Physics Reports vol.312

TOKAMAK PLASMA DIAGNOSTICS BASED ON MEASURED NEUTRON SIGNALS B. WOLLE Institut fu( r Angewandte Physik, Universita( t H...

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TOKAMAK PLASMA DIAGNOSTICS BASED ON MEASURED NEUTRON SIGNALS

B. WOLLE Institut fu( r Angewandte Physik, Universita( t Heidelberg, D-69120 Heidelberg, Germany

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 312 (1999) 1—86

Tokamak plasma diagnostics based on measured neutron signals B. Wolle Institut f u( r Angewandte Physik, Universita( t Heidelberg, D-69120 Heidelberg, Germany Received July 1998; editor: D.L. Mills Contents 1. Introduction 1.1. Neutron diagnostics on tokamak fusion experiments 1.2. General overview 2. Basic theory 2.1. Theoretical background of the fusion neutron emission 2.2. Velocity distributions of ions in magnetically confined plasmas 3. Simulation of D—D neutron emission 3.1. General remarks 3.2. Physics input 3.3. Computation

3 3 4 8 9 16 27 28 29 38

4. Plasma parameters deduced from neutron measurements 4.1. Basic properties of the neutron source strength 4.2. Derivation of ion densities 4.3. Derivation of plasma temperatures 4.4. Information on ion diffusivities 4.5. Studies of MHD activity and fast-ion confinement 5. Discussion Acknowledgements References

49 50 55 58 63 66 70 73 73

Abstract Neutron diagnostics are of increasing importance for future fusion devices. Consequently, efforts are being made to improve the accuracy of underlying experimental and computational methods. The present article reviews the modelling and the analysis of measured neutron signals relevant for plasma diagnostics on tokamaks. The underlying numerical simulation of neutron signals involves various aspects. Firstly, a realistic characterization of the plasma as a neutron source is needed. Secondly, detailed knowledge about changes in energy spectra and total number of the initially emitted neutrons due to scattering and absorption in the volume between the neutron source and the detector system is required. Finally, the detection properties of the measuring systems have to be taken into account. Presently, a sophisticated numerical procedure which directly relates detector signals to physics properties of the emitted neutrons from the plasma is not available and progress is found to be incremental rather than revolutionary. This is mainly attributable to problems with modelling the plasma neutron source based on measured plasma data and modelling difficulties for the neutron transport. However, more recent results of plasma parameters derived from neutron measurements provide evidence for the improvements in the measurement, simulation and analysis procedures over the past two decades. 1999 Elsevier Science B.V. All rights reserved. PACS: 52.55.Fa; 52.70.!m; 52.65.Ff; 52.70.Nc Keywords: Neutron diagnostics; Plasma; Fokker—Planck

0370-1573/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 8 4 - 2

B. Wolle / Physics Reports 312 (1999) 1—86

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1. Introduction During the past few decades intensive research has been undertaken in plasma physics, and in particular in the field of thermonuclear fusion. As a result, the field includes a very substantial body of knowledge ranging from the most theoretical to the most practical topics. Progress has been made most effectively when an early confrontation between theory and experiment has been possible. However, such comparisons require both, theoretical calculations for realistic configurations and conditions, as well as rather detailed and accurately measured plasma properties. For this reason much of the effort in experimental plasma physics is devoted to developing, providing and testing the experimental techniques and the associated theoretical and computational techniques for diagnosing the properties of fusion plasmas. Fusion plasmas can be divided into two kinds: those produced by rapid compression of small fuel pellets by light or ion beams (inertial confinement fusion), and those confined by strong magnetic fields (magnetic confinement fusion). The largest toroidal magnetic confinement device built at the Kurchatov Institute in the 1960s was the T-3 tokamak. The measured plasma data obtained from its pioneered diagnostics equipment showed that the tokamak was capable of confining plasma at temperatures of several hundred keV. This established the tokamak as the leading contender for a thermonuclear confinement system and, hence, over the past decades the international magnetic confinement fusion research programme has been mainly focusing on tokamak fusion devices. The world’s largest magnetic fusion experiments, present and planned, are of this type. Therefore, this review shall be concerned only with tokamak fusion plasmas. 1.1. Neutron diagnostics on tokamak fusion experiments Measurements of the neutron emission were carried out at very early stages in thermonuclear fusion research as the number of produced neutrons is a direct measure of the progress towards the achievement of thermonuclear reactor conditions. Since other more conventional diagnostic systems are incapable of operating quasi-continuously under the high neutron and gamma-ray fluences of a thermonuclear reactor or require substantial radiological shielding, neutron diagnostics are considered to be of increasing importance for future fusion devices [1]. For instance, at the planned international thermonuclear experimental reactor ITER [2—4], neutron diagnostics will play a prominent role in the control and evaluation of thermonuclear plasmas [4—9]. Thus, efforts are being made to improve the accuracy of underlying experimental and computational methods. Neutron measurement techniques suitable for nuclear fusion devices are based on sophisticated developments originally made for fission reactors and in experimental neutron physics [10]. In this connection, the reader is also referred to the very instructive book by Knoll [11], where references to the relevant literature can be found. However, on fusion experiments neutron detectors have to operate in somewhat unusual conditions for neutron physics and require the following additional specifications: 1. 2. 3. 4.

insensitivity to the magnetic field, low sensitivity to gamma rays, ability to operate from the lowest to the highest neutron fluences, tolerance of electromagnetic disturbances,

4


5. ability to withstand mechanical vibrations, 6. rapid processing of the measurement signals, and 7. ability to operate within the wide energy range of the neutron field. The variety of neutron measurements that can be made on fusion experiments is limited to measuring the total emission strength, the relative spatial emissivity in the plasma, and the energy spectra of neutrons emitted from small plasma volumes or from selected lines-of-sight through the plasma. Furthermore, for obtaining a time-resolution suitable for diagnostics on fusion experiments some relevant measurement techniques require neutron fluxes which are achievable only with the larger experimental devices. Although neutron measurements for plasma diagnostics have essentially been pioneered at the T-3 tokamak in the late 1960s and the early 1970s [12], historically, the year 1981 can be regarded as a turning point or maybe even marks the birth of modern neutron diagnostics on tokamak experiments. In this year, the TRANSP code [13] became available, the first activation measurements were carried out at PLT [14], and a time-of-flight spectrometer suitable for diagnostics of extended fusion plasmas was presented [15]. Then, the third version of the neutron transport code MCNP was released in 1983 which was the first version of this code that has been internationally distributed [16]. Neutron measurements could be used efficiently for plasma diagnostic purposes with the operation of larger fusion experiments of type tokamak such as PLT, T-10 or ASDEX in the 1970s and TFTR, JET, JT-60 or DIII-D in the 1980s [17]; see in this connection the book of Wesson [18] which contains an overview of current tokamak experiments and some references to the relevant literature. In the article by Jarvis [19] the pioneering experimental works relevant for routinely using neutron diagnostics on tokamaks are reviewed. 1.2. General overview In the plasma physics literature, results of neutron diagnostics are normally organized by experimental technique or by predictive numerical calculations of fusion neutron production. Furthermore, in practice, at many tokamak experiments neutron measurements are viewed primarily as fast-ion diagnostics and only secondarily as plasma diagnostics which to some extent obscures the view on the general progress. Therefore, the main motivation of this article is to try to provide a logical link between the basic physics of a tokamak plasma as a neutron source and the diagnosticians who are mainly interested in deducing characteristic plasma parameters from neutron measurements. Since experimental results for 14 MeV neutrons from D—T operation are, as yet, only available from TFTR [20—26] and JET [27—31], this article is devoted to reviewing the numerical modelling and analysis techniques of 2.5 MeV D—D neutron measurements for inference of relevant plasma data on tokamaks. Therefore, it is not appropriate to include other areas of nuclear fusion research, such as e.g. inertial confinement fusion, in which neutron measurements and simulation analysis are carried out (see e.g. Refs. [32—34]). However, readers interested in the variety of fusion product measurements on different types of magnetic confinement experiments such as stellarators, mirror or plasma focus experiments will find most valuable references in the comprehensive bibliographic compilation by Bosch [35]. This bibliographic database contains more than 1000 references on fusion reaction cross sections, diagnostics and plasma physics studies


5

related to fusion products including a-particles. In the earlier bibliography of fusion product physics in tokamaks by Hively and Sigmar [36] almost 690 citations have been compiled. This bibliographic review contains many of the citations with more experimental focus that are contained in the compilation by Bosch. In addition, many citations on the theory related to single-particle effects, collective processes, neoclassical transport or burning plasmas have also been included. In plasma experiments neutrons are being produced by nuclear reactions of the fuel ions. The absolute number and the energy spectrum of these neutrons is, in a somewhat complicated manner, related to the plasma conditions where the neutrons are being born. Being uncharged the neutrons instantly leave the plasma in their original direction of emission. Then, they hit structural components of the experimental device and its measuring systems. The neutrons will be scattered and to some extent absorbed. As a result, the initial direction of emission, the initial energy spectrum, and the number of neutrons are altered. Finally, some neutrons reach a neutron detector system and, with a certain probability, produce a signal. The properties of this signal depends on the properties of the incident neutrons as well as on the detector properties. It is the objective of neutron diagnostics to obtain as much information as possible on the properties of the plasma fuel ions by analysing these measured neutron signals. Optimal performance and use of different neutron diagnostic systems involves various aspects and require three different layers of computational procedures. Firstly, there are the fast dedicated computer programs for the primary data evaluation and for processing the directly measured neutron signals of each detector system. Such computer programs are usually regarded to be part of the measurement systems and will, therefore, not be discussed in this article. Secondly, computer codes for neutron transport calculations are needed in order to assess the influence of neutron scattering and absorption on the measured neutron signals. Neutron transport simulation plays an important role for the calibration of the different detector systems, in particular the neutron flux detectors. The normal procedure is to simulate the effects of neutron scattering and absorption on the detector signals once, and then simply correct the actual measurement. This implies that the corrections are sufficiently small and do not change from measurement to measurement. Therefore, and because of the errors in the simulations, much effort in experimental neutron diagnostics is devoted to minimize the influence of these effects on the actual measurement. Neutron transport simulations and the associated methods provide more than enough material for a separate review. In the present article only a brief overview on the basic concepts has been included. Therefore, readers interested in the most important achievements of Monte Carlo particle calculations for solving neutron and photon transport problems are referred to the comprehensive book by Lux and Koblinger [37]. This book gives useful information for both beginners and experienced readers and contains valuable references to relevant literature. Thirdly, there have to be computer codes for interpretation of the measurements. Interpretation codes are needed to deduce plasma parameters such as deuteron densities and temperatures out of the measured neutron signals. Routine interpretations of neutron signals aimed at obtaining information on basic plasma parameters such as densities and temperatures require fast dedicated computer codes. For routine analysis of neutron signals, there are mainly two different approaches. First, one can build up a database by calculating the expected measured neutron signals for various plasma properties and experimental conditions. By comparing the measured signal with

6


the pre-calculated results, one can narrow down the physical parameter space and deduce relevant plasma parameters. However, the results are, in general, ambiguous because different input parameters can yield indistinguishable neutron signals. In the second approach, by employing relevant physical boundary conditions, the analysis is restricted to the most relevant of the possible physical parameter spaces. Further restriction is obtained by choosing a set of plasma data which are sufficient to describe the most important features of the plasma neutron source. Then, using this likely set of plasma data, the neutron signal is calculated. By comparing measured and calculated results in an iterative procedure, the values of the plasma parameters can be found for which consistency between the measurement and the physical assumptions can be achieved. Clearly, in each of the two different approaches given above, the neutron production has to be calculated. Therefore, in Section 2 the necessary underlying theoretical background for the plasma as a neutron source is provided. It starts with a brief overview on fusion reactions and cross sections, the calculation of fusion reactivities, neutron spectra and the neutron transport equation. Only foundations are laid out and a few worked out examples are given. For more technical details the reader is referred to the substantial body of original papers cited in Section 2. The key assumption for interpretation calculations using the measured neutron signals directly as input in order to extract plasma parameters of interest is that the ion velocity distribution can be modelled with sufficient accuracy. For thermal plasmas modelling the ion velocity distribution and inferring plasma data from the neutron signals is a straightforward procedure. In order to treat auxiliary heated plasmas, it is important to use models which describe non-Maxwellian velocity distributions with sufficient accuracy but, at the same time, are not too time-consuming. In the case of neutral-beam heated plasmas, on many tokamak devices the absolute magnitude of the fusion neutron emission has been compared in detail with calculations that assume classical beam deposition and thermalization. In these calculations, the fast ions are usually assumed to have negligible spatial transport. As shown in Fig. 1, over presently eight orders of magnitude, the measured and calculated neutron emissions typically agree within the quoted uncertainties. The accuracy has steadily improved, and is now mainly given by the accuracy in the calibration of the neutron counter systems (+10—15%). This indicates that numerical modelling for neutral beamheated plasmas is well developed. Therefore, the remainder of Section 2 is mainly concerned with the calculation of ion velocity distributions in the presence of neutral-beam heating by means of a Fokker—Planck formalism. Section 3 is devoted to the simulation of the D—D neutron emission in tokamaks. Firstly, the basic physics input requirements for a sophisticated interpretation calculation are summarized. Accurately measured basic plasma data and neutron signals are needed as input data. As a direct link with the experimental practice, known sources of systematic errors for emission rate and interpretation calculations are discussed. Some other input data such as the neutral beam deposition profile have to be calculated by means of fast dedicated codes which also use measured plasma data as input. Therefore, an overview on the concepts and available computer codes for calculating neutral-beam deposition data are given. The remainder of Section 3 is concerned with computer codes for calculating D—D neutron rates, the numerical modelling of neutron spectra, and, briefly, neutron transport simulations. The last part of the article is devoted to reviewing the plasma physics quantities that can be inferred by analysing measured neutron signals. Emphasis is placed on simulation and analysis of


7

Fig. 1. Ratio of the experimental to the calculated neutron emission for plasmas with neutral-beam-heating. The data points for ISX-B (£ — 21 discharges), TFR (䉭 — four discharges), PDX (䉫 — 14 discharges), TFTR (䢇 — 28 discharges from 1988 and 1989; 䊏 — 118 discharges from 1990) and JET (䊊 — single discharge) are from the Heidbrink and Sadler review [38]. The points for the PLT data (;— 37 discharges), the TFTR D—D data (#— about 200 discharges averaged), and the TFTR D—T data (䊐 — 65 discharges) are from the summary paper by Strachan [22]. The data point for DIII-D (夹 — about 130 discharges averaged) is taken from Ref. [39].

neutron source strength measurements, as the majority of inferred plasma data pertains to this area. Clearly, the key assumption for extracting plasma parameters from neutron measurements is that the fast ions behave classically. Thus, violations of this assumption can lead to large systematic errors in the inferred plasma data. It is, therefore, important to map out operation regimes where the fast-particle slowing-down process is classical and where this key assumption is likely to be violated. The results of experimental studies of fast ions in tokamaks covering a period of more than two decades are reviewed in the comprehensive article by Heidbrink and Sadler [38]. Their review includes approximately 430 papers, laboratory reports and conference proceedings based upon fusion product, neutral particle analysis and various other plasma diagnostics measurements. The published results discussed in their review provide evidence that in most operating regimes, fast-ion confinement approaches the classical limit and, thus, validate the simplified kinetic models used for simulation. Therefore, the beginning of Section 4 concerns with results of neutron diagnostics for testing the classical nature of the fast-particle slowing-down process. Next, results for deriving deuteron densities, ion temperatures and electron temperatures out of measured neutron signals for ohmically and neutral-beam-heated plasmas are summarized. In addition, the determination of the minority ion concentration and ion diffusivities out of neutron measurements are discussed. As an example of more general plasma physics results, the impact of neutron diagnostics on the study of magnetohydrodynamic (MHD) activities is outlined. Furthermore, several tokamak plasma regimes where the assumption of classical slowing-down is violated are briefly described. Finally, in Section 5, the progress of the field towards the use of neutron diagnostics as the standard diagnostics for future fusion experiments and reactors is summarized and the likely directions of future research are indicated.

8


2. Basic theory Many important measurable macroscopic plasma quantities can be written as moments of the velocity distribution function f () as

1I2" f ()I d , where the quantity 1I2 is the kth-order moment. If the moments for all k"0 to R are known, then the velocity distribution is completely determined. However, often knowledge of only the lower-order moments is sufficient to provide information about the plasma. In particular, if the plasma is close to local thermodynamic equilibrium, then the local distribution function is approximately Maxwellian and the measurable lower-order moments, viz., density, average velocity, pressure, describe the plasma with sufficient detail. If the plasma is not close to thermal equilibrium, which is e.g. the case when high power auxiliary heating is employed, then the moments still provide valuable information, but the complete description of the plasma then requires knowledge about the non-Maxwellian distribution function explicitly. Such non-Maxwellian velocity distributions depend on many quantities such as the character of ion sinks and sources, different plasma parameters, and the magnetic field configuration. Furthermore, a knowledge of the non-Maxwellian velocity distributions of the plasma particles is required for studying problems connected with plasma heating and stability or for calculating important quantities such as the collisional power transfer to the background plasma, fusion reaction rates and fusion products spectra. The fusion products of nuclear reactions occurring within the plasma can be used as a convenient diagnostic for the ions. For this purpose the neutron is the reaction product of most interest since, being uncharged, it is able to escape immediately from the plasma and, hence, can be detected. Neutron diagnostics involve the experimental neutron detection techniques and the (computational) techniques for extracting relevant information about the velocity functions of the fusing ions out of the measured neutron signals. In this section the necessary theoretical background for describing the plasma as a neutron source is provided. In the first part of this section, a brief overview over the fusion reactions, the cross sections, the calculation of fusion reactivities, spectra and the basic concepts used in neutron transport simulations is given. As illustrative examples, analytic results for Maxwellian deuterium plasmas have been included. In the case of non-Maxwellian plasmas the neutron rates or neutron spectra are no longer simple analytic functions of the plasma temperature. Thus, analytic treatment is not suited for theoretical calculations or the analysis of actual measured neutron signals. Instead, the problem has to be tackled numerically involving the calculation of realistic velocity distributions taking into account the necessary physical properties of the plasma. Therefore, the remainder of this section is devoted to the kinetic description of a tokamak plasma by means of a Fokker—Planck formalism which is sufficiently accurate and allows fast numerical solution.


