located in the middle between the two spheres, > is zero since d1 = d2 . On the other hand, if it sits at the surface of sphere 1, d1 = 0, and > becomes one. In Fig. 35 we plot the free energy F^ versus >. In curve 1, where the small spheres are farthest apart from each other (z1 = 5), we clearly 0nd the defect close to sphere 1. This veri0es that the dipole is existing. It is stable against >uctuations since a rough estimate of the thermally induced mean displacement of the defect yields 0.01. The estimate is performed in full analogy to Eq. (4.6) of Section 4.2. When sphere 1 is approaching the center (curve 2: z1 = 4 and curve 3: z1 = 3:5), the defect moves away from the droplet until it nearly reaches the middle between both spheres (curve 4: z1 = 3). This means, the dipole vanishes gradually until the hyperbolic hedgehog is shared by both water droplets. An interesting situation occurs when sphere 1 and 2 are placed symmetrically about z = 0. Then, the defect has two equivalent positions on the positive and negative part of the z axis. In Fig. 36 we plot again the free energy F^ versus the position > of the defect. From curve 1 to 3 (z1 = z2 = 4; 3; 2:5) the minimum in F^ becomes broader and more shallow. The defect moves closer towards the center until at z1 = −z2 ≈ 2:3 (curve 4) it reaches >=0. This is reminiscent to a symmetry-breaking second-order phase transition [27,124] which occurs when, in the course of moving the water droplets apart, the dipole starts to form. We take > as an order parameter, where >=0 and > = 0 describe, respectively, the high- and the low-symmetry phase. A Landau expansion of the free energy yields ^ F(>) = F^0 (z1 ) + a0 [2:3 − z1 ]>2 + c(z1 )>4 ;
(7.1)
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Fig. 35. The free energy F^ as a function of > = (d2 − d1 )=(d1 + d2 ). Sphere 2 is placed at z2 = 0. The position z1 of sphere 1 is the parameter. Curve 1: z1 = 5, curve 2: z1 = 4, curve 3: z1 = 3:5, and curve 4: z1 = 3. The radii are r1 = r2 = 1. From Ref. [220]. Fig. 36. The free energy F^ as a function of > = (d2 − d1 )=(d1 + d2 ). The small spheres are placed symmetrically about z = 0. Curve 1: z1 = −z2 = 4, curve 2: z1 = −z2 = 3, curve 3: z1 = −z2 = 2:5, curve 4: z1 = −z2 = 2:3, curve 5: z1 = −z2 = 2. The radii are r1 = r2 = 1. From Ref. [220].
where z1 = −z2 plays the role of the temperature. Odd powers in > are not allowed because ^ ^ −>). This free energy qualitatively describes the curves of the required symmetry, F(>) = F( in Fig. 36. It should be possible to observe such a “second-order phase transition” 4 with a method introduced recently by Poulin et al. [179] to measure dipolar forces in inverted nematic emulsion. We already explained the method in Section 5.2 after Eq. (5.12). Two small droplets 0lled with a magnetorheological >uid are forced apart when a small magnetic 0eld of about 100 G is applied perpendicular to the z axis. When the magnetic 0eld is switched o9, the two droplets move towards each other to reach the equilibrium distance. In the course of this process the phase transition for the dipole should be observable. 7.3.3. The dipole in a bipolar con4guration It is possible to change the anchoring of the director at the outer surface of the large nematic drop from homeotropic to planar by adding some amount of glycerol to the surrounding water phase [182]. Then the bipolar con0guration for the director 0eld appears [26,120], where two boojums [145], i.e., surface defects of charge 1 are situated at the north and south pole of the large nematic drop (see con0guration (1) in Fig. 37). The topological point charge of the interior of the nematic drop is zero, and every small water droplet with homeotropic boundary condition has to be accompanied by a hyperbolic hedgehog. In the experiment the hedgehog sits close to the water droplet, i.e., the dipole exists and it is attracted by the strong splay deformation 4
There is strictly speaking no true phase transition since our investigated system has 0nite size. However, we do not expect a qualitative change in Fig. 36, when the nematic drop is much larger than the enclosed water droplets (r3 r1 ; r2 ), i.e., when the system reaches the limit of in0nite size.
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Fig. 37. Planar boundary conditions at the outer surface of the large sphere create boojums, i.e., surface defects at the north and the south pole. A water droplet with homeotropic boundary conditions nucleates a hyperbolic hedgehog. Two con0gurations exist that are either stable or metastable depending on the position of the water droplet; (1) the dipole, (2) the hyperbolic hedgehog sitting at the surface. From Ref. [220]. Fig. 38. The free energy F^ as a function of the position z1 of the water droplet for the con0gurations (1) and (2). For z1 ¿ − 3:5, (1) is stable, and (2) is metastable. The situation is reversed for −4:3 ¡ z1 ¡ − 3:5. Con0guration (1) loses its metastability at z1 = −4:3. From Ref. [220].
close to the south pole [182], as predicted by the phenomenological theory of Section 5 and Refs. [182,140]. A numerical analysis of the free energy F^ is in agreement with experimental observations but also reveals some interesting details which have to be con0rmed. In Fig. 38 we plot F^ as a function of the position z1 of the small water droplet with radius r1 =1. The diagram consists of curves (1) and (2), which correspond, respectively, to con0gurations (1) and (2) in Fig. 37. The free energy possesses a minimum at around z1 = −5:7. The director 0eld assumes con0guration (2), where the hyperbolic hedgehog is situated at the surface of the nematic drop. Moving the water droplet closer to the surface, induces a repulsion due to the strong director deformations around the point defect. When the water droplet is placed far away from the south pole, i.e., at large z1 , the dipole of con0guration (1) forms and represents the absolute stable director 0eld. At z1 = −3:5 the dipole becomes metastable but the system does not assume con0guration (2) since the energy barrier the system has to overcome by thermal activation is much too high. By numerically calculating the free energy for di9erent positions of the hedgehog, we have, e.g., at z1 = −4:0, determined an energy barrier of K3 a ≈ 1000kB T , where kB is the Boltzmann constant, T the room temperature, and a ≈ 1 m. At z1 = −4:3, the dipole even loses its metastability, the hyperbolic defect jumps to the surface at the south pole and the water droplet follows until it reaches its energetically preferred position. On the other hand, if it were possible to move the water droplet away from the south pole, the hyperbolic hedgehog would stay at the surface, since con0guration (2) is always metastable for z1 ¿ −3:5. The energy barrier for a transition to the dipole is again at least 1000kB T . We have also investigated the distance d1 of the defect from the surface of the water droplet. For z1 ∈ [ − 2; 4], d1 >uctuates between 0.3 and 0.35. For z1 ¡ − 2, it increases up to 0.5 at z1 = −4:3, where the dipole loses its metastability.
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Fig. 39. An alternative, metastable con0guration. Both droplets are surrounded by a −1=2 disclination ring which compensates the topological charge +1 of each droplet. An additional +1=2 disclination ring close to the surface of the nematic drop satis0es the total topological charge +1. From Ref. [220]. Fig. 40. The free energy F^ as a function of the distance d of the droplets. A repulsion for d ¡ 0:6 is clearly visible. From Ref. [220].
7.3.4. Repulsion without defect We return to the 0rst geometry with two water droplets and homeotropic boundary conditions at all the surfaces. When we take either a uniform director 0eld or randomly oriented directors as a starting con0guration, our system always relaxes into the con0guration sketched in Fig. 39. Both water droplets are surrounded in their equatorial plane by a −1=2 disclination ring which compensates the point charge +1 carried by each droplet. That means, each droplet creates a Saturn-ring con0guration around it, which we introduced in Section 4.1 (see also Refs. [225,119]). To obtain the total point charge +1 of the nematic drop there has to be an additional topological defect with a point charge +1. In the numerically relaxed director 0eld, we 0nd a +1=2 disclination ring close to the outer surface. This con0guration has a higher energy than the one with the hyperbolic hedgehog. It is only metastable. Since a transition to the stable con0guration needs a complete rearrangement of the director 0eld, the energy barrier is certainly larger than K33 a ≈ 1000kB T . We, therefore, expect the con0guration of Fig. 39 to be stable against thermal >uctuations. It would be interesting to search for it in an experiment. We use the con0guration to demonstrate that even without the hyperbolic hedgehog the two water droplets experience some repulsion when they come close to each other. In Fig. 40 we plot the free energy F^ versus the separation d of the two spheres. For large d, the free energy oscillates which we attribute to numerical artifacts. For decreasing d, the free energy clearly increases, and the water droplets repel each other due to the strong deformation of the director 0eld lines connecting the two droplets. 7.4. Coda: twist transition in nematic drops Already thirty years ago, in connection with nematic emulsions, the two main director con0gurations in a nematic drop were discussed both experimentally and theoretically [147,61]: for
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homeotropic boundary conditions, a radial hedgehog at the center of the drop appears, whereas tangential surface anchoring leads to the bipolar structure already discussed above. The simple picture had to be modi0ed when it was found that nematic drops in both cases also exhibit a twisted structure [26]. For the bipolar con0guration, a linear stability analysis of the twist transition was performed [237]. A numerical study of the twisting in the radial structure of capillaries was presented in Refs. [185,186]. Lavrentovich and Terentjev proposed that the twisted director 0eld in a nematic drop with homeotropic surface anchoring is given by a combination of a hyperbolic hedgehog at the center of the drop and a radial one at its periphery [126] as illustrated in Fig. 7 of Section 2.4.1. This con0guration was analyzed by means of an ansatz function, and a criterion for the twist transition was given [126]. In this subsection we focus on the director 0eld between two concentric spheres with perpendicular anchoring at both the surfaces and present a stability analysis for the radial con0guration against axially symmetric deformations. In particular, we will derive a criterion for the twist transition, and we will show that even small spheres inside a large one are suNcient to avoid twisted con0gurations. This has been recently observed in the experiments on multiple nematic emulsions [182,183]. Throughout the paper we assume rigid surface anchoring of the molecules. In nematic emulsions it can be achieved by a special choice of the surfactant [182,183]. For completeness we note that in a single droplet for suNciently weak anchoring strength an axial structure with an equatorial disclination ring appears [66,60]. In the following three subsections, we 0rst expand the Frank free energy into small deviations from the radial con0guration up to second order. Then, we formulate and solve the corresponding eigenvalue equation arising from a linear stability analysis. The lowest eigenvalue leads to a criterion for the twist transition. We close with a discussion of our results. 7.4.1. Expansion of the elastic energy We consider the defect-free radial director con0guration between two concentric spheres of radii rmin and rmax and assume rigid radial surface anchoring at all the surfaces. If the smaller sphere is missing, the radial director con0guration exhibits a point defect at the center. We will argue below that this situation, rmin = 0, is included in our treatment. The twist transition reduces the SO(3) symmetry of the radial director con0guration to an axial C∞ symmetry. In order to investigate the stability of the radial con0guration n0 = er against a twist transition, we write the local director in a spherical coordinate basis, allowing for small deviations along the polar (5) and the azimuthal (7) direction: n(r; 5) = (1 − 12 b2 f2 − 12 a2 g2 )er + age5 + bfe7 :
(7.2)
f(r; 5) and g(r; 5) are general functions which do not depend on 7 due to our assumption of axial symmetry. The amplitudes a and b describe the magnitude of the polar and azimuthal deviation from the radial con0guration. The second-order terms in a and b result from the normalization of the director. The radial director 0eld between the spheres only involves a splay distortion, and its Frank free energy is Fradial = 8K11 (rmax − rmin ) ;
(7.3)
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where we did not include the saddle-splay energy. If an azimuthal (b = 0) or a polar component (a = 0) of the director is introduced, the splay energy can be reduced at costs of non-zero twist and bend contributions depending on the values of the Frank elastic constants K1 , K2 , and K3 . We expand the Frank free energy of the director 0eld in Eq. (7.2) up to second order in a and b and obtain 2 VF = 2b dr d cos 5[ − 4K1 (f2 + rfr f) + K2 (cot 5f + f5 )2 + K3 (f + rfr )2 ] 2 + 2a dr d cos 5[ − 4K1 (g2 + rgr g) + K1 (cot 5g + g5 )2 + K3 (g + rgr )2 ] (7.4) as the deviation from Fradial . The respective subscripts r and 5 denote partial derivatives with respect to the corresponding coordinates. Note that there are no linear terms in a or b, i.e., the radial director 0eld is always an extremum of the Frank free energy. Furthermore, there is no cross-coupling term ab in Eq. (7.4), and the stability analysis for polar and azimuthal perturbations can be treated separately. For example, for any function f(r; 5) leading to a negative value of the 0rst integral in Eq. (7.4), the radial con0guration (a = b = 0) is unstable with respect to a small azimuthal deformation (b = 0), which introduces a twist into the radial director 0eld. Therefore, we will call it the twist deformation in the following. An analogous statement holds for g(r; 5) which introduces a pure bend into the radial director 0eld. We are now determining the condition the elastic constants have to ful0l in order to allow for such functions f(r; 5) and g(r; 5). As we will demonstrate in the next subsection, the solution of this problem is equivalent to solving an eigenvalue problem. 7.4.2. Formulating and solving the eigenvalue problem In a 0rst step, we focus on the twist deformation (b = 0). We are facing the problem to determine for which values of K1 , K2 , and K3 the functional inequality dr d x {K2 (1 − x2 )[xf=(1 − x2 ) − fx ]2 + (K3 − 4K1 )f2 + (2K3 − 4K1 )rfr f + K3 r 2 fr2 } ¡ 0
(7.5)
possesses solutions f(r; x). The left-hand side of the inequality is the 0rst integral of Eq. (7.4) after substituting x = cos 5. After some manipulations (see Ref. [197]), we obtain
dr d x (K2 fD(x) f + K3 fD(r) f)
¡ 2K1 ; (7.6) dr d x f2 where the second-order di9erential operators D(x) and D(r) are given by D(x) = (1 − x2 )
92 9 1 + 2x + 2 9x 9x 1 − x2
and
D(r) = −r 2
92 9 − 2r : 2 9r 9r
(7.7)
The inequality in Eq. (7.6) is ful0lled the best when the left-hand side assumes a minimum. According to the Ritz principle in quantum mechanics, this minimum is given by the lowest
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eigenvalue of the operator K2 D(x) + K3 D(r)
(7.8)
on the space of square-integrable functions with f(rmin ; 5) = f(rmax ; 5) = 0 for 0 6 5 6 (0xed boundary condition) and f(r; 0) = f(r; ) = 0 for rmin 6 r 6 rmax . The eigenvalue equation of the operator K2 D(x) +K3 D(r) separates into a radial and an angular part. The radial part is an Eulerian di9erential equation [20] with the lowest eigenvalue 2 1 (r) &0 = + (7.9) 4 ln(rmax =rmin ) and the corresponding eigenfunction ln(r=rmin ) 1 (r) f (r) = √ sin : ln(rmax =rmin ) r
(7.10)
The angular part of the eigenvalue equation is solved by the associated Legendre functions Pnm=1 . The lowest eigenvalue is &0(x) = 2, and the corresponding eigenfunction is f(x) (5) = P11 (5) = sin 5. With both these results, we obtain the instability condition for a twist deformation: 2 1 K3 1 K2 + ¡1 : (7.11) + 2 K1 4 ln(rmax =rmin ) K1 This inequality is the main result of the paper. If it is ful0lled, the radial director 0eld no longer minimizes the Frank free energy. Therefore it is a suNcient condition for the radial con0guration to be unstable against a twist deformation. It is not a necessary condition since we have restricted ourselves to second-order terms in the free energy, not allowing for large deformations of the radial director 0eld. Hence, we cannot exclude the existence of further con0gurations which, besides the radial, produce local minima of the free energy. To clarify our last statement, we take another view. The stability problem can be viewed as a phase transition. Let us take K3 as the “temperature”. Then condition (7.11) tells us that for large K3 the radial state is the (linearly) stable one. If the phase transition is second-order-like, the radial state loses its stability exactly at the linear stability boundary, while for a 0rst-order-like transition the system can jump to the new state (due to non-linear >uctuations) even well inside the linear stability region. Thus, as long as the nature of the transition is not clear, linear stability analysis cannot predict for sure that the radial state will occur in the linear stability region. Furthermore, if the transition line is crossed, the linear stability analysis breaks down, and there could be a transition from the twisted to a new con0guration. However, there is no experimental indication for such a new structure. Keeping this in mind, we will discuss the instability condition (7.11) in the next subsection. We 0nish this subsection by noting that the elastic energy for a bend deformation (a = 0) has the same form as the one for the twist deformation (b = 0), however, with K2 replaced by K1 . Therefore, we immediately conclude from (7.11) that the instability condition for a polar component (a = 0) in the director 0eld (7.2) cannot be ful0lled for positive elastic constants. A director 0eld with vanishing polar component is always stable in second order.
