Physics Reports vol.397

Physics Reports 397 (2004) 1 – 62 www.elsevier.com/locate/physrep Development of turbulence in subsonic submerged jets ...

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Physics Reports 397 (2004) 1 – 62 www.elsevier.com/locate/physrep

Development of turbulence in subsonic submerged jets Polina S. Landaa , P.V.E. McClintockb;∗ a

Department of Physics, Lomonosov Moscow State University, 119899 Moscow, Russia b Department of Physics, Lancaster University, Lancaster LA1 4YB, UK Accepted 13 March 2004 editor: I. Procaccia

Abstract The development of turbulence in subsonic submerged jets is reviewed. It is shown that the turbulence results from a strong ampli3cation of the weak input noise that is always present in the jet nozzle exit section. At a certain distance from the nozzle the ampli3cation becomes essentially nonlinear. This ampli3ed noise leads to a transition of the system to a qualitatively new state, which depends only slightly on the characteristics of the input noise, such as its power spectrum. Such a transition has much in common with nonequilibrium noise-induced phase transitions in nonlinear oscillators with multiplicative and additive noise. The Krylov–Bogolyubov method for spatially extended systems is used to trace the evolution of the power spectra, the root-mean-square amplitude of the turbulent pulsations, and the mean velocity, with increasing distance from the nozzle. It is shown that, as turbulence develops, its longitudinal and transverse scales increase. The results coincide qualitatively and also, in speci3c cases, quantitatively, with known experimental data. c 2004 Elsevier B.V. All rights reserved. PACS: 47.27.−i; 47.20.Ft; 05.40.−a; 47.27.Wg Keywords: Turbulence; Submerged jet; Noise; Nonequilibrium phase transition

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Brief review of the evolution of views of turbulence as an oscillatory process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Some experimental results concerning turbulence development in jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The main properties of jet ?ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Evolution of power spectra of the pulsations of ?uid velocity and pressure with the distance from the nozzle exit section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. E-mail address: [email protected] (P.V.E. McClintock).

c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2004.03.004

2 4 7 7 9

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P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

3.3. A jet as an ampli3er of acoustic disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Suppression and intensi3cation of turbulence in jets by a weak periodic forcing . . . . . . . . . . . . . . . . . . . . . . . . 4. The analogy between noise-induced oscillations of a pendulum with randomly vibrated suspension axis and turbulent processes in a jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The main equations and dynamics of a plane jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The derivation of truncated equations for the amplitude of stochastic constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Generative solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. The 3rst approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Region I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Region II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. The second approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 12 18 22 29 30 48 50 52 53 57 58 58

1. Introduction There is an abundance of published works relating to hydrodynamic turbulence problems. It is interesting that the 3rst experimental work where the transition to turbulence was observed as the ?uid viscosity decreased, due to heating, was reported in 1839 by Hagen [1]. Over the years the volume of experimental works has increased to such an extent that it cannot even be listed in a review of this kind. As examples, we mention only a fundamental paper by Reynolds [2], where elegant experiments with stained liquid were described and intermittent behavior was 3rst discovered, the Compte-Bellot’s book [3], wherein a detailed comparison is made between turbulence power spectra in a plane channel and Kolmogorov’s spectra, and the book by Ginevsky et al. [4] in which experiments with jets are reviewed. A wide variety of books is devoted to the problem of hydrodynamic instability playing the major role in the transition to turbulence (see e.g. [5–10]). Among the many general texts we mention [11–17]. A number of books and a plethora of papers are devoted to numerical calculations of turbulence by both direct and indirect methods (see e.g. the books [18–21]). An important place in the literature is occupied by studies in which the general properties of so-called fully developed turbulence are derived and investigated. Thus Kolmogorov and Obukhov [22–24], for example, derived the power spectra of developed isotropic turbulence starting from simple dimensional arguments (see also [25]). DiIerent generalizations and re3nements of these results were achieved by Novikov [26], Procaccia et al. [27–36], Amati et al. [37] and many other researchers. Recent works, developing an approach to turbulence in the context of contemporary theoretical physics, including 3eld- and group-theoretic methods, can be also assigned to this class. Among these we mention [38–40]. It is known that, as distinct from ?ows in channels, jet ?ows are rarely, if ever, laminar. Over a wide range of Reynolds numbers, so-called hydrodynamic waves are excited and ampli3ed in the body of the jet. The amplitude of these waves decreases exponentially outside the jet shear layer. Undamped hydrodynamic waves can propagate only downstream with a velocity of the order of the ?ow velocity. The distinctive feature of hydrodynamic waves is their random character. Nevertheless, against the background of this randomness there are comparatively regular large-scale patterns known as coherent structures.


3

It is very important to note that, when hydrodynamic waves interact with an obstacle or inhomogeneity, they do not undergo simple re?ection. Rather, they induce acoustic waves that propagate upstream. The acoustic waves coming up against an obstacle or an inhomogeneity, in their turn, induce hydrodynamic waves propagating downstream, and so on. Owing to these transformations feedback occurs in jet ?ows, and can excite self-oscillations. Just such a phenomenon arises in jets impinging upon e.g. a ?at plate, a wedge, a coaxial ring or a coaxial pipe [41–45]. In free jets inhomogeneities formed by vortices also induce acoustic waves, again resulting in feedback [46]. But this feedback is nonlinear, and it cannot cause the self-excitation of oscillations. Nonetheless, it exerts an in?uence on the development of turbulence and coherent structures. As will be shown below, the turbulent character of jet ?ows is caused by strong ampli3cation of the random disturbances which are always present at the jet nozzle exit section. 1 At a certain distance from the nozzle the ampli3cation becomes inherently nonlinear. The ampli3cation transforms the system to a qualitatively new state which depends only slightly on the power spectrum or other characteristics of the input disturbances. The system behaves much as though it had undergone a phase transition. The hypothesis that the onset of turbulence can usefully be considered as a noise-induced phase transition was 3rst oIered in [47]. It was based on the existence of profound parallels between turbulent processes in nonclosed ?uid ?ows and noise-induced oscillations in a pendulum with a randomly vibrated suspension axis, which undergoes such a phase transition [44,45,48–51]. Note that this hypothesis is in contradistinction with the widespread belief that the transition to turbulence arises through the excitation of self-oscillations, 3rst periodic and then chaotic [11,12]; but the latter idea does not explain the origin of the feedback mechanism responsible for exciting the self-oscillations. It is well known that instability in a nonclosed ?uid ?ows is of a convective character, but not absolute. Such an instability cannot excite self-oscillations because all disturbances drift downstream. 2 An extremely interesting manifestation of nonlinear eIects in jets lies in the possibility of exploiting them to control turbulence with the aid of acoustic waves applied at some appropriate frequency [4,52–57]. Similar control of noise-induced oscillations was demonstrated for the harmonically driven pendulum [58,51]. Through an approximate solution of the Navier–Stokes equations based on the Krylov–Bogolyubov asymptotic method, we will show that explicit consideration of the ampli3cation of the input noise allows us to account for many known experimental results within the initial part of a jet [59]. Moreover, it follows from our theory that the commonly accepted [60–65,21,4] explanation for the well-known shift of velocity pulsation power spectra towards the low-frequency region is in fact erroneous. According to this explanation, the shift of the power spectra occurs because of feedback via an acoustic wave nascent where vortex pairing occurs, as seen in experiments. We will show that the reason for the spectral shift lies in the jet’s divergence; and that this shift causes the increase of spatial scale with increasing distance from the nozzle, and results in the observed vortex pairing.

1

It should be noted that random sources are present at all points of a jet, even with no external disturbances—i.e. the so called natural ?uctuations [16]. But their in?uence is signi3cantly less than that of disturbances at the jet nozzle exit section and they can therefore be ignored. 2 It should be noted, however, that the instability of a jet ?ow in counter-current stream is of an absolute character and can result in self-excited oscillations.