9

2.1. Theoretical background of the fusion neutron emission The emitted neutron rate from a plasma is a weighted average of the velocity distribution with cross section and relative velocity and the neutron spectrum is simply the energy-dependent probability of neutron emission per steradian. In present-day fusion devices with magnetically confined plasmas experiments are usually carried out with deuterium fuel for simulation of reactor plasmas and the employment of neutron diagnostics. Therefore, the present article focuses mainly on the simulation and interpretation of neutron signals from D—D reactions and other fusion reactions are only briefly described for completeness. 2.1.1. Fusion reactions The main fusion reactions in a tokamak plasma relevant for neutron diagnostics are the following: D#DPt (1.01 MeV)#p (3.02 MeV) ,

Q"4.03 MeV ,

(1)

D#DPHe (0.82 MeV)#n (2.45 MeV) ,

Q"3.27 MeV ,

(2)

D#TPHe (3.56 MeV)#n (14.03 MeV),

Q"17.59 MeV ,

(3)

D#tPHe#n ,

Q"17.59 MeV ,

(3a)

T#TPHe (3.78 MeV)#2n (7.56 MeV) , Q"11.34 MeV .

(4)

Here, the hydrogenic species are represented by upper-case letters for reacting ions and by lower-case letters for fusion products. In the above given equations, the reaction Q-values and, in addition, the particle energies for zero-energy reactants are given where appropriate. The two branches of the D—D reaction occur with nearly equal probability. Eq. (3a) relates to fusion reactions undergone by the fusion product tritons from the first branch of the D—D reaction and is commonly referred to as triton burn-up reaction. The study of triton burn-up is of particular interest since the emission of 2.5 MeV neutrons is indicating the birth of the 1.0 MeV tritons and the signal of the 14 MeV neutrons provides information on the confinement, slowing-down and radial migration of these particles [40,41]. The 1.0 MeV tritons have Larmor radii close to those of 3.5 MeV alpha particles from the D—T reaction and exhibit similar slowing-down and confinement properties. Therefore, in plasmas with large D—D reaction rates, triton burn-up measurements provide a frequently used method to investigate the single-particle behaviour of alpha particles in tokamaks without having to introduce tritium into the experimental devices. Triton burn-up has been studied on a variety of tokamak experiments, such as TFTR [41—48] PDX [48], ASDEX [49], PLT [50—52], FT [53—56], JET [19,28,57—68], DIII-D [69], JT-60U [70—75], ASDEX Upgrade [171] or TEXTOR [76]. It should be mentioned that under these experimental conditions, the neutron production due to the T—T reaction can be neglected since the triton population is by several orders of magnitude smaller than that of the deuterons in a deuterium plasma. 2.1.2. Fusion cross sections The interpretation of neutron source strength measurements in present fusion devices or the prediction of the fusion power gain of future experiments requires accurate knowledge of the relevant fusion cross sections. In particular, since measured fusion rates are of increasing importance

10


for plasma diagnostics on large fusion experiments, the uncertainties in the cross sections are required to be of the order or less than the errors in neutron source strength measurements. Since about 1945, many measurements of the fusion cross sections have been carried out. However, reliable experimental data are not available for energies below about 10 keV and even for the limited experimental energy range available the measurements are not always in agreement. Therefore, it is necessary to extrapolate downwards using theoretical formulae. Furthermore, analytical representations of the fusion cross sections are desirable for calculations of fusion reaction rates. As shown in Fig. 2, the cross section varies over more than 10 orders of magnitude over the energy range 1—500 keV. Due to the strong dependence on the particle energy it has been found most convenient to represent the cross section as (5) p(E)"S(E)(1/E)exp(!B /(E) , % where E denotes the energy in the centre-of-mass frame and B "paZ Z (2k c is the Gamov % constant for reacting particles with atomic numbers Z and Z . Here, k "m m /(m #m ) is the reduced mass and a is the fine structure constant. The exponential term in Eq. (5) describes simply the tunnelling probability and was first given by Gamov [77]. The factor 1/E results from the quantum mechanical description of the fusion probability, and S is the astrophysical S-function [78]. Thus, the cross section is factorized into terms describing the well-known and strongly energy-dependent quantum mechanical processes and a term which refers solely to nuclear processes of the fusion reaction. For energies below about 90 keV in the case of D—D reactions and about 30 keV for D—T reactions, the S-function can be written as S(E)+b exp(!cE) .

(6)

The parameters b, c and B are given in Table 1. %

Fig. 2. Fusion cross sections for the fusion reactions D(D,n)He (——) and D(T,n)He (- - - -) as a function of energy in the centre-of-mass frame.


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Table 1 Low-energy parameterization of the cross section in the centre-of-mass system

Reaction

b (barn keV)

c (keV\)

B % (keV)

D—D D—T

52.6 9821

!5.8;10\ !2.9;10\

31.3970 34.3827

Two approximate analytical representations of the fusion cross sections for a wide energy range have been widely used until recently. The first has been derived by Duane [79], which is also given in the NRL formulary [80], and the other has been derived by Peres [81]. Recently, improved formulae for the cross sections have been given by Bosch and Hale [82] which provide a higher degree of accuracy than the previous analytical representations. The paper by Bosch and Hale also contains a useful survey over the relevant literature concerning the measured cross section data and different evaluations. The improvement in the cross section representation could be achieved by fitting the S-function data obtained from R-matrix analysis [83] with a Pade´ polynomial as a #E(a #E(a #E(a #a E))) . (7) S(E)" b #E(b #E(b #E(b #b E))) New parameterizations were given for the reactions D(D,n)He, D(T,n)He, D(D,p)t and He(D,p)He. The fit results for the neutron producing D—D and D—T reactions are shown in Table 2. For the reaction T(T,2n)He, a mass-6 R-matrix analysis has also been carried out [84]. The result agrees well with new accurate measurements [85]. 2.1.3. Fusion reactivities The local fusion reaction rate R for a plasma containing ion species of types A and B is given by n n 1pv2 , (8) R" 1#d where n and n are the particle densities and d is the Kronecker symbol. The reactivity 1pv2 is in general given by the six-dimensional integral

1pv2 "

f ( ) f ( )p(" ! ")" ! " d d ,

(9)

where f , f are the normalized velocity distributions of the reacting particles, p is the cross section and " ! " is the velocity of impact. For this integration considerable analytic simplification can be achieved by choosing a spherical coordinate system for velocity space and assuming azimuthal symmetry [86]. With the assumption of azimuthal symmetry, the distribution functions are of the form f (v, k), where v""" and k"v /v"cos h is the cosine of the pitch angle, and can be ) , expanded in Legendre polynomials P as L f (v, k)" k (v)P (k) . ) L L L

12


Table 2 Parameters for the cross section fit in the centre-of-mass system in units of mb for the D(D,n) He and the D(T,n)He fusion reactions, respectively (from Ref. [82]) Reaction Coefficient

D(D,n)He

D(T,n)He

a a a a a b b b b b

5.3701;10 3.3027;10 !1.2706;10\ 2.9327;10\ !2.5151;10\ 1 0.0 0.0 0.0 0.0

6.927;10 7.454;10 2.050;10 5.2002;10 0.0 1 6.38;10 !9.95;10\ 6.981;10\ 1.728;10\

E range (keV)

0.5—4900

0.5—550

(*S) (%)

2.5

1.9

One obtains the following simplified expression:

2 1pv2 "4p a (v ) b (v ) P (k ) p(u)u dk dv dv . (10) L L L 2n#1 \ L Here, a and b are the nth-order coefficient functions from the expansion of the ion distributions L L in Legendre polynomials (thus reducing the number of integrations to be performed), k "cos(h !h ) and u"v #v!2v v k . Eq. (10) is valid for arbitrary two-dimensional distribution functions. However, it is of interest to consider two special cases for which expression (10) can be further simplified. Firstly, for the interaction of a fast monoenergetic particle population with velocity v with a Maxwellian plasma with temperature ¹ one obtains [87,88]

m (v!v ) m (v#v ) 1 !exp ! dv . (11) 1pv2 " p(v)v exp ! v v (n 2¹ 2¹ Secondly, for a thermal plasma with two interacting Maxwellian ion species of the same temperature, expression (10) reduces to [88—90]

k v 2 k dv . (12) 1pv2 " vp(v)exp ! 2¹ (n 2¹ On using the approximate low-temperature formula (6) for the S-function in expression (5) of the cross section, the Maxwellian fusion reactivity simplifies to [19,80] 1pv2 "k ¹\exp(!j ¹\)+kH ¹G .

(13)


13

It should be noted that the overall agreement of reactivities calculated using this expression with correct results using the rather accurate cross section fit given by Bosch and Hale is barely sufficient for reasonable estimates. The following more complicated, but rather accurate parameterized form for the thermal reactivities has been given by Bosch and Hale [82]: 1pv2"c h(m/k c¹) exp(!3m) , \ ¹(c #¹(c #¹c )) , (14) h"¹ 1! 1#¹(c #¹(c #¹c )) m"(B /4h) . % Here, the reactivity is in cm s\ and the parameters resulting from this fit are shown in Table 3 for the temperature range 0.2—100 keV. Other published results can be found in Refs. [91—93]. However, by minor empirical modification of the simplified expression (13), a much better approximate expression can be obtained and is given by

(15) 1pv2 "k (¹ exp(!3g ¹\) . Here, the temperature is in keV and the reactivity is again in cm s\. Furthermore, k "2.33;10\ cm s\, g "6.27, k "6.68;10\ cm s\ and g "6.66, respectively. "" "" "2 "2 On using this slightly different expression (15) the agreement with the fit results from Bosch and Hale, Eq. (14), is for the D—D reaction in the temperature range from 3.5 to 37 keV better than 5%. In the temperature range from 5 to 28 keV the agreement is better than 2%. Only below 1 keV, down to 0.1 keV, the error is increasing from about 18% to about 43% for the D—D reaction. For the D—T reaction this approximate expression is accurate to less than 20% in the temperature range from 0.1 to 22 keV and in the temperature range from 8 to 19.5 keV the agreement is better than 11%. Table 3 Parameters for the thermal reactivity fit for the D(D,n)He and the D(T,n)He fusion reactions, respectively (from Ref. [82]) Reaction Coefficient

D(D,n)He

D(T,n)He

k c (keV)

937 814

1 124 656

c c c c c c c

5.43360;10\ 5.85778;10\ 7.68222;10\ 0.0 !2.96400;10\ 0.0 0.0

1.17302;10\ 1.51361;10\ 7.51886;10\ 4.60643;10\ 1.35000;10\ !1.06750;10\ 1.36600;10\

¹ range (keV)

0.2—100

0.2—100

14


2.1.4. Neutron energy spectra The starting point for calculating fusion neutron energy spectra is the kinematics of the binary neutron producing reactions (2), (3) and (3a). Using classical kinematics the energy of the fusion neutron from the reaction A(B,n)a can in the laboratory frame be written (see e.g. Refs. [94,95]) as

2m m m ? (Q#K) , ? (Q#K)#» cos u (16) E "m v"m »# m #m m #m ? ? where m is the neutron mass, v is its velocity in the laboratory frame, » is the centre-of-mass velocity of the colliding particles, m is the mass of the second reaction product, u is the angle ? between the centre-of-mass velocity and the neutron velocity in the centre-of-mass frame and K is the relative energy given by 1 m m ( ! ) , K" 2 m #m where m , m and , are the masses and velocities of the reacting particles, respectively. The local neutron energy spectrum for a given direction of emission is

n n dp dN " f ( ) f ( ) d(E!E )" ! " d d . dX dE dX dE 1#d The differential cross section, dp/dX, can be expanded in Legendre polynomials, P , as L dp "p (A #A P (cos 0)#A P (cos 0)#2) , dX

(17)

(18)

where 0 is the emission angle in the centre-of-mass frame and p is the differential cross section for 0"0. The expansion coefficients are tabulated for energies above 10 keV [96]. If the particle velocities , and the emission direction are given, the neutron energy, E , is determined according to Eq. (16). For calculating the neutron energy spectra, Eq. (17) has to be evaluated for the given velocity distributions f and f . The energy spectrum of neutrons produced in fusion plasmas provides information on the production mechanisms of the emitted neutrons and the energy distributions of the reacting ions. For thermonuclear plasmas, various authors [95,97—99] have shown analytically that the energy distribution of the emitted neutrons is approximately given by a Gaussian as

dN 1 (E!1E 2) " exp ! , (19) dX dE ¼(p ¼ where 1E 2 denotes averaging of Eq. (16) over the angle u and 4m 1E 2¹ ¼" . (m #m ) Thus, the width, *E"2¼(ln 2, of the spectra is a direct measure of the plasma temperature ¹. The analytical results are *E (keV)"82.5(¹ and *E (keV)"177(¹, respectively. Maxwel"" "2 lian neutron spectra serve as an important test case for the numerical spectra simulation. Therefore,


15

detailed numerical analysis has also been carried out [100—107]. Results for the Maxwellian neutron spectra are summarized in Table 4 for various plasma temperatures. The different results agree well for temperatures of 10 keV and below. However, differences occur for the higher temperatures at 20 keV and above. In this temperature range the more recent numerical results are comparable and agree well with the analytic analysis while the previous numerical analyses deviate by about 2%. As mentioned above, several authors have related the energy distribution of neutrons produced by thermonuclear reactions to the plasma temperature. However, few analytical treatments relating the distribution function of non-thermal reactants to the energy spectrum of the reaction products have been published. The first analytical formulae have been given by Lehner and Pohl [99]. Unfortunately, as Heidbrink [108] noted, the expressions for the width of the fusion spectra are incorrect. In his paper, Heidbrink has extended the work by Lehner and Pohl by explicitly taking into account the effect of a strong magnetic field on the fusion spectra produced in ‘beam-target’ reactions. The derived analytical expressions are useful for calculating the spectrum of 15 MeV protons produced in the D(He,p)a reactions [109,110]. 2.1.5. Neutron transport equation The behaviour of individual neutrons emitted from fusion experiments cannot be predicted. However, the average behaviour of a statistically large population of neutrons can be described quite accurately by extending the concepts of neutron particle densities, nuclear cross sections and reaction rates. The basic concepts are briefly outlined below. A complete mathematical representation of the neutron population requires knowledge of seven variables, viz., position in space r, velocity (usually broken into energy E and direction x) and time t, for which the coordinates r, E and x are appropriate. Fusion neutron transport problems are usually considered as stationary problems, i.e. time-independent. The neutron transport equation may formally be written as a Fredholm-type integral equation:

t(r, E, x)" drQ(r, E, x)¹(rPr"E, x)

#

dr dE dxt(r, E, x)C(E, xPE, x"r)¹(rPr"E, x) .

(20)

Table 4 Calculated widths, *E , of Maxwellian D—D neutron spectra for various plasma temperatures ¹. Given are analytic "" results [99] and numerical results from FSPEC [100], NSPEC [101], NSOURCE [102] and BALLABIO [103], respectively *E (keV) ""

¹ (keV)

2 4 10 20

Analytic

FSPEC

NSPEC

NSOURCE

BALLABIO

116.7 165.0 260.9 369.0

115.3 163.6 261.4 376.0

117.1 166.1 264.6 378.1

116.7 163.5 261.9 371.2

116.8 165.2 261.3 370.0

16


Here, t(r, E, x) is commonly called ‘outcoming collision density’ though it is not a density function by the definition used in probability theory. It relates to the expected number of particles coming out of a collision in a volume element of the six-dimensional phase-space and is, thus, directly connected to the particle flux. Q(r, E, x) is the source term which describes the emission of particles at r with energy E and direction x. When interaction with matter takes place at a point r, the energy and the direction of motion of the neutron will be changed if the neutron is scattered. There are, however, also collisions which lead to absorption of the neutron, or to multiplication. The total effect of all types of possible interaction is described by the collision kernel L K 1 (21) l p (r"E, xPE, x) , C(E, xPE, x"r)" GH GH p (r, E) G H where the summations are over the n possible elements in the material considered and the m possible types of interactions with l expected numbers of outcoming neutrons. Furthermore, GH p is the differential cross section for element i and interaction j and p is the total macroscopic GH cross section. When a neutron has just left a collision, until its next interaction, its energy and direction remain unchanged. This is described by the transition kernel

1 r!r p (r, E) ds d x !1 , (22) "r!r" "r!r" r r Y where rPr represents the integration along a straight line from r to r. Numerically, Monte Carlo methods are being effectively used for solving neutron transport problems. A comprehensive and detailed overview on the Monte Carlo particle transport methods is given in the book by Lux and Koblinger [37]. ¹(rPr"E, x)"p (r, E)exp !

2.2. Velocity distributions of ions in magnetically confined plasmas The exact description of a magnetically confined plasma containing N particles with N equations of motion coupled through electro-magnetic fields is not practicable. The transition from the 6N-dimensional phase space to the six-dimensional phase space (r, ) is leading to the kinetic equations which describe the evolution for the distribution function f (t, r, ) for each particle species A in the presence of particle sinks ¸ and sources S : j j F j # ) # ) f "C( f )#S !¸ . (23) jt jr m j Here, m is the mass of the species and F "eZ (E#c\[;B]) is the external force acting on them. The collision operator C( f ) represents the rate of change of f (t, r, ) due to the collisions between the plasma particles. For instance, if the collision operator is calculated on the basis of binary collisions, then Eq. (23) is called the Boltzmann equation. The kinetic equation (23) is called the Fokker—Planck equation if the collision term is calculated on the assumption that there are simultaneous random, small changes in the momentum of a particle due to its interaction with the other particles in the plasma. In this connection, as representative and very instructive literature, the books of Wu [111] and Balescu [112] are quoted. The Fokker—Planck equation is the appropriate equation for simulation and analysis of neutron signals from fusion experiments.