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Fig. 41. (a) Stability diagram for the twist transition [cf. Eq. (7.11)]. The dark grey corresponds to the ratios of Frank constants where the radial con0guration is unstable for a ratio rmax =rmin = 50. The light grey triangle is the region where the radial con0guration is unstable for rmax =rmin ¿ 50. The circles represent the elastic constants for the liquid crystal compounds MBBA, 5CB, and PAA. (b) A comparison between the regions of instability for a radial director 0eld against twisting derived in this work (full line) and by Lavrentovich and Terentjev (dashed line) for rmax =rmin → ∞. The regions di9er by the areas I and II.
7.4.3. Discussion5 The instability condition (7.11) indicates for which values of the elastic constants K1 , K2 , and K3 the radial con0guration is expected to be unstable with respect to a twist deformation. The instability domain is largest for rmax =rmin → ∞ and decreases with decreasing ratio rmax =rmin , i.e., a water droplet inside a nematic drop can stabilize the radial con0guration. In Fig. 41a, the instability condition (7.11) is shown. If the ratios of the Frank elastic constants de0ne a point in the grey triangles, the radial con0guration can be unstable depending on the ratio rmax =rmin . The dark grey area gives the range of the elastic constants where a twisted structure occurs for rmax =rmin = 50. With increasing ratio rmax =rmin the instability domain enlargens until it is limited by K3 =(8K1 )+K2 =K1 =1 for rmax =rmin → ∞. The light grey triangle is the region where the radial con0guration is unstable for rmax =rmin ¿ 50 but where it is stable for rmax =rmin ¡ 50. The circles in Fig. 41a, represent, respectively, the elastic constants for the liquid crystal compounds MBBA, 5CB, and PAA. For 5CB the elastic constants are in the light grey domain, i.e., a twisted structure is expected for rmax =rmin → ∞ (no inner sphere) but not for rmax =rmin ¡ 50. Such a behavior has been recently observed in multiple nematic emulsions [182]. It has been found that a small water droplet inside a large nematic drop prevents the radial con0guration from twisting. Two examples of nematic drops observed under the microscope between crossed polarizers can be seen in Fig. 42. In the left image the director con0guration is pure radial, in the right one it is twisted. The left drop contains a small water droplet that stabilizes the radial con0guration according to Eq. (7.11). The water droplet is not visible in this image because of the limited resolution. A better image is presented in [182]. We have calculated the polarizing microscope picture of the twisted con0guration by means of the 2 × 2 Jones matrix formalism [60]. We took the director 0eld of Eq. (7.2) and used the eigenfunction of Eq. (7.10) with an amplitude 5
Reprinted with permission from A. RUudinger, H. Stark, Twist transition in nematic droplets: A stability analysis, Liq. Cryst. 26 (1999) 753. Copyright 1999 Taylor and Francis, http:==www.tandf.co.uk.
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Fig. 42. Radial (left) and twisted (right) con0guration of the director 0eld in a nematic drop (diameter ≈ 20 m) of 5CB observed under the microscope between crossed polarizers. In the radial con0guration there is a small isotropic liquid droplet in the center of the nematic drop (invisible in this image).
Fig. 43. Calculated transmission for the twisted con0guration of the director 0eld in a nematic drop whose diameter is 20 m. The transmission amplitude was obtained by summing over 20 wave lengths between 400 and 800 nm. The amplitude b of the twist deformation was set to 0.15. This 0gure has to be compared to the right image of Fig. 42. Fig. 44. Radial dependence of f(r) (r) [cf. Eq. (7.10)] for rmax =rmin = 50. The function is strongly peaked close to rmin .
b = 0:15. The result shown in Fig. 43 is in qualitative agreement with the experimental image on the right in Fig. 42. In Fig. 44 we plot the radial part f(r) (r) [see Eq. (7.10)] of the eigenfunction f(r; 5) = f(r) (r)f(x) (5) governing the twist deformations. For large values of rmax =rmin it is strongly
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peaked near rmin . The maximum of f(r) (r) occurs at a radius r0 which is given by ln
r0 ln(rmax =rmin ) 2 = arctan : rmin ln(rmax =rmin )
(7.12)
Hence, for rmax =rmin 1 the maximal azimuthal component bf(r0 ; 5) of the director 0eld is located at r0 =rmin = e2 ≈ 7:39, i.e., close to the inner sphere. From the polarizing microscope pictures it can be readily seen that the twist deformation is largest near the center of the nematic drop. In the opposite limit, rmax =rmin ≈ 1, the position of maximal twist is at the geometric mean of rmin and rmax : r0 = (rmin rmax )1=2 . In the limit rmin → 0, where the inner sphere is not existing, a point defect with a core radius rc is located at r = 0. In this case our boundary condition, f(r) (rmin ) = 0, makes no sense since the director is not de0ned for r ¡ rc . Fortunately, for rmin → 0 the lowest eigenvalue of the operator (7.8) and therefore the instability condition is insensitive to a change of the boundary condition. Furthermore, the shape of the eigenfunction is also independent of the boundary condition, in particular its maximum is always located close to rmin . A last comment concerns the work of Lavrentovich and Terentjev [126]. In Fig. 41b, we plot as a dashed line the criterion, K3 =(4K1 ) + K2 =(2K1 ) = 1, which the authors of Ref. [126] derived for the twist transition in the case rmax =rmin → 0. They constructed an ansatz function which connects a hyperbolic hedgehog at the center via a twist deformation to a radial director 0eld at the periphery of a nematic drop. Then they performed a stability analysis for an appropriately chosen order parameter. The region of instability calculated in this article and their result di9er by the areas I and II. This is due to the complementarity of the two approaches. While the authors of Ref. [126] allow for large deviations with respect to the radial con0guration at the cost of 0xing an ansatz function, we allow the system to search the optimal con0guration (i.e., eigenfunction) for small deformations. We conclude that both results together give a good approximation of the region of instability for the radial con0guration against twisting. However, we cannot exclude that a full non-linear analysis of the problem leads to a change in the stability boundaries. In conclusion, we have performed a stability analysis of the radial con0guration in nematic drops with respect to a twist deformation. Assuming strong perpendicular anchoring at all the surfaces, we have derived an instability condition in terms of the Frank constants. We could show that a small water droplet inside the nematic drop stabilizes the radial con0guration.
8. Temperature-induced +occulation above the nematic-isotropic phase transition Ping Sheng [210,211] was the 0rst to study the consequences of surface-induced liquid crystalline order above the nematic-isotropic phase transition. He introduced the notion paranematic order in analogy to the paramagnetic phase, in which a magnetic 0eld causes a non-zero magnetization. He realized that the bounding surfaces of a restricted geometry in>uence the bulk transition temperature Tc . In nematic 0lms, e.g., the phase transition even vanishes below a critical thickness [210]. Sheng’s work was extended by Poniewierski and Sluckin [177], who studied two plates immersed in a liquid crystal above Tc and who calculated an attractive force
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between the two plates due to the surface-induced order. This force was investigated in detail \ by Bor\stnik and Zumer [18]. The work presented in this section explores the liquid crystal mediated interaction between spherical particles immersed into a liquid crystal above Tc . It has to be added to the conventional van der Waals, electrostatic, and steric interactions as a new type of interparticle potential. Its strength can be controlled by temperature, and close to the clearing temperature Tc , it can induce a >occulation transition in an otherwise stabilized colloidal dispersion. In Section 8.1 we review the Landau–de Gennes theory, which describes liquid crystalline order close to the phase transition, and we present Euler–Lagrange equations for the director and the Maier–Saupe order parameter to be de0ned below. Section 8.2 illustrates paranematic order in simple plate geometries and introduces the liquid crystal mediated interaction of two parallel plates. In Section 8.3 we extend it to spherical particles and investigate its consequences when combined with van der Waals and electrostatic interactions. 8.1. Theoretical background We start with a review of the Landau–de Gennes theory and then formulate the Euler–Lagrange equations for restricted geometries with axial symmetry. 8.1.1. Landau–de Gennes theory in a nutshell The director n, a unit vector, only indicates the average direction of the molecules. It tells nothing about how well the molecules are aligned. To quantify the degree of liquid crystalline order, we could just vary the magnitude of n, i.e., choose a polar vector as an order parameter. However, all nematic properties are invariant under inversion of the director, thus every polar quantity has to be zero. The next choice is any second-rank tensor, e.g., the magnetic susceptibility tensor . The order parameter Q is de0ned by the relation 9 1 − 1 tr ; Q= (8.1) 2 tr 3 where tr = ii stands for the trace of a tensor, and Einstein’s summation convention over repeated indices is always assumed in the following. We subtract the isotropic part 1 tr =3 from , in order that Q vanishes in the isotropic liquid. The prefactor is convention. The order parameter Q describes, in general, biaxial liquid crystalline ordering through its eigenvectors and eigenvalues. The uniaxial symmetry of the nematic phase demands that two eigenvalues of Q are equal, which then assumes the form 3( − ⊥ ) 1 3 Q= S n⊗n− 1 with S = : (8.2) 2 3 2⊥ + The Maier–Saupe or scalar order parameter S indicates the degree of nematic order through the magnetic anisotropy V = − ⊥ . It was 0rst introduced by Maier and Saupe in a microscopic treatment of the nematic phase [142]. The microscopic approach was generalized by Lubensky to describe biaxial order [138].