4


It should be noted that interesting phenomena similar to those for ordinary hydrodynamic turbulence are also observed in ?ows of super?uid helium [66]. 2. Brief review of the evolution of views of turbulence as an oscillatory process It is well known that ?uid ?ow in channels is laminar for small ?ow velocities and turbulent for large ?ow velocities [11,12,9]. The problem of how turbulence originates has long attracted the considerable attention of researchers. As is known from the Rytov memoirs [67], the Russian physicist Gorelik believed that : : :turbulence with its threshold of ‘self-excitation’, with typical hysteresis in its appearance or disappearance as the ?ow velocity increases or decreases, with paramount importance of nonlinearity for its developed (stationary) state—is self-oscillations. Their speci3c character lies in that they are self-oscillations in a continuous medium, i.e. in a system with very large number of degrees of freedom. Landau held implicitly the same viewpoint. According to Landau turbulence appears in the following manner: 3rst the equilibrium state corresponding to laminar ?ow becomes unstable and self-oscillations with a single frequency are excited. To describe the amplitude of these self-oscillations, based on physical considerations, Landau wrote a phenomenological equation similar to the truncated van der Pol equation for the amplitude of self-oscillations in a vacuum tube generator, commenting [68]: “With further increase of the Reynolds number new periods appear sequentially. As for the newly appeared motions, they have increasingly small scales”. As a result, multi-frequency self-oscillations with incommensurate frequencies, i.e. quasi-periodic motion, must set in. An attractor in the form of a multi-dimensional torus in the system phase space has to be associated with these self-oscillations. For a large number of frequencies such quasi-periodic self-oscillations diIer little in appearance from chaotic ones, which is why developed turbulence is perceived as a random process. In spite of the fact that Landau’s theory was phenomenological, and did not follow from hydrodynamic equations, it was accepted without question for a long time by almost all turbulence researchers. Moreover, this theory was further developed by Stuart [69–72] who proposed a technique for calculating the coePcients involved in the Landau equations, based on an approximate solution of the Navier–Stokes equations. However, the approximate solution sought by Stuart in the form of A(jt)ei(!t −kx) is, from a physical standpoint, incorrect. It describes a wave that is periodic in space, with a given wave number k and with a slowly time varying amplitude A(jt). Strictly speaking such a solution is true only for a ring ?ow of length L = 2 n=k, where n is an integer, i.e. for a ?ow with feedback. We note that a similar approach to hydrodynamic instability was used by many scientists, beginning from Heisenberg [73]. In the 1970s, after the discovery of the phenomenon of deterministic chaos and the realization that a multi-dimensional torus is unstable [74], the Landau theory became open to question, but the conception of self-oscillations was retained. The diIerence lay only in that, instead of quasi-periodic self-oscillations, they became spoken of as chaotic ones. Thus, according to these new ideas, the onset of turbulence is the sudden birth of a strange attractor in the phase space of certain dynamical variables [74,75]. We note that similar ideas were repeatedly expressed by Neimark (see [76]). Using


5

the concept of turbulence as self-oscillations, Gaponov–Grekhov and co-workers published several articles on the simulation of turbulence, modelled in a chain of coupled oscillators [77,78]. However, we believe that turbulence arising in nonclosed ?uid ?ows is not a self-oscillatory process. As already mentioned above, the instability of nonclosed laminar ?ows is of a convective character but not absolute. This means that a disturbance arising at some point of the ?ow will not increase inde3nitely with time, but will drift downstream. It follows from this property of convectively unstable systems that they are not self-oscillatory, but are ampli3ers of disturbances. 3 For such a system to become self-oscillatory, global feedback must be introduced, e.g. by closing the system in a ring. 4 Disturbances are necessarily present in all real systems, both from external sources (technical ?uctuations) and as a result of the molecular structure of a substance (natural ?uctuations). The disturbances can be included as external forces in equations describing the system behavior. The calculation of the forces caused by the natural ?uctuations in hydrodynamic ?ows, based on the ?uctuation–dissipation theorem, was performed by Klimontovich [16]. In hydrodynamic ?ows the presence of ?uctuations, especially at the input, is crucial because they are precisely what lead eventually to the turbulent disturbances observed. It follows from this that an approach to turbulence within the framework of (deterministic) dynamical systems theory is not always appropriate. Naturally, the question arises as to how to treat the features of turbulence which, as pointed out by Gorelik, are seemingly precisely those that are inherent in self-oscillatory systems. First, the term “self-excitation” should be replaced by “loss of stability”. Furthermore, the hysteresis of turbulence, its “appearance or disappearance as the ?ow velocity increases or decreases” can be explained in terms of the speci3c character of the nonlinearity of the gain factor. Finally, the “paramount importance of nonlinearity for its fully developed (stationary) turbulent state” is quite possible in ampli3ers too, because nonlinearity of the ampli3er can have considerable in?uence on its output power spectrum. One piece of evidence suggesting that turbulence is not a self-oscillatory process comes from the numerical experiments of Nikitin [80,81]. He simulated ?uid ?ow in a circular pipe of a 3nite length and radius R with a given velocity at the input cross-section, and with so-called ‘soft’ boundary conditions at the output cross-section; these latter are 9 2 u 92 92 = 2 = 2 =0 ; (2.1) 9x2 9x 9x where u is the longitudinal velocity component, and are the radial and angular components of vorticity = rot u, u = {u; v; w} is the ?ow velocity vector in cylindrical coordinates x, r and . Under these conditions a re?ected wave apparently does not appear, or is very weak. At the input cross-section of the pipe the longitudinal velocity component was taken to be in the form of the Poiseuille pro3le u0 (1 − r 2 =R2 ), weakly disturbed by a harmonic force at the frequency ! = 0:36u0 =R, i.e., r2 u = u0 1 − 2 + A Re(u (r)e−i!t ) cos ; R v = A Re(v (r)e−i!t ) cos ; 3 4

w = A Re(w (r)e−i!t ) sin ;

(2.2)

This fact was 3rst mentioned by Artamonov [79]. In essence, this is exactly what occurs in the process of numerical simulation with periodic boundary conditions.

6


Fig. 1. Instantaneous distributions of the longitudinal velocity component u in a steady regime for A=u0 = 0:04: (a) along the pipe axis (r=R = 0:02) and (b) near the pipe wall (r=R = 0:93). After [81].

where u (r), v (r) and w (r) are the components of the Orr–Sommerfeld vector-eigenfunction at frequency !, R is the pipe radius, and A is the disturbance amplitude. The velocity u0 and the pipe radius R were set such that the Reynolds number Re was equal to 4000. As the amplitude A exceeded a certain critical value (A ¿ Acr ), random high-frequency pulsations appeared in the ?ow after a short time interval. They occupied all the lower part of the pipe from x = x0 , where x0 depended only weakly on the distance r from the pipe axis. It turned out that the value of x0 decreased as A became larger. The appearance of turbulent pulsations was accompanied by corresponding deformation of the pro3le of the longitudinal constituent of the mean velocity: at the pipe axis the mean velocity decreased, whereas near the pipe wall it increased. We note that a similar deformation of the mean velocity pro3le with increasing turbulent pulsations occurs in jet ?ows as well. The instantaneous distributions of the longitudinal velocity component in a steady regime for A=u0 = 0:04 are shown in Fig. 1 [81]. As the amplitude A gradually decreased, the turbulent region drifted progressively downstream and disappeared at a certain value of A. It is known [82,8] that Poiseuille ?ow in a circular pipe, in contrast to that in a plane channel, possesses the property that laminar ?ow is stable with respect to small perturbations for any Reynolds number. However, in the case of suPciently large Reynolds numbers, such a ?ow is unstable with respect to 3nite perturbations. If an attractor existed corresponding to the turbulent mode, and if the role of the harmonic disturbance was to lead phase trajectories into the attractor basin, then turbulence should not disappear following cessation of the harmonic disturbance. It may be inferred from Fig. 1 that the development of turbulence for A ¿ Acr is associated with a peculiar phase transition at the point x = x0 induced by an ampli3cation of the noise that is always present in any numerical experiment owing to rounding errors. The harmonic disturbance plays


7

Fig. 2. View of the turbulent velocity pulsations in a pipe (a) with periodic boundary conditions and (b) with the boundary conditions (2.1) and (2.2). After [80].

a dual role. First, it causes the appearance of instability and, secondly, it initiates the phase transition, much as occurs in a pendulum with a randomly vibrated suspension axis [58], or in jets under low-frequency acoustic forcing [4]. It is no accident that the transition to turbulence was observed by Nikitin only for low-frequency disturbances (for Strouhal numbers of order 0.1). Possible counter-arguments against the above ideas lie in the fact that numerical simulation results obtained with periodic boundary conditions are very close to those observed experimentally. But the data obtained by Nikitin in the numerical experiment described above are also close to numerical data for periodic boundary conditions [80]. The visual similarity of turbulent pulsations calculated for periodic conditions, and for the boundary conditions (2.1) and (2.2), is illustrated in Fig. 2 [45]. This similarity may be explained by the fact that many nonlinear oscillatory systems possess such pronounced intrinsic properties that they exhibit these properties independently of the means of excitation. Some examples of such (nonhydrodynamic) systems are described in [83]. Note that our discussion is not related to so called closed ?ows, e.g. to the Couette ?ow between two rotating cylinders or spheres (see [44]). In closed ?ows there is always feedback linking the output of the ampli3er to its input, so that they consequently become self-oscillatory. 3. Some experimental results concerning turbulence development in jets 3.1. The main properties of jet 6ows Issuing from a nozzle, a ?uid jet always noticeably diverges. This is associated with the fact that, owing to viscosity, neighboring ?uid layers are increasingly drawn into the motion. This phenomenon has come to be known as entrainment. The pro3le of the ?ow velocity changes essentially in the process. At the nozzle exit, it is nearly rectangular, whereas away from the nozzle it becomes bell-shaped: see Fig. 3a. The ?uid layer within which the mean velocity changes signi3cantly is called the shear layer or the mixing layer (see, for example, [84,86,87]). It can be seen from Fig. 3a that, within the initial part of the jet (x 6 xin ), the thickness of the mixing layer increases with increasing distance from the nozzle. At x = xin the thickness of the internal part of the mixing layer 1 becomes equal to the half-width of the nozzle outlet for a plane jet, or the nozzle radius for a circular jet, whereupon a continuous boundary layer is formed. In the vicinity of the

8


Fig. 3. (a) Schematic diagram of a diverging free jet illustrating the change of its mean velocity pro3le and widening of the mixing layer. Curves 1 and 2 correspond to the internal and external boundaries of the mixing layer, respectively. (b) Schematic dependence of the relative mean velocity U=U0 along the jet axis on the distance x from the nozzle exit section.