17

Rosenbluth, MacDonald and Judd first derived an expression of the second-order Taylor expansion of the Fokker—Planck collision operator for Coulomb interaction [113]. However, the Fokker—Planck equation in the general form given by Rosenbluth et al. is extremely difficult to solve since it is a non-linear integro-differential equation in six phase-space variables and time. The only practicable method for solving this equation directly would be by using discrete particle simulation methods. However, a discrete particle method for proper treatment of magnetically confined fusion plasmas would inevitably be extremely expensive in computer time. Presently, the state of the art for tokamak applications are non-linear 3D Fokker—Planck codes. Several of these codes are briefly discussed in the review article by Arter [114]. However, despite their usefulness, non-linear 3D codes are too time-consuming for routine analysis. Therefore, the Fokker—Planck equation has to be simplified sufficiently in order to make numerical solution practicable. This is briefly outlined below. Since for tokamaks, toroidal symmetry can be assumed, it is sufficient to discuss the particle dynamics only in two directions, i.e. parallel and perpendicular to the magnetic field. Thus, the velocity can be written as " # , where contains drift velocities and the gyro-velocity. In , , , the direction parallel to the field, the particle dynamics is influenced by the electric field component E , while perpendicular to the field the particle dynamics is determined by the fast gyro-motion , and the slower drift motions. The normal procedure for obtaining the distribution function of ions in magnetically confined plasmas is based on the existence of different characteristic time scales and spatial scales. Assuming that XiC\HO)D"o2 . X#iC!H

(45)

This equation yields the following general dynamical representation for W (q), which is valid for all 2 values of the complex time ¹; W (q)"lim Q¹(1!¹*/*t)\(t(t)!e\O2t(q#t)) , 2 R W (q)"Q¹(t(0)#¹t(0)#2!e\O2+t(q)#¹t(q)#2,) . (46) 2 Such a time-dependent representation differs conceptually from the original representation (39) and allows to predict directly what time domain gives the main contribution to the stationary RXS amplitude F"F(R) (38) with the stationary wave packet W (R)"lim Q¹(1!¹*/*t)\t(t) . (47) 2 R This is most easily understood by considering the important special case of short RXS duration (see below). It should be pointed out that expression (47) is equivalent to the following differential equation: * 1

(t)" (t)!Qt(t), W (R)" (0) 2 *t ¹

(48)

with solution (39). The right-hand side of this equation consists of a decay-dephasing part, (C!ιX) (t), and a source, !Qt(t). 4.4.2. Fast RXS If the excitation energy is tuned to the wing of the photoabsorption band the duration of the scattering (40) can be shorter than the inverse width q "1/*u of the photoabsorption band close M to u "¹";q . (49) M The time q characterizes the quantum beats of the wave packets t(t) and W (q) and gives the main M 2 time scale for problems concerning the RXS duration. *u:1 eV in the problems connected with the nuclear dynamics, while this spectral width has the order 1—10 eV or more when the dynamics of the relaxation of the electronic shells due to the creation of a core hole is studied (see below). Keeping the first two terms in the expansion over ¹ in Eq. (46), the following remarkable result is immediately obtained: FK!ι¹1 f "Qt(¹)2,

W (R)KQ¹t(¹), 2

"¹";q . M

(50)

F. Gel+mukhanov, H. As gren / Physics Reports 312 (1999) 87—330

125

One sees that the spectral shape of fast RXS is given by a simple projection of the wave packet Qt(¹) at the complex time ¹ (40) onto the final state " f 2. Eqs. (50) generalize the corresponding results from Ref. [102]. It is worthwhile to compare W (R) (50) and W (q) (46); 2 2 W (q)KQ¹(t(¹)!e\O2t(q#¹)), "¹";q . (51) 2 M One finds, contrary to intuition, that the wave packets W (R) (47), (50) and W (q) (46), (51) do not 2 2 coincide even when q essentially exceeds the RXS duration time "¹". A coincidence takes place only if the time of the wave packet evolution q is larger than the lifetime C\. One can say that the fast limit (50) for the RXS amplitude is obtained only when the wave packet (50) must go through the long time evolution (longer than the lifetime, q and R\ RIXS bands of CO (Fig. 73) calculated for different excitation energies [117]. S S The calculations are carried out for the ground XR , core-excited 1p\2p; P and final valence E E S S excited 1p\2p; BR\, 1p\2p; DR\, and 1p\2p; ER> electronic states of the CO E S S S S E E S S molecule. 10.3.3. Excitation with very large detuning The s-function given by Eq. (182) is in practice valid only fairly close to the absorption resonance being considered. When the incident photon frequency is tuned very far from this resonance many other core-excited states, centered at different energies and of different symmetries, will simultaneously be excited with comparable probabilities due to tail excitation. When these core-excited states are of different symmetry the symmetry character of the emission will also be mixed, and the emission will become more like non-resonant X-ray emission [95]. So the symmetry breaking parameter s (182) will in practice have a non-monotonic behaviour. At first, the s-function decreases (solid line, Fig. 71a) and then it increases (dashed line, Fig. 71) as the absolute value of the detuning increases. Fig. 71a shows also the behaviour of the s-function for positive detuning frequencies. The frequency dependence of s is qualitatively the same if the detuning is negative but is for “real molecules” different due to the non-homogeneous distribution of core-excited states


205

Fig. 73. Calculated (2p ) resonant X-ray emission spectra of CO assuming narrow-bandpass (c"0.1 eV) excitation. S Excitation energies: (a) u"536.00 eV, (b) u"536.30 eV, (c) u"536.43 eV, (d) u"536.59 eV, (e) u"536.75 eV, (f) u"536.91 eV, (g) u"537.22 eV, (h) u"538.00 eV. The intensities of the vibronic bands R\ and R> have to be E S summed together for the final total spectrum. The D$D$ and R$R$ final states are assumed degenerate [117]. S E S E

above and below the resonance and due to the difference in the FC factors above and below the turning point. 10.3.4. Experimental investigations of breakdown and restoration of electronic selection rules The experiments with the CO molecule were made at the undulator beamline 7.0 at ALS in Berkeley (see Section 2.2), see Figs. 74 and 75. A comparison between Figs. 73 and 74 shows that theory reproduces all main observed spectral features [117]. Moreover, both theory and experiment demonstrate the breakdown of the electronic selection rules close to the photoabsorption resonance and their restoration with the detuning away from the vertical photoabsorption energy. The effect of the selection rule restoration is seen directly in Fig. 75. A list of other experimental

206


Fig. 74. Oxygen K2p resonant X-ray emission spectra of CO with different detuning energies below the 2p resonance. S S The resonant spectra show two spectator peaks and a weak high-energy elastic peak due to a participator transition [103,117]. Fig. 75. Ratio between high-energy (“forbidden”) and low-energy (“allowed”) spectator peaks (Fig. 74) of the 2p S resonant oxygen K emission spectra as a function of detuning energy measured from the vertical O1sP2p absorption S energy. i+D +1.4 eV. Monochromator resolution is estimated to 0.65 eV. Solid line shows results of theoretical simulations [103,117].

investigations of non-adiabatic effects in RIXS of more complex systems was presented at the beginning of Section 10.3.

11. X-ray resonant scattering involving dissociative states With the development of tunable, narrowband, synchrotron-radiation sources, the studies of the resonant X-ray scattering process are no longer limited to systems with discrete bound states, but can and do now also involve systems with states that are unbound along the nuclear degrees of freedom. The diatomic hydrides served as the original prototypes for non-radiative RXS spectra involving dissociative states. Decay channels with dissociation preceding electronic decay were first identified in the spectrum of HBr recorded at the 3dPpH excitation energy [245]. The HCl


207

2pPpH resonance was also found to decay predominantly by dissociation followed by the electronic decay [246]. The time scales of the dissociation and the Auger decay were estimated to be of the same order of magnitude [245]. The H S molecule served as the first polyatomic species showing similar features [247,248]. Studies of this species clearly indicated that the character of the core-excited state determines the relaxation path, and that dissociation before decay indeed is possible even for short-lived core hole states. The 2p absorption spectra in this molecule, as in HCl, exhibits a pre-edge structure [249,247] consisting of a broad band due to excitations to the first unoccupied molecular orbital — 6a and 3b in the case of H S, 6p or “pH” for HCl — followed by a series of sharp peaks corresponding to excitations to Rydberg orbitals. The identification of the Auger spectra for the various excitation energies indicated that the first type of excited states relaxed through Auger transitions in dissociative fragments, while excitations to the bound Rydberg states showed resonant Auger decay in the molecular environment. Calculations on core-excited adiabatic interatomic potentials of different molecules, for example O [250], HBr [251], HCl [246], H S [247], have confirmed that intermediate or final states with dissociative character are indeed relevant to consider for the RXS process. The experimental conclusions about the relaxation paths of the core-excited states thus followed from energy assignments of the Auger decay spectra. These spectra were interpretable in terms of diagram levels of the fragments. The assignments, the excitation energy dependences as well as mass spectroscopic data gave hints of a mechanism in which dissociation is faster than the electronic decay of the excited fragment. From further experimental progress with synchrotron radiation it has also been possible to use line shapes and the Auger resonance Raman effect [24,42,86] to draw conclusions on the character of the intermediate and also of the final states [252,253,88,102]. Only the bound states showed the expected resonance narrowing of the bands (Raman effect), while the Auger transitions to final dissociative states lacked such narrowing and were determined by their lifetime broadening only. In this section the theory for RXS involving dissociative states is reviewed with emphasis on the time-independent description. The time-dependent theory for the RXS process is further elaborated in the following section, Section 12.

11.1. Decay channels involving continuum and bound states Many, if not most, molecular core-excited states are dissociative or predissociative, and it is desirable to include these in a general theoretical treatment. Interference effects will be a central concept also in such a treatment. The main peculiarities of the problem can be understood for the special case when the incoming X-ray photon excites the molecule to the adiabatic interatomic potential º (R) of a dissociative intermediate state, or above the dissociation threshold if º (R) has A A minimum. As is shown in Fig. 76 two qualitatively different channels for the radiative decay exist. One is the decay from an intermediate continuum nuclear state "c2"uA A(R) to a bound vibrational # state " f 2"uD (R) of the final internuclear potential º (R). Here E is the molecular energy of the K D A continuum state uA A(R). This channel is thus a “continuum—bound” channel. The second type of # channel is given by the decay into final dissociative states " f 2"uDD(R) with the molecular energy # E , the “continuum—continuum” channel. In the considered case, the cross section (95) of the D resonant scattering of a narrowband X-ray beam is given by the sum of cross sections p (u,u) and A@


208

Fig. 76. Excitation and decay schemes. The ground (o), intermediate (c) and final (f) states are displaced relative to each other. All notations are explained in the text.

p (u,u) for the continuum—bound and continuum—continuum decay channels: AA p(u,u)"p (u,u)#p (u,u) , A@ AA p (u,u)"(u/u) "F "d(u!u#u ) , (184) A@ DK KM K p (u,u)"(u/u)"F ", E "u!u#E . AA D D M The RXS cross section (184) has two qualitatively different contributions, one of them, p , is sharp A@ in frequency, while the other, p , has a smooth frequency dependence. The spectral width of the AA incoherent part of the cross section p is defined by the width of the continuum scattering AA amplitude F . The spectral distribution of F is given by the spectral distribution of the D D Franck—Condon amplitudes as discussed further in the next subsection. Eqs. (184) show directly that for the continuum final state the positions of the emission resonances do not follow the Raman—Stokes law (9) and that the width of the emission peak cannot be made smaller than lifetime broadening [91]. In the common Born—Oppenheimer and Condon approximations, the electronic transition matrix elements are treated as constants instead of as functions of the nuclear coordinates, and the scattering amplitudes (110) for the continuum—continuum F and for the continuum—bound D F channels read: DK 1uD "uA 21uA A"u 2 # M , (185) F "a dE #DK #A A u!u #iC DDK AM where u "E !E . The continuum nuclear wave functions of states i"c, f are here normalized AM A M to a d-energy function. The sum on the right-hand side of Eq. (185) implies that for a bound intermediate state "c2 one needs to integrate over the energy E or to sum over the vibrational states A uA if the incoming photon frequency is tuned above or below dissociation threshold of the L intermediate state, respectively. The scattering amplitude (185) is defined by Franck—Condon (FC) amplitudes (the overlap integrals) between the vibronic wave function u (R) of the ground state and M the continuum nuclear wave functions uA A(R), and between uA A(R) and the final state nuclear wave # # functions uDD(R). #


209

11.2. Space correlation between absorption and emission The interference between different intermediate continuum states plays a fundamental role in the damping of emission as the internuclear distance increases. To explicitly show this fact it is convenient to rewrite the scattering amplitude (185) F "1uDD"u 2 D # A

(186)

in terms of the stationary wave packet u (R) and the time-independent Green’s function A u "aG u , A # M

uAH(R)uA A(R) # G (R,R)" dE #A # A E!E #iC A

(187)

with E"u#E . The stationary lifetime broadened Green’s function G (R,R) describes the M # propagation of nuclei on a decaying potential surface, º (R), from the internuclear distance R, A where the molecule was core excited, up to R, where the emission transition took place. When the duration of RXS q (40) is short (61), the Green’s function [102] A d(R!R) G (R,R)+ # E!º (R )#iC A M

(188)

shows that the emission and absorption transitions take place at the same point (*R"R!R"0) (Fig. 77). This result also follows directly from Eq. (59). Thus, in the limit of a small lifetime C\ or large absolute value of detuning X"u!u (R ), the molecule has no time to spread from the AM M point of absorption. Here R is the equilibrium internuclear distance in the ground state, M u (R )"º (R )!E . The finite RXS duration time means that the core excitation cuts off the GM M G M M coherent superposition u (187) of the core excited states residing in an energy band width given by A the inverse duration time "E !E"4q\ (Fig. 77). When q is short all intermediate states A A A ("E !E"4R) give coherent contributions to the wave packet u (maximum interference between A A core excited states). In this case the point of emission is known exactly (R"R, *RP0) according to Eq. (188). The X-ray excitation cuts off only a small part ("E !E"4q\P0) of the continuum A A intermediate states if q is long (Fig. 77). Hence, according to the uncertainty principle the A d(R!R)-function in Eq. (188) is broadened and the explicit information about the emission point R is lost (*RPR) (Fig. 77). A deeper understanding of the case of finite q can be obtained with help of the lifetime A broadened Green’s function (187). In the relevant region, the nuclei move with an energy E larger A than the potential height º (R). So the criterion of applicability of the quasiclassical approximation A is fulfilled everywhere and the quasi-classical wave function can be written as

0 A exp !i p (R) dR . uA A(R)" A # (p (R) A A

(189)

Here p (R)"(2k(E !º (R)) is the momentum, c+R is the classical turning point where M A A A p (c)"0, and A is the normalization constant. The small correction term containing the wave A reflected by inhomogeneities of the potential is neglected in Eq. (189). In the classically accessible


210

Fig. 77. Illustration of E-interference (upper panel) and t-interference (lower panel). *R"R!R is the distance between absorption (R) and emission (R) points, v is the characteristic nuclear velocity, and q is the duration of RXS. All A notations are explained in the text.

region (R,R5c and R5R), the lifetime broadened Green’s function shows strong space correlation between the absorption and emission processes G (R,R)"GM(R,R)e\CO0Y0, GM(R,R)"!2ipA # # #

exp(!i0Yp(R) dR) 0 , (p (R)p (R)

(190)

where p (R)"p(R)!iC/v(R), p(R)"(2k(E!º (R), and v(R)"p(R)/k is the relative velocity of A the nuclei at the point R. The lifetime broadening C is assumed here to be small in comparison with (E!º (R)). As follows from the factor exp(!Cq(R,R)) in Eq. (190) the emission intensity is A negligible if the time of propagation between the absorption point (R) and the emission point (R)

0Y dR (191) v(R) 0 exceeds the lifetime; q(R,R)5C\. Indeed, the emission takes place only as long as the population of the core excited state remains unexhausted. A second reason for small contributions from far “emission points” R to the scattering amplitude is given by large detuning frequencies X"u!(º (R )!º (R )). This contribution is A M M M small for large positive detunings (X'0) because of the strong oscillation of the Green’s function G (R,R)JeιN0Y\0 in the integral (186), where p"("2kX". When detuning is negative (X(0) the # amplitude of excitation is exponentially small GM(R,R)Je\N0Y\0 for core excitation in the # q(R,R)"


211

classically forbidden region (E(º (c)). One can conclude that both positive and negative detunA ings “cut off” the far emission transitions with the sudden limit (188) for large "X" (61). It can be pointed out that the scattering amplitude F tends to the fast (or vertical) limit (188) for different D reasons for positive and negative detunings. It is easy to understand that the internuclear distance "R!R" between absorption and emission points cannot exceed the distance (vq ) passed by the nuclei during the RXS duration (Fig. 77): A "R!R"4vq . (192) A The small contribution of the “far” emission transitions is caused by the interference between the core excited continuum states uA A(R) coherently excited into the band "E !E"4q\. So this A A # interference between continuum states and the finite value of q plays the key role in the damping of A X-ray emission at large internuclear distances. It should be noted that the time-dependent representation gives an alternative view on this problem [102], see Section 4.8, and also Section 12. 11.3. Franck—Condon amplitudes The measured X-ray lineshape results from the interplay between the shapes of several functions; the photon function, the lifetime broadening function, and the vibrational (discrete or continuous) distribution function. This interplay will in turn be dependent on the character of the participating states, if they are bound or dissociative. The spectral shape of the RXS cross section is defined by the spectral shape of the FC factors. Only three types of the FC factors can enter the scattering amplitude, namely, the bound—bound, the bound—continuum and the continuum—continuum FC factors. The properties of the bound—bound FC factors are well known and they do not connect directly with the discussed dissociative problem, and therefore only the properties of the latter two kinds of the FC factors will be reviewed in the following. 11.3.1. Bound—continuum Franck—Condon amplitudes Consider the transitions between a dissociative state uG G(x) and a bound state uH (y), where K # x"R!R and y"R!R are the deviations of the internuclear distance R from an equilibrium G H position R and R of the electronic states i and j, respectively. The characteristic length a of the G H G bound wave function and characteristic scale a of oscillations of the continuum wave function are GH (193) a "( /ku , a "( /2kF ) GH GH H H with i, j"+o, c, f , and F "!(dº (R)/dR) H as the interatomic force at the equilibrium point GH G 0 R of bound state j. Very often the scale of oscillations of the continuum wave function is larger H than the size of the bound state a /a ion correspond to these atomic states, S:R>, D:*,R>,P, and P:R\,R\,P [257]. To give the semiquantitative description of these spectra a simplified model was used with only three final molecular states [106]. The “molecular” part of the RXS cross section was approximated by the asymptotic formula (199). The result of the simulations (Fig. 81) show that the total RXS profile is the sum of the three bands, each consisting of the narrow atomic-like contribution with the lifetime width C"70 meV and a “molecular” background with the full-width at half-maximum (FWHM) 2c (ln 2+1.8 eV. The ab initio time-dependent calculations of the DM Auger resonant Raman spectrum of the HCl molecule (see Section 12.2.3) confirm the main spectral features of the experimental spectra shown in Fig. 81. The quantity g"p (u)/p (u)"z/(1#z) refers to the relative contribution of the molecular AA AA part defined as the ratio of integral cross sections p (u) and p (u)"p (u)#p (u), see Fig. 82. AA AA AA AA Both theory and experiment demonstrate the enchancement of a molecular-like broad background by detuning the excitation frequency, and that, in contrast to the atomic-like resonances, it follows approximately a Raman—Stokes dispersion law [101]. A recent experiment by Magnuson et al. [258] at the sulphur ¸-edge of OCS clearly demonstrated the possibility to observe atomic fragmentation also in resonant X-ray emission (radiative RXS). A very strong emission band from atomic sulphur was observed following excitation to the