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In his seminal publication (see Ref. [48]) de Gennes was interested in pretransitional e9ects above the nematic-isotropic phase transition. He constructed a free energy in Q and ∇i Qjk in the spirit of Landau and Ginzburg, commonly known as Landau–de Gennes theory: FLG = d 3 r (fb + f∇Q ) ; (8.3) with fb = 12 a0 (T − T ∗ ) tr Q 2 − 13 b tr Q 3 + 14 c(tr Q 2 )2 ;
(8.4)
f∇Q = 12 L1 (∇i Qjk )2 + 12 L2 (∇i Qij )2 :
(8.5)
The quantity fb introduces a Landau-type free energy density which describes a 0rst-order phase transition, and f∇Q is necessary to treat, e.g., >uctuations in Q , as noticed by Ginzburg. Both free energy densities are Taylor expansions in Q and ∇i Qjk , and each term is invariant under the symmetry group O(3) of the isotropic liquid, i.e., the high-symmetry phase. The Landau parameters of the compound 5CB are a0 = 0:087 × 107 erg=cm3 K; b = 2:13 × 107 erg=cm3 ; c = 1:73 × 107 erg=cm3 , and T ∗ = 307:15 K [38]. The elastic constants L1 and L2 are typically of the order of 10−6 dyn. It can be shown unambiguously that fb is minimized by the uniaxial order parameter of Eq. (8.2), for which the free energies fb and f∇Q take the form 3 9 1 fb = a0 (T − T ∗ )S 2 − bS 3 + cS 4 ; 4 4 16
(8.6)
9 3 f∇Q = L1 (∇i S)2 + L1 S 2 (∇i nj )2 : 4 4
(8.7)
To arrive at Eq. (8.7), we set L2 = 0 in order to simplify the free energy as much as possible for our treatment in Sections 8.2 and 8.3. L2 = 0 merely introduces some anisotropy, as shown by de Gennes [48]. Assume, e.g., that S is 0xed to a non-zero value at a space point rs in the isotropic >uid, then the nematic order around rs decays exponentially on a characteristic length scale called nematic coherence length. If L2 = 0, the respective coherence lengths along and perpendicular to n are di9erent. In Fig. 45 we plot fb as a function of S using the parameters of 5CB. Above the superheating temperature T † = T ∗ + b2 =(24a0 c), there exists only one minimum at S = 0 for the thermodynamically stable isotropic phase. At T † a second minimum for the metastable nematic phase evolves, which becomes absolutely stable at the clearing temperature Tc =T ∗ +b2 =(27a0 c). A 0rst-order phase transition occurs, and the order parameter as a function of temperature assumes the form
1b 2a † S(T ) = (8.8) + (T − T ) : 6c 3c Finally, at the supercooling temperature T ∗ the curvature of fb at S = 0 changes sign, and the isotropic >uid becomes absolutely unstable. For the compound 5CB, we 0nd Tc − T ∗ = 1:12 K and T † − Tc = 0:14 K.
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Fig. 45. The free energy density fb in units of 1000a0 T ∗ as a function of the Maier–Saupe order parameter S for various temperatures. The Landau coeNcients of the compound 5CB are employed. A 0rst-order transition occurs at Tc .
8.1.2. Euler–Lagrange equations for restricted geometries In the following, we determine the surface-induced liquid crystalline order above Tc . As usual, it follows from a minimization of the total free energy, F = FLG + Fsur ;
(8.9)
where we have added a surface term Fsur to the Landau–de Gennes free energy FLG . We restrict ourselves to uniaxial order and employ a generalization of the Rapini–Papoular potential, introduced in Section 2.1, 3 Fsur = dA (WS (S − S0 )2 + 3Wn SS0 [1 − (n · ˆ)2 ]) ; (8.10) 4 where dA is the surface element. The quantity S0 denotes the preferred Maier–Saupe parameter at the surface, and ˆ is the surface normal since we always assume homeotropic anchoring. The surface-coupling constants WS and Wn penalize a respective deviation of S from S0 and of the director n from ˆ. In recent experiments, anchoring and orientational wetting transitions of liquid crystals, con0ned to cylindrical pores of alumina membranes, were analyzed [42,43]. It was found that WS and Wn vary between 10−1 and 5, with the ratio Wn =WS not being larger than 0ve. If WS = Wn = W , the intergrand in Eq. (8.10) is equivalent to the intuitive form W tr(Q − Q0 )2 =2 with the uniaxial Q from Eq. (8.2) and Q0 = 32 S0 (ˆ ⊗ ˆ − 13 1). It was introduced by Nobili and Durand [165]. In formulating the elastic free energy density f∇Q of Eq. (8.5), one also identi0es a contribution which can be written as a total divergence, ∇i (Qij ∇k Qjk − Qjk ∇k Qij ). When transformed into a surface term and when a uniaxial Q is inserted, it results in the saddle-splay energy of Eq. (2.4). To simplify our calculations, we will neglect this term. It is not expected to change the qualitative behavior of our system for strong surface coupling. In what follows, we assume rotational symmetry about the z axis. We introduce cylindrical coordinates and write the director in the local coordinate basis, n(6; z) = sin (6; z)e6 + cos (6; z)ez , restricting it to the (6; z) plane. The same is assumed for the surface normal ˆ(6; z) = sin 0 (6; z)e6 + cos 0 (6; z)ez . Expressing and minimizing the total free energy under
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all these premises, we obtain the Euler–Lagrange equations for S and the tilt angle in the bulk, 1 3c 3 b 2 sin2 2 2 ∇ S− 2S+ S − S − 3S (∇) + =0 ; (8.11) 2L1 2L1 62 N sin cos ∇2 − =0 ; (8.12) 62 and the boundary equations are 1 3 (ˆ ·∇)S − (S − S0 ) − S0 sin2 ( − 0 ) = 0 ; %S N 2%n N 1 S0 (ˆ ·∇) − sin[2( − 0 )] = 0 : 2%n N S The meaning of the nematic coherence length N = L1 =[a0 (T − T ∗ )]
(8.13) (8.14)
(8.15)
will be clari0ed in the next subsection. At the phase transition, NI = N (Tc ) is of the order of 10 nm, as can be checked by the parameters of 5CB. The surface-coupling strengths WS and Wn are characterized by dimensionless quantities a0 (T − T ∗ )L1 a0 (T − T ∗ )L1 1 L1 1 L1 = and %n = = ; (8.16) %S = N WS WS N Wn Wn which compare the respective surface extrapolation lengths L1 =WS and L1 =Wn to the nematic coherence length N . For W = 1 erg=cm2 and L1 = 10−6 dyn, the extrapolation lengths are of the same order as N at Tc , i.e., 10 nm. 8.2. Paranematic order in simple geometries In the 0rst two subsections we study the paranematic order in a liquid crystal compound above Tc for simple plate geometries. It is induced by a coupling between the surfaces and the molecules. We disregard the non-harmonic terms in S in Eq. (8.11) to simplify the problem as much as possible and to obtain an overall view of the system. In Section 8.2.3 the e9ect of the non-harmonic terms is reviewed. 8.2.1. One plate We assume that an in0nitely extended plate, which induces a homeotropic anchoring of the director, is placed at z = 0. Its surface normals are ±ez , and its thickness should be negligibly small. A uniform director 0eld along the z axis obeys Eqs. (8.12) and (8.14), and the Maier–Saupe order parameter S follows from a solution of Eqs. (8.11) and (8.13), S0 S(z) = exp[ − |z |=N ] : (8.17) 1 + %S The order parameter S decays exponentially along the z axis on a characteristic length scale given by the nematic coherence length N . The value of S at z = 0 depends on the strength %S
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of the surface coupling, i.e., on the ratio of the surface extrapolation length L1 =W and N . The plate is surrounded by a layer of liquid crystalline order whose thickness N decreases with increasing temperature since N ˙ (T − T ∗ )−1=2 . The total free energy per unit surface, F=A, consisting of the Landau–de Gennes and the surface free energy, is F 3 %S : (8.18) = WS S02 A 2 1 + %S √ Note that the energy increases with temperature since %S ˙ T − T ∗ . The whole theory certainly becomes invalid when N approaches molecular dimensions. For 10 K above Tc , we 0nd N ≈ 3 nm, i.e., the theory is valid several Kelvin above Tc . Finally, we notice that a nematic wetting layer can be probed by the evanescent wave technique [214]. 8.2.2. Two plates If two plates of the previous subsection are placed at z = ±d=2, the order parameter pro0le S(z), determined from Eqs. (8.11) and (8.13), is cosh(z=N ) S(z) = S0 : (8.19) cosh(d=2N ) + %S sinh(d=2N ) For separations d2N , the layers of liquid crystalline order around the plates do not overlap, as illustrated in the inset of Fig. 46. 6 If d 6 2N , the whole volume between the plates is occupied by nematic order, which induces an attraction between the plates. The interaction energy per unit area, VF=A, is de0ned as VF=A = [F(d) − F(d → ∞)]=A. It amounts to VF F(d) − F(d → ∞) 3 tanh(d=2N ) 1 2 − : (8.20) = = WS S0 %S A A 2 1 + %S tanh(d=N ) 1 + %S In Fig. 46 we plot VF=A versus the reduced distance d=2NI for di9erent temperatures at Tc and above Tc . The material parameters of 5CB are chosen; WS = 1 erg=cm2 , and S0 = 0:3. The energy unit 3WS S02 =2=104 kB T is determined at room temperature. Note, that NI is the coherence length at Tc . If dN , the interaction energy decays exponentially in d; VF=A ˙ exp(d=N ). The interaction is always attractive over the whole separation range. This can be understood by a simple argument. Above Tc , the nematic order always possesses higher energy than the isotropic liquid. Therefore, the system can reduce its free energy by moving the plates together. The minimum of the interaction energy occurs at d = 0, i.e., when the liquid with nematic order between the plates is completely removed. This simple argument explains the deep potential well in Fig. 46. It extends to a separation of 2N where the nematic layers start to overlap. Since N ˙ (T − T ∗ )−1=2 , the range of the interaction decreases with increasing temperature, and the depth of the potential well becomes smaller. 8.2.3. E<ect of non-harmonic terms In this subsection we review the e9ects on the two-plate geometry when the complete Landau–de Gennes theory including its non-harmonic terms in S is employed. A wealth of \ Figs. 46, 47, 51 and 52 are reprinted with permission from A. Bor\stnik, H. Stark, S. Zumer, Temperature-induced >occulation of colloidal particles above the nematic-isotropic phase transition, Prog. Colloid Polym. Sci. 115 (2000) 353. Copyright 2000 Springer Verlag. 6
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Fig. 46. Interaction energy per unit area, VF=A, as a function of the reduced distance d=2NI for various temperatures. For further explanation see text.