1 0.9 0.8 U / U0

0.7 0.6 0.5 0.4 0.3 0.2 0

5

10

15

20

25

30

x/D

Fig. 4. Experimental dependence of the relative mean velocity U=U0 along the jet axis on the relative distance x=D from the nozzle exit section, for three intensities of the disturbance at the nozzle exit section: ju (0)=0:015, 0.093 0.209 (curves marked by open circles, 3lled circles and stars, respectively). After [50].

jet axis, the mean velocity 3rst decreases very slowly with increasing distance x from the nozzle. This part of the jet is called the initial part: see I in Fig. 3b. Further on, the decrease of the mean velocity becomes signi3cant. This part of the jet is called the main part: see III in Fig. 3b. Parts I and III are separated by the so-called transient part II. The length of the initial part decreases with increasing intensity of disturbances at the nozzle exit section. This can be seen in Fig. 4, where experimental dependences of the relative mean velocity U=U0 on the relative distance x=D from nozzle are plotted (D is the nozzle diameter). Results are shown for three values of the intensity of


9

the disturbances at the nozzle exit section ju (0) = u(0)2 − U (0)2 =U0 , where U0 and u(0) are the mean velocity and longitudinal component of total ?ow velocity at the center of nozzle exit section, respectively [4,21]. The main parts of plane and axially symmetric jets possess approximately the property of selfsimilarity, i.e. at all jet cross-sections the velocity pro3les are aPne-similar [14]. For a plane jet the property of self-similarity means that the jet velocity can be presented in the form u(x; y) = x− F(y=x ), where x and y are longitudinal and transverse coordinate respectively, and are certain numbers and F is a function of y=x . The processes in the jet main part are studied in considerable detail (see e.g. [85–93]). We will consider only the processes in the initial part of a jet. It is interesting that coherent structures are formed just in the mixing layer of initial part of the jet. They are vortex formations (bunches of vorticity). Their sizes are of the order of the thickness of the shear layer, and they are moderately long lived. The presence of coherent structures in a jet shear layer results in the intermittent behavior of a jet ?ow, especially in the neighborhood of the external boundary of a jet, where turbulent and laminar phases alternate [94]. 3.2. Evolution of power spectra of the pulsations of 6uid velocity and pressure with the distance from the nozzle exit section The randomness of the hydrodynamic waves excited in a jet manifests itself, in particular, as continuous power spectra of the pulsations of ?uid velocity and pressure. Within the initial part of the jet, these spectra are of a resonant character. Experiments show that the frequency fm corresponding to the maximum of the power spectrum within the initial part of the jet decreases as the distance from the nozzle exit increases [95–97,50]. Within the main part of the jet, the power spectra decrease monotonically with frequency. Fig. 5 shows examples of how the power spectra of the velocity pulsations evolve with distance from the nozzle exit, along the jet axis, and along a line oIset by R from the axis [50]. The abscissa in each case plots the frequency expressed in terms of the Strouhal numbers St = fD=U0 . As mentioned above, most studies of the diIerent processes in jets attribute such behavior of the pulsation power spectrum within the mixing layer to a pairing of vortices. When pairing takes place, the vortex repetition rate must be halved. Within the initial part of the jet, depending on the conditions of out?ow, from 3 to 4 pairings of vortices are usually observed [61]. The frequency fm at the end of the jet’s initial part should therefore decrease by factor of between 8 and 16, a conclusion that con?icts with experimental data. Experiments show that the frequency fm is not a step, but a smooth function of distance from the nozzle exit (see Fig. 6, where the experimental dependences of the Strouhal number Stm on the relative distance from the jet nozzle exit x=D are plotted [45]). In an attempt to resolve this con?ict, the researchers holding this viewpoint speculate that there is a statistical spread in the sites of pairing, but without explaining why there should be such a spread. The faster decrease of Stm within the mixing layer, compared to what happens on the jet axis, may result from the in?uence of nonlinear feedback caused by acoustic waves induced by vortices within the jet mixing layer. The presence of such waves is indirectly supported by the experimental data of Laufer [61]. According to these data high-frequency pulsations of ?uid velocity within

10


Fig. 5. Evolution of the spectral density Sp (in decibels) of velocity pulsations u with increasing distance from the nozzle exit x=D along the jet axis (r = 0), and along a line oIset by R from the axis (r = 1). At the bottom, the spectral density for x=D = 0:5, r = 1 is shown on a larger scale. After [50].

a mixing layer near the nozzle exit are modulated by low-frequency pulsations with frequencies corresponding to Strouhal numbers St from 0.3 to 0.5. This fact can be also illustrated by the power spectrum of velocity pulsations on a line oIset by R from the jet axis for x=D = 0:5 (see Fig. 5, at the bottom). We see that the spectrum peaks at the main frequency corresponding to the Strouhal number St=3:2 and the two side frequencies corresponding to St1 =2:7 and St2 =3:7. This means that the modulation frequency corresponds to the Strouhal number 0.5. Owing to the nonlinear feedback, each jet cross-section can be considered as an oscillator with a natural frequency depending on the distance from that cross-section to the nozzle exit. It is evident that the strongest pulsations at the


11

Fig. 6. The experimental dependence of the Strouhal number Stm on the relative distance x=D from the jet nozzle exit along the jet axis and within the mixing layer: (a) Petersen’s data for the mixing layer [96]; and (b) the data of [50]. In (b) the dependence on distance along the jet axis, and along a line oIset by R from the axis, are shown by squares and circles, respectively. The solid lines show the dependences Stm = C1 x−1=3 and Stm = C2 x−1 , where C1 ≈ 0:67 and C2 ≈ 1.

cross-section at coordinate x have to occur at a frequency fm that is related to x by the resonant relation xfm xfm =N ; + Uv a where Uv is the velocity of the vortex motion, 5 a is the sound velocity, and N is an integer. From this it follows at once that fm ∼ x−1 . This is precisely the dependence which was found experimentally by Petersen [96] (Fig. 6a). Outside the boundary layer where, within the initial part, inhomogeneities are very weak and nonlinear feedback is nearly absent, the decrease of fm with increasing x follows from the linear theory and is explained by the jet’s divergence as is shown below. 3.3. A jet as an ampli:er of acoustic disturbances Owing to its strong instability, a ?uid jet acts as an ampli3er of disturbances whose frequencies lie within a certain range. It is an ampli3er with a high spatial gain factor. A small acoustic disturbance at some frequency fa within this range near the nozzle transforms into a growing hydrodynamic wave. There is evidence for this in the experimental results of Crow and Champagne [98] and Chan [99]. It follows from the experimental data in Fig. 7 [98] that, above a certain value of the acoustic wave amplitude, the dependence of therelative root-mean-square pulsation of the longitudinal component of hydrodynamic velocity ju = u2 =U0 acquires a resonant character. Here u is the deviation of the longitudinal component of hydrodynamic velocity from its mean value at the acoustic wave frequency fa , measured in terms of Strouhal numbers. The latter authors consider that the resonance is caused by a combination of linear ampli3cation and nonlinear saturation. The latter increases as the frequency of the disturbance rises. For jua = ua2 =U0 = 0:02, where ua is the oscillatory velocity in the acoustic wave, the dependence of ju on Sta = fa D=U0 is shown in Fig. 8. We see that ju is maximal for Sta ≈ 0:3. 5

It follows from visual observations and measurements of spatio-temporal correlations that Uv ≈ 0:5–0:7U0 .

12


Fig. 7. The experimental dependence of ju = near the corresponding curves. After [98].

u2 =U0 on jua = ua2 =U0 for x=D and values of the Strouhal number marked

Fig. 8. The dependence of ju on Sta for jua = 0:02, x=D = 4 constructed from the data given in Fig. 7. In the absence of acoustical disturbance ju ≈ 0:04. After [98].