217

Fig. 81. Experimental resonant Auger spectra, showing the 2p 3pP3p, 2p 3pP3p region of the atomic Auger decay transitions in HCl as function of detuning [106]. Fig. 82. Comparison of the experimental and theoretical relative integral contribution of the molecular background [106].

lowest core excited resonances in OCS, while higher excitations lead to molecular like emission spectra. The dipole selection acts to simplify the atomic emission features compared to the non-radiative case. It seems that strong coupling among many possible close-lying dissociation channels also lead to a simplification of the spectral outcome. 11.4.5. Bound—continuum and continuum—bound channels. Mapping of vibrational wave functions Only the bound vibrational states uA (x) are populated if the excitation energy u is tuned below K the dissociation threshold º (R) of the core excited state, see Fig. 76. In this case both the A


218

bound—bound and bound—continuum decay channels are open, and the total cross section becomes the sum of the corresponding cross sections; p(u,u)"p (u,u)#p (u,u). The properties of @@ @A p (u,u) were described above in Sections 6.4 and 8.1. @@ In the general case of several coherently excited intermediate vibrational states, the scattering channels will interfere through these discrete states, Section 6.4. In accordance with Eqs. (185) and (194) the contribution of the bound—continuum decay channel to the total RXS cross section is equal to

uA (x) K . (206) p (u,u)" q Ku!u (R )!e #iC @A AM M K K when q\;c ,c . Here A DM DA (207) x"(1/F )(u!u!u (R )#F a ), q "a(2ka u/u 1uA "u 2 . K M DA DM A DA A K DA It is remarkable that the dependence of the cross section (206) on the emission frequency u copies the space distribution of the vibrational wave function uA (x) of the core-excited state. Indeed, when K C(u one can tune the frequency into exact resonance with some vibrational state m. In this case A the RXS cross section (206) simply becomes p (u,u)J(uA (x)). This equation thus shows how the @A K vibrational wave function can be mapped, and that the cross section is equal to zero in the m points where uA (x) is equal to zero. The latter expression for p (u,u) leads to the simple geometrical K @A consideration given in Fig. 83. This consideration is based on the physical meaning of x (207) as the classical turning point for propagation on the potential surface º of the final dissociative state (see D Eq. (194)). According to this physical meaning the spectral shape of the RXS cross section reflects the square of the vibrational wave function (uA (x)) of the core excited state by the linearized K potential of the final state (Fig. 83). It is important to note that Eq. (194) and its geometrical interpretation (Fig. 83a) are based on the linear approximation of the potential º near RA . Thus D

Fig. 83. A geometrical illustration of the proportionality of the bound—continuum RXS cross section p (u,u) (206) to @A the square of the vibrational wave function uA (x) of the core excited state at the classical turning point x (207). The linear K approximation of the final state potential º is depicted with a thick dashed line; the exact potential º is depicted with D D a thin solid line. The cross sections are obtained by reflection on (a) a linear potential and (b) a nonlinear potential.


219

only in the case of such a linear potential, the mapping p (u,u)J(uA (x)) is linear (see Fig. 83a). In @A K the general case of a non-linear potential the mapping is evidently non-linear (see Fig. 83b), and the application of the reflection method will not produce a direct copy of the squared wave function as given by p (u,u)J(uA (x)), but a deformation of this wave function depending on the particular @A K shape of the potential asymptote. The description of the continuum—bound decay channel is very similar to the case reviewed here and can be found in Ref. [102].

12. Time-dependent theory of resonant X-ray Raman scattering As stated in the previous text there are two basic representations to describe resonant Raman scattering; the time-independent and the time-dependent representations. These give different interpretational content to the RXS process despite that they obviously lead to identical results. For example, although RXS through dissociative states has an obvious “dynamic flavor”, a fully time-independent derivation could explain all known characteristics, as reviewed in the previous section. Despite this fact and despite the fact that studies of RXS at the present time have involved only stationary experiments, time-dependent treatments have gained an increasing popularity on the theoretical side owing to their inherent interpretability and our inclination to relate spectral features to processes rather than to states. The time-dependent representation allows a penetration into the physics of the scattering process because of to two important notions; the time evolution of the electro-vibrational wave packet and the duration of the scattering process. The results reviewed in Section 4 demonstrated that the duration time is a main physical concept which can be exploited for an active manipulation by the RXS spectral shape. This concept also forms a basis for the evaluation of the time-dependent RXS cross section on which the present section focuses. 12.1. Time-dependent representation of the RXS cross section The time-dependent representation for the RXS cross section in the case of narrowband excitation was considered for the first time in Refs. [83,84] (see also [101,102]). Lee and Heller [259] introduced the concept of time-dependent wave packets in the time-dependent representation for the amplitude of optical Raman scattering. Further development of this technique has occurred for many problems connected with RXS [260,190,102,117,261,114,113,110,262] (see also Section 4). As in the time-independent case we account for the situation when an incoming X-ray photon with the frequency u is absorbed, core exciting the molecule to the state "c2. Due to the Coulomb interaction and vacuum fluctuations this intermediate core excited state decays by emitting Auger electrons and X-ray photons with the energy E to the final state " f 2. The radiative and nonradiative RXS amplitudes have the same structure near the resonant region [42] 1 f "Q"c21c"D"o2 . F" E!u #iC AD A

(208)


220

The operator D describes the interaction of the target with the incident X-ray photon. In the case of non-radiative RXS, Q is the Coulomb operator, and Q"DYH when the emitted particle is the final X-ray photon [42]. The half-Fourier transform of the denominator on the right-hand side of Eq. (208) yields the time-dependent representation for the scattering amplitude [263,102]

O F"F(R), F(q)"#i dtei#>#D>iCR1 f " (t)2 .

(209)

The wave packet reads

(t)"Qe\i&ARD"o2.

(210)

Let us note that in strict theory the molecular Hamiltonian H is the same for all electronic states j (except the final state in non-radiative RXS). However, we will use the notation H with index j due H to two reasons; firstly to identify the electronic shell in which the wave packet evolves, and, secondly, to apply directly the general theory to nuclear degrees of freedom with the nuclear Hamiltonian depending on the electronic state j. A corresponding time-dependent representation for the RXS cross section can be obtained by a Fourier transform of the spectral function

1 duU(u,c)e\iSR . U(u,c)" Re dtu(t,c)eiSR, u(t,c)" p \

(211)

Since the U-function is real, its Fourier transform has the property: uH(t,c)"u(!t,c). We note that u(t,c)"d(t) and u(t,c)"const correspond to the cases of having white and monochromatic incident light beams, respectively. Very often the spectral function U is approximated by a Gaussian (18). In this case

tc . u(t,c)"exp ! (2 ln 2)

(212)

To receive the time-dependent representation for the RXS cross section we review the method outlined in Refs. [84,102,113]. Substituting (209) and (211) in Eq. (16) the following dynamical representation for the RXS cross section is obtained

1 p(E,u)" Re dqp(q)u(q,c)eiS\#>#MO p

(213)

in terms of the autocorrelation function p(q)"1t (0)"t (q)2 . # #

(214)

Here

t (q)"e\i&DOt (0), t (0)" # # #

dtei#\CRt(t) .

(215)


221

The wave packet t (q) with the initial value t (0) is the solution of the non-stationary Schro¨dinger # # equation with the final state Hamiltonian H , whereas the wave packet D (216) t(t)"ei&DRQe\i&ARD"o2 admits two different interpretations and computational strategies. 12.1.1. Two-step evolution of the wave packet The two-propagator representation (216) prompts that t(t)"t(0,t), t(q,t)"ei&DR\OQe\i&ARD"o2

(217)

is the result of a two-step evolution: The initial wave packet D"o2 propagates after core excitation in the core excited state from time equal to 0 up to t. After the decay transition at moment t the wave packet evolves in the final state in the opposite time direction from moment t up to 0. The evolution of the wave packet t(q,t) is given by two coupled Schro¨dinger equations i(j/jq)t(q,t)"H t(q,t), t(t,t)"Qt (t) , D A i(j/jt)t (t)"H t (t), t (0)"D"o2 , A A A A

(218)

12.1.2. One-step dynamics A conceptually different interpretation of the wave packet (216) is based on the one-step dynamics with one effective time-dependent hamiltonian *»(t). By differentiation of Eq. (216) with respect to t the following Schro¨dinger equation is obtained: (219) i(j/jt)t(t)"*»(t)t(t), *»(t)"ei&DR*»e\i&DR, *»"QH Q\!H A D with the initial condition t(0)"QD"o2. The solution of the latter equation is straightforward

R t(t)"U(t,t )t(t ), U(t,t )"¹exp !i dt *»(t ) (220) M M M RM where ¹ is the time-ordering operator. At this point it is worth pointing out the following striking property of the wave packet t(t): The evolution of t(t) is completely halted in the dissociative region (RPR) where *»Pu (R), AD º (R)!º (R) since the evolution operator U(t,t ) (220) then becomes equal to the c-number: A D M (221) *º(R)Ku (R), U(t,0)Ke\iSADR, t(t)Ke\iSADR"o2 . AD The same result follows immediately also from the two-step representation (216), since in the dissociative region H KH #u (R) and hence again t(t)Kexp(!ιu (R)t)"o2. This result is A D AD AD important also from the viewpoint of numerical simulations since it makes it possible to avoid the integration of Eq. (219) as well of Eqs. (218) in the region of dissociation. The two-step and one-step approaches for the evaluation of the wave packet t(t) (216) leads to two qualitatively different numerical techniques. The two step technique requires more computational time because the solution of the time-dependent Schro¨dinger equation (218) for t(q,t) requires in advance the solution of the time-dependent Schro¨dinger equation for t (t). The one-step A method is free from this disadvantage, but contains on the other hand the complicated operator *»(t) which needs to be diagonalized.


222

We review also a third method for a time-dependent evaluation of the RXS cross section, namely one which is based on the evolution of the RXS cross section for monochromatic excitation with the forthcoming convolution of this cross section with the spectral function. This version can be preferred over the one- and two-step techniques. 12.1.3. Convolution of the cross section for narrowband excitation The main characteristic features of RXS can be unravelled when the spectral width c of the incident light beam is small. The spectral function U for the narrowband incident beam (c;C) can be replaced in Eq. (16) by the Dirac d-function. Hence, the denominator in Eq. (208) becomes equal to (u!(E !E )#ιC). The autocorrelation function then takes the form A M p(q)Pp (q)"1W(0)"W(q)2 . (222) M Here

W(q)"e\i&DOW(0), W(0)"

dt eiS>#M\CRQt (t) . A

(223) Once more a two-step technique is retained; (1) the solution of the Schro¨dinger equation (218) for t (t)"exp(!iH t)D"o2 with the initial condition t (0)"D"o2, and (2) the solution of the A A A Schro¨dinger equation for W(t) with the final state Hamiltonian H and with the initial condition D W(0). However, this two-step technique is advantageous over the first one commented above, since the initial condition W(0) does not depend on time. Having settled the question of evaluation of the RXS cross section p (E,u) for the monochroM matic incident X-ray beam, one can evaluate the RXS cross section p(E,u) for arbitrary spectral distribution of incoming radiation as the following convolution [95,113]:

1 (224) p(E,u)" du p (E,u )U(u!u ,c), p (E,u)" Re dq p (q)eiS\#>#MO . M M M p From the computational point of view the method given by these equations is both simpler and faster in comparison with the two techniques described above (see also below). One of the computational advantages of this approach is the possibility to use a parallel algorithm. 12.2. One-step nuclear dynamics We turn again to the important special case in RXS when only the nuclear degrees of freedom need to be taken into account — which often is motivated when the different electronic transitions in the RXS spectra are well separated — and assume the validity of the Born—Oppenheimer (BO) approximation, separating the nuclear and electronic degrees of freedom. We will also neglect the dependence of the electronic transition matrix elements D and Q on the nuclear coordinates. This approximation is sufficiently good for the photoabsorption matrix element D since the ground state nuclear wave function is localized close to the equilibrium molecular geometry. The same assumption concerning the amplitude of the decay transition Q can be justified for the bound—bound decay transitions. However, this dependence of Q can be more essential for the continuum—continuum decay transitions since a large span of internuclear distances then is involved, and this dependence


223

might influence the ratio of the molecular and atomic-like contributions. On the other hand, for hydrides with one heavy atom, like the HCl molecule, numerically evaluated in a following subsection (Section 12.2.3), the total rate is probably dominated by the heavy atom. Investigations on the partial and the total Auger decay rates for the H O molecule indeed indicates that most of the internuclear dependence of these rates is allocated at distances shorter than the equilibrium, so that the “constant resonance width” approximation holds [264]. When the R dependence of the transition matrix elements is weak, D and Q can be factored out, and one can thus simplify the RXS amplitude by putting D"Q"1 .

(225)

(We notice that the approximation D"const, Q"const most often is used in the numerical simulations and in the expressions for the Franck—Condon (FC) factors. All other results reviewed here are free from this assumption). The question of interest is posed by the pure nuclear problem with the Hamiltonians H "K#º (R), a"o,c, f , (226) ? ? where K is the nuclear kinetic energy operator. It is relevant to note the physical meaning of the wave packet t(t) (216) in the Q"const approximation. The one-step (219) evolution of this wave packet is given exactly by the interaction picture with H as the unperturbed Hamiltonian and D *»"*º(R) (227) as the perturbation. One obtains the remarkable result that the one-step dynamics is determined by the difference *º(R) between the potentials of the core exited º (R) and final º (R) states. A D 12.2.1. Mapping of the core excited wave packets and potentials. The reflection technique Another physical interpretation of the wave packet W(0) is obtained by first considering RXS with a monochromatic incident X-ray beam. In this case the RXS amplitude (208) becomes the projection of the coherent superposition of the core excited states W(0) on the final state Q"c21c"D"o2 . (228) F "#i1 f "W(0)2, W(0)"i D u!u #iC AM A This representation immediately shows that W(0) (223) is the coherent superposition or “wave train” of the eigenstates of the core excited Hamiltonian H created due to the core excitation. A The ground is now prepared for expressing an entirely different representation for the RXS cross section, namely p (E,u)"1W(0)"d(E!H )"W(0)2"tr d(E!H )o, o""W(0)21W(0)" (229) M D D with E"u#E !E. Through this expression a mapping of the squared wave packet "W(0)" and M internuclear potentials can be obtained. The easiest path to the desired semiclassical result is given by the Wigner transform (see Ref. [113]). To proceed further we use the conventional separation of the internuclear domain into two regions, the “molecular” (R(R ) and the “dissociative” (R'R ) B B regions. Here R is the minimal internuclear distance where º (R) and º (R) are close to the B A D


224

corresponding dissociation limits. Using this partitioning the RXS cross section becomes the sum of the “molecular” and “atomic-like” contributions p(E,u)"p

(E,u)#p (E,u) .

(230)

Here

0B dR d(E!X!*º(R))"W(0)" , (231) p (E,u)K dR d(E!*º(R))"W(0)", *º(R)"º (R)!º (R) . A D 0B This general reflection technique contains an important special case which allows to map *º(R)"º (R)!º (R). When the slope *º(R)"d *º(R)/dR is not equal to zero one can write A D "W(0)" , p(E,u)K "*º(R)" p (E,u)K

*º(R)"E!X if R(R , B *º(R)"E if R'R , (232) B where the prime denotes a derivative with respect to position. What strikes the eye here is the simple relation between the RXS cross section and the slope *º(R). This relation is important for the inverse problem of finding potentials from the spectroscopic measurements. However, the simplicity is ephemeral in the molecular region where the wave packet W(0) is a complicated function of R. The situation is more promising in the dissociative region where the R-dependence of "W(0)" "W(0)"K"W(0,R )"e\ODO, q"1/C B is caused [102] only by the time of flight

q " D

0 dR R K , v(R) v

v(R)"

(233)

2 , v"v(R) [º (R )!º (R)] A k A M

(234)

and by the lifetime q"1/C. Though "W(0)" is a smooth function of R, W(0) shows the fast oscillations typical for continuum states (Fig. 85). The atomic-like resonance and the corresponding near wing can dominate only when X"0 [102,106,114,113]. Therefore to find *º(R) it is natural to tune u in exact resonance, X"0. Apparently, the reflection method maps the longrange part of *º(R) on the nearest wings of the atomic-like profile since the dependence of the potentials on R is slow in the dissociative region (see Section 12.2.2). The extension of the reflection technique to the whole spectral region allows to map the ratio "W(0)"/"*º(R)" (232). This gives information about the space distribution of the squared wave packet (see Fig. 83). The comparison of the exact cross section with the one obtained in the reflection approximation indicates that the semiclassical approximation (232) is not perfect. The deviation with the exact cross section is stronger in the vicinity of the atomic-like resonance where the factor 1/"*º(R)" diverges and the semiclassical approximation breaks down.