phenomena exists, which we illustrate step by step [210,211]. Their in>uence on the interaction \ of two plates was studied in detail by Bor\stnik and Zumer [18]. First, we assume rigid anchoring at the nematic-plate interfaces, i.e., S(±d=2) is 0xed to S0 [210]. For d → ∞, there is a phase transition at the bulk transition temperature Tc∞ = Tc from the nematic to the surface-induced paranematic phase, as expected. When the plates are moved together, the transition temperature Tcd increases until the 0rst-order transition line in crit ). For d ¡ d , no phase transition a d–T phase diagram ends in a critical point at (dcrit ; Tcd crit between the nematic and the paranematic phase is observed anymore. This is similar to the gas–liquid critical point in an isotropic >uid. For S0 = S(±d=2) = 0:5 − 1 and typical values of crit is situated approximately 0.2 K above T the Landau parameters, Tcd c∞ = Tc and 0.1 K above the superheating temperature T † . Secondly, we concentrate on a basically in0nite separation, dNI , and allow a 0nite surfacecoupling strength WS [211]. For suNciently small WS , both the boundary [S(±d=2)] and the bulk [S(0)] value of the scalar order parameter exhibit a jump at Tc . That means, the surface coupling is so small that S(±d=2) follows the bulk order parameter. However, in a 0nite interval WS0 ¡ WS ¡ WScrit , the discontinuity of S(±d=2), which Sheng calls a boundary-layer phase transition, occurs at temperatures Tbound above Tc . Beyond the critical strength WScrit , the boundary transition vanishes completely. Sheng just used the linear term ˙ WS S of our surface potential for his investigation. The separate boundary-layer transition occurred in the approximate interval 0:01 erg=cm2 ¡ WS ¡ 0:2 erg=cm2 . We do not expect a dramatic change of this interval for the potential of Eq. (8.10). Thirdly, we combine the 0nite separation of the plates with a 0nite surface-coupling strength WS . The boundary-layer transition temperatures Tbound and the interval WS0 ¡ WS ¡ WScrit are not e9ected by a 0nite d. In addition, a jump of S(±d) occurs at the bulk transition temperature Tcd 6 Tbound . It evolves gradually with decreasing d. When Tcd becomes larger than Tbound
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in the course of moving the plates together, the separate boundary-layer transition disappears. Finally, at a critical thickness dcrit the nematic-paranematic transition vanishes altogether. crit − T All these details occur close to Tc∞ = Tc within a range of Tcd c∞ = 0:5 K [211]. The calculations are non-trivial. Since we do not want to render our investigation in the following subsection too complicated, we will skip the non-harmonic terms in the Landau–de Gennes theory. Furthermore, we use a relatively high anchoring strength of about WS = 1 erg=cm2 , so crit − T that Tcd c∞ is even smaller than 0.5 K. The simpli0cations are suNcient to bring out the main features of our system. 8.3. Two-particle interactions above the nematic-isotropic phase transition In this subsection we present the liquid crystal mediated interaction above Tc as a new type of two-particle potential. We combine it with the traditional van der Waals and electrostatic interaction and explore its consequences, namely the possibility of a temperature-induced >occulation. We start with a motivation, introduce all three types of interactions, and 0nally discuss their consequences. Our presentation concentrates on the main ideas and results (see also Ref. [17]). Details of the calculations can be found in Refs. [15,16]. 8.3.1. Motivation In Section 3 we already mentioned that the stability of colloidal systems presents a key issue in colloid science since their characteristics change markedly in the transition from the dispersed to the aggregated state. There are always attractive van der Waals forces, which have to be balanced by repulsive interactions to prevent a dispersion of particles from aggregating. This is achieved either by electrostatic repulsion, where the particles carry a surface charge, or by steric stabilization, where they are coated with a soluble polymer brush. Dispersed particles approach each other due to their Brownian motion. They aggregate if the interaction potential is attractive, i.e., if it possesses a potential minimum Umin ¡ 0 at 0nite separations. Two situations are possible. In the case of weak attraction, where |Umin | ≈ 1–3kB T , an equilibrium phase separation of a dilute and an aggregated state exists. The higher interaction energy of the dispersed particles is compensated by their larger entropy in comparison to the aggregated phase. Strong attraction, i.e., |Umin | ¿ 5 − 10kB T , causes a non-equilibrium phase with all the particles aggregated. They cannot escape the attractive potential in the observation time of interest of, e.g., several hours. Due to Chandrasekhar, the escape time tesc can be estimated as [28] tesc =
a2 D0 exp(−Umin =kB T )
with D0 =
kB T : 6)a
(8.21)
D0 is the di9usion constant of a non-interacting Brownian particle with radius a, and ) is the shear viscosity of the solvent. The quantity tesc approximates the time a particle needs to di9use a distance a in leaving a potential well of depth Umin . More re0ned theories suggest that the complete two-particle potential has to be taken into account when calculating tesc [131,100]. Here, we study the in>uence of liquid crystal mediated interactions on colloidal dispersions above Tc , which are stabilized by an electrostatic repulsion. We demonstrate that the main e9ect of the liquid crystal interaction ULC is an attraction at the length scale of N , whose strength can be controlled by temperature. If the electrostatic repulsion is suNciently weak, ULC induces
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Fig. 47. Two particles at a separation d2N do not interact. At d ≈ 2N both a strong attraction and repulsion set in.
a >occulation of the particles within a few Kelvin close to the transition temperature Tc . It is completely reversible. A similar situation is found in polymer stabilized colloids. There, the abrupt change from a dispersed to a fully aggregated state within a few Kelvin is called critical >occulation [162,202]. The reversibility of >occulation has interesting technological implications. For example, in “instant” ink, the particles of dried ink redisperse rapidly when put into water [162]. So far, experiments on colloidal dispersions above the clearing temperature Tc are very rare [19,178]. They would help to explore a new class of colloidal interactions. Furthermore, they could provide insight into wetting phenomena above Tc with all its subtleties close to Tc , which we reviewed in Section 8.2.3. Also, experiments by Mu\sevi\c et al. [158,159], who probe interactions with the help of an atomic force microscope, are promising. 8.3.2. Liquid crystal mediated interaction One particle suspended in a liquid crystal above the clearing temperature Tc is surrounded by a layer of surface-induced nematic order whose thickness is of the order of the nematic coherence length N . The director 0eld points radially outward when a homeotropic anchoring at the particle surface is assumed. Two particles with a separation d2N do not interact. When the separation is reduced to d ≈ 2N , a strong attraction sets in since the total volume of nematic order is decreased as in the case of two plates (see Fig. 47). In addition, a repulsion due to the elastic distortion of the director 0eld lines connecting the two particles occur. In this subsection we quantify the two-particle interaction mediated by a liquid crystal. In principle, the director 0eld and the Maier–Saupe order parameter S follow from a solution of Eqs. (8.11) – (8.14). Since the geometry of Fig. 48a cannot be treated analytically, we employ two simpli0cations. First, we approximate each sphere by 72 conical segments, whose cross sections in a symmetry plane of our geometry are illustrated in Fig. 48a. In the following, we assume a particle radius a = 250 nm, and, therefore, each line segment has a length of
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Fig. 48. (a) Two spheres A and B are approximated by conical segments as illustrated in the blowup. From Ref. [15]. (b) At separations d ≈ 2N , the director is chosen as a tangent vector nc of a circular segment whose radius is determined by the boundary condition (8.14).
26 nm. Secondly, we construct appropriate ansatz functions for the 0elds S(r) and n(r). To arrive at an ansatz for S(r), we approximate the bounding surfaces Ai and Bi of region i by two parallel ring-like plates and employ the order parameter pro0le of Eq. (8.19), where d is replaced by an average distance di of the bounding surfaces. Since the particle radius is an order of magnitude larger than the interesting separations, which do not exceed several coherence lengths, the analogy with two parallel plates is justi0ed. Furthermore, we expect that only a few regions close to the symmetry axis are needed to calculate the interaction energy with a suNcient accuracy. In the limit of large separations (d2N ), the director 0eld around each sphere points radially outward. In the opposite limit (d ≈ 2N ), the director 0eld lines are strongly distorted, and we approximate them by circular segments as illustrated in Fig. 48b, for the third region. The radius of the circle is determined by the boundary condition (8.14) of the director. With decreasing separation of the two particles, the director 0eld should change continuously from n∞ at d2N to the ansatz nc at small d. Hence, we choose n(r) as a weighted superposition of nc and n∞ : n(r ) ˙ 'i nc + (1 − 'i )n∞ ;
(8.22)
where the free parameter 'i follows from a minimization of the free energy in region i with respect to 'i . As in the case of two parallel plates, the interaction energy is de0ned relative to the total free energy of in0nite separation: ULC (d) = F(d) − F(d → ∞) :
(8.23)
In calculating ULC , we employ the free energy densities of Eqs. (8.6) and (8.7) and the surface potential of Eq. (8.10), neglecting the non-harmonic terms in S. The volume integrals cannot be performed analytically without further approximations which we justi0ed by a comparison with a numerical integration. The 0nal expression of ULC is very complicated, and we refer the reader to Ref. [15] for its explicit form. We checked that regions i = 1; : : : ; 9 are suNcient to
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Fig. 49. The liquid crystal mediated interaction ULC in units of kB T as a function of the particle separation d. The interaction is shown at Tc ; Tc + 1 K; Tc + 3 K; and Tc + 11 K. It strongly depends on temperature. Large inset: ULC is composed of an attractive and repulsive part. Small inset: A weak repulsive barrier occurs at d ≈ 60 nm.
calculate ULC . The contribution of region 9 to the interaction energy is less than 5%. Hence, the orientational order outside these nine regions is not relevant for ULC . We subdivide the interaction energy in an attractive part which results from all terms in the free energy depending on the order parameter S or its gradient, only. The repulsive part is due to the elastic distortion of the director 0eld and a deviation from the homeotropic orientation at the particle surfaces. All the graphs, which we present in the following, are calculated with the Landau parameters of the compound 8CB [38], i.e., a0 = 0:12 × 10−7 erg=cm3 K; b = 3:07 × 10−7 erg=cm3 ; c=2:31×10−7 erg=cm3 , and L1 =1:8×10−6 dyn, which gives Tc −T ∗ =b2 =(27a0 c)= 1:3 K. The surface-coupling constants are WS = 1 erg=cm2 and Wn = 5 erg=cm2 . In the large inset of Fig. 49 we plot the attractive and repulsive contribution at the clearing temperature Tc in units of the thermal energy kB T . As in the case of two parallel plates, the total interaction energy exhibits a deep potential well with an approximate width of 2NI . At larger separations, it is followed by a weak repulsive barrier whose height is approximately 1:5kB T , as indicated by the small inset in Fig. 49. If d2N ; ULC decays exponentially: ULC ˙ exp(−d=N ). Fig. 49 illustrates further that the depth of the potential well, i.e., the liquid crystal mediated attraction of two particles decreases considerably when the dispersion is heated by several Kelvin. That means, the interaction can be controlled by temperature. It is turned o9 by heating the dispersion well above Tc . The same holds for the weak repulsive barrier. As expected, both the depth of the potential well and the height of the barrier decrease with the surface-coupling constants, where WS seems to be more important [15]. 8.3.3. Van der Waals and electrostatic interactions The van der Waals interaction of two thermally >uctuating electric dipoles decays with the sixth power of their inverse distance, 1=r 6 . To arrive at the interparticle potential of two macroscopic objects, a summation over all pair-wise interactions of >uctuating charge distributions is performed. In the case of two spherical particles of equal radii a; the following,
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always attractive, van der Waals interaction results [202]: d(d + 4a) 2a2 A 2a2 UW = − + ln : + 6 d(d + 4a) (d + 2a)2 (d + 2a)2
461
(8.24)
Here d is the distance between the surfaces of the particles, and A is the Hamaker constant. For equal particles made of material 1 embedded in a medium 2, it amounts to [202] 3 '1 − '2 2 3hBuv (n21 − n22 )2 A = kB T + √ ; (8.25) 4 '1 + '2 16 2 (n21 + n23 )3=2 where '1 and '2 are the static dielectric constants of the two materials, and n1 and n2 are the corresponding refractive indices of visible light. The relaxation frequency Buv belongs to the dominant ultraviolet absorption in the dielectric spectrum of the embedding medium 2. Typical values for silica particles immersed into a nematic liquid crystal are '1 = 3:8; n1 = 1:45; '2 = 11; n2 = 1:57; and Buv = 3 × 1015 s−1 [15]. As a result, the Hamaker constant equals A = 1:1 kB T . Note, that for separations da the particles are point-like, and the van der Waals interaction decays as 1=d6 . In the opposite limit, da; it diverges as a=d. We stabilize the colloidal dispersion against the attractive van der Waals forces by employing an electrostatic repulsion. We assume that each particle carries a uniformly distributed surface charge whose density qs does not change under the in>uence of other particles. Ionic impurities in the liquid crystal screen the surface charges with which they form the so-called electrostatic double layer. For particles of equal radius a embedded in a medium with dielectric constant '2 ; the electrostatic two-particle potential is described by the following expression [202]: UE = −kB T
aqs2 ln(1 − e−Cd ) : z 2 e02 np
(8.26)
Here, e0 is the fundamental charge, and z is the valence of the ions in the solvent, which have a concentration np . The range of the repulsive interaction is determined by the Debye length (8.27) C−1 = '2 kB T=(8e02 z 2 np ) ; whereas the surface-charge density qs controls its strength. The potential UE decays exponentially at dC−1 . Expression (8.26) is derived via the Derjaguin approximation [53,202], which is only valid for d; C−1 a. In the following, we take a monovalent salt (z = 1); choose '2 = 11; and vary np between 10−4 and 10−3 mol=l. Then, at room temperature the Debye length C−1 ranges from 10 to 3.5 nm. Together with typical separations d not larger than a few coherence lengths N and a = 250 nm; the Derjaguin approximation is justi0ed. Furthermore, we adjust the surface-charge density around 104 e0 = m2 . The ranges of np and qs are well accessible in an experiment. In Fig. 50 we plot the electrostatic and the van der Waals interactions and their sum in units of kB T . The surface-charge density qs is 0:5 × 104 e0 = m2 and C−1 = 8:3 nm. All further parameters besides the Hamaker constant A are chosen as mentioned above. We increased A from 1.1 to 5.5. Even then it is clearly visible that the strong electrostatic repulsion determines the interaction for d ¡ 30 nm; the dispersion of particles is stabilized. At about 55 nm, UE + UW exhibits a shallow potential minimum (see inset of Fig. 50), and at dC−1 ; the algebraic decay
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Fig. 50. The electrostatic (dashed) and van der Waals (dotted) interaction and their sum UE +UW (full line) in units of kB T as a function of particle separation d. The parameters are chosen according to the text. Inset: A shallow potential minimum appears at d ≈ 55 nm.