Fig. 9a taken from [99] shows that the gain factor depends nonmonotonically on distance from the nozzle exit: it has a maximum at x=D = (0:75–1:25)=Sta . A theoretical dependence similar to that shown in Fig. 9a was found by Plaschko [100] by approximate solution of the linearized Euler equations for a slowly diverging jet. It is depicted in Fig. 9b. By doing so, Plaschko showed that the decrease of the gain factor away from the nozzle exit is caused by jet divergence, and not by nonlinear eIects as was claimed by a number of researchers. 3.4. Suppression and intensi:cation of turbulence in jets by a weak periodic forcing An interesting consequence of the nonlinear eIects in a jet is the possibility they provide for controlling the turbulence level and the length of the jet’s initial part by application of a weak acoustic wave, or by vibration of the nozzle, at an appropriate frequency [101,4]. In the case of


13

Fig. 9. (a) The experimental dependence on x˜ = (x=D)Sta of the root-mean-square pulsation of hydrodynamical pressure

p˜ = p2 (in decibels), in the middle of the mixing layer, for a 3xed amplitude of acoustic disturbance and for diIerent Strouhal numbers. After [99]. (b) The theoretical dependence on the relative distance from the jet nozzle x=R of the gain factor K for axially symmetric pulsations of hydrodynamical pressure in a circular jet, for r=R = 1:05, Sta = 0:5. After [100].

high-frequency forcing, the hydrodynamic pulsations are suppressed, whereas at low frequencies, vice versa, there is intensi3cation of pulsations and turbulence. The experiments show that marked intensi3cation or suppression of turbulence within the initial part of a jet, induced by a periodic forcing, is accompanied by changes in the aerodynamic, thermal, diIusive and acoustic properties of the jet. All of these phenomena have been observed by diIerent researchers. It should be noted that the in?uence of acoustic forcing was 3rst studied by Ginevsky and Vlasov [102–106]. Let us consider their main results. In the case of a low-frequency harmonic acoustic forcing at a frequency f corresponding to a Strouhal number in the range 0.2–0.6, the vortices in the jet’s mixing layer are enlarged within the initial part. In turn, this results in an intensi3cation of the turbulent intermixing, thickening the mixing layer, shortening of the initial part and an increase in entrainment; at the same time, the longitudinal and radial velocity pulsations at the jet axis rise steeply. These eIects are observed independently of the direction of the jet irradiation, provided that the amplitudes of the longitudinal and radial components of oscillatory velocity in the sound wave at the jet axis near the nozzle lie in the range 0.05–2% of U0 . For the eIects to occur, the amplitude of the acoustic wave must exceed a certain threshold value. As the wave amplitude rises above this threshold, turbulent intermixing at 3rst intensi3es and then saturates. A further increase of the wave amplitude has little or no eIect on the jet. For high-frequency acoustic forcing of the jet at a frequency corresponding to a Strouhal numbers in the range 1.5–5.0, the vortices in the jet mixing layer become smaller. This results in an attenuation of the turbulent intermixing, a reduction in the thickness of the mixing layer, a lengthening of the initial part, and a decrease of entrainment. Correspondingly, the longitudinal and radial velocity pulsations on the jet axis decrease. In contrast to the eIect of low-frequency forcing, high-frequency forcing does not lead to saturation with increasing amplitude; moreover, an increase in the amplitude beyond a certain value causes, not suppression of the turbulence, but its intensi3cation (see Fig. 10 [65,107]). These eIects are observed universally for jets over a wide range of Reynolds numbers (Re = 102 –106 ), both for initially laminar and for turbulent boundary layers with a level of initial turbulence less than 10%. The foregoing can be illustrated by the experimental dependences of the ?ow relative mean velocity, and the relative root-mean-square pulsation of the longitudinal (ju ) and radial jv components of

14


Fig. 10. The experimental dependences of the relative root-mean-square pulsation of the suppression factors (a) ju =j(0) u and (b) jv =j(0) v of the longitudinal and radial components of hydrodynamical velocity on the relative amplitude of acoustic pressure p˜ a measured in decibels, for Sta = 2:35, x=D = 8; j(0) and j(0) are relative pulsations of the longitudinal and u v radial velocity components in the absence of acoustic excitation. After [65].

hydrodynamic velocity on the distance from the nozzle exit along the jet axis for 3xed values of the Strouhal number (Fig. 11). All the dependences shown correspond to a 3xed value of the acoustic forcing intensity. We see that the mean velocity decreases essentially in the case of low-frequency forcing (0:2 ¡ Sta ¡ 1:5) and increases in the case of high-frequency forcing (Sta ¿ 1:5). It should be noted that, as the acoustic forcing intensity at low-frequency increases, the initial part of the jet decreases in length right down to the point where it disappears [108]. EIects similar to those described above are also observed for other means of periodic forcing of the jet: e.g. longitudinal or radial vibration of the nozzle, or a pulsating rate of ?uid out?ow from the nozzle [65,4]. Detailed experimental and numerical studies of turbulence suppression in jet ?ows were also performed by Hussain and collaborators [109–112]. We concentrate in particular on a single result of these works: that the suppression of turbulence by acoustic forcing of constant amplitude depends on its frequency nonmonotonically: it is maximal at a value of the forcing frequency that depends on the amplitude (see Fig. 12 taken from [112]). 6 Periodic forcing of a jet changes markedly the form of its power spectra. For low-frequency forcing, the power spectra of the velocity pulsations near the nozzle contain discrete constituents at the forcing frequency and its higher harmonics. An example of such a spectrum is given in Fig. 13. In the case of high-frequency forcing (see Fig. 14), the power spectra of the velocity pulsations within the jet mixing layer in the immediate vicinity of the nozzle exit also contain discrete components at the forcing frequency and its higher harmonics. At a short distance from the nozzle the second subharmonic appears in the spectrum. Next the fourth, eighth and successively higher subharmonics appear in the spectra. At suPciently large distances from the nozzle exit the spectra are decreasing almost monotonically. Kibens [113] obtained the dependences on distance from the nozzle of the Strouhal number corresponding to the spectral line of highest intensity, both along the jet axis and along a line oIset 6

We note that authors of [112] used, not the conventional Strouhal number St, but St =St=D, where is the so-called boundary layer momentum thickness at the nozzle exit (see [14]).


15

Fig. 11. Experimental dependences: of (top) the relative mean ?ow velocity along the jet axis U=U (0) and of the relative root-mean-square pulsation of the longitudinal (middle) (ju ) and radial (bottom) (jv ) components of hydrodynamic velocity (in %) on x=D under a longitudinal acoustic forcing at Sta = 0:25 (light circles), Sta = 2:75 (3lled circles). When the acoustic forcing is absent, the corresponding curves are marked by triangles. The amplitude of the oscillatory velocity in the acoustical wave on the jet axis near the nozzle exit constitutes 0.07% of U0 . After [65].

from the axis by R, for high-frequency acoustic forcing with a Strouhal number of 3.54 (Fig. 15 [113,45]). We see that these dependences are step-like, with distinct hysteresis phenomena. Adherents to the viewpoint that the decrease of Stm with distance from the nozzle for a free jet is caused by vortex pairing attribute the step-like character of the dependences to localization of the sites of pairing caused by the acoustic forcing [65,4]. In this explanation, the causes of the localization are ignored and the hysteresis phenomena are not discussed. The picture presented in Fig. 15 can also be interpreted as the successive occurrence of subharmonic resonances of higher and higher order as x increases. The transition from subharmonic resonance of one order to the next can clearly be accompanied by hysteresis, if within a certain range of x both of the resonances are stable. In the

16


Fig. 12. The experimental dependences on the Strouhal number St = (=D)St of the suppression factor ju =j(0) u , where j(0) u is the relative intensity of the longitudinal velocity pulsations in the absence of acoustic forcing, for x= = 200. The plots are constructed for four values of the amplitude of the oscillatory velocity in the acoustic wave, namely 0.5% of U0 (circles), 2.5% (pluses), 3.5% (crosses) and 4.5% (squares). After [112].

Fig. 13. The power spectrum of the velocity pulsations in response to low-frequency acoustic forcing of a circular jet for Sta = 0:25, x=D = 0:5. After [50].

transition to a subharmonic resonance of higher order, the frequency has to be halved. This can manifest itself as vortex pairing. We can thus infer that the experimentally observed localization of the sites of vortex pairing, when an acoustic wave acts upon a jet, is a consequence, but not a cause, of the indicated behavior of the power spectra.


17

Fig. 14. The evolution of power spectra of the velocity pulsations with increasing relative distance x=D from the nozzle exit under high-frequency acoustic forcing at Sta = 2:5. After [50].