225

12.2.2. Red and blue wings of the atomic-like resonance The atomic-like resonance is mainly formed by a spectral transition in one of the dissociation fragments. The central part of the atomic-like resonance band is close to a Lorentzian, while the wing (the outer part) of the band has a different profile [102,106,113]. We are now prepared to describe the “near wings” (Fig. 84) more explicitly, focusing on the dissociative region where the long-range forces dominate. To reduce the molecular contribution the case with X"0 is considered (Fig. 85). The difference *º(R) of the core excited and final state potentials is easily approximated as *º(R)"u (R)#a(R /R)L, u (R)"º (R)!º (R) . (235) AD B AD A D The solution of Eq. (232), E"*º(R), is straightforward: R/R "(a/(E!*º(R))L. The final B results follow directly from Eqs. (231) and (233). Hence, the red or blue wings of the atomic-like resonance are described by Eq. (232) and in the case of a power potential (235) as p (E,u)J"*/*E">L . (236) The spectral shape of the atomic-like resonance in the vicinity of the resonant frequency u (R) AD [102,91,106,113], is a Lorentzian exp(!(*E!X)/c) A , "*E"("*" , p (E,u)J *E#C

(237)

Fig. 84. The conventional partition of the spectral domain in the central (Lorentzian) part, and near and far wings. *E"E!(º (R)!º (R)). A D Fig. 85. The space distributions of the wave packet (223) in the unbound core excited state (2p\pH) of HCl for different excitation energies (X"0 and 3 eV). ReW(0) and Im W(0) are depicted as solid lines while the absolute value ("W(0)") of the wave packet is shown as a dashed line.

226


since in the soft X-ray region c ion correspond to these atomic-like lines: B: P, R> and C: P, R\, R\, P. To understand the main spectral features of the resonant Auger electron decay spectra for the 2p\pH core excited state, the three most important final states were accounted for [106], the bound R> and the dissociative R\ and P states. The corresponding potential surfaces (Fig. 86) were calculated by the MCSCF method [265] as described in Ref. [113]. The evaluation of the electron matrix elements of the decay transitions Q was then neglected, meaning that the RXS cross section for different final states were given for the same value of Q (the calculations in Ref. [113] refer to one of the two spin—orbit split core excited 2p\ pH states).

Fig. 86. Potential surfaces for ground, core excited (2p\pH) and final (P,R\,R>) states of the HCl molecule. Fig. 87. The RXS cross section of HCl for bound R> and dissociative R\ and P final states. The case of monochromatic, resonant excitation (X"0).


227

Fig. 87 shows the RXS profiles for different final states for X"0. One can see that the onsets of the molecular bands can be found both at the shortwave (R>) or the longwave (R\, P) sides of the atomic-like resonance. As clarified above, blue and red tails are formed if the potential surfaces of the core excited and final states converge or diverge, respectively, when R increases and approaches the dissociation region. The atomic-like resonance for the same final state can in general simultaneously have both blue and red tails [102,266]. It depends on the behaviour of the potentials º (R) and º (R) close to the equilibrium geometry. A D The spectral shape of the molecular tails strongly depends on the shape of º (R) and º (R). If the A D final (or core excited) state is bound the molecular contribution consists of a vibrational band and a smooth continuum—continuum tail (see upper panel in Fig. 87). Notice that the molecular contribution with the vibrational structure in this panel corresponds to the experimental band assigned in Ref. [106] as 5p\. The next question concerns the the role of the detuning: As is well established [102,101, 104,106,114] the molecular and atomic-like contributions depend on the excitation energy in qualitatively different manners. Indeed, the position of the atomic-like resonance does not depend on the excitation energy [102,91] while the center of gravity of the molecular band depends nonlinearly on X inside of the region of strong photoabsorption and follows a linear Raman law at the wings of the photoabsorption band [101]. Moreover, the weight of the atomic-like resonance tends to zero for large "X" faster than the molecular contribution. In this limit of sudden RXS only the molecular band contributes to the spectral shape of RXS. It is simply given by the FC factor between the ground and the final nuclear state [102,117] (see also Section 12.2.4). The commented spectral features are shown in Figs. 88, 90, 91 and 93. The quenching of the RXS cross section when u is tuned below or above the photoabsorption band is shown in Fig. 89 in comparison with the spectral shape of photoabsorption. One can see that neither the RXS nor the absorption cross sections are symmetrical functions of the detuning. It is necessary also to emphasize that both these cross sections decrease slowly, more like Lorentzians than Gaussians (see also Section 12.2.4). The switching over from Gaussian to Lorentzian behavior takes place for E(200 eV and is therefore not seen in the long wave region (Fig. 89). The RXS cross sections are depicted in Figs. 90, 91, and 93, with attention focused on the X-dependence of the RXS spectral shape. According to the reflection approximation (see Section 12.2.1) the RXS spectral profile maps the space distribution of the wave packet. This leads to the appearance of additional fine structure in the RXS profile [102,114] (Figs. 90 and 91) caused by the inhomogeneous space distribution of the core excited wave packet W(0) in the molecular region (see Fig. 85, X"3 eV, R(3 a.u.). 12.2.4. Interference between molecular and dissociative scattering channels Fig. 91 demonstrates the appearance of an additional spectral feature of the RXS profile when the narrow atomic-like resonance is embedded in a smooth molecular background. One can see that the atomic-like resonance converts into a spectral hole when u is tuned from the photoabsorption resonance [113]. This anomaly is the result of an interference between the molecular background and the atomic-like resonance channels. The possibility for such kind of interference between resonant and off-resonant contributions was mentioned earlier [102,113]. Here we give a more extended explanation; we note that the manifestation of this phenomenon is not the same as the Fano profile [58], although it reminds thereof. For the sake of transparency we assume that

228


Fig. 88. The RXS cross section for the unbound R\ final state of HCl for different excitation energies u. The resonant frequency (º (R )!E ) of the vertical photoabsorption transition is equal to 202.58 eV. E is the energy of the Auger A M M electron. One can see the fast decrease of the RXS cross section when the excitation energy is tuned out of the resonant frequency of the vertical photoabsorption transition. c"0.

both the core excited and the final states are dissociative and that the incident light beam is monochromatic. This circumstance leads to p(E,u)""F", e "u#E !º (R)!E . (238) D M D It is convenient to consider here the real core excited "c2""e 2 and final " f 2""e 2 continuum A D nuclear states with the released dissociation energies e and e , respectively. Since the conA D tinuum—continuum Franck—Condon factor 1e "e 2 is singular one can conclude that D A 1e "e 21e "o2"r(e ,*e )#s(e ,*e )d(e !e ) . (239) D A A D A A A D A Here *e "e !*º and *º "º (R )!º (R). The smooth function s(e ,*e ) shows the weight A A A A A M A A A of the narrow atomic-like contribution while the smooth profile r(e ,*e ) is responsible for the D A molecular background [102,106,113]. Since the continuum wave functions are real functions,


229

Fig. 89. The RXS cross section integrated over E (solid line) and the Cl 2p X-ray photoabsorption spectrum (dashed line) of HCl. The core excited and final states are 2p\pH and R\ respectively. The Gaussian profile is depicted as a dot-dashed line. The cross sections have maximum for u+202.58 eV. The half-width at half-maximum (HWHM) of the Gaussian is equal to 0.84 eV.

Fig. 90. The RXS cross section for the unbound R\ final state of HCl for different excitation energies. The RXS cross sections are normalized; the integral cross sections are the same for different excitation energies. c"0.

230


Fig. 91. The RXS cross section for the unbound P final state of HCl for different excitation energies. The RXS cross sections are normalized; the integral cross sections are the same for different excitation energies. c"0. The lower panel shows the region near the spectral hole in more detail.

s(e ,*e ) and r(e ,*e ) are also real quantities. Eqs. (208) and (225), (239) infer that also the scattering A A D A amplitude is the sum of molecular and atomic-like contributions F"g(X,*E)#s(e ,X)/*E#ιC . (240) D The interference between the molecular and the atomic-like contributions emerges here naturally. One can observe a resemblance of the spectral shape of the atomic-like resonance with the well-known Fano profile [58] which describes the interference between continuum states and a discrete state embedded in this continuum. However, qualitative differences exist. In the case of interest the discrete state is missing both in the core excited and the final nuclear states. The reason for the appearance of the discrete resonance in the X-ray Raman spectrum is the cancellation of the


231

Fig. 92. Spectral profile versus spectral width c of incident radiation. The final HCl state is P (see Fig. 91). The RXS cross sections are normalized. The spectral function is approximated by a Gaussian (18) with the pilot frequency 206 eV and the following values of HWHM: c"0.01 eV, 0.3 eV, 0.5 eV, 1 eV. The RXS cross section for monochromaticexcitation is depicted by a dashed line. Fig. 93. The RXS cross section for the bound R> final state of HCl for different excitation energies. The RXS cross sections are normalized; the integral cross sections are the same for different excitation energies. c"0.

kinetic energies in the resonant frequency of the decay transition, º (R)#e !º (R)!e " A A D D º (R)!º (R), due to the effective conservation of the kinetic energy under decay in the A D dissociative region. This conservation relation is reflected by the singular partJd(e !e ) in the D A continuum—continuum overlap integral 1e "e 2 (239). D A Apart from the evident asymmetry of the atomic-like profile, the interference leads to new striking spectral features: One is the total suppression of the atomic like resonance, Fig. 91. When

232


the weight of the molecular pedestal exceeds the amplitude of the atomic-like part an atomic-like “hole” emerges instead of the resonance peak (Fig. 91). This “hole” disappears for large detunings where the RXS profile coincides with the photoabsorption band of the direct oPf transition (see lower panel in Fig. 91). A naive picture prompts that any narrow spectral feature is blurred when the spectral width c of the incident radiation increases. Simulations show something different, however, see Fig. 92. Indeed, one can see that the width of the spectral hole as well as of the atomic-like feature is practically independent of c [102,91,113]. Moreover, Fig. 92 demonstrates another unexpected effect, namely, the transformation of the hole into a peak when c increases. This atomic-like feature can be understood if one recalls that broadband excitation implies a summation of the RXS cross sections for different u. If the particular u with the sharp peak is included in the summation, then it might dominate for large c and there will be a peak instead of a hole. 13. Direct versus resonant X-ray Raman scattering Up to this point the present paper has reviewed the consequences of second-order perturbation theory for the interaction between matter and light for a variety different experimental situations and findings that have concerned 2nd and 3rd generation synchrotron sources. In a more general multichannel resonance scattering framework the radiative and non-radiative processes can be given a unified formulation [57,40]. However, also in the perturbation theory ansatz the radiative and non-radiative RXS processes have remarkably many common features [42], something that has been extensively utilized so far in this review. The main crossing point of these two phenomena is the one-step model with the golden rule for the scattering cross section and the Kramers—Heisenberg (KH) expression for the scattering amplitude. The obvious difference is the interaction leading to the non-radiative decay, i.e. the discrete—continuum Coulomb interaction between the core-excited state and the many continua into which it is embedded, while the radiative spectrum occurs as a result of spontaneous emission. The implication of these differences for selection rules and the resolution of fine structures has already been reviewed, see Section 9. Likewise, from the experimental point of view the important advances in synchrotron radiation sources of the 2nd and 3rd generations, has promoted measurements of both radiative and non-radiative Raman effects. This holds for atoms, molecules as well as for solids. In fact, it was in the non-radiative Raman mode that the various “Raman effects”, such as linear dispersion and resonance narrowing first were described. Another important difference between the radiative and non-radiative processes is that non-resonant scattering is more important in the case of nonradiative RXS. The “direct” transitions can thus quantitatively change the dependence of the spectral width on the detuning. 13.1. Resonant photoemission The resonant non-radiative RXS leads to the same final state as the “direct” transition — or direct photoemission. In the case when the excited electron “remains” during the transition — a spectator decay — the final state is a 2-hole 1-particle state which is identical to a “direct” photoemission satellite. In the case the excited electron participates in the process — a participator decay — the final


233

state is a 1-hole state and identical to the ones reached by normal photoemission. In both cases the direct transition contributes to the total cross section and interferes with the resonant transition. From the experimental point of view the latter type, the participator decay, is the important subclass of the non-radiative Raman process. Since the final states are identical to those obtained by a normal photoemission experiment the process often goes by the name resonant photoemission (RPE). Although RPE has been studied for a long time it is evident that the current availability of narrowband polarized and tunable X-ray beams and the accompanying strong improvement of spectral resolution has promoted the studies of this process just as much as of general X-ray Raman scattering processes. Resonant photoemission has been analyzed in the framework of various theoretical approaches [42,165,44,267—271] most of them connecting to the theory of Fano resonances [58,82]. It is pertinent here to review some advances in RPE and in particular the interference with the direct photoemission process and the frequency-dependent features of RPE that now can be studied. The systematic investigation of the spectral shape versus excitation frequency u and scattering angle seems thus to be a new branch of RPE research. Very recent experiments on argon [272], on the C1s resonant photoemission in CO [273], on metallic Ni [38] (see Section 16.3.4) and on weakly interacting systems on a platinum surface [39], have shown a rather sharp dependency of the RPE profiles on u when the photon energy is tuned close to the resonant excitation of the core level. Camilloni et al. [272] (see also [274]) showed that the main reason for this spectral anomaly for the 2p\3d excitation of Ar is the interference between direct and resonant photoemission. The same conclusion was reached by Weinelt et al. [38]. According to the recent measurements by Piancastelli et al. [273] and Carravetta et al. [75] the physical picture of the similar phenomenon in the RPE spectra of carbon monoxide is not so unambiguous. We thus focus on some recent advances of frequency-dependent molecular RPE spectra and their description using many-channel scattering (and Fano) theory, again using CO as the primary testbed. 13.1.1. Resonant vibronic photoemission It is pertinent in this review to discuss RPE theory of gas-phase molecules with resolved vibrational structure. The general theory will be reviewed for the special case of RPE by the CO molecule to illustrate its key features. In the process considered here, the photon frequency u is tuned close to the X-ray C1sPnH transition from the K-shell of carbon to the first unoccupied level (nH) of the CO molecule. This highly excited “discrete” "U 2""1s\nH2 state has an energy A lying above the ionization thresholds of all the occupied molecular levels and is energetically well isolated (*E'5 eV) from any other core excited state. Due to the Coulomb interaction with many continuum states, this discrete state decays through several autoionization channels with emission of photoelectrons t Pt . The final electronic state of energy E can be written as H JC "W 2""t\t 2, where the index f"j,l identifies the ion electronic wave function "t\2 and the D# H JC H symmetry of the continuum wave function t of the photoelectron with energy e (for example JC t , t V , t W ). NC L C L C Only few valence shell ionization channels t\(4p\,1n\,5p\ for CO) among the many H possible autoionization channels, will be explicitly considered in the following application of the theory since the experiment usually only resolves the outermost valence ionized states. The processes corresponding to “participator Auger decay” is then considered. The number of continua interacting with the "U 2""1s\nH2 state is much larger and some of them can provide decay A

234


paths even more efficient then the participator channels (the spectator resonant Auger channels). However, their presence can be taken into account considering the total lifetime of the core excited state. Radiative decay channels are also present, but they are less effective for the decay of core excited states of light elements as CO. For photon energies u close to the discrete C1sPnH level the photoemission consists of two qualitatively different channels, namely, direct photoionization of a valence orbital t and JC a resonant process involving core excitation of the 1s electron to the molecular orbital nH with A forthcoming autoionization to the final electronic state "W 2""t\t 2. D# H JC If the incident X-ray beam is linearly polarized, the RPE cross section (13) for the j-channel summed over the final vibrational states "m 2 of the ion has the following structure: D b (241) p (u,e)" p D(u) 1# HKD (3 cos 0!1) U(u#E !e!E D,c) , HK HK H 2 KD 4pa p D(u)" u "1W "D"W D 2", E"u#E HK HK J# 3 J where p D(u) is the RPE cross section averaged over 0 ; 0 is the angle between the photon HK polarization vector e and the photoelectron momentum k. The electron angular distribution parameter b D depends on j and m . The label m must here be considered an “asymptotic” label D D HK because, as will be shown in the following, the total wave function of the final state cannot be simply represented as the product of an electronic and a vibronic wave function owing to the resonant contribution "U 2"m2 which depends on the vibrational wave function "m2 of the intermediate electronic state "U 2. The total energies, E "E #e and E D"E #e D, of ground H K HK and ionized final states, respectively, consist of the total electronic energies (E and E ) and the H corresponding vibrational energies e and e D. K

13.2. Fano problem for electro-vibrational transitions The solution of the Schro¨dinger equation HW "EW for E in the continuum can be obtained $# $# by the Fano approach [58,82,60,275—277,274]; F"f, m is a collective label for the electronic and D vibrational state of the ion and the continuum degeneracy. It is reasonable to use the Born—Oppenheimer (BO) approximation and to approximate the total Hamiltonian H"H # C H by the sum of the electronic and the nuclear Hamiltonians. The interchannel interaction between discrete and continuum molecular states » "1U "H "U 2 (242) D# C D# mixes the electronic discrete states "U 2"m2 and the electronic continuum states "U 2"m 2 [75] D# H » D# "UI 2#ip » "U 2 1m"m 2"m2 , "W 2""U 2"m 2# (243) $# D# D D# D# D E!E K K D » "UI 2""U 2# dE DY#Y "U 2 . E!E DY#Y DY


235

The profile of the cross section (241) of the RPE process in molecules may be rather complex due to the vibrational structure of both intermediate and final states and due to lifetime vibrational interference. A remarkable simplification is obtained if, as often done in experimental measurements, the cross section is integrated over the photoelectron energy, i.e. summed up over all the vibrational final states. In the case of a single electronic resonant state the so integrated cross section can be simply expressed as a sum over the intermediate vibrational states of contributions that are given by the product of an electronic factor, including both direct and resonant terms, and the FC factor between initial and intermediate vibrational state. In the case of narrowband excitation (c;C) the condition of completeness of the vibrational states and Eqs. (241) and (243) lead to the following expression for the energy integrated RPE cross section:

4pa » HJ# Z #ip » Z u i Z # , (244) p (u)" dep (u,e)" H K HJ# DY# DY# H 3 X #iC K K J DY where X "u!u is the detuning from the frequency u "E #e #D!E of the resonant K K K K photoabsorption transition to the discrete state "U 2"m2, and i ""10"m2", Z "1W "D"UI 2, Z "1W "D"U 2 , K D# D#

(245)

dE C"n "» ", D" "» " . D# D#Y E!E D D This equation is valid for the angularly averaged RPE cross section or for the scattering at the magic angle 0 (70). Eq. (244) shows that the lifetime vibrational interference is absent in the cross K section integrated over the photoelectron energy and that the vibrational structure of the intermediate electronic state can be taken into account simply through the FC factor i and the K vibrational energy in the resonant denominator of the cross section [75]. A similar result was obtained in [179]. In spite of the strong mixing of the vibrational states of closed and open channels in the final state wave function (243), the BO approximation leads to an effective separation of electronic and nuclear degrees of freedom in the energy integrated partial cross section in Eq. (244). Very often the rate of direct photoionization is much smaller than the rate of the resonant channel "Z C" ;1 . m" J HJ# "Z » " J HJ#