Fig. 51. The total two-particle interaction ULC + UE + UW as a function of particle separation d for various temperatures. A complete >occulation of the particles occurs within a temperature range of about 0.3 K. qs =0:5 × 104 e0 = m2 ; C−1 = 8:3 nm, and further parameters are chosen according to the text.
of the van der Waals interaction takes over. In the following subsection, we investigate the combined e9ects of all three interactions for the Hamaker constant A = 1:1. 8.3.4. Flocculation versus dispersion of particles In Fig. 51 we plot the total two-particle interaction ULC + UE + UW as a function of particle separation d for various temperatures. We choose qs = 0:5 × 104 e0 = m2 and C−1 = 8:3 nm. At 4.5 K above the transition temperature Tc ; the dispersion is stable. With decreasing temperature, a potential minimum at 0nite separation develops. At TFD = Tc + 0:54 K; the particle doublet or aggregated state becomes energetically preferred. We call TFD the temperature of >occulation
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Fig. 52. In comparison to Fig. 51 the surface-charge density is increased to 0:63 × 104 e0 = m2 . As a result, >occulation does not occur.
transition. Below TFD ; the probability of 0nding the particles in the aggregated state is larger than the probability that they are dispersed. Already at Tc + 0:3 K the minimum is 7 kB T deep, and all particles are condensed in aggregates. That means, within a temperature range of about 0.3 K there is an abrupt change from a completely dispersed to a fully aggregated system, reminiscent to the critical >occulation transition in colloidal dispersions employing polymeric stabilization [162,202]. Between d = 30 and 50 nm, the two-particle interaction exhibits a small repulsive barrier of about 1:5 kB T . Such barriers slow down the aggregation of particles, and one distinguishes between slow and rapid >occulation. The dynamics of rapid >occulation was 0rst studied by Smoluchowski [215]. Fuchs extended the theory to include arbitrary interaction potentials [82]. However, only after Derjaguin and Landau [54] and Verwey and Overbeek [229] incorporated van der Waals and electrostatic interactions into the theory, became a comparison with experiments possible. In our case, the repulsive barrier of 1:5 kB T slows down the doublet formation by a factor of three, i.e., it does not change very dramatically if the barrier is reduced to zero. If the surface-charge density qs is increased to 0:63 × 104 e0 = m2 ; the dispersed state is thermodynamically stable at all temperatures above Tc ; as illustrated in Fig. 52. An increase of the Debye length C−1 ; i.e., the range of the electrostatic repulsion, has the same e9ect. In Fig. 53 we present >occulation phase diagrams as a function of temperature and surfacecharge density for various Debye lengths C−1 . The inset shows one such diagram for C−1 = 8:3 nm. The full line represents the >occulation temperature TFD as a function of qs . For temperatures above TFD ; the particles stay dispersed while for temperatures below TFD the system is >occulated. To characterize the aggregated state further, we have determined lines in the phase diagram of C−1 = 8:3 nm; where the escape time tesc of Eq. (8.21) is, respectively, ten (dash-dotted) or hundred (dotted) times larger than in the case of zero interaction. These lines are close to the transition temperature TFD ; and indicate again that the transition from the dispersed to a completely aggregated state takes place within less than one Kelvin. The large plot of Fig. 53 illustrates the >occulation temperature TFD as a function of qs for various Debye
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Fig. 53. Flocculation phase diagrams as a function of temperature and surface-charge density for various Debye lengths C−1 . Inset: Phase diagram for C−1 = 8:3 nm. The full line represents the >occulation temperature TFD as a function of qs . The dash-dotted and dotted lines indicate escape times from the minimum of the interparticle potential which are, respectively, ten or hundred times larger than in the case of zero interaction. From Ref. [16].
lengths C−1 . TFD increases when the strength (qs ) or the range (C−1 ) of the electrostatic repulsion is reduced. The intersections of the transition lines with the T = Tc axis de0ne the “>occulation end line”. In the parameter space of the electrostatic interaction (surface charge density qs versus Debye length C−1 ), this line separates the region where we expect the >occulation to occur from the region where the system is dispersed for all temperatures above Tc (see Ref. [16]). 8.3.5. Conclusions Particles dispersed in a liquid crystal above the nematic-isotropic phase transition are surrounded by a surface-induced nematic layer whose thickness is of the order of the nematic coherence length. The particles experience a strong liquid crystal mediated attraction when their nematic layers start to overlap since then the e9ective volume of liquid crystalline ordering and therefore the free energy is reduced. A repulsive correction results from the distortion of the director 0eld lines connecting two particles. The new colloidal interaction is easily controlled by temperature. In this section we have presented how it can be probed with the help of electrostatically stabilized dispersions. For suNciently weak and short-ranged electrostatic repulsion, we observe a sudden >occulation within a few tenth of a Kelvin close to Tc . It is reminiscent to the critical >occulation transition in polymer stabilized colloidal dispersions [202]. The >occulation is due to a deep potential minimum in the total two-particle interaction followed by a weak repulsive barrier. Thermotropic liquid crystals represent polar organic solvents, and one could wonder if electrostatic repulsion is realizable in such systems. In Ref. [101] complex salt is dissolved in nematic liquid crystals and ionic concentrations of up to 10−4 mol=l are reported which give
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rise to Debye lengths employed in this section. Furthermore, when silica spheres are coated − with silanamine, the ionogenic group [ = NH+ 2 OH ] occurs at the particle surface with a density of 3 × 106 mol= m2 . It dissociates to a large amount in a liquid crystal compound [59]. In addition, the silan coating provides the required perpendicular boundary condition for the liquid crystal molecules. These two examples illustrate that electrostatic repulsion should be accessible in conventional thermotropic liquid crystals, and we hope to initiate experimental studies which probe the new colloidal force. Our work directly applies to lyotropic liquid crystals [51], i.e., aqueous solutions of non-spherical micelles, when the nematic-isotropic phase transition is controlled by temperature [181,180]. They are appealing systems since electrostatic stabilization is more easily achieved. When the phase transition is controlled by the micelle concentration 6m ; as it is usually done, then our diagrams are still valid but with temperature replaced by 6m . In polymer stabilized dispersions, we 0nd that the aggregation of particles sets in gradually when cooling the dispersion down towards Tc . This is in contrast to electrostatic stabilization where >occulation occurs in a very narrow temperature interval (see Ref. [16]). 9. Final remarks In this article we have demonstrated that the combination of two soft materials, nematic liquid crystals and colloidal dispersions, creates a novel challenging system for discovering and studying new physical e9ects and ideas. Colloidal dispersions in a nematic liquid crystal introduce a new class of long-range twoparticle interactions mediated by the distorted director 0eld. They are of either dipolar or quadrupolar type depending on whether the single particles exhibit the dipole, Saturn-ring or surface-ring con0guration. The dipolar forces were veri0ed in an excellent experiment by Poulin et al. [179]. Via the well-known >exoelectric e9ect [147], strong director distortions in the dipole con0guration should induce an electric dipole associated with each particle. It would be interesting to study, both theoretically and experimentally, how this electric dipole contributes to the dipolar force. On the other hand, there exists a strong short-range repulsion between particles due to the presence of a hyperbolic point defect which prevents, e.g., water droplets from coalescing. Even above the nematic-isotropic phase transition, liquid crystals mediate an attractive interaction at a length scale of 10 nm. Its strength is easily controlled by temperature, and it produces an observable e9ect since it can induce >occulation when the system is close to the phase transition. To understand colloidal dispersions in nematics in detail, we have performed an extensive study of the three possible director con0gurations around a single particle. These con0gurations are ideal objects to investigate the properties of topological point and line defects. The dipolar structure should exhibit a twist in conventional calamitic compounds. The transition from the dipole to the Saturn ring can be controlled, e.g., by a magnetic 0eld which presents a means to access the dynamics of topological defects. Furthermore, we have studied how the strength of surface anchoring in>uences the director con0guration. Surface e9ects are of considerable importance in display technology, and there is fundamental interest in understanding the coupling between liquid crystal molecules and surfaces. In addition, we have clari0ed the mechanism due to which the saddle-splay term in the Frank free energy promotes the formation of the
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surface-ring structure. The Stokes drag and Brownian motion in nematics have hardly been studied experimentally. Especially the dipole con0guration with its vector symmetry presents an appealing object. We have calculated the Stokes drag for a 0xed director 0eld. However, we have speculated that for small Ericksen numbers (Er 1) >ow-induced distortions of the director 0eld result in corrections to the Stokes drag which are of the order of Er. Preliminary studies support this conclusion. Furthermore, for growing Er they reveal a highly non-linear Stokes drag whose consequences seemed to have not been explored in colloidal physics. Finally, we have demonstrated that the dipole, consisting of the spherical particle and its companion point defect, also exists in more complex geometries, and we have studied in detail how it forms. Acknowledgements I am grateful to Tom Lubensky, Philippe Poulin and Dave Weitz for their close and inspiring collaboration on nematic emulsions at the University of Pennsylvania which initiated the present work. I thank Anamarija Bor\stnik, Andreas RUudinger, Joachim Stelzer, Dieter Ventzki, \ and Slobodan Zumer for collaborating on di9erent aspects of nematic colloidal dispersions. The results are presented in this review article. A lot of thanks to Eugene Gartland, Thomas Gisler, Randy Kamien, Axel Kilian, R. Klein, G. Maret, R.B. Meyer, Michael Reichenstein, E. Sackmann, Thorsten Seitz, Eugene Terentjev, and H.-R. Trebin for fruitful discussions. Finally, I acknowledge 0nancial assistance from the Deutsche Forschungsgemeinschaft through grants Sta 352=2-1=2 and Tr 154=17-1=2. References [1] A.A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5 (1957) 1174–1182. [2] A. Ajdari, B. Duplantier, D. Hone, L. Peliti, J. Prost, “Pseudo-Casimir” e9ect in liquid crystals, J. Phys. II France 2 (1992) 487–501. [3] A. Ajdari, L. Peliti, J. Prost, Fluctuation-induced long-range forces in liquid crystals, Phys. Rev. Lett. 66 (1991) 1481–1484. [4] E. Allahyarov, I. D’Amico, H. LUowen, Attraction between like-charged macroions by Coulomb depletion, Phys. Rev. Lett. 81 (1998) 1334–1337. [5] D.W. Allender, G.P. Crawford, J.W. Doane, Determination of the liquid-crystal surface elastic constant K24 , Phys. Rev. Lett. 67 (1991) 1442–1445. [6] D. Andrienko, G. Germano, M.P. Allen, Computer simulation of topological defects around a colloidal particle or droplet dispersed in a nematic host, Phys. Rev. E., to be published. [7] R.J. Atkin, Poiseuille >ow of liquid crystals of the nematic type, Arch. Rational Mech. Anal. 38 (1970) 224–240. [8] R.J. Atkin, F.M. Leslie, Couette >ow of nematic liquid crystals, Q. J. Mech. Appl. Math. 23 (1970) S3–S24. [9] G. Barbero, C. Oldano, Derivative-dependent surface-energy terms in nematic liquid crystals, Nuovo Cimento D 6 (1985) 479–493. [10] J.L. Billeter, R.A. Pelcovits, Defect con0gurations and dynamical behavior in a Gay-Berne nematic emulsion, Phys. Rev. E 62 (2000) 711–717. [11] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. [12] C. Blanc, M. Kleman, The con0nement of smectics with a strong anchoring, Eur. Phys. J. E., to be published.
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[13] G. Blatter, M.V. Feigel’mann, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Vortices in high-temperature superconductors, Rev. Mod. Phys. 66 (1994) 1125–1388. [14] L.M. Blinov, A.Y. Kabayenkov, A.A. Sonin, Experimental studies of the anchoring energy of nematic liquid crystals, Liq. Cryst. 5 (1989) 645–661. \ [15] A. Bor\stnik, H. Stark, S. Zumer, Interaction of spherical particles dispersed in liquid crystals above the nematic–isotropic phase transition, Phys. Rev. E 60 (4) (1999) 4210 – 4218. ∗ \ [16] A. Bor\stnik, H. Stark, S. Zumer, Temperature-induced >occulation of colloidal particles immersed into the isotropic phase of a nematic liquid crystal, Phys. Rev. E 61 (3) (2000) 2831–2839. ∗∗ \ [17] A. Bor\stnik, H. Stark, S. Zumer, Temperature-induced >occulation of colloidal particles above the nematic-isotropic phase transition, Prog. Colloid Polym. Sci. 115 (2000) 353–356. \ [18] A. Bor\stnik, S. Zumer, Forces in an inhomogeneously ordered nematic liquid crystal, Phys. Rev. E 56 (1997) 3021–3027. [19] A. BUottger, D. Frenkel, E. van de Riet, R. Zijlstra, Di9usion of Brownian particles in the isotropic phase of a nematic liquid crystal, Mol. Cryst. Liq. Cryst. 2 (1987) 539 –547. ∗ [20] W.E. Boyce, R.C. Di Prima, Elementary Di9erential Equations, Wiley, New York, 1992. [21] A. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43 (1994) 357–459. [22] F. Brochard, P.G. de Gennes, Theory of magnetic suspensions in liquid crystals, J. Phys. (Paris) 31 (1970) 691–708. ∗∗ [23] R. Bubeck, C. Bechinger, S. Neser, P. Leiderer, Melting and reentrant freezing of two-dimensional colloidal crystals in con0ned geometry, Phys. Rev. Lett. 82 (1999) 3364–3367. [24] S.V. Burylov, Y.L. Raikher, Orientation of a solid particle embedded in a monodomain nematic liquid crystal, Phys. Rev. E 50 (1994) 358–367. [25] H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd Edition, Wiley, New York, 1985. [26] S. Candau, P.L. Roy, F. Debeauvais, Magnetic 0eld e9ects in nematic and cholesteric droplets suspended in an isotropic liquid, Mol. Cryst. Liq. Cryst. 23 (1973) 283–297. [27] P. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1995. [28] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943) 1–89. [29] S. Chandrasekhar, Liquid Crystals, 2nd Edition, Cambridge University Press, Cambridge, 1992. [30] S. Chandrasekhar, G. Ranganath, The structure and energetics of defects in liquid crystals, Adv. Phys. 35 (1986) 507–596. [31] S.-H. Chen, N.M. Amer, Observation of macroscopic collective behavior and new texture in magnetically doped liquid crystals, Phys. Rev. Lett. 51 (1983) 2298–2301. [32] S. Chono, T. Tsuji, Numerical simulation of nematic liquid crystalline >ows around a circular cylinder, Mol. Cryst. Liq. Cryst. 309 (1998) 217–236. [33] A.J. Chorin, A numerical method for solving incompressible viscous >ow problems, J. Comput. Phys. 2 (1967) 12–26. [34] I. Chuang, R. Durrer, N. Turok, B. Yurke, Cosmology in the laboratory: defect dynamics in liquid crystals, Science 251 (1991) 1336–1342. [35] P.E. Cladis, M. KlXeman, Non-singular disclinations of strength S =+1 in nematics, J. Phys. (Paris) 33 (1972) 591–598. [36] P.E. Cladis, M. KlXeman, P. PiXeranski, Sur une nouvelle mXethode de dXecoration de la mXesomorphe du p, n-mXethoxybenzilid_ene p-bXetylaniline (MBBA), C. R. Acad. Sci. Ser. B 273 (1971) 275–277. [37] P.E. Cladis, W. van Saarloos, P.L. Finn, A.R. Kortan, Dynamics of line defects in nematic liquid crystals, Phys. Rev. Lett. 58 (1987) 222–225. [38] H.J. Coles, Laser and electric 0eld induced birefringence studies on the cyanobiphenyl homologues, Mol. Cryst. Liq. Cryst. Lett. 49 (1978) 67–74. [39] P. Collings, Private communication, 1995. [40] G.P. Crawford, D.W. Allender, J.W. Doane, Surface elastic and molecular-anchoring properties of nematic liquid crystals con0ned to cylindrical cavities, Phys. Rev. A 45 (1992) 8693–8708. [41] G.P. Crawford, D.W. Allender, J.W. Doane, M. Vilfan, I. Vilfan, Finite molecular anchoring in the escaped-radial nematic con0guration: A 2 H-NMR study, Phys. Rev. A 44 (1991) 2570–2576.