Fig. 15. The dependence on distance from the nozzle of the Strouhal number corresponding to the spectral line of highest intensity, in the presence of high-frequency acoustic forcing for Strouhal number 3.54, along the jet axis (light circles) and along a line oIset by R from the axis (3lled circles). After [113].

18


4. The analogy between noise-induced oscillations of a pendulum with randomly vibrated suspension axis and turbulent processes in a jet It seems at 3rst sight very surprising that there should exist any analogy between the development and control of turbulence in a jet, and the noise-induced oscillations of a pendulum with a randomly vibrated suspension axis. These latter oscillations and their control were 3rst studied in [114–117,58,51]. We believe that the analogy arises because the onset of turbulence in jets is a noise-induced phase transition, and the pendulum with a randomly vibrated suspension axis is an appropriate model for illustrating just such a transition [48]. In the simplest case, when additive noise is neglected, the equation describing the oscillations of a pendulum with a randomly vibrated suspension axis is ’U + 2(1 + ’˙ 2 )’˙ + !02 (1 + (t)) sin ’ = 0 ;

(4.1)

where ’ is the pendulum’s angular deviation from the equilibrium position, 2(1 + ’˙ 2 )’˙ is proportional to the moment of the friction force which is assumed to be nonlinear, !0 is the natural frequency of small oscillations, and (t) is a comparatively wide-band random process with nonzero power spectral density at the frequency 2!0 . When the intensity of the suspension axis vibration 7 exceeds a certain critical value proportional to the friction factor , excitation of pendulum oscillations occurs, and the variance of the pendulum’s angular deviation becomes nonzero. The evolution of such oscillations, and their power spectra with increasing noise intensity, found by the numerical simulation of Eq. (4.1), are shown in Fig. 16. It can be seen from the 3gure that, close to the excitation threshold, the pendulum oscillations possess the property of so-called on–oI-intermittency. This notion was 3rst introduced by Platt et al. [118], although a similar phenomenon was considered earlier in [119]. It was noted in [118] that intermittency of this kind is similar to the intermittency in turbulent ?ows. It is of importance that on–oI intermittency is possible, not only in dynamical systems, but in stochastic ones as well [120]. It results from ?uctuational transitions through the boundary of excitation [121,123]. External manifestations of on–oI intermittency are similar to those of ordinary intermittency (see e.g. [124]), i.e. over prolonged periods the pendulum oscillates in the immediate vicinity of its equilibrium position (‘laminar phases’); these slight oscillations alternate with short random bursts of larger amplitude (‘turbulent phases’). Away from the excitation threshold the duration of the laminar phases decreases and that of the turbulent ones increases, with the laminar phases ultimately disappearing altogether [121]. The variance of the pendulum’s angular deviation increases in the process. Comparing the evolution of the power spectra shown in Figs. 5 and 16, we can see that they have much in common. As described in [122], high-pass 3ltering of turbulent velocity pulsations reveals their intermittent behavior. We have studied this phenomenon both for experimental velocity pulsations in a jet measured by one of us [50] and also for the pendulum oscillations considered above. In each case we have observed on–oI intermittency after high-pass 3ltering. This fact can be considered as an additional argument in support of the parallels between noise-induced pendulum oscillations and turbulent processes in jets. 7

By intensity of the suspension axis vibration is meant the spectral density of (t) at frequency 2!0 (%(2!0 )).


19

Fig. 16. Numerical simulations showing the pendulum oscillations (lower plot in each case) and their power spectra (upper plots) with increasing noise intensity for !0 = 1, = 0:1, = 100 and: (a) %(2!0 )=%cr = 1:01; (b) %(2!0 )=%cr = 1:56; (c) %(2!0 )=%cr = 2:44: and (d) %(2!0 )=%cr = 6:25. After [45].

It is important to note that the response of the pendulum to a small additional harmonic force (additional vibration of the suspension axis) is similar to the response of a jet to an acoustic force. For example, in the case when the intensity of the random suspension axis vibration is close to its threshold value, the dependence of the intensity of pendulum oscillations on the frequency of the additional harmonic forcing is of a resonant character, very much like a jet subject to an acoustic force (cf. Figs. 17 and 8). Just as in the case of turbulent jets, the noise-induced pendulum oscillations under consideration can be controlled by a small additional harmonic force. The inclusion of the additional force can be eIected by substitution into Eq. (4.1) of +a cos !a t in place of , where a and !a are, respectively, the amplitude and frequency of the additional vibration of the suspension axis. If the frequency of the additional forcing is relatively low, then this forcing intensi3es the pendulum oscillations and lowers the excitation threshold; vice versa, a relatively high-frequency forcing suppresses the pendulum

20


Fig. 17. The dependence of & = ’2

1=2

on !a for !0 = 1, = 0:1, = 100, %(2!0 )=%cr = 1:01, a = 0:5. After [45].

Fig. 18. The dependences of & on the amplitude a of low-frequency vibration for !0 = 1, = 0:1, = 100 and: (a) %(2!0 )=%cr = 1:89, !a = 0:3; (b) %(2!0 )=%cr = 2:23, !a = 1:5. After [45].

oscillations and increases the excitation threshold. The intensi3cation of the pendulum oscillations by a low-frequency additional vibration is illustrated in Fig. 18 for two values of the vibration frequency. We see that the lower the forcing frequency is, the larger the variance of the oscillation becomes. Just as for jets [125], when the forcing amplitude becomes relatively large, the pendulum’s oscillation amplitude saturates. We now consider in detail the possibility of suppressing noise-induced pendulum oscillations by the addition of a high-frequency vibration. Numerical simulation of Eq. (4.1) with +a cos !a t in place of , where !a ¿ 2, shows that such suppression can occur. The results of the simulation are presented in Figs. 19 and 20. It is evident from Fig. 20 that, for small amplitudes of the high-frequency vibration, this vibration has little or no eIect on the noise-induced oscillations (see Fig. 19a). As the amplitude increases, however, the intensity of the noise-induced oscillations decreases rapidly and the duration of the ‘laminar’ phases correspondingly increases (see Figs. 19b–e). When the amplitude exceeds a certain critical value (for !a = 19:757 it is equal to 42) the oscillations are


21

Fig. 19. Time series of ’(t) and ’(t) ˙ for !0 = 1, = 0:1, = 100, %(2!0 )=%cr = 5:6, !a = 19:757 and: (a) a = 5; (b) a = 15; (c) a = 30; (d) a = 40. After [45].

suppressed entirely. As the amplitude increases further the oscillations reappear, but now because the conditions required for parametric resonance come into play. For smaller frequencies !a , the behavior of the pendulum oscillations is diIerent. The dependences of the variance of the angle ’ on a for a number of values of the vibration frequency are shown in Fig. 20. It is evident that the variance of ’ at 3rst decreases, passes through a certain minimum value, and then increases again. It is important to note that this minimum value becomes smaller with increasing forcing frequency, but that it is attained for larger forcing amplitudes at higher frequencies. For suPciently high frequencies the oscillations can be suppressed entirely (the case illustrated in Fig. 19). The dependence shown in Fig. 20a closely resembles the corresponding dependence for a jet presented in Fig. 10. The dependences of & on !a for a number of 3xed amplitudes of the additional vibration are illustrated in Fig. 21. Again, these dependences closely resemble the corresponding ones for a jet shown in Fig. 12. The presence of a small additive noise, in addition to the multiplicative one in Eq. (4.1), does not change the behavior of the pendulum qualitatively, but there are large quantitative diIerences. The principal one is the impossibility of achieving full suppression of the pendulum oscillations. Nevertheless, a very marked attenuation of the oscillation intensity occurs. This is illustrated in Fig. 22. It should be emphasized that, in the case of turbulence, full suppression is of course also impossible.

22


Fig. 20. The dependences of & on a for !0 = 1, = 0:1, = 100, %(2!0 )=%cr = 5:6 and: (a) !a = 3:5; (b) !a = 6; (c) !a = 11; (d) !a = 19:757. After [45].

Fig. 21. The dependence on !a of &=&0 , where &0 is the value of & in the absence of additional vibration, for %(2!0 )=%cr = 5:6, a = 2:5 (3lled circles), a = 5 (pluses), a = 10 (squares), and a = 20 (crosses). After [45].