(246)

This ratio has been estimated from the experimental measurements of CO (see below) to be, in the average, m+0.5% for the X,A, and B final states. Because of this small parameter one can neglect » Z in Eq. (244); this corresponds to neglecting the small coupling of the continua through DY DY# DY# their interaction with the discrete state. By this approximation the partial RPE cross section (244) can be easily converted to one of the parametrized forms of the Fano profile commonly used [75]:

(X #q ) H p (u)"pM(u) i 1!o#o K H H K H H X #C K K

(247)

236


with "» Z " " » (Z ) Z )" J HJ# q" . (248) o" J HJ# HJ# , H H » (Z ) Z ) z "» Z " J HJ# HJ# H# J HJ# The Fano parameters q and 1!o show the position of the Fano minimum and the depth of this H H minimum, respectively [58,275]. 13.2.1. Dependence of branching ratios on excitation frequency RPE experiments [272,273,75] have shown the dependence of the RPE spectral shape on the excitation energy. To understand the physical reason for this dependency, the RPE spectra of carbon monoxide serve as a good illustration [75]. The measurements of these spectra were carried out at the MAX I storage ring in Lund, Sweden (see Section 2). Deexcitation spectra including the three outermost final (1h) participator states X"5p\(R>), A"1n\(P), B"4p\(R>)

(249)

were recorded at various photon energies around the C1sPnH resonance. These single holes states have a significant probability of being populated via the direct photoionization channel, leaving the molecule in the same final state as would be the case for the resonant process. At 0"55° the ratio of the direct to resonant population was estimated to be on average +0.5% from comparison of spectra recorded at a photon energy corresponding to the maxima of the resonance and spectra recorded 4.5 eV below the resonance. Since !4.5 eV detuning corresponds to more than 50 times the lifetime width of the resonance (FWHM), 85(10) meV [173], the resonance population should be negligible. The C1sPnH resonance is separated by more than 5 eV from the next higher resonance, 3s Rydberg, and the lowest vibrational level, m"0, is more than 6 times more intense than the next higher vibrational level, m"1 [193]. This makes the deexcitation from the C1sPnH resonance ideal for studies of the resonant Auger process with respect to frequency detuning. Spectra recorded at 0"0° with a detuning from the C1sPnH resonance between 0 and !4.5 eV are presented in Fig. 94. The total experimental resolution is +150 meV, therefore transitions to different vibrational levels can be studied separately. From these spectra two different types of detuning effects are observed. Firstly, the vibrational progression strongly mimics the vibrational progression of the direct photoemission spectrum (!4.5 eV detuning) already at a moderate photon energy detuning. This collapse of vibrational structure upon frequency detuning [105] was described already in Section 8.3. The other striking feature of these spectra is the change of branching ratios of the participator peaks upon frequency detuning. The variation of branching ratios p(u)" p (u) (250) H H is presented in Fig. 95. In this figure the branching ratio of a particular (1h) participator state is obtained as the ratio between the intensity corresponding to the (1h) state and the total intensity of the three outermost participator states. The branching ratios of this figure show asymmetries with respect to the center of the resonance. Firstly, one observes an asymmetry of the branching ratios of the A and B states at a moderate detuning frequency, less than 1 eV. This asymmetry can be B (u)"p (u)/p(u), H H


237

Fig. 94. Frequency detuned C1s\nH RPE spectra of CO leading to three main final states of CO>; the X 5p\ R>, A 1n\ P, and B 4p\ R> states [75]. Fig. 95. Comparison between experimental (symbols) and computed (lines) branching ratios vs photon energy (eV) for the channels: X(5p\) (open circles), A (1n\) (filled circles), B (4p\) (filled squares). The theoretical branching ratios have been obtained by the model neglecting the interference of direct and resonant photoemission channels (top panel) and by the model including the effect of the interference (bottom panel) [75].

explained with the higher vibrational levels of the resonance located at the positive detuning side, and can be considered to be of a “trivial” origin. However, there is also a “non-trivial” asymmetry affecting the branching ratios for a detuning of more than 1 eV. For instance, the crossing of branching ratios for the A and B states is located at +1.7 eV for negative detuning whereas on the positive side this crossing is clearly located further from the resonance, at +3.5 eV. This asymmetry cannot be explained by high vibrational levels, since the highest vibrational level unambiguously observed in a photoabsorption measurement at the C1sPnH resonance is the m"3 level, which is more than 1000 times less intense than the m"0 level. Fig. 96 shows a comparison between spectra recorded at 0° and 55° for frequency detuning between 0 and !200 meV. Also, as an inset to Fig. 96, the variation in relative intensity for energies between 0 and !700 meV is presented. Within the dipole approximation, the angular distribution of emitted electrons for completely linearly polarized light is expressed as Eq. (241). It should be noted that near the magic angle 0 (+55°) the intensity depends only on the integral K cross section and at 0° the intensity increases with an increasing b D parameter. The most obvious HK

238


Fig. 96. Frequency detuned C1s\nH RPE spectra of CO. The spectra have been recorded at 0°, broken lines, and 55°, solid lines. As inset, the variation of relative intensity for 0°, open markers, and 55°, filled markers, are presented as function of photon energy detuning [75].

observation from Fig. 96 is that the relative intensities vary more rapidly at 0° than at 55°. This observation contradicts what can be expected from the strict two-step model, in which the angular anisotropy parameter b D is given by the product, b D"A ) c , where A is the so-called alignment HK ? HK parameter and c the intrinsic Auger decay parameter [278]. From ion yield studies of the ? C1sPnH resonance in CO it is known that A"!1 at resonance maximum and approaches 0 when tuning the photon energy away from the resonance maximum [279,280]. The b DHK parameter has been found to be close to 1 for all the (1h) participator states with the transitions corresponding to the A state having the lowest value of the three participator states b D+0.75 HK [281]. All the states in the C1sPnH deexcitation spectra are necessarily described by the same alignment parameter A. The results for the A band show clearly that the resonant process in a detuning experiment cannot be described by the strict two-step model in which the excitation step is assumed to be separated from the deexcitation step. One obvious difference between 0° and 55° spectra is that the direct population of (1h) final states are +50% stronger at 0° than at 55° (the angular dependence of the direct transitions are described by b D-parameters close to 2 [282], HK whereas the resonant transitions have b D-parameters +1). Thus, the angular dependence of the HK branching ratios is affected by interference or direct populations of the (1h) final states via the


239

Table 4 Parameters of the Fano profile (see Eq. (247)) for the ionization channels: X(5p\), A(1n\), B(4p\). The values of o have been obtained by a fitting of experimental and theoretical branching ratios, while p (arbitrary units) and q have H H H been obtained directly from the experimental branching ratios at large and 0 detuning, see Section 13.2.1 for details Channel j

X

A

B

o H q (eV) H p H

0.27 2.58 0.102

0.27 3.27 0.194

0.21 1.35 0.215

corresponding direct channel. It can be noted that the vibrational collapse effect could be fully understood within the resonant part of the scattering formula whereas for the variations of the branching ratios it thus seems necessary to include also the direct population part. The results of the semiempirical simulations [75] are shown in Fig. 95 and the corresponding Fano parameters are collected in Table 4. 13.3. Role of interference between direct and resonant photoemission The dependency of the RPE branching ratios on the excitation energy is remarkable. It is easily understood that the branching ratio is constant if the spectral shape of the partial cross sections close to the resonance is a simple Lorentzian, i.e. if the direct photoemission is completely negligible. The deviation of p (u) from a Lorentzian profile is due to both the non-resonant H contribution of direct photoemission and to the interference of this process with the resonant photoemission. If the interference (JX q ) of direct and resonant channels is neglected, the K H approximate cross section of the photoionization channel j can be derived from Eq. (247) as

i K , p,'(u)KpM(u) 1#oq H H X #C H H K K

(251)

where the resonant contribution is described by Lorentzian profiles, one for each intermediate vibrational state, superimposed on the smooth background of the direct photoemission cross section p(u). H The p,'(u) and B,'(u) factors computed with neglect of the interference effect (by the expression H H in Eq. (251)) for the three ionization channels X (5p\), A (1n\) and B (4p\) are collected in the top panels of Fig. 95. The comparison between experimental and theoretical branching ratios in Fig. 95 shows that the no-interference model can explain the gross structure of the observed branching ratios. In this model, however, the crossing (for a single vibrational intermediate state) of two cross section curves can only occur at two points symmetric with respect to the zero detuning energy. An asymmetry of the crossing points can only be introduced by the presence of several close lying vibrational peaks of different intensity. The calculations [75] show (see top panel of Fig. 95) that the asymmetry obtained in this way is evidently much weaker than that present in the experimental data. This is a clear indication of the importance of the interference effect.

240


Considering that the symmetric Lorentzian function is the one generally adopted to describe absorption and emission line profiles in excitation and ionization of core electrons, the observed behaviour of the branching ratios with the excitation energy can be regarded as anomalous. The CO example chosen here gives a good illustration that, in spite of its small nominal strength, the direct photoionization channel plays a strong role for the ionization branching ratios even for excitation close to the first absorption resonance. The strong dependence of the branching ratios on the photon energy and their asymmetry around the resonance energy can thus not be explained without taking into account the direct photoionization channels. The contributions to the crosssection of both the direct and the resonant photoemission channels can still describe the main features of the energy dependency of the branching ratios even if the interference between them is neglected. This simplified model, however, fails to reproduce the asymmetry observed in some of the branching ratios and which cannot be quantitatively explained in terms of only the vibrational structure of the core excited state. The asymmetry can thus safely be traced to the interference between the direct and resonant ionization channels, and can in fact be considered to constitute a direct manifestation of the interference effect.

14. Doppler effects In this section we forestall some anticipated developments in high-resolution X-ray Raman scattering spectroscopy by reviewing theory for Doppler effects in RXS and by demonstrating a variety of new physical phenomena related to this effect [266]. The theory is general, covering both radiative and non-radiative X-ray scattering and any character of the states involved, but most emphasis is put on the situation where the Doppler effects are most conspicuous, namely for non-radiative scattering of molecules core excited above the dissociation threshold. As is well-known — and reviewed in the foregoing — an RXS spectrum involving dissociative states consists of two qualitatively different parts, the so-called “molecular” and “atomic-like” parts, Fig. 80. According to the theory in Refs. [103,113] the “atomic-like” contribution caused by the decay transitions in the one of the dissociation fragments has the width equal to the lifetime broadening, something confirmed by the experimental investigations of the resonant Auger spectra of HCl [91,106], that is the same example as discussed in Section 11.4.4. We review a generalization of this result and show that for large release energies following dissociation and accompanying large electron Doppler effects, the atomic-like Auger resonance can be strongly non-Lorentzian. The Doppler shift can thus exceed the lifetime broadening for a kinetic energy release in the region e&1—10 eV, which is not uncommon for dissociating molecules [283], but which is substantially larger than the thermal energy k ¹K0.03 eV. In the case of heteroatomic molecules the electron Doppler shift will be smaller for RXS of the heavy atom because the released kinetic energy is transferred mainly to the light atoms. The HCl molecule investigated in Refs. [91,106] is here a typical example, with the chlorine Auger resonance showing a Lorenzian profiles despite comparatively large release energy. In molecules with comparable atomic masses the electron Doppler shift for the dissociative resonance is sufficiently large to exceed the lifetime broadening by several times (for example, k &0.2!0.5 eV for a molecule like O with C"0.09 eV). Here k is the momentum of the Auger electron.


241

14.1. Shortly about Doppler effects in X-ray spectra In the soft X-ray region the thermal motion of molecules can give additional broadening. Indeed, the total initial and final energies of the photon#molecule system for the Auger process are u#E #M»/2 and (MV#p !k)/2M#E#E , respectively. Here V is the velocity of M D a center of gravity of a molecule with mass M, u and k are the photoelectron velocity and momentum in the laboratory frame of reference, p is the incoming photon momentum, E and M E are total internal molecular energies for the ground and final states. According to the energy D conservation law the photoelectron energy E"u/2 has a shift E"u!u !e!( p !k) ) V DM

(252)

caused by molecular motion, where e is the kinetic energy released under dissociation of the molecule. The term p ) V is the well-known Doppler shift caused by the photon momentum p . This shift is negligibly small in the soft X-ray region due to the small value of the photon momentum p "u/cKu/137 (for example p K0.14 a.u. for O ). This means that ordinary Doppler broadening often can be neglected in X-ray spectroscopy. One can also anticipate that Doppler effects are difficult to identify in the hard X-ray region due to large lifetime broadening and poor spectral resolution. The shift k ) V (252) is analogous to the Doppler shift but is larger and arises from the combination with the electron momentum k [284] (for example kK6 a.u. for O ). The electron Doppler broadening D "k»M (ln 2 depends on the temperature ¹ and on the excitation energy 2 through the thermal velocity u "(2k ¹/M and k"(2E, respectively. Here k is the Boltzmann constant. To get an idea about the Doppler broadening one can consider the X-ray resonant photoemission spectra (RPE — or resonant Auger Raman) of carbon monoxide. The comparison of different broadenings for the C1sPnH RPE spectra of the CO molecule (D K 20 meV, CK42.5 meV) [75] 2 shows that the broadening D caused by thermal motion of molecules must be taken into account 2 in the analysis of highly resolved RPE spectra. This example demonstrates the typical case when the electronic Doppler broadening D caused by thermal motion at room temperature (k ¹K0.03 2 eV) is smaller than the lifetime broadening. Recalling the large kinetic energy (1—10 eV) released under dissociation one can understand that the electron Doppler effect in dissociative states is the largest among the cases mentioned. 14.2. Phase analysis of the scattering amplitude and the Doppler effect For the sake of transparency we first review the case of resonant X-ray scattering (RXS) by a simple three-level diatomic molecule AB with the reduced mass k"m m /(m #m ) (Fig. 97) and with core excitation of atom A (X-ray scattering by homonuclear diatomic molecules is considered in Section 14.5). As discussed above, the Doppler effect in radiative RXS is usually small in comparison with the lifetime broadening C, while it is more important in the non-radiative case (Raman Auger). Due to this fact we focus the review in this section on resonant Auger scattering, but remind that the material covered easily can be extended to the radiative RXS case.


242

We consider the case when an X-ray photon with energy u, wave vector p and polarization vector e is passed during the scattering to an Auger electron of energy E"k/2 and momentum k and when the molecule is excited from the ground "o2 to the final electronic state " f 2. We take special interest in core excitation above the dissociation threshold of the potential surface º (x), G thus when specific “atomic-like” narrow resonances appear [102,106,113]. A qualitative picture of the formation of narrow atomic-like resonances with broad short- and long-wave wings [102,106] is given in Fig. 80. Contrary to the wings that follow the Raman—Stokes dispersion law, the energy position of atomic-like resonance does not depend on the excitation energy [102,91,106]. As one can see in Fig. 97 the nuclear states "e,i2 and "e, f 2 for the core excited and final electronic states are lying in the continuum having nuclear kinetic energies at infinite separation e"p/2k and e"p/2k, respectively. The continuum nuclear wave functions are here normalized to a dfunction: 1 j,e"e , j2"d(e!e ). The double differential RXS cross section for a fixed molecular orientation and monochromatic excitation reads [42,95] p(E,u)""F",

e"u!E!u !(p !k) ) V , DM

(253)

where u "º (R)!º (R )!u /2 ( j"i, f ), R is the ground state equilibrium interatomic HM H M M M M distance, º (x) is the interatomic potential of the jth electronic state, and u is the frequency of the H M vibrational state "o2 of the ground electronic state. For brevity the notation p(E,u) is used here instead of dp(E,u)/dE dO. The lifetime broadening of the final state C is often small in comparison D with the lifetime broadening C of the core excited states and in comparison with the spectral width of incident radiation. Therefore C is neglected in Eq. (253). D

Fig. 97. Scheme of spectral transitions.