468
H. Stark / Physics Reports 351 (2001) 387–474
\ [42] G.P. Crawford, R. Ondris-Crawford, S. Zumer, J.W. Doane, Anchoring and orientational wetting transitions of con0ned liquid crystals, Phys. Rev. Lett. 70 (1993) 1838–1841. \ [43] G.P. Crawford, R.J. Ondris-Crawford, J.W. Doane, S. Zumer, Systematic study of orientational wetting and anchoring at a liquid-crystal–surfactant interface, Phys. Rev. E 53 (1996) 3647–3661. \ [44] G.P. Crawford, S. Zumer (Eds.), Liquid Crystals in Complex Geometries, Taylor & Francis, London, 1996. [45] J.C. Crocker, D.G. Grier, When like charges attract: The e9ects of geometrical con0nement on long-range colloidal interactions, Phys. Rev. Lett. 77 (1996) 1897–1900. [46] P.K. Currie, Couette >ow of a nematic liquid crystal in the presence of a magnetic 0eld, Arch. Rational Mech. Anal. 37 (1970) 222–242. [47] P.K. Currie, Apparent viscosity during viscometric >ow of nematic liquid crystals, J. Phys. (Paris) 40 (1979) 501–505. [48] P.G. de Gennes, Short range order e9ects in the isotropic phase of nematics and cholesterics, Mol. Cryst. Liq. Cryst. 12 (1971) 193–214. [49] P.G. de Gennes, Nematodynamics, in: R. Balian, G. Weill (Eds.), Molecular Fluids, Gordon and Breach, London, 1976, pp. 373–400. [50] P.G. de Gennes, Interactions between solid surfaces in a presmectic >uid, Langmuir 6 (1990) 1448–1450. [51] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd Edition, International Series of Monographs on Physics, Vol. 83, Oxford Science, Oxford, 1993. [52] S.R. de Groot, Thermodynamics of Irreversible Processes, Selected Topics in Modern Physics, North-Holland, Amsterdam, 1951. [53] B.V. Derjaguin, Friction and adhesion. IV: The theory of adhesion of small particles, Kolloid Z. 69 (1934) 155–164. [54] B.V. Derjaguin, L. Landau, Theory of the stability of strongly charged lyophobic sols and the adhesion of strongly charged particles in solutions of electrolytes, Acta Physicochim. URSS 14 (1941) 633–662. [55] A.D. Dinsmore, D.T. Wong, P. Nelson, A.G. Yodh, Hard spheres in vesicles: Curvature-induced forces and particle-induced curvature, Phys. Rev. Lett. 80 (1998) 409–412. [56] A.D. Dinsmore, A.G. Yodh, D.J. Pine, Entropic control of particle motion using passive surface microstructures, Nature 383 (1996) 239–244. [57] A.C. Diogo, Friction drag on a sphere moving in a nematic liquid crystal, Mol. Cryst. Liq. Cryst. 100 (1983) 153–165. \ [58] J.W. Doane, N.A. Vaz, B.G. Wu, S. Zumer, Field controlled light scattering from nematic microdroplets, Appl. Phys. Lett. 48 (1986) 269–271. [59] I. Dozov, Private communication, 1999. [60] P.S. Drzaic, Liquid Crystal Dispersions, Series on Liquid Crystals, Vol. 1, World Scienti0c, Singapore, 1995. X [61] E. Dubois-Violette, O. Parodi, Emulsions nXematiques. E9ects de champ magnXetiques et e9ets piXezoXelectriques, J. Phys. (Paris) Coll. C4 30 (1969) 57–64. [62] R. Eidenschink, W.H. de Jeu, Static scattering in 0lled nematic: new liquid crystal display technique, Electron. Lett. 27 (1991) 1195. ∗ U [63] A. Einstein, Uber die von der molekularkinetischen Theorie der WUarme geforderte Bewegung von in ruhenden FlUussigkeiten suspendierten Teilchen, Ann. Phys. (Leipzig) 17 (1905) 549–560. [64] A. Einstein, Eine neue Bestimmung der MolekUuldimensionen, Ann. Phys. (Leipzig) 19 (1906) 289–306. [65] A. Einstein, Zur Theorie der Brownschen Bewegung, Ann. Phys. (Leipzig) 19 (1906) 371–381. \ [66] J.H. Erdmann, S. Zumer, J.W. Doane, Con0guration transition in a nematic liquid crystal con0ned to a small spherical cavity, Phys. Rev. Lett. 64 (1990) 1907–1910. [67] J.L. Ericksen, Anisotropic >uids, Arch. Rational Mech. Anal. 4 (1960) 231–237. [68] J.L. Ericksen, Theory of anisotropic >uids, Trans. Soc. Rheol. 4 (1960) 29–39. [69] J.L. Ericksen, Conservation laws of liquid crystals, Trans. Soc. Rheol. 5 (1961) 23–34. [70] J.L. Ericksen, Continuum theory of liquid crystals, Appl. Mech. Rev. 20 (1967) 1029–1032. [71] J.L. Ericksen, Continuum theory of liquid crystals of nematic type, Mol. Cryst. Liq. Cryst. 7 (1969) 153–164. [72] A.C. Eringen (Ed.), Continuum Physics: Vols. I–IV, Academic Press, New York, 1976. [73] A.C. Eringen, C.B. Kafadar, Part I: Polar 0eld theories, in: A.C. Eringen (Ed.), Continuum Physics: Polar and Nonlocal Field Theories, Vol. IV, Academic Press, New York, 1976, pp. 1–73.
H. Stark / Physics Reports 351 (2001) 387–474
469
[74] J. Fang, E. Teer, C.M. Knobler, K.-K. Loh, J. Rudnick, Boojums and the shapes of domains in monolayer 0lms, Phys. Rev. E 56 (1997) 1859–1868. [75] A.M. Figueiredo Neto, M.M.F. Saba, Determination of the minimum concentration of ferro>uid required to orient nematic liquid crystals, Phys. Rev. A 34 (1986) 3483–3485. [76] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, Frontiers in Physics: A Lecture Note and Reprint Series, Vol. 47, W.A. Benjamin, Massachusetts, 1975. [77] D. Forster, T. Lubensky, P. Martin, J. Swift, P. Pershan, Hydrodynamics of liquid crystals, Phys. Rev. Lett. 26 (1971) 1016–1019. [78] S. Fraden, Phase transitions in colloidal suspensions of virus particles, in: M. Baus, L.F. Rull, J.P. Ryckaert (Eds.), Observation, Prediction, and Simulation of Phase Transitions in Complex Fluids, NATO Advanced Studies Institute Series C: Mathematical and Physical Sciences, Vol. 460, Kluwer, Dordrecht, 1995, pp. 113–164. [79] F.C. Frank, I. Liquid Crystals: On the theory of liquid crystals, Discuss. Faraday Soc. 25 (1958) 19–28. [80] G. Friedel, F. Grandjean, Observation gXeomXetriques sur les liquides aX coniques focales, Bull. Soc. Fr. Mineral 33 (1910) 409–465. [81] G. Friedel, Dislocations, Pergamon Press, Oxford, 1964. U [82] N. Fuchs, Uber die StabilitUat und Au>adung der Aerosole, Z. Phys. 89 (1934) 736–743. [83] J. Fukuda, H. Yokoyama, Director con0guration and dynamics of a nematic liquid crystal around a spherical particle: numerical analysis using adaptive grids, Eur. Phys. J. E., to be published. ∗ [84] C. GUahwiller, Direct determination of the 0ve independent viscosity coeNcients of nematic liquid crystals, Mol. Cryst. Liq. Cryst. 20 (1973) 301–318. [85] P. Galatola, J.B. Fournier, Nematic-wetted colloids in the isotropic phase, Mol. Cryst. Liq. Cryst. 330 (1999) 535–539. [86] A. Garel, Boundary conditions for textures and defects, J. Phys. (Paris) 39 (1978) 225–229. [87] E.C. Gartland, Private communication, 1998. [88] E.C. Gartland, S. Mkaddem, Instability of radial hedgehog con0gurations in nematic liquid crystals under Landau–de Gennes free-energy models, Phys. Rev. E 59 (1999) 563–567. [89] A.P. Gast, W.B. Russel, Simple ordering in complex >uids, Phys. Today 51 (1998) 24–30. [90] A.P. Gast, C.F. Zukoski, Electrorheological >uids as colloidal suspensions, Adv. Colloid Interface Sci. 30 (1989) 153–202. [91] A. Glushchenko, H. Kresse, V. Reshetnyak, Yu. Reznikov, O. Yaroshchuk, Memory e9ect in 0lled nematic liquid crystals, Liq. Cryst. 23 (1997) 241–246. [92] J. Goldstone, Field theories with “superconductor” solutions, Nuovo Cimento 19 (1) (1961) 154–164. [93] J. Goldstone, A. Salam, S. Weinberg, Broken symmetries, Phys. Rev. 127 (1962) 965–970. [94] J.W. Goodby, M.A. Waugh, S.M. Stein, E. Chin, R. Pindak, J.S. Patel, Characterization of a new helical smectic liquid crystal, Nature 337 (1989) 449–452. [95] J.W. Goodby, M.A. Waugh, S.M. Stein, E. Chin, R. Pindak, J.S. Patel, A new molecular ordering in helical liquid crystals, J. Am. Chem. Soc. 111 (1989) 8119–8125. [96] E. Gramsbergen, L. Longa, W.H. de Jeu, Landau theory of the nematic–isotropic phase transition, Phys. Rep. 135 (1986) 195–257. [97] Y. Gu, N.L. Abbott, Observation of saturn-ring defects around solid microspheres in nematic liquid crystals, Phys. Rev. Lett. 85 (2000) 4719 – 4722. ∗∗ [98] M.J. Guardalben, N. Jain, Phase-shift error as a result of molecular alignment distortions in a liquid–crystal point-di9raction interferometer, Opt. Lett. 25 (2000) 1171–1173. [99] Groupe d’Etude des Cristaux Liquides, Dynamics of >uctuations in nematic liquid crystals, J. Chem. Phys. 51 (1969) 816 –822. [100] P. HUanggi, P. Talkner, M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62 (1990) 251–341. [101] I. Haller, W.R. Young, G.L. Gladstone, D.T. Teaney, Crown ether complex salts as conductive dopants for nematic liquids, Mol. Cryst. Liq. Cryst. 24 (1973) 249–258. [102] W. Helfrich, Capillary >ow of cholesteric and smectic liquid crystals, Phys. Rev. Lett. 23 (1969) 372–374.