5. The main equations and dynamics of a plane jet Let us consider a plane jet issuing from a nozzle of width 2d. Neglecting compressibility, we may describe the processes in such a jet by the two-dimensional Navier–Stokes equation for the stream


23

Fig. 22. Time series ’(t) and ’(t) ˙ in the presence of additive noise in Eq. (4.1) with multiplicative noise (t) of variance 0.05 for: (a) a = 40; and (b) a = 50. The other parameters are the same as in Fig. 20. After [45].

function ((t; x; y) [11]: 9W( 9( 9W( 9( 9W( − + − )WW( = 0 ; 9t 9x 9y 9y 9x

(5.1)

where W = 92 =9x2 + 92 =9y2 is the Laplacian, ) is the kinematic viscosity, x is the coordinate along the jet axis, and y is the transverse coordinate. The stream function ((t; x; y) is related to the longitudinal (U ) and transverse (V ) components of the ?ow velocity by 9( 9( ; V (t; x; y) = − : (5.2) U (t; x; y) = 9y 9x We can conveniently rewrite Eq. (5.1) in terms of the stream function ((t; x; y) and the vorticity ˜ x; y) which is de3ned by +(t; ˜ x; y) = W((t; x; y) : +(t;

(5.3) x

y

In dimensionless coordinates = x=d, function and vorticity become

= y=d and time

t

= U0 t=d, the equations for the stream

+˜ (t ; x ; y ) = W ( (t ; x ; y ) ;

(5.4)

9+˜ (t ; x ; y ) 9( (t ; x ; y ) 9+˜ (t ; x ; y ) − 9t 9x 9y

+

9( (t ; x ; y ) 9+˜ (t ; x ; y ) 2 ˜ W + (t ; x ; y ) = 0 ; − 9y 9x Re

(5.5)

where W is the Laplacian in terms of x and y , Re = 2U0 d=) is the Reynolds number, and U0 is the mean ?ow velocity in the nozzle center. From this point onwards the primes will be dropped. It should be noted that in so deciding on a dimensionless time, the circular frequencies ! = 2 f are measured in units of S = !d=U0 ≡ St, where St = 2fd=U0 is the Strouhal number. In accordance with the ideas presented above, the onset of turbulence is caused by random disturbances (noise) in the nozzle exit section. The authors of most of the works devoted to the stability of these small disturbances [126–128,132] split the solution into mean values and small random disturbances. In our opinion this procedure is inappropriate for two reasons: 3rstly, exact equations for the mean values are unknown; and, secondly, the random disturbances make a signi3cant contribution to the mean values. Therefore we split the solution of Eqs. (5.4) and (5.5) into dynamical

24


and stochastic constituents. The dynamical constituents are described by stationary Navier–Stokes equations and diIer from the mean values of the corresponding variables because, owing to the quadratic nonlinearity, the stochastic constituents also contribute to the mean values. Ignoring noise sources everywhere except in the nozzle exit (for x = 0) we set U (t; 0; y) = ud (0; y) + 1 (t; y);

V (t; 0; y) = vd (0; y) + 2 (t; y) ;

(5.6)

where ud (0; y) and vd (0; y) are the dynamical constituents of the longitudinal and transverse velocity components, respectively, and 1 (t; y) and 2 (t; y) are random processes. It should be noted that ud (0; y) and vd (0; y), as well as 1 (t; y) and 2 (t; y), are not independent, but are related by the continuity equation. We consider 3rst the dynamical constituents of the velocity and vorticity. It follows from Eqs. (5.4) and (5.5) that the dynamical constituents ud (x; y), vd (x; y) and +d (x; y) are described by the equations 9ud (x; y) 9vd (x; y) +d (x; y) = − ; (5.7) 9y 9x 9ud (x; y) 9vd (x; y) + =0 ; (5.8) 9x 9y 9+d (x; y) 9+d (x; y) 2 92 +d (x; y) 92 +d (x; y) ud (x; y) =0 : (5.9) + + vd (x; y) − 9x 9y Re 9x2 9y2 It is very diPcult, if not impossible, to solve these nonlinear equations exactly. Therefore we choose ud (x; y) in the form of a given function of y with unknown parameters depending on x. The shape of this function must depend on whether the out?ow from the nozzle is laminar or turbulent. For simplicity, we restrict our consideration to laminar nozzle ?ow. In this case we can set ud (x; y) so that, at the nozzle exit section, the boundary layer is close in form to that described by the Blasius equation (see [14,11]) F() for |y| 6 1 ; uBl (y) = (5.10) 0 for |y| ¿ 1 ; where a(1 − |y|) ; (5.11) 00 √ 00 = 1=(b0 Re) is the relative thickness of the boundary layer at the nozzle exit, b0 is determined by the conditions of out?ow from the nozzle, a is a parameter which depends on the de3nition of the boundary layer thickness, 8 F() is the derivative with respect to of the Blasius function f(), described by the equation =

d3 f f d2 f + =0 d3 2 d2

(5.12)

with initial conditions f(0) = 0, df(0)=d = 0, d 2 f(0)=d2 ≈ 0:332 [14,11]. 8

If the boundary layer thickness is de3ned so that at its boundary the relative velocity is equal to 0.99, then a ≈ 5.


25

Taking account of the entrainment of the ambient ?uid, we set the velocity pro3le close to (5.10) for x = 0 to the form |y| − 1 1 1 − tanh q ud (x; y) = − r(x) ; (5.13) 1 + tanh(q=00 + r0 ) 0 (x) where 0 (x) and r(x) are unknown functions of x, and 0 (x) is the boundary layer thickness which is equal to 00 for x = 0, r0 = r(0). We note that pro3le (5.13) was 3rst suggested in [46]; it is similar to that given for the mean velocity in [126–128,132]. The thicknesses of inner and external boundary layers (1 (x) and 2 (x)) are de3ned by the relations: ud (x; 1 − 1 (x)) = ;

ud (x; 1 + 2 (x)) = 1 − ;

where is a number close to 1. As follows from (5.13) and (5.14) 1 (x) 2 (x) q + r(x) = arc tanh(2 − 1); q − r(x) = arc tanh(2 − 1) : 0 (x) 0 (x) Adding Eqs. (5.15) we obtain the relation between q and : q = 2 arc tanh(2 − 1) :

(5.14) (5.15)

(5.16)

For = 0:95 we 3nd q ≈ 3. The substitution of (5.16) into (5.15) gives 1 (x) 1 r(x) 2 (x) 1 r(x) = − = + ; : (5.17) 0 (x) 2 q 0 (x) 2 q The form of the velocity pro3le determines the so-called shape-factor H [14], which is equal to the ratio between the displacement thickness 1+2 (0) 1 (1 − ud (0; y)) dy ∗ (0) = 0 (0) 0 and the thickness of momentum loss 1+2 (0) 1 ud (0; y)(1 − ud (0; y)) dy : (0) = 0 (0) 0 For a turbulent boundary layer, the shape-factor H has to lie in the range 1.4–1.6 [14], whereas for a laminar boundary layer it has to be signi3cantly more. Using the values of parameters calculated above we can calculate ∗ (0), (0) and H (0) for our velocity pro3le. As a result, we 3nd ∗ (0) ≈ 0:5081, (0) ≈ 0:1588 and H (0) ≈ 3:2. Thus, our velocity pro3le does correspond to a laminar boundary layer. To 3nd the unknown functions in expression (5.13), we use the conservation laws for the ?uxes of momentum and energy. Usually, they are derived for the mean values of these ?uxes starting from the Reynolds equations [86,21], and therefore contain the so-called turbulent viscosity. We derive them directly starting from Eqs. (5.7)–(5.9) for dynamical constituents. For this we transform Eqs. (5.7)–(5.9) in the following way. Substituting +d (x; y) from Eq. (5.7) into Eq. (5.9), and taking into account that within the jet’s initial part 92 ud =9x2 and 92 vd =9x2 are negligibly small, we obtain 92 ud (x; y) 92 ud (x; y) 2 93 ud (x; y) + vd (x; y) ud (x; y) − =0 : (5.18) 9x9y 9y2 Re 9y3

26


Taking account of the continuity equation (5.8) we can rewrite Eq. (5.18) as 2 92 ud (x; y) 9 9ud2 (x; y) 9(ud (x; y)vd (x; y)) + − =0 : 9y 9x 9y Re 9y2

(5.19)

By integrating Eq. (5.19) over y, we obtain the following approximate equation: 9ud2 (x; y) 9(ud (x; y)vd (x; y)) 2 92 ud (x; y) + − =0 : 9x 9y Re 9y2

(5.20)

The conservation law for the dynamical constituent of the momentum ?ux is found by integrating Eq. (5.20) over y from −∞ to ∞, taking into account that ud (x; ±∞) = 0 and 9ud (x; ±∞)=9y = 0. We thus obtain ∞ 9 u2 (x; y) dy = 0 : (5.21) 9x −∞ d To derive the conservation law for the dynamical constituent of the energy ?ux, we multiply Eq. (5.20) by 2ud (x; y) and transform it to the form 9ud3 9 2 ud 9(ud2 vd ) 4 ud : + = 9x 9y Re 9y2

(5.22)