243

The “atomic-like” decay transition iPf in the radiative X-ray Raman scattering is dictated by the dipole selection rules Q "eH ) D e\ip R . (254) DG DG In the dissociative region the spectral transition is essentially atomic-like and all electronic wave functions involved to this decay are very close to those of the atom A. We consider here the case of high-energy Auger electrons, which allows to express the wave function of this electron, tk(r), relative to the nucleus A as tk(r)"tk(r!R )eik · R . This means that the Coulomb matrix element has a phase factor

(255)

(256) Q "Q e\ik · R , DG DG since the Auger transition in the considered case can be assumed to take place in the isolated atom A. The Coulomb matrix element Q is here calculated with tk(r!R ). DG 14.2.1. Generalized Franck—Condon factors Due to the nuclear motion, the coordinate R "RM #dR of atom A is shifted relative to the equilibrium site RM at the distance dR . To calculate the nuclear matrix elements we need the expression for R through the normal coordinate x"R !R for the relative motion and through the center of gravity of the molecule R"(m R #m R )/(m #m ) R "ax#R, R "!bx#R , a"k/m , b"k/m . (257) The Born—Oppenheimer approximation makes it possible to rewrite the RXS amplitude as follows

FJand(P!P!k #p) de

1 f,e"e\ik x?"e,i21i,e"eip x?"o2 . X!e!ιC

(258)

The amplitude of the resonant X-ray scattering by atom B is also given by this expression if exp(!ik ) xa) and exp(ip ) xa) are replaced by exp(ik ) xb) and exp(!ip ) xb), respectively. The momentum conservation law (P "P#p !k) describing the photon and electron recoil effects yields immediately the Doppler shift in Eqs. (252) and (253). One here recalls that for the final description of the Doppler effect caused by the thermal motion, the cross section (5) must be convoluted with the Maxwellian distributionJexp(!M»/2k ¹). The d-function in Eq. (258) and the small photon and electron Doppler shifts (p !k) ) V will not further be taken into account due to the smallness of the Doppler effect caused by the thermal motion of molecules. The factor an"eD Q depends on the unit vector n along the molecular axis (x"xn) through GM DG the dipole moment D . The dependence of an on the internuclear distance x enters mainly via the GM decay amplitude Q . This dependence is not so important for the here discussed dissociative or DG atomic-like resonances which are formed by the spectral transitions in one of the isolated atoms (A). So the decay factor Q (x) can be approximated by its asymptotic value Q KQ (R). DG DG DG


244

The FC factors connected with the photoabsorption transition can easily be evaluated assuming the harmonic approximation for the ground-state vibrational wave function 1i,e"eip x?"o2"G(p)ei?p RM ,

2kb 1 *e exp ! G(p),1i,e"o2K2k na 2 c M G

(259) if b;a , M

where *e"e!*º , *º "º (R )!º (R), c "F a , b"(2kF )\, and F "!(dº /dx) is G G G M G G G M G G G M the slope of the interatomic potential º (x) for the core excited state at the equilibrium point R . G M The phase factor exp(ιap ) R ) is very important for hard RXS by symmetrical molecules with M identical atoms [95], since the phase multiplier then destroys the coherence between the scattering channels through the identical atoms. In the case of radiative RXS this destructive interference leads to the violation of the selection rules for the X-ray scattering tensor [95]. Here, though, we will consider only the case of soft X-rays (p R ;1) for which the phase factor exp(iap ) R ) can be M M replaced by unity. The dissociating atoms are moving in a constant potential º (R) with the plane wave function H "e, j2"(2k/pp)sin(p(x!x )#u ) , H H

u" H

p (p !p) dx# . H 4 VH

(260)

Here j"i, f, p "[2k(e!(º (x)!º (R)))], and x is the classical turning point where p "0. H H H H H We will assume below that "e, j2"0 in the classically inaccessible region x(x . The exact value of H the scattering phase can be found directly from the Schro¨dinger equation or from the semiclassical formula (260). The classical turning points x , x as well as the scattering phases u , u for the core G D G D excited and final state potentials depend on the energies e and e, respectively (the energy labels for these quantities are dropped here since we will need the values of x and u only for e"X). H H According to the Condon principle the value of x is close to R . G M From the continuum wave functions (260) the second generalized FC factor can be evaluated analytically as

k e\ιOVG eiPd(p!p#q)#e\iPd(p!p!q) 1 f,e"e\ik x?"e,i2" 2(pp)

1 1 1 #ι eiP # 2sin(u ) p#p p!p!q n !e\iP

1 p!p#q

, x 5x , G D

(261)

where q"ak cos h, u"u !u #pD, D""x !x " , D G G D

(262)


245

u "u #u #pD, h is the angle between the momentum k of the Auger electron and the G D molecular axis x, P is the principal value. The continuum FC factor for x (x is again given by G D Eq. (262) after permutations: I & f, p & p. It is relevant to mention that the generalized FC factors can be important also in bound—bound transitions due to a large momentum of the Auger electron. 14.3. Anomalous anisotropy of Auger electron and ion yields To succeed in the evaluation of the scattering amplitude (258) we take into account only resonance contributions in (261). In this approximation the non-resonant term (1/(p#p)) is thus neglected and one can continue the integration over p up to !R. The expression (258) for the scattering through the core excited state in atom A then becomes [266] F"f e\iOVG, u"u !u #p D if x 5x , D G M G D F"f e\iOVD, u"u !u #p D if x 'x G D M D G

(263)

with e\iP+G(p)!G(p ), eiPG(p ) M M fJan # if x 5x , G D l#kv cos h!iC l!kv cos h!iC eiP fJanG(p )e\CDTM if x 'x , (264) M D G l#kv cos h!iC l"E!u , and u "º (R)!º (R). The electron Doppler shift kv cos h depends on the GD GD G D velocity v "ap/k of the atom A after emission of the Auger electron. Here p"[2k(X!(E!u ))], v "(2X/k) is the relative atomic velocity corresponding to the GD M kinetic energy e"X. In the latter expression for f only the main contribution is retained. 14.3.1. Resonant cone of dissociation At this stage we emphasize the main contribution to the RXS cross section (253) 1 (265) (E!u #kv cos h)#C GD which concerns electron—ion coincident spectroscopies with the experimental fixation of the molecular axis (via a direction of dissociation) and the direction of the Auger electron propagation. Experimental investigations [128,285,286] have clearly shown the anisotropy of dissociation in angle-resolved photoelectron—photoion spectroscopy. This anisotropy, hidden in the anisotropy factor an, is smoother in comparison with the strongly resonant ion yield (265) caused by the Doppler effect. The last equation shows that the electron—ion coincidence signal changes drastically if the Auger electron energy E lies in the Doppler band !kv (E!u (kv . When the GD lifetime broadening C is smaller than the electron Doppler shift kv the photoions or the fragment of dissociation propagate in the narrow angular interval dhKC/kv

(266)


246

Fig. 98. Resonant cone of dissociation.

close to the cone surface (Fig. 98) hKh P u !E u !E if GD 41 . (267) cos h " GD P kv kv The appearance of the narrow resonant cone of dissociation with “resonant” angle h is one of the P important features of the studied problem. Eq. (266) clearly demonstrates the strong correlation between the propagation directions of the Auger electron and the ion A>.

14.4. Averaging of the cross section over molecular orientations We consider the spectral shape of atomic-like resonances in the Auger spectrum of atom A in molecule AB with different atoms (the principally different case of identical atoms is treated in Section 14.5.2). In this case the cross section emanates entirely from the scattering amplitude for atom A (to be specific only the case x 'x will be considered in this section). For ordinary G D resonant Auger measurements with the gas-phase molecules, the cross section must be averaged over molecular orientations. Two qualitatively different physical reasons are responsible for the dependence of the scattering amplitude (263) on the molecular orientation. As shown in Section 14.3 the first reason is the Doppler effect which leads to a sharper resonant dependence of the cross section on the molecular orientation (265). The second reason is the orientation of the molecular axis n relative to the polarization vector e and the Auger electron momentum k. This smooth polynomial dependence is hidden in the factor an. This allows to extract the factor "anM" from the integral over the molecular orientation at the resonant angle (267), leading to an additional averaging of this factor over all nM of the cone. Using this averaging together with Eqs. (253) and (263), one obtains

C p(E,u)"p o G(p )#(G(p)!G(p ))# s cos(2u)G(p )(G(p)!G(p )) M M M M M E!u GD

.

(268)


247

All unessential quantities are collected in p J1"anM"22p/C. The spectral shape of the cross section M is defined by the two functions

o"(1/2pkv ) arctan((E!u #kv )/C)!arctan GD

E!u !kv GD C

,

(269)

1 (E!u #kv )#C GD s" ln . 2pkv (E!u !kv )#C GD The Doppler shift

2 cos h , kv cos h"ka (X!(E!u )) GD k

(270)

depends on the Auger electron energy E (see Fig. 100). The energy dependence of the electron wave number k"(2E can be neglected when E is large. To understand the main spectral features of the RXS cross section one can for a while neglect the E-dependence of kv . The o function is in this case a symmetrical function relative to the resonance E!u "0 normalized to unity: dEo"1. The integral of s is equal to zero (dEs"0) since s is GD an antisymmetrical function relative to the resonant energy E"u . The asymptots of the GD o function

E!u 1 GD H if C;kv , o" kv 2kv C o" if CH>? in the one-phonon approximation. This results in the D selection rules F' "f [1!(!1)J>H>?] . D M

(347)

16.5. Zero-phonon line in RIXS The RXS band is formed as a sum of the subbands corresponding to the electron—vibrational transitions from different occupied MOs t . Every subband can have a so-called zero phonon (ZP) H line corresponding to the decay transitions without change of vibrational quantum numbers (Fig. 124): X"w , m"mD . H

(348)

292


In the framework of the reviewed model — the harmonic approximation with the same vibrational frequencies for ground, core excited, and final states — the ZP line is a single resonance. It is then assumed that the adiabatic potentials of the core excited and final states are not shifted relative to the ground state potential surface. This single line is of course broadened or split in a “real” system due to the anharmonicity and the different vibrational frequencies for different electronic states. 16.5.1. Broadband excitation The positions of the RIXS resonances do not depend on the excitation frequency, u, when the width of the spectral function is larger or comparable with the spacings between electronic levels

Fig. 123. The RIXS cross section of a N"6 polyene versus the momentum exchange of valence electrons with X-ray phonons and photons. (a) accounting only for momentum exchange between electrons and photons: g "0, f"0.175; (b) M accounting only for momentum exchange between electron and phonons: g "0.9 and f"0; (c) accounting for M momentum exchange of electron with both photons and phonons: g "0.9 and f"0.175. The cross section is decreased M by 10 times. C"0.09 eV, c"0.05 eV, I"290 eV, u "0.5 eV. The value of f"0.175 in (a) and (c) corresponds to M 0"60° and "0. The abbreviations HUMO and LUMO#1 mean the core excitation (X"w ) to the highest J unoccupied MO (l"6) and to the MO with l"5, respectively. The arrows in panel (a) mark the positions of the zero-phonon lines for the decay transitions from occupied MO with j"1,2,3. The dashed lines in the panels (b) and (c) show the excitonic bands.


293

(c9b) and u is tuned to the photoabsorption band. As one can see from Fig. 124 the ZP line has blue and red vibronic sidebands. The effective width of these phonon sidebands can be estimated with help of the reflection method a "1/(ku (349) C "(2u D /a . ? ?L ? ? ? $! Apparently, it is difficult to select the ZP line from the quasicontinuum vibrational spectra of long polyenes or solids. 16.5.2. Narrowband excitation In the case of RXS being initiated by a narrowband X-ray beam, c;u , the RXS spectral profile ? is formed according to the energy conservation, or Raman—Stokes law, for the whole RXS process X"X!w #e(mM)#w !e(mD) . (350) J H The lowest vibrational levels (mM"0) of the ground state are the only populated ones if the temperature is low. According to the energy conservation law (350) (which “couples” only ground and final states) the ZP line corresponds to the emission m"0PmD"0

(351)

and possesses a long wave phonon sideband (the Stokes sideband, see Figs. 123 and 124). The effective width of this broad sideband is given by Eq. (349) since the final vibrational states are populated through the core-excited state in accordance with the corresponding FC factors. Contrary to the ZP line, the phonon sideband is formed by decay transitions that change the

Fig. 124. Qualitative picture of the formation of the zero-phonon line in the RIXS spectrum for one vibrational mode. Left panel — broadband excitation: The decay transitions without change of vibrational quantum numbers mPmD (348) form the ZP line. Right panel — narrowband excitation: The ZP line 0P0 (351) is formed according to the energy conservation law (350).

294


vibrational quantum numbers, mPmD$Dm. It is interesting to note that the position of the ZP line (350) does not depend on l X"w #e(0)!e(mD)4w if X"w (352) H H J if the incident photon frequency is tuned to resonance with the lth unoccupied MO. The ZP line for higher temperatures ¹ has also a blue phonon sideband (the anti-Stokes sideband), since other vibrational levels of the ground state (mMO0) are populated, see Eq. (350). Apparently, the width of the anti-Stokes sideband is proportional to the temperature for small ¹. Comparison of Eqs. (348) and (352) allows to conclude that mainly m"0PmD"0 vibrational transitions contribute to the ZP line for narrowband excitation. 16.5.3. Selection rules for the zero-phonon line It is not hard to see that the RIXS amplitude for the ZP line is proportional to 1#(!1)H>J [325]. This results immediately in the following selection rules for the ZP line: F "0 if j#l"odd . (353) M These selection rules for occupied, t , and unoccupied, t , MOs are valid only for narrowband H J excitation. As it was pointed out above, the 0P0 vibrational decay transition gives then the main contribution to the ZP line. This is shown by Figs. 123b and c, which also confirm the selection rules (353). The contribution to the ZP line from transitions with m"mDO0 is not suppressed when the width of the incident radiation is comparable or larger than the smallest vibrational frequency u . Hence, the selection rules (353) are not valid for broadband excitation. Let us return to the narrowband excitation and the example given by Figs. 123a—c. In this case only the ZP lines contribute to the spectrum if VC is absent (g "0), see Fig. 123a. When the M incident radiation is tuned in resonance with the even unoccupied MO (LUMO: l"4 or HUMO: l"6) the ZP line is seen only for j"2 in accordance with the selection rules (353). And vice versa, the ZP line is forbidden for j"2 and allowed for j"1,3 under core excitation to the odd unoccupied MO, l"5, (upper panel, Fig. 123b). All three ZP lines ( j"1,2,3) are allowed when the photon momentum is taken into account, see Fig. 123a (without VC) and Fig. 123c (with VC). 16.6. RIXS in the dipole approximation over molecular size To complete our discussion of selection rules, the present subsection offers a brief outline of RIXS in the dipole approximation versus the molecular size fN&2p¸/j;1 ,

(354)

where the molecular length ¸"aN is assumed to be smaller than wave length j of the X-ray photon. This approximation is valid also for long molecules when the scattering angle 0 is small. The dipole approximation breaks down for polyenes with N910 if the scattering angle is fairly large. One should note the qualitative distinction of this approximation from the dipole approximation for intraatomic transitions. Eq. (354) allows to expand the photon factor e\ιDL in the series over the small parameter fn. The zero-order term does not contribute to the amplitude because of the orthonormality of the MO coefficients (329), while the non-zero contribution to the RIXS


295

amplitude appears in the first order over fn as , f c c n. (355) JL HL X!w #iC H L In fact, this term is the quadrupole contribution since the dipole approximation for intraatomic transitions was taken into account already for the atomic transition moment d . The implemenQN tation of the inversion symmetry and Eq. (331) results in the desired selection rules F' "0 if l#j"even . (356) D Fig. 125 shows clearly these selection rules (fNK0.1). The violation of the selection rules (due to fNK1) is seen in Fig. 123a. 16.7. RXS for detuned incident radiation 16.7.1. Restoration of the selection rules and collapse of vibrational structure in polymers RXS spectra (Fig. 123) are generally broadened by the vibrational structure of the intermediate and final states. The vibrational band broadening is often comparable with the spacings between electronic levels. We have shown in Section 8.3 that vibrational broadening as well as VC of core excited states can be suppressed by shortening of the RXS duration q , see also Section 4. Both A effects were illustrated in these sections for small molecules; CO and CO . The same picture is obtained for polyenes (Figs. 126—128). We recall that the vibrational structure in elastic RXS collapses totally to a single resonance with the width defined by c [105,104,325]. In the inelastic, RIXS, case a large detuning eliminates the vibrational structure due to the core excited state, but not the vibrational structure caused by the final state. The total vibrational collapse in RIXS takes place only when the potential surfaces of ground and final states are the same [105,104,325] (Figs. 126b and c). It is not hard to see that a large detuning quenches VC and, hence, that the symmetry breaking thereby restores the selection rules also for RIXS of polymers. Indeed, according to the orthogonality condition (329) F' "0 if the photon momenta are neglected. Fig. 126c shows the D coincidence of the RIXS spectral profile (343) with the one obtained from calculations without electron—phonon interaction (except the evident Raman—Stokes shift). Due to the small photon momentum, only the highest occupied MO (HOMO), close to the Fermi level, participates in the RIXS process, see Fig. 126. As mentioned above, all occupied MOs become active in RIXS for large photon momenta, see Fig. 127 which also clearly illustrates the purification of the RIXS spectrum due to the collapse effect for each occupied MO. The collapse effect takes place here since the shifts between all adiabatic potentials are neglected. In this case the spectral profile of a certain electronic transition collapses to single resonance. The spectral shape of the RXS resonance copies the spectral function of the incident radiation with a width c for narrowband excitation. The RIXS spectral profile is broadened by the vibrational structure of the final state if the potential surfaces of the ground and final states are different (Fig. 128). 16.7.2. Restoration of selection rules and collapse effect in solids It was shown above that the electron—phonon interaction is strong with an accompanying breakdown of the momentum conservation for the electrons (318) when the excitation energy is

296


Fig. 125. The RIXS cross section of an N"6 polyene in the dipole approximation over molecular size (355), f"0.0175 (fN"0.105). The incident radiation is tuned in resonance with unoccupied MO, l"5, in panel (a), and with MO l"6 in panel (b). The approximate selection rules (356) suppress the resonances in occupied band when l#j"even.

tuned to the photoabsorption band. For solids with continuum electronic spectra referring to a conduction band, the notion of a detuning is not well defined. Nevertheless, the fast RXS (61) can be reached by tuning the excitation energy below the photoabsorption threshold. For solids the translational invariance of the Bloch function for electrons of the ith band: t k(r)"eιk ru k(r), G G u k(r#R )"u k(r), appears as a new symmetry element. The asymptotic expressions for the RIXS ? G G and REXS amplitudes in this case read F' J[1/(X#ÓC)]1mD, f "mM,o2 DHk D kd(k!k!q!G) , HY J D G F#Jdm f mo fq(X)d(q!G) . u

(357)

The latter equation results in a collapse of the phonon structure to a single resonance for the elastic band [105,104,325]. p#(u,u)JU(u!u,c) .

(358)

It is relevant to consider RIXS in the soft X-ray region where one can neglect the change of the photon momentum q (see, however, previous sections). According to Eq. (358) the inelastic scattering takes place with conservation of the electron momentum, k"k. The RIXS cross section


297

Fig. 126. Collapse of vibrational structure in RIXS of an N"6 polyene due to the detuning of incident X-ray radiation from the absorption band. The potential surfaces of initial and final states are the same. The detuning of incident radiation from the LUMO (l"4) is defined as XM "X!w , M"3. (a) The resonant excitation of the LUMO +> (X"w ) leads to vibrational broadening of the RIXS profile (the cross section is decreased by 10 times). Panels (b) +> and (c) show the collapse effect corresponding to a detuning of incident radiation below the LUMO with XM "!5 eV and XM "!8 eV, respectively. The dashed line in panel (c) shows the RIXS cross section for resonant excitation of the LUMO and without electron—phonon interaction (g "0, XM "0). The value of the VC parameter is the same, g "0.9, M M except for the RIXS profile shown by a dashed line with g "0. All other parameters are the same as for Fig. 123. M Fig. 127. Same as for Fig. 126, but for an artificially increased X-ray photon frequency, f"2.625. Here XM "X!w is , the detuning from the HUMO. (a) Exact resonance with HUMO (l"6); the cross section is decreased by 10 times. (c) incident radiation is tuned above the HUMO, XM "8 eV. The comparison of (a) and (b) shows the “purification” of spectrum (b) due to the collapse effect.

then becomes

p'(u,u)J dk. (mM)"DHk D k1mD, f "mM,o2"U(X!X#u ,c), HY J DM f o JH m m u "u (k)#e(mD)!e(mM), u (k)"e (k)!e (k) . (359) DM JH JH J H Of particular interest here is the conservation of the electron momentum as a direct manifestation of the restoration of the translational invariance of the electronic subsystem in the core excited state. This is obviously the same effect as the symmetry purification effect.