470
H. Stark / Physics Reports 351 (2001) 387–474
[103] G. Heppke, D. KrUuerke, M. MUuller, Surface anchoring of the discotic cholesteric phase of chiral pentayne systems, in: Abstract Book, Vol. 24, Freiburger Arbeitstagung FlUussigkristalle, Freiburg, Germany, 1995. [104] H. Heuer, H. Kneppe, F. Schneider, Flow of a nematic liquid crystal around a cylinder, Mol. Cryst. Liq. Cryst. 200 (1991) 51–70. [105] H. Heuer, H. Kneppe, F. Schneider, Flow of a nematic liquid crystal around a sphere, Mol. Cryst. Liq. Cryst. 214 (1992) 43–61. [106] R.G. Horn, J.N. Israelachvili, E. Perez, Forces due to structure in a thin liquid crystal 0lm, J. Phys. (Paris) 42 (1981) 39–52. [107] H. Imura, K. Okano, Friction coeNcient for a moving disclination in a nematic liquid crystal, Phys. Lett. A 42 (1973) 403–404. [108] A. JXakli, L. AlmXasy, S. BorbXely, L. Rosta, Memory of silica aggregates dispersed in smectic liquid crystals: E9ect of the interface properties, Eur. Phys. J. B 10 (1999) 509–513. [109] G.M. Kepler, S. Fraden, Attractive potential between con0ned colloids at low ionic strength, Phys. Rev. Lett. 73 (1994) 356–359. [110] R. Klein, Interacting brownian particles – the dynamics of colloidal suspensions, in: F. Mallamace, H.E. Stanley (Eds.), The Physics of Complex Systems, IOS Press, Amsterdam, 1997, pp. 301–345. [111] M. KlXeman, Points, Lines and Walls: in Liquid Crystals, Magnetic Systems, and Various Ordered Media, Wiley, New York, 1983. [112] H. Kneppe, F. Schneider, B. Schwesinger, Axisymmetrical >ow of a nematic liquid crystal around a sphere, Mol. Cryst. Liq. Cryst. 205 (1991) 9–28. [113] K. Ko\cevar, R. Blinc, I. Mu\sevi\c, Atomic force microscope evidence for the existence of smecticlike surface layers in the isotropic phase of a nematic liquid crystal, Phys. Rev. E 62 (2000) R3055 –R3058. ∗ [114] S. Komura, R.J. Atkin, M.S. Stern, D.A. Dunmur, Numerical analysis of the radial–axial structure transition with an applied 0eld in a nematic droplet, Liq. Cryst. 23 (1997) 193–203. \ [115] S. Kralj, S. Zumer, FrXeedericksz transitions in supra-m nematic droplets, Phys. Rev. A 45 (1992) 2461–2470. [116] M. Krech, The Casimir E9ect in Critical Systems, World Scienti0c, Singapore, 1994. \ [117] M. Kreuzer, R. Eidenschink, Filled nematics, in: G.P. Crawford, S. Zumer (Eds.), Liquid Crystals in Complex Geometries, Taylor & Francis, London, 1996, pp. 307–324. [118] M. Kreuzer, T. Tschudi, R. Eidenschink, Erasable optical storage in bistable liquid crystal cells, Mol. Cryst. Liq. Cryst. 223 (1992) 219 –227. ∗ [119] O.V. Kuksenok, R.W. Ruhwandl, S.V. Shiyanovskii, E.M. Terentjev, Director structure around a colloid particle suspended in a nematic liquid crystal, Phys. Rev. E 54 (1996) 5198–5203. ∗∗ [120] M. Kurik, O. Lavrentovich, Negative-positive monopole transitions in cholesteric liquid crystals, JETP Lett. 35 (1982) 444–447. [121] M.V. Kurik, O.D. Lavrentovich, Defects in liquid crystals: Homotopy theory and experimental studies, Sov. Phys. Usp. 31 (1988) 196–224. [122] E. Kuss, pVT -data and viscosity-pressure behavior of MBBA and EBBA, Mol. Cryst. Liq. Cryst. 47 (1978) 71–83. [123] L.D. Landau, E.M. Lifschitz, Electrodynamics of Continuous Media, Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford, 0rst English Edition, 1960. [124] L.D. Landau, E.M. Lifschitz, Statistische Physik, Teil 1, Lehrbuch der Theoretischen Physik, Vol. 5, 6th Edition, Akademie, Berlin, 1984. [125] A.E. Larsen, D.G. Grier, Like-charge attractions in metastable colloidal crystallites, Nature 385 (1997) 230–233. [126] O. Lavrentovich, E. TerentXev, Phase transition altering the symmetry of topological point defects (hedgehogs) in a nematic liquid crystal, Sov. Phys. JETP 64 (1986) 1237–1244. [127] O.D. Lavrentovich, Topological defects in dispersed liquid crystals, or words and worlds around liquid crystal drops, Liq. Cryst. 24 (1998) 117–125. [128] F.M. Leslie, Some constitutive equations for anisotropic >uids, Quart. J. Mech. Appl. Math. 19 (1966) 357–370. [129] F.M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28 (1968) 265–283.
H. Stark / Physics Reports 351 (2001) 387–474
471
[130] B.I. Lev, P.M. Tomchuk, Interaction of foreign macrodroplets in a nematic liquid crystal and induced supermolecular structures, Phys. Rev. E 59 (1999) 591–602. [131] S. Lifson, J.L. Jackson, On the self-di9usion of ions in a polyelectrolyte solution, J. Chem. Phys. 36 (1962) 2410–2414. [132] D. Link, N. Clark, Private communication, 1997. [133] J. Liu, E.M. Lawrence, A. Wu, M.L. Ivey, G.A. Flores, K. Javier, J. Bibette, J. Richard, Field-induced structures in ferro>uid emulsions, Phys. Rev. Lett. 74 (1995) 2828–2831. [134] H. LUowen, Kolloide – auch fUur Physiker interessant, Phys. Bl. 51 (3) (1995) 165–168. [135] H. LUowen, Solvent-induced phase separation in colloidal >uids, Phys. Rev. Lett. 74 (1995) 1028–1031. [136] H. LUowen, Phase separation in colloidal suspensions induced by a solvent phase transition, Z. Phys. B 97 (1995) 269–279. [137] J.-C. Loudet, P. Barois, P. Poulin, Colloidal ordering from phase separation in a liquid-crystalline continuous phase, Nature 407 (2000) 611– 613. ∗∗ [138] T.C. Lubensky, Molecular description of nematic liquid crystals, Phys. Rev. A 2 (1970) 2497–2514. [139] T.C. Lubensky, Hydrodynamics of cholesteric liquid crystals, Phys. Rev. A 6 (1972) 452–470. [140] T.C. Lubensky, D. Pettey, N. Currier, H. Stark, Topological defects and interactions in nematic emulsions, Phys. Rev. E 57 (1998) 610 – 625. ∗∗∗ [141] I.F. Lyuksyutov, Topological instability of singularities at small distances in nematics, Sov. Phys. JETP 48 (1978) 178–179. [142] W. Maier, A. Saupe, Eine einfache molekular-statistische Theorie der nematischen kristallin>Uussigen Phase. Teil I, Z. Naturforsch. Teil A 14 (1959) 882–889. [143] S.P. Meeker, W.C.K. Poon, J. Crain, E.M. Terentjev, Colloid-liquid-crystal composites: An unusual soft solid, Phys. Rev. E 61 (2000) R6083–R6086. ∗∗ [144] S. Meiboom, M. Sammon, W.F. Brinkman, Lattice of disclinations: The structure of the blue phases of cholesteric liquid crystals, Phys. Rev. A 27 (1983) 438–454. [145] N. Mermin, Surface singularities and super>ow in 3 He-A, in: S.B. Trickey, E.D. Adams, J.W. Dufty (Eds.), Quantum Fluids and Solids, Plenum Press, New York, 1977, pp. 3–22. [146] N.D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (1979) 591–648. [147] R.B. Meyer, Piezoelectric e9ects in liquid crystals, Phys. Rev. Lett. 22 (1969) 918–921. [148] R.B. Meyer, Point disclinations at a nematic-isotropic liquid interface, Mol. Cryst. Liq. Cryst. 16 (1972) 355 –369. ∗ [149] R.B. Meyer, On the existence of even indexed disclinations in nematic liquid crystals, Philos. Mag. 27 (1973) 405–424. [150] M. MiZesovicz, In>uence of a magnetic 0eld on the viscosity of para-azoxyanisol, Nature 136 (1935) 261. [151] MiZesowicz, The three coeNcients of viscosity of anisotropic liquids, Nature 158 (1946) 27. [152] V.P. Mineev, Topologically stable defects and solitons in ordered media, in: I.M. Khalatnikov (Ed.), Soviet Scienti0c Reviews, Section A: Physics Reviews, Vol. 2, Harwood, London, 1980, pp. 173–246. [153] O. Mondain-Monval, J.C. Dedieu, T. Gulik-Krzywicki, P. Poulin, Weak surface energy in nematic dispersions: saturn ring defects and quadrupolar interactions, Eur. Phys. J. B 12 (1999) 167–170. ∗∗ [154] L. Moreau, P. Richetti, P. Barois, Direct measurement of the interaction between two ordering surfaces con0ning a presmectic 0lm, Phys. Rev. Lett. 73 (1994) 3556–3559. [155] H. Mori, H. Nakanishi, On the stability of topologically non-trivial point defects, J. Phys. Soc. Japan 57 (1988) 1281–1286. [156] A.H. Morrish, The Physical Principles of Magnetism, Wiley Series on the Science and Technology of Materials, Wiley, New York, 1965. [157] V.M. Mostepanenko, N.N. Trunov, The Casimir E9ect and its Application, Clarendon Press, Oxford, 1997. [158] I. Mu\sevi\c, G. Slak, R. Blinc, Observation of critical forces in a liquid crystal by an atomic force microscope, in: Proceedings of the 16th International Liquid Crystal Conference, Abstract Book, Kent, USA, 1996, p. 91. [159] I. Mu\sevi\c, G. Slak, R. Blinc, Temperature controlled microstage for an atomic force microscope, Rev. Sci. Instrum. 67 (1996) 2554–2556. [160] F.R.N. Nabarro, Theory of Crystal Dislocations, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1967.
472
H. Stark / Physics Reports 351 (2001) 387–474
[161] H. Nakanishi, K. Hayashi, H. Mori, Topological classi0cation of unknotted ring defects, Commun. Math. Phys. 117 (1988) 203–213. [162] D.H. Napper, Polymeric Stabilization of Colloidal Dispersions, Colloid Science, Vol. 3, Academic Press, London, 1983. [163] J. Nehring, A. Saupe, On the elastic theory of uniaxial liquid crystals, J. Chem. Phys. 54 (1971) 337–343. [164] D.R. Nelson, Defect-mediated Phase Transitions, in: C. Domb, J. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 7, Academic Press, New York, 1983, pp. 1–99. [165] M. Nobili, G. Durand, Disorientation-induced disordering at a nematic-liquid-crystal–solid interface, Phys. Rev. A 46 (1992) R6174–R6177. [166] C. Oldano, G. Barbero, An ab initio analysis of the second-order elasticity e9ect on nematic con0gurations, Phys. Lett. 110A (1985) 213–216. [167] C.W. Oseen, The theory of liquid crystals, Trans. Faraday Soc. 29 (1933) 883–899. [168] M.A. Osipov, S. Hess, The elastic constants of nematic discotic liquid crystals with perfect local orientational order, Mol. Phys. 78 (1993) 1191–1201. [169] S. Ostlund, Interactions between topological point singularities, Phys. Rev. E 24 (1981) 485–488. [170] O. Parodi, Stress Tensor for a nematic liquid crystal, J. Phys. (Paris) 31 (1970) 581–584. [171] E. Penzenstadler, H.-R. Trebin, Fine structure of point defects and soliton decay in nematic liquid crystals, J. Phys. France 50 (1989) 1027–1040. [172] V.M. Pergamenshchik, Surfacelike-elasticity-induced spontaneous twist deformations and long-wavelength stripe domains in a hybrid nematic layer, Phys. Rev. E 47 (1993) 1881–1892. [173] V.M. Pergamenshchik, P.I. Teixeira, T.J. Sluckin, Distortions induced by the k13 surfacelike elastic term in a thin nematic liquid-crystal 0lm, Phys. Rev. E 48 (1993) 1265–1271. [174] V.M. Pergamenshchik, K13 term and e9ective boundary condition for the nematic director, Phys. Rev. E 58 (1998) R16–R19. [175] D. Pettey, T.C. Lubensky, D. Link, Topological inclusions in 2D smectic C 0lms, Liq. Cryst. 25 (1998) 579 –587. ∗ [176] P. Pieranski, F. Brochard, E. Guyon, Static and dynamic behavior of a nematic liquid crystal in a magnetic 0eld. Part II: Dynamics, J. Phys. (Paris) 34 (1973) 35–48. [177] A. Poniewierski, T. Sluckin, Theory of the nematic-isotropic transition in a restricted geometry, Liq. Cryst. 2 (1987) 281–311. [178] P. Poulin, Private communication, 1999. [179] P. Poulin, V. Cabuil, D.A. Weitz, Direct measurement of colloidal forces in an anisotropic solvent, Phys. Rev. Lett. 79 (1997) 4862– 4865. ∗∗ [180] P. Poulin, N. Franc_es, O. Mondain-Monval, Suspension of spherical particles in nematic solutions of disks and rods, Phys. Rev. E 59 (1999) 4384 – 4387. ∗ [181] P. Poulin, V.A. Raghunathan, P. Richetti, D. Roux, On the dispersion of latex particles in a nematic solution. I. Experimental evidence and a simple model, J. Phys. II France 4 (1994) 1557–1569. ∗ [182] P. Poulin, H. Stark, T.C. Lubensky, D.A. Weitz, Novel colloidal interactions in anisotropic >uids, Science 275 (1997) 1770 –1773. ∗∗∗ [183] P. Poulin, D.A. Weitz, Inverted and multiple emulsions, Phys. Rev. E 57 (1998) 626 – 637. ∗∗∗ [184] P. Poulin, Novel phases and colloidal assemblies in liquid crystals, Curr. Opinion in Colloid & Interface Science 4 (1999) 66–71. [185] M.J. Press, A.S. Arrott, Theory and experiment of con0gurations with cylindrical symmetry in liquid-crystal droplets, Phys. Rev. Lett. 33 (1974) 403–406. [186] M.J. Press, A.S. Arrott, Elastic energies and director 0elds in liquid crystal droplets, I. Cylindrical symmetry, J. Phys. (Paris) Coll. C1 36 (1975) 177–184. [187] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran: The Art of Scienti0c Computing, Cambridge University Press, Cambridge, 1992. [188] V.A. Raghunathan, P. Richetti, D. Roux, Dispersion of latex particles in a nematic solution. II. Phase diagram and elastic properties, Langmuir 12 (1996) 3789–3792. [189] V.A. Raghunathan, P. Richetti, D. Roux, F. Nallet, A.K. Sood, Colloidal dispersions in a liquid-crystalline medium, Mol. Cryst. Liq. Cryst. 288 (1996) 181–187.