Integrating further Eq. (5.22) over y from −∞ to ∞ and taking into account the boundary conditions indicated above we 3nd ∞ ∞ 9ud (x; y) 2 4 9 3 u (x; y) dy = − dy : (5.23) 9x −∞ d Re −∞ 9y Because ud (x; y) is an even function of y, we obtain from (5.21) and (5.23) the following approximate equations: ∞ ∞ 2 ud (x; y) dy = ud2 (0; y) dy ; (5.24) 0

3

0

0

∞

ud2 (x; y)

4 9ud (x; y) dy = − 9x Re

0

∞

9ud (x; y) 9y

2

dy :

(5.25)

Substituting (5.13) into Eq. (5.24) and taking into account that ∞ ud2 (0; y) dy ≈ 1 0

we obtain a relationship between r(x) and 0 (x): q q q 1 1 + tanh ln 2 cosh + r(x) − + r(x) − + r(x) = 0 : 0 (x) 0 (x) 2 0 (x) Within the jet part, where q 1 ; 0 (x)

(5.26)

(5.27)


27

relationship (5.26) reduces to 2r(x) ≈ 1, i.e. r(x) ≈ r0 = 0:5. It follows from this and (5.19), (5.17) that 1 (x) ≈

0 (x) ; 3

2 (x) ≈

20 (x) : 3

(5.28)

Substituting (5.13) into Eq. (5.25), and taking account of (5.26), we 3nd the diIerential equation for 0 (x): q q 2q q −2 + r0 − 4 − cosh + r0 − 1 − tanh + r0 5 tanh 0 (x) 0 (x) 0 (x) 0 (x)

q q d0 (x) 1 −2 3 + tanh cosh + + r0 + r0 4 0 (x) 0 (x) dx q 4q2 1 + tanh + r0 = 3 Re 0 (x) 0 (x) q q −2 : × 1 + tanh + r0 + cosh + r0 0 (x) 0 (x)

(5.29)

A solution of this equation can be found analytically only for condition (5.27). In this case Eq. (5.29) becomes 16q2 d0 (x) = : 3 Re 0 (x) dx

(5.30)

It follows from Eq. (5.30) that 0 (x) =

200 + 2kx;

k d0 (x) = ; dx 0 (x)

(5.31)

where k = 16q2 =(3 Re). We note that the dependence 0 (x) found here from the Navier–Stokes equations diIers from that found from the Reynolds equations [86,21] and containing the turbulent viscosity )t . Since, by Prandtl’s hypothesis [13], )t is proportional to the boundary layer thickness (x), the dependence (x) was found to be linear. The expressions for vd (x; y) and +d (x; y) can be found by exact solution of Eqs. (5.7), (5.8). As a result, we obtain q(|y| − 1) 16q sign y q(|y| − 1) vd (x; y) = − tanh − r0 3 Re 0 (x)(1 + tanh(q=00 + r0 )) 0 (x) 0 (x) q cosh(q(|y| − 1)=0 (x) − r0 ) q tanh + r0 − ln ; − 0 (x) 0 (x) cosh(q=0 (x) + r0 )

(5.32)

28


256q4 (|y| − 1)2 q sign y −2 q(|y| − 1) 1+ cosh +d (x; y) = − − r0 0 (x)(1 + tanh(q=00 + r0 )) 0 (x) 940 (x) Re2 2 q q 256q2 −2 + r0 − 2 cosh 0 (x) 90 (x) Re2 20 (x) q(|y| − 1) q (|y| − 1) tanh − r0 − 0 (x) 0 (x)

cosh(q(|y| − 1)=0 (x) − r0 ) q + r0 : (5.33) + ln − tanh 0 (x) cosh(q=0 (x) + r0 ) For condition (5.27), from (5.32) and (5.33) we 3nd the following approximate asymptotic expressions for vd and +d : vd (x; ±∞) ≈ ∓

16qr0 ; 30 (x) Re

+d (x; ±∞) ≈ ∓

256q3 r0 : 930 (x) Re2

(5.34)

Fig. 23 shows plots of ud (x; y), vd (x; y), +d (x; y) versus y for b0 = 0:1, q = 3, r0 = 0:5, Re = 25; 000, x = 0 and 8. We see that for all values of y, except for narrow intervals near y = ±1, ud (x; y), vd (x; y) and +d (x; y) are nearly constant. The constant transverse velocity component for |y| ¿ 1 directed towards the jet axis accounts for the entrainment of ambient ?uid with the jet ?ow. It should be emphasized that the results obtained here concern only the dynamical constituents of the velocity and vorticity. Stochastic constituents greatly in?uence the thickness of the boundary layer, its dependence on the distance from the nozzle, and values of the mean velocities (see below). Substituting further U (t; x; y) = ud (x; y) +

9 (t; x; y) ; 9y

V (t; x; y) = vd (x; y) −

9 (t; x; y) ; 9x

˜ x; y) = +d (x; y) + +(t; x; y) +(t;

(5.35)

into Eqs. (5.4), (5.5) and taking into account (5.7)–(5.9), we 3nd the equations for the stochastic constituents (t; x; y) and +(t; x; y), which we write in the form +−W =0 ;

(5.36)

2 9+ 9+ 9+ 9 9 + ud (x; y) + vd (x; y) − +dy (x; y) + +d x (x; y) − W+ 9t 9x 9y 9x 9y Re =

9 9+ 9 9+ − ; 9x 9y 9y 9x

(5.37)

where +d x (x; y) =

9+d (x; y) ; 9x

+dy (x; y) =

9+d (x; y) : 9y

1

1

0.8

0.8

0.6

0.6

ud

ud


0.4

0.4

0.2

0.2

0

-1 -0.5

0.006 0.004 0.002 0 -0.002 -0.004 -0.006

0 0.5

1

y

0.001 0

-0.001 -1 -0.5

0 0.5

-0.002

1

y

(e)

-1 -0.5

(b)

0.002

(c)

-1 -0.5

0 0.5

1

y

(d)

10 5

Ωd

30 20 10 0 -10 -20 -30

0

1

vd

vd

0.5

y

(a)

Ωd

0

29

0 -5

-1 -0.5

0

y

0.5

-10

1

-1 -0.5

(f)

0 0.5

1

y

Fig. 23. Plots of various quantities versus y for b0 = 0:1, q = 3, r0 = 0:5, Re = 25; 000: (a) ud (0; y); (b) ud (8; y); (c) vd (0; y); (d) vd (8; y); (e) +d (0; y); and (f) +d (8; y).

According to (5.6) the boundary conditions for Eqs. (5.36) and (5.37) are 9 9 = 1 (t; y); = −2 (t; y) : 9y x=0 9x x=0

(5.38)

6. The derivation of truncated equations for the amplitude of stochastic constituents To describe the development of turbulence, we can assume that the right-hand side of Eq. (5.37) is of the order of a small parameter j. In this case Eqs. (5.36) and (5.37) can be solved approximately by a method similar to the Krylov–Bogolyubov method for spatially extended systems [131]. We therefore seek a solution in the form of a series in j: +(t; x; y) = +0 (t; x; y) + jr1 (t; x; y) + j2 r2 (t; x; y) + : : : ; (t; x; y) =

0 (t; x; y)

+ js1 (t; x; y) + j2 s2 (t; x; y) + : : : ;

30


u(t; x; y) =

9 (t; x; y) = u0 (t; x; y) + jq1 (t; x; y) + j2 q2 (t; x; y) + · · · ; 9y

(6.1)

where +0 (t; x; y) and 0 (t; x; y) are generative solutions of Eqs. (5.36) and (5.37), u0 (t; x; y) = 9 0 (t; x; y)=9y; r1 (t; x; y); r2 (t; x; y); : : : ; s1 (t; x; y); s2 (t; x; y); : : : are unknown functions, and q1 (t; x; y)= 9s1 (t; x; y)=9y; q2 (t; x; y) = 9s2 (t; x; y)=9y; : : : : It should be emphasized that because of the quadratic nonlinearity the contribution of nonlinear terms into turbulent processes can be estimated only by using the second approximation of the Krylov–Bogolyubov method. Thus in expansion (6.1) we have to retain the terms up to the second order with respect to j. 6.1. Generative solutions Putting the right-hand side of Eq. (5.37) to zero and eliminating the stochastic constituent of vorticity, we obtain the generative equation for the stochastic constituent of the stream function: 9W 9t

0

+ ud (x; y)

− +dy (x; y)

9W 0 9W 0 + vd (x; y) 9x 9y

9 0 9 0 2 + +d x (x; y) − WW 9x 9y Re

0

=0 :

(6.2)