298

Fig. 128. Incomplete narrowing of the RIXS profile due to the different potential surfaces of the ground and final states. N"6 polyene, f"0.175. The dimensionless shift, gD"DD/(2a "0.3, of the final and ground state potentials is the L L L same for all vibrational modes in this calculation. All other parameters are the same as for Fig. 123. XM "X!w is the +> detuning of incident radiation from the LUMO. (a) XM "0. (b) XM "!0.5 eV. (c) XM "!2.0 eV. (d) XM "!5.0 eV.

16.7.3. The joint density of states One can see from Eq. (359) two broadening mechanisms for the RIXS profile. The first one is given by the different dispersion laws for the valence, e (k), and conduction, e (k), bands. H J The second broadening mechanism is related to the FC factors caused by the many phonon transitions (mDOmM). In this context it is relevant to examine a solid with the same potential surfaces of the ground and final states. The FC factor 1mD, f "mM,o2 then reduces to dmf mo, and one obtains

p'(u,u)J dk"DHk D k"U(X!X#u (k),c) (360) HY J JH JH making use of Tr . "1. Inspection of this equation shows that the spectral profile is free from any phonon broadening. It can be converted to an integration over a surface of constant energy dS X!X"u (k) (361) JH in the case of narrowband excitation, U(X,c)Kd(X). As an approximation, one may suppose that ¼ (X!X),"DHk D k" does not vary strongly with the angle on a surface of constant energy, so JH HY J


299

that it may be extracted outside the integral. This allows to connect the RIXS profile

dS (362) p'(u,u)J ¼ (X!X)o (X!X), o (X!X)" JH JH JH " ku (k)" JH JH with the joint (or combined) density of states o (E) for bands l and j. Such a “joint density of states” JH appears, for instance, in the theory of optical photoabsorption [332]. The essential physical result here is that the singularities, ku (k)"0, of the joint density of states follow the Raman—Stokes JH dispersion law in the sense of Eq. (361), something that allows to map the band structure. One can here recall that the RIXS spectral profile is described in the one-electron approximation by the interband density of states (322) [195,202], which, according to Eq. (362) reduces to the joint density of states if the detuning is large. Eqs. (358) and (362) show that the RXS profile is free from phonon broadening when the detuning or when the lifetime broadening C is large. This means a narrowing of the RXS spectral features caused by the singularities of the joint density of states for large X. As a special case when the bands e (k) and e (k) are parallel, the RIXS profile collapses to the single resonance; J H JU(X!X#u (0),c), with the width c [325]. JH 17. X-ray Raman scattering by surface adsorbates Most X-ray spectroscopy studies of surface adsorbates have been carried out in the absorption mode. As for other aggregates, however, the availability of 2nd and 3rd synchrotron radiation sources with high brightness has initiated a concomitant development of the more arduous X-ray emission spectroscopy, producing good-quality spectra also in this mode [333—335]. This has made it possible to exploit the many complementary features of the emission and absorption spectroscopies also for surface adsorbates and to study features like adsorbate orientation, bonding and internal reconstruction in fundamentally new ways. For instance, unlike valence band photoemission for which one obtains signals that represent the delocalized states, it has been verified that the X-ray emission signals indeed reflect the part of the electronic structure that is localized to the adsorbate [334]. The fact that XES probes the electronic structure of the occupied, rather than of the unoccupied levels as in NEXAFS, has been utilized not only to obtain information about bonding and for proposing new bonding models [336], but also for studying the geometric corruption, and the concomitant loss of bond order, of the molecule when adsorbed on the surface [337—339]. As for NEXAFS, X-ray emission spectroscopy has special symmetry, orientational and polarization selective properties when applied to surfaces. For resonantly excited spectra these properties are strongly enhanced, and qualitatively different, owing to the fact that channel interference effects then are of primary importance. In the resonant case the spectral properties are therefore closely related to the excitation frequency, to the symmetry and precise organization of the intermediate core-excited states prepared by the resonant excitation. The tuning of the incoming X-ray radiation to the discrete resonances of the adsorbate target may thus lead to scattering spectra that show very specific polarization and symmetry dependences which can provide evidence of symmetry assignments of the participating electronic levels and of orientation of the adsorbate with respect to the surface.

300


The number of X-ray emission studies of surface adsorbates are still relatively few in numbers, in particular those which explicitly utilize the resonant nature of the core excitation in the analysis (in many cases resonant excitation has been used only as a means to enhance the total cross section and to remove satellite contributions). As for gases and crystalline solids one can expect the main features associated with resonant X-ray emission spectra of surface adsorbates to be understood from second-order perturbation theory between light and matter, as manifested by the Kramers—Heisenberg equation for the X-ray scattering amplitude — the consequences of which have been extensively reviewed in this work — although comparatively little is yet known about its actual consequences for resonant X-ray emission spectra of surface adsorbates with varying degrees of (dis)order; 1-, 2-, and 3-dimensional (dis)order. Some of the recent advances [214,340], in this respect are reviewed below. An evident difference for strictly ordered adsorbates with respect to 3-dimensional disordered systems as free molecules is that orientational averaging of the sample is no longer appropriate, making the orientational and polarization dependences so much sharper. Symmetry and momentum selection rules, appropriate for gases and solids, respectively, will have a different meaning, due to the quasi-continuous set of levels that often show up for surface adsorbate systems. One can anticipate that, just as for gas phase systems and solids, there is a strong connection between the channel interference and the actual observation of the symmetry selection rules. Towards the harder X-ray region, K1000 eV and beyond, one might anticipate a close connection between the channel interference and the selection rules on the one hand and the photon phase factors and the Bragg conditions on the other, making the scattering cross sections strongly anisotropic and oscillatory. Different dephasing mechanisms, such as orientational disorder, vibrational motion and vibronic coupling, may destroy the interchannel coherence and eliminate the selection rules [214]. Upon adsorption, molecules orient themselves in order to minimize the total energy. The most favorable orientation varies with the coverage and nature of the adsorbate and the substrate. The outcome of the RXS spectrum is evidently dependent on the adsorption strength between the molecule and the surface, whether the adsorption systems can be classified as physisorbed or chemisorbed. In physisorption the bonding is week, with adsorption energies in the order of 0.1 eV, whereas chemisorption energies are in the order of 1 eV. In systems with weak adsorbate—substrate interaction, the adsorbate—adsorbate interaction may also be of importance, depending on the coverage. The richness of the structural phase diagram encountered in physisorption systems arises from the subtle balance between the corrugation of the substrate potential and the adsorbate—adsorbate interaction [341] (Fig. 129). At low coverage only adsorbate—substrate interaction is important. As the coverage increases, the molecules can form a monolayer with several orientational phases [341] (Fig. 129). The present section reviews RXS for surface adsorbates in two aspects; firstly, Section 17.1 describes how the theoretical formulation using point groups for the one-step resonant X-ray emission cross section of randomly oriented molecules (Section 5.3) — applicable to any type of polarization for the absorbed and emitted photons in the soft X-ray region — carries over to samples with fixed order [141]. Some recent results for benzene and ethylene adsorbed to copper surfaces are reviewed in order to illustrate the symmetry and polarization information and the role of the channel interfence effects [293]. In the second part to follow, Section 17, radiative and nonradiative RXS following “hard” X-ray excitation of surface adsorbates is also considered, then focusing on various dephasing mechanisms for the X-ray scattering.


301

Fig. 129. Four orientationally ordered phases for molecules A physisorbed on a triangular lattice [341]. (a) Two sublattice in-plane herringbone phase. (b) Four-sublattice pinwheel phase where the circles with dots indicate the molecules perpendicular to the surface. (c) Two-sublattice out-of-plane herringbone phase. (d) Ferrorotational phase where all molecules are free to rotate uniformly by a constant phase angle . A systematic out-of-plane tilt of the molecular axes is shown by arrows.

17.1. Theory for polarized RXS from adsorbed molecules The general theory for symmetry and polarization selection from randomly oriented molecules [95,141,122,342] reviewed in Section 5, forms a good reference point for the consideration of completely ordered systems, such as surface adsorbates of 1-dimensional order, see work of Triguero et al. in Ref. [293], which gives the basic material to the present subsection. A necessary ingredient in that theory was a general transformation of the transition dipole matrix elements d and d — naturally given in the molecular frame x, y, z — to the laboratory coordinates X, ½ and JA AH Z — in which the polarization vectors are expressed — through the directional cosine transformation ¹6"¹?t , see Fig. 130 and Section 5.3. Using these transformations the total X-ray scattering A A ?6 amplitude is obtained through the expressions (71), (72), (73), (74), in which the polarization, orientational and molecular information is collected in three different factors, Eq. (71). Tractable expressions for systems of fixed order — without orientational averaging — are obtained by rearranging Eq. (72) as F " ¸ ¸ F@A , JH @ A JH @A where

(363)

¸ " eHt , @ @

(364)

¸ " e t A A

(365)


302

in which the summation has been reintroduced. ¸ and ¸ are the directional cosines of @ A polarization vectors e and e in the molecular frame. Therefore,

"F "" ¸ ¸ F@A . JH @ A JH @A The total cross section is given by

(366)

p(u,u )"r uu ¸ ¸ F@A U(u#u !u,c) . (367) M @ A JH JH JH @A For adsorbed molecules with degenerate and quasi-degenerate core orbitals, it is a good approximation to assume u "u for all core MOs c. Eq. (73) can thus be written as HA H (C/n) 1 u u d@ dA " K@A (368) F@A" JA HA JA AH u!u #iC JH JH u!u #iC H H A and "F ""D(u!u ,C)"K ", K " ¸ ¸ K@A . JH H JH JH @ A JH @A

Fig. 130. Transformations between the laboratory and molecular coordinate frames. From Ref. [293].

(369)


303

The cross section is then simply obtained as p(u,u)" p (u,u) . (370) JH JH p (u,u)"r(u/u)"K "D(u!u ,C)U(u#u !u,c) . (371) JH M JH H JH It is relevant in this context to review prototype systems in terms of few-states models. We discuss three such systems assuming narrowband excitation in a strongly absorbing n-system orthogonal to the substrate with the incident polarization e-vector along the surface normal: ¸ "d . (372) @ 8X These typical systems will also be utilized in the two examples of surface adsorbate RXS spectra, reviewed in Section 17.3 below. 17.1.1. Isolated core levels: Two-step model In the case with a single, isolated, core orbital involved, one has for the strict resonance u"u J (373) "F ""D(u!u ,C)"IX " ¸ IA , A AH JH H JA A where I@ "u d@ , I@ "u d@ (b"z,c) JA JA JA AH HA AH (374) p(u,u )"r "IX " ¸ I A D(u!u ,C)U(u!u ,c) . H H J M JA A IL J H A The absorption and emission processes are completely decoupled in this case, within a frozen orbital picture (no-screening approximation, see Section 15), as is the case in the traditional two-step model of RXS.

17.1.2. Core levels of different symmetry: Polarization selective excitation The situation changes qualitatively with two initial and near-degenerate core orbitals, c and c (here assumed to be of C symmetry a and b ). Under the given circumstances only unoccupied T orbitals with symmetries a and b will be involved in the absorption processes, and p(u,u)" "IX " ¸ IA D(1)U(1)# "IX " ¸ IA D(2)U(2) , (375) JA A A H JA A A H J H J A A where D(i)"D(u!u G(l ),C), U(i)"U(u#u G !u,c), u G(l )"E(c\l)!E( j\l). If the screenHA G HA G JH ing effect is the same, u (l )"u (l )"u , one has HA H HA (376) p(u,u)" D(u!u ,C) "IX " ¸ IA U(1)# "IX " ¸ IA U(2) . JA A A H JA A A H H J J A A H Only in this case will the resonant X-ray emission have the same spectral lineshape as that of the non-resonant spectrum, and one should expect to see some difference between resonantly and non-resonantly excited X-ray emission spectra already for a system with two core levels, even

304


though of different symmetry. This difference is not due to the interference effect, but rather due to the polarization selective excitation which necessitates an explicit consideration of the intermediate state. Eq. (376) can be obtained from the so-called generalized two-step model [122], i.e. a model which accounts for the character of the intermediate states but which ignores channel interference. 17.1.3. Core levels of same symmetry: Channel interference The role of interference is most clearly described when the molecule has two near-degenerate core levels of the same symmetry (a , as in most chemically shifted species). With a polarization vector along z, only unoccupied orbitals with symmetry a will then be excited. For a system with one such unoccupied orbital l, the excitation can involve either of the core levels and, as a consequence, the total amplitude must be computed as a summation of two individual channels (i"1,2): (377) FXA"FXA (c )#FXA(c ) , JH JH JH K(c ) G FXA(c )" , (378) JH G u!u G#iC HA (379) K(c )"u Gu GdX GdAG G J HA JA A H with the cross section given by Eqs. (307) and (308). The cross section produced by the interference of the two channels, p (308), is dependent on the energy separation between the two core levels. If one assumes u"u i.e. resonance with one decay channel, then D "u !u "u is the HA A HA HA AA energy difference between the two core-excited states, or the energy difference between the two core levels in the frozen picture. The interference term (308) can then be written as p "2K K 1/(u #C) , AA A A i.e. the larger the energy separation, u , the smaller is the interference effect. AA

(380)

17.2. Dephasing of X-ray Raman scattering by surface adsorbed molecules Radiative RXS by symmetrical molecules with identical orientation (1D system) shows strict selection rules (156) for radiative RXS, as demonstrated in Section 9.3. As briefly reviewed below, parity selection rules are retained also in the non-radiative (Auger Raman) process of such systems. These selection rules break down in the hard X-ray region due to orientational dephasing. The effects of the dephasing and violation of selection rules were reviewed in Section 9 for free molecules (3D disorder), and we consider here the similar problem — vibrational and librational dephasing for surface adsorbed molecules. The dephasing depends on the degree of disorder which evidently can be different for different systems. A strict theory of the orientational dephasing covering 1D, 2D, 3D (dis)order in radiative and non-radiative RXS were given in Refs. [214] and [340], respectively. To obtain a compact review of this theory, we consider here only the dephasing in 1-dimensional systems, i.e. systems with the same orientation of all molecules, as certainly is realizable by adsorption on a surface. To be specific, it is assumed that the molecules lie in the substrate plane (Fig. 131). When a diatomic homonuclear molecule is perfectly aligned the radiative RXS cross section is given by Eq. (153). The cross section for the resonant Auger scattering (Fig. 132) has the same structure [340].


305

17.2.1. Librational dephasing Surface adsorbed molecules can be considered perfectly aligned only in a classical physics picture at low temperatures. In reality, the zero-point quantum librations of the molecules in their potential wells considerably broaden the orientational distribution. For example, the calculated root-mean-square (RMS) amplitude of these librations is 14° for the herringbone phase of N adsorbed on graphite [341] (Fig. 129a). Apparently, the effect of librational dephasing is stronger than the vibrational dephasing [214,340] due to the weaker van der Waals interaction responsible for the librations. For example, the large librational dephasing for 1D ordered N molecules dominates in comparison with the very small vibrational dephasing [214,340] (Fig. 133). The radiative and nonradiative RXS cross sections averaged over librations can be expressed in the following way [214,340]: p(u,u)"p0(1#Ps0) , M (381) p(e,u)"p,0(1!(!1)lPs,0), p ) R"ln. M The cross section for non-radiative RXS is given for the photon interference factor satisfying to the Bragg condition with l"0,1,2,2. l"0 corresponds to the soft X-ray region. We consider here the case of high-energy Auger electrons with the energy e and momentum k. Both MOs t and J t involved in the Auger decay (Fig. 132) are assumed to be n orbitals with negligible contributions H from d orbitals. Without librational dephasing the s functions, s0"cos(q ) R) and s,0"cos(k ) R), result in the following selection rules for both radiative (156) and non-radiative RXS [340] p(e,u)"0 if f"g,

(k#p) ) R "even , n

p(e,u)"0 if f"u,

(k#p) ) R "odd . n

(382)

Fig. 131. Geometry of the substrate with physisorbed molecule. Non-radiative RXS is shown; the same geometry is valid for radiative RXS if we do the following replacements: kPq, 0Ph , and uPu . O O Fig. 132. Scheme of spectral transitions for non-radiative RXS.

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The dephasing caused by the librations leads to a violation of the parity selection rules (156) and (382). The physical reason for this was discussed already in Section 9.3. The formal reason for this violation of the selection rules is the averaging of the interference terms cos(q ) R) and cos(k ) R) over librations. The average of these terms over zero-point and thermal librations is conveniently obtained using the density operator technique 1 1 s0" Re Tr(. p0(RK )eiq R), s,0" Re Tr(. p,0(RK )eik R) , M M p,0 p0 M M . "e\@&/Tr(e\@&), b"1/k ¹

(383)

with H as the libron Hamiltonian and the density operator normalized to unit Tr(. )"1. We consider here 1D ordering with small librations relative to the equilibrium molecular direction (Fig. 131). In the general case, the librations in the surface plane ("") and those “tilted” out of the plane (N) have different amplitudes and frequencies. Due to the smallness of librations the density operator factorizes as

. ". ) . . (384) , The harmonic approximation leads to the well-known expression for the density operator in configuration space [343] . (d,¹)"1/(2p1d2exp(!d/21d2), j"N,"" . H H H This immediately results in the following expression for the interference term (383): s0"cos(q ) R)e\50, s,0"cos(k ) R)e\5,0 .

(385) (386)

The libron Debye—Waller (DW) factors are strongly anisotropic ¼0"¼0 #¼0, ¼,0"¼,0#¼,0 , , , ¼0"qR1d2cos0, ¼,0" kR1d2cosh , j"N , H H H H O ¼0"qR1d2sin0 cosu, ¼,0" kR1d2sinh cosu , j""" . (387) H H H H O O The mean square of libron angles d depends on the temperature and on the amplitudes of the H zero-point librons dH (j"N,"") M ¹H if ¹/¹ H ;1 , 1d2"dHcoth "dH M H M 2¹

¹H 1d2"dHcoth "dH 2¹/¹H , H M M 2¹

if ¹/¹ H

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