H. Stark / Physics Reports 351 (2001) 387–474
473
[190] S. Ramaswamy, R. Nityananda, V.A. Raghunathan, J. Prost, Power-law forces between particles in a nematic, Mol. Cryst. Liq. Cryst. 288 (1996) 175 –180. ∗∗ [191] J. Rault, Sur une mXethode nouvelle d’Xetude de l’orientation molXeculaire aX la surface d’un cholestXerique, C. R. Acad. Sci. Ser. B 272 (1971) 1275–1276. [192] M. Reichenstein, T. Seitz, H.-R. Trebin, Numerical simulations of three dimensional liquid crystal cells, Mol. Cryst. Liq. Cryst. 330 (1999) 549–555. [193] S.R. Renn, T.C. Lubensky, Abrikosov dislocation lattice in a model of the cholesteric–to–smectic-A transition, Phys. Rev. A 38 (1988) 2132–2147. [194] V.G. Roman, E.M. Terentjev, E9ective viscosity and di9usion tensor of an anisotropic suspension or mixture, Colloid J. USSR 51 (1989) 435–442. [195] R. Rosso, E.G. Virga, Metastable nematic hedgehogs, J. Phys. A 29 (1996) 4247–4264. [196] D. Rudhardt, C. Bechinger, P. Leiderer, Direct measurement of depletion potentials in mixtures of colloids and nonionic polymers, Phys. Rev. Lett. 81 (1998) 1330–1333. [197] A. RUudinger, H. Stark, Twist transition in nematic droplets: A stability analysis, Liq. Cryst. 26 (1999) 753–758. [198] R.W. Ruhwandl, E.M. Terentjev, Friction drag on a cylinder moving in a nematic liquid crystal, Z. Naturforsch. Teil A 50 (1995) 1023–1030. [199] R.W. Ruhwandl, E.M. Terentjev, Friction drag on a particle moving in a nematic liquid crystal, Phys. Rev. E 54 (1996) 5204 –5210. ∗∗ [200] R.W. Ruhwandl, E.M. Terentjev, Long-range forces and aggregation of colloid particles in a nematic liquid crystal, Phys. Rev. E 55 (1997) 2958–2961. ∗ [201] R.W. Ruhwandl, E.M. Terentjev, Monte Carlo simulation of topological defects in the nematic liquid crystal matrix around a spherical colloid particle, Phys. Rev. E 56 (1997) 5561–5565. ∗∗ [202] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, 1995. [203] G. Ryskin, M. Kremenetsky, Drag force on a line defect moving through an otherwise undisturbed 0eld: Disclination line in a nematic liquid crystal, Phys. Rev. Lett. 67 (1991) 1574–1577. [204] S.D. Sarma, A. Pinczuk (Eds.), Perspectives in Quantum Hall Fluids, Wiley, New York, 1997. [205] M. Schadt, Optisch strukturierte FlUussigkristall-Anzeigen mit groaem Blickwinkelbereich, Phys. Bl. 52 (7=8) (1996) 695–698. [206] M. Schadt, H. Seiberle, A. Schuster, Optical patterning of multi-domain liquid-crystal displays with wide viewing angles, Nature 318 (1996) 212–215. [207] N. Schopohl, T.J. Sluckin, Defect core structure in nematic liquid crystals, Phys. Rev. Lett. 59 (1987) 2582–2584. [208] N. Schopohl, T.J. Sluckin, Hedgehog structure in nematic and magnetic systems, J. Phys. France 49 (1988) 1097–1101. [209] V. Sequeira, D.A. Hill, Particle suspensions in liquid crystalline media: Rheology, structure, and dynamic interactions, J. Rheol. 42 (1998) 203–213. [210] P. Sheng, Phase transition in surface-aligned nematic 0lms, Phys. Rev. Lett. 37 (1976) 1059–1062. [211] P. Sheng, Boundary-layer phase transition in nematic liquid crystals, Phys. Rev. A 26 (1982) 1610–1617. [212] P. Sheng, E.B. Priestly, The Landau–de Gennes theory of liquid crystal phase transition, in: E.B. Priestly, P.J. Wojtowicz, P. Sheng (Eds.), Introduction to Liquid Crystals, Plenum Press, New York, 1979, pp. 143–201. [213] S.V. Shiyanovskii, O.V. Kuksenok, Structural transitions in nematic 0lled with colloid particles, Mol. Cryst. Liq. Cryst. 321 (1998) 45–56. [214] R. Sigel, G. Strobl, Static and dynamic light scattering from the nematic wetting layer in an isotropic crystal, Prog. Colloid Polym. Sci. 104 (1997) 187–190. [215] M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationkinetik kolloider LUosungen, Z. Phys. Chem. (Leipzig) 92 (1917) 129–168. [216] K. Sokalski, T.W. Ruijgrok, Elastic constants for liquid crystals of disc-like molecules, Physica A 113A (1982) 126–132. [217] A. Sommerfeld, Vorlesungen uU ber Theoretische Physik II. Mechanik der deformierbaren Medien, 6th Edition, Verlag Harri Deutsch, Frankfurt, 1978.
474
H. Stark / Physics Reports 351 (2001) 387–474
[218] A. Sonnet, A. Kilian, S. Hess, Alignment tensor versus director: Description of defects in nematic liquid crystals, Phys. Rev. E 52 (1995) 718–722. [219] H. Stark, Director 0eld con0gurations around a spherical particle in a nematic liquid crystal, Eur. Phys. J. B 10 (1999) 311–321. ∗∗ [220] H. Stark, J. Stelzer, R. Bernhard, Water droplets in a spherically con0ned nematic solvent: A numerical investigation, Eur. Phys. J. B 10 (1999) 515 –523. ∗ [221] J. Stelzer, M.A. Bates, L. Longa, G.R. Luckhurst, Computer simulation studies of anisotropic systems. XXVII. The direct pair correlation function of the Gay–Berne discotic nematic and estimates of its elastic constants, J. Chem. Phys. 107 (1997) 7483–7492. [222] K.J. Strandburg, Two-dimensional melting, Rev. Mod. Phys. 60 (1988) 161–207. [223] B.D. Swanson, L.B. Sorenson, What forces bind liquid crystal, Phys. Rev. Lett. 75 (1995) 3293–3296. [224] G.I. Taylor, The mechanism of plastic deformation of crystals, Proc. R. Soc. London, Ser. A 145 (1934) 362–415. [225] E.M. Terentjev, Disclination loops, standing alone and around solid particles, in nematic liquid crystals, Phys. Rev. E 51 (1995) 1330 –1337. ∗∗ [226] H.-R. Trebin, The topology of non-uniform media in condensed matter physics, Adv. Phys. 31 (1982) 195–254. [227] E.H. Twizell, Computational Methods for Partial Di9erential Equations, Chichester, Horwood, 1984. [228] D. Ventzki, H. Stark, Stokes drag of particles suspended in a nematic liquid crystal, in preparation. [229] E.J.W. Verwey, J.T.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [230] D. Vollhardt, P. WUol>e, The Phases of Helium 3, Taylor & Francis, New York, 1990. [231] G.E. Volovik, O.D. Lavrentovich, Topological dynamics of defects: Boojums in nematic drops, Sov. Phys. JETP 58 (1983) 1159–1168. [232] T.W. Warmerdam, D. Frenkel, R.J.J. Zijlstra, Measurements of the ratio of the Frank constants for splay and bend in nematics consisting of disc-like molecules—2,3,6,7,10,11-hexakis(p-alkoxybenzoyloxy)triphenylenes, Liq. Cryst. 3 (1988) 369–380. [233] Q.-H. Wei, C. Bechinger, D. Rudhardt, P. Leiderer, Experimental study of laser-induced melting in two-dimensional colloids, Phys. Rev. Lett. 81 (1998) 2606–2609. [234] A.E. White, P.E. Cladis, S. Torza, Study of liquid crystals in >ow: I. Conventional viscometry and density measurements, Mol. Cryst. Liq. Cryst. 43 (1977) 13–31. [235] J. Wilks, D. Betts, An Introduction to Liquid Helium, Clarendon Press, Oxford, 1987. [236] C. Williams, P. PieraXnski, P.E. Cladis, Nonsingular S = +1 screw disclination lines in nematics, Phys. Rev. Lett. 29 (1972) 90–92. [237] R.D. Williams, Two transitions in tangentially anchored nematic droplets, J. Phys. A 19 (1986) 3211–3222. [238] K. Zahn, R. Lenke, G. Maret, Two-stage melting of paramagnetic colloidal crystals in two dimensions, Phys. Rev. Lett. 82 (1999) 2721–2724. [239] M. Zapotocky, L. Ramos, P. Poulin, T.C. Lubensky, D.A. Weitz, Particle-stabilized defect gel in cholesteric liquid crystals, Science 283 (1999) 209 –212. ∗∗ \ [240] P. Ziherl, R. Podgornik, S. Zumer, Casimir force in liquid crystals close to the nematic-isotropic phase transition, Chem. Phys. Lett. 295 (1998) 99–104. [241] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 3rd Edition, International Series of monographs on Physics, Vol. 92, Oxford Science, Oxford, 1996. [242] H. ZUocher, The e9ect of a magnetic 0eld on the nematic state, Trans. Faraday Soc. 29 (1933) 945–957.
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CONTENTS VOLUME 351 B. Gumhalter. Single- and multiphonon atom}surface scattering in the quantum regime
1
H. Heiselberg. Event-by-event physics in relativistic heavy-ion collisions
161
G.E. Volovik. Super#uid analogies of cosmological phenomena
195
C. Ronning. E.P. Carlson, R.F. Davis. Ion implantation into gallium nitride
349
H. Stark. Physics of colloidal dispersions in nematic liquid crystals
387
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476
FORTHCOMING ISSUES D.G. Yakovlev, A.D. Kaminker, O.Y. Gnedin, P. Haensel. Neutrino emission from neutron stars B. Ananthanarayan, G. Colangelo, J. Gasse, H. Leutwyler. Roy analysis of pi-pi scattering R. Alkofer, L. von Smekal. The infrared behaviour of QCD Green's functions J.-P. Minier, E. Peirano. The PDF approach to turbulent polydispersed two-phase #ows D.F. Measday. The nuclear physics of muon capture C. Schubert. Perturbative quantum "eld theory in the string-inspired formalism M. Bordag, U. Mohideen, V.M. Mostepanenko. New developments in the Casimir e!ect G.E. Mitchell, J.D. Bowman, S.I. PenttilaK , E.I. Sharapov. Parity violation in compound nuclei: experimental methods and results G. Grynberg, C. Robilliard. Cold atoms in dissipative optical lattices I. Pollini, A. Mosser, J.C. Parlebas. Electronic, spectroscopic and elastic properties of early transition metal compounds R. Fazio, H. van der Zant. Quantum phase transitions and vortex dynamics in superconducting networks D.I. Pushkarov. Quasiparticle kinetics and dynamics in nonstationary deformed crystals in the presence of electromagnetic "elds V.M. Shabaev. Two-time Green's function method in quantum electrodynamics of high-Z few-electron atoms G. Bo!etta, M. Cencini, M. Falcioni, A. Vulpiani. Predictability: a way to characterize complexity A. Wacker. Semiconductor superlattices: a model system for nonlinear transport I.L. Shapiro. Physical aspects of the space}time torsion Ya. Kraftmakher. Modulation calorimetry and related techniques M.J. Brunger, S.J. Buckman. Electron}molecule scattering cross sections. I. Experimental techniques and data for diatomic molecules S.Y. Wu, C.S. Jayanthi. Order-N methodologies and their applications
PII: S0370-1573(01)00056-4