It should be noted that 3nding the generative solution is similar to the well-known problem of the linear instability of a jet ?ow. During the last three decades, this problem was studied primarily by Crighton and Gaster [126,132], Michalke [128] and Plaschko [127]. In these works a pro3le of the mean ?ow velocity for a circular jet was given, and the problem was solved approximately, mainly within the framework of linearized Euler equations. Because the coePcients of these equations depend on the coordinates, an exact analytic solution could not be found. Numerical calculations performed by these authors are in qualitative agreement with experimental data. Here we 3nd the generative solution for a plane jet based on the linearized Navier–Stokes equation (6.2) and using the dynamical constituents of velocity and vorticity calculated above. We emphasize that viscosity should be taken into account, because all terms in Eq. (6.2) are of the same order over the region of the boundary layer. We chose a plane jet, rather than a circular one, by virtue of the same reasoning as in [92]: its simple geometry and boundary conditions. We seek a partial solution of Eq. (6.2) in the form of a sum of running waves of frequency S with a slowly varying complex wave number Q(S; x): ∞ x 1 (S) (t; x; y) = f (x; y) exp i St − Q(S; x) d x dS : (6.3) 0 2 −∞ 0 Taking into account that the jet diverges slowly, we can represent the√function f(S) (x; y) and the wave number Q(S; x) as series in a conditional small parameter 2 ∼ 1= Re: f(S) (x; y) = f0 (S; x; y) + 2f1 (S; x; y) + : : : ;

Q(S; x) = Q0 (S; x) + 2Q1 (S; x) + : : : ;

(6.4)


31

where f0 (S; x; y); f1 (S; x; y); : : : are unknown functions vanishing, along with their derivatives, at y = ±∞. Substituting (6.3), in view of (6.4), into Eq. (6.2) and retaining only terms containing 3rst derivatives with respect to x we obtain the following equations for f0 (S; x; y) and f1 (S; x; y): L0 (Q0 )f0 = 0 ;

(6.5)

L0 (Q0 )f1 = iQ1 L1 (Q0 )f0 − L2 (Q0 )f0 ;

(6.6)

where

L0 (Q0 ) = i(S − ud (x; y)Q0 )

92 − Q02 9y2

+ vd (x; y)

93 9 − Q02 3 9y 9y

4 2 2 9 9 2 9 4 + iQ0 +dy (x; y) + +d x (x; y) − − 2Q0 2 + Q0 ; 9y Re 9y4 9y 2 9 9 L1 (Q0 ) = ud (x; y) − 3Q02 + 2SQ0 − 2iQ0 vd (x; y) 9y2 9y 2 8iQ0 9 − Q02 ; − +dy (x; y) + Re 9y2 3 9Q0 9Q0 9 9 9 L2 (Q0 ) = S 2Q0 + + ud (x; y) + − ivd (x; y) − 3Q0 Q0 9x 9x 9x9y2 9x 9x 9Q0 9 9 92 + − +dy (x; y) × 2Q0 9x9y 9x 9y 9x 9Q0 93 9 9Q0 92 4i 2 2Q0 +3 : + − Q0 2Q0 + Re 9x9y2 9x 9y2 9x 9x

(6.7)

(6.8)

(6.9)

Eq. (6.5), with the boundary conditions for function f0 and its derivatives so as to be vanishing at y = ±∞, describes a non-self-adjoint boundary-value problem, where Q0 plays the role of an eigenvalue. Similar boundary-value problems, but on a 3nite interval, were studied by Keldysh [129]. Consistent with Fredholm’s well-known theorem [130] about linear boundary-value problems described by an inhomogeneous equation, Eq. (6.6) has a nontrivial solution only if ∞ ∞ iQ1 3(S; X x; y)L1 (Q0 )f0 (S; x; y) dy − 3(S; X x; y)L2 (Q0 )f0 (S; x; y) dy = 0 ; (6.10) −∞

−∞

where 3(S; X x; y) is a complex conjugate eigenfunction of the adjoint boundary-value problem described by the equation: 3 2 9 9 2 2 9 (vd (x; y)3) − Q0 [(S − ud (x; y)Q0 )3] X − − Q0 X i 9y2 9y3 9y 2 2 94 3X X 9(+d x (x; y)3) 2 9 3X 4 − 2 + iQ0 +dy (x; y)3X − − 2Q + Q 3 X =0 : (6.11) 0 0 9y 4 9y4 9y2 Condition (6.10) allows us to 3nd the small correction Q1 (S; x) to the eigenvalue Q0 (S; x).

32


Over the region of |y| 6 y1 (x) (region I), where y1 (x) is the internal boundary of the boundary layer, ud (x; y) ≈ 1, vd (x; y) ≈ 0, +d x (x; y) ≈ 0 and +dy (x; y) ≈ 0. For this region the general solution of Eqs. (6.5) and (6.11) is f0 (y) = A1 sinh(B11 y) + A2 sinh(B12 y) + A3 cosh(B11 y) + A4 cosh(B12 y) ; 3(y) X = A˜ 1 sinh(B11 y) + A˜ 2 sinh(B12 y) + A˜ 3 cosh(B11 y) + A˜ 4 cosh(B12 y) ; where

B11 = Q0 ;

B12 =

Q02 +

i(S − Q0 ) Re 2

(6.12)

(6.13)

are the roots of the characteristic equation corresponding to Eqs. (6.5) and (6.11), and A1 , A2 , A3 , A4 , A˜ 1 , A˜ 2 , A˜ 3 and A˜ 4 are arbitrary constants. Any arbitrary disturbance can be represented as a linear combination of even and odd constituents. We consider the case of odd disturbances. In this case we can solve Eqs. (6.5) and (6.11) only for positive values of y, seeking a solution of these equations in the form f0 (y) = A1 f01 (y) + A2 f02 (y);

3X0 (y) = A˜ 1 3X01 (y) + A˜ 2 3X02 (y) ;

(6.14)

where, for |y| 6 y1 (x), the functions f01 (y), 3X1 (y) transform to sinh(B11 y), and f02 (y), 3X2 (y) transform to sinh(B12 y). It follows that f01 (y), 3X01 (y), f02 (y) and 3X02 (y) must satisfy the following initial conditions: 9f01 93X01 = = B11 ; f01 (0) = 3X01 (0) = 0; 9y y=0 9y y=0 92 3X01 93 f01 93 3X01 92 f01 3 = = 0; = = B11 ; 9y2 y=0 9y2 y=0 9y3 y=0 9y3 y=0 9f02 93X02 = = B12 ; f02 (0) = 3X02 (0) = 0; 9y y=0 9y y=0 92 3X02 93 f02 93 3X02 92 f02 3 = = 0; = = B12 : (6.15) 9y2 y=0 9y2 y=0 9y3 y=0 9y3 y=0 Over the region of |y| ¿ y2 (x) (region II), where y2 (x) is the external boundary of the boundary layer, ud (x; y) ≈ 0, vd (x; y) ≈ vd (x; ∞), +d x (x; y) ≈ +d x (x; ∞) and +dy (x; y) ≈ 0. For this region the solutions of Eqs. (6.5) and (6.11) must behave as partial solutions of the equations 3 2 9 f0 9 f0 2 2 9f0 − Q0 f0 + vd (x; ∞) − Q0 iS 9y2 9y3 9y 4 2 2 9 f0 9f0 2 9 f0 4 + +d x (x; ∞) − − 2Q0 + Q0 f0 = 0 ; 9y Re 9y4 9y2


iS

92 3X0 − Q02 3X0 9y2

− vd (x; ∞)

2 93X0 − +d x (x; ∞) − 9y Re

93 3X0 93X0 − Q02 9y3 9y

33

2 94 3X0 2 9 3X0 − 2Q + Q04 3X0 0 9y4 9y2

=0 ;

(6.16)

satisfying the condition of vanishing at y = ∞. Such partial solutions can be written as f0 (y) = C21 exp[B21 (y − y2 (x))] + C22 exp[B22 (y − y2 (x))] ; 3X0 (y) = C˜ 21 exp[B˜ 21 (y − y2 (x))] + C˜ 22 exp[B˜ 22 (y − y2 (x))] ;

(6.17)

where C21 , C22 , C˜ 21 and C˜ 22 are arbitrary constants, and B21 , B22 , B˜ 21 and B˜ 22 are roots of the characteristic equations corresponding to Eqs. (6.16) with negative real parts. The characteristic equations for region II take the form B4 − a10 B3 − a20 B2 − a30 B + a40 = 0 ; B˜ 4 + a10 B˜ 3 − a20 B˜ 2 + a30 B˜ + a40 = 0 ;

(6.18)

where a10 = ±

vd (x; ∞) Re ; 2

a30 = ±

(+d x (x; ∞) − Q02 vd (x; ∞)) Re ; 2

a20 = 2Q02 +

iS Re ; 2 a40 = Q04 +

iSQ02 Re ; 2

(6.19)

the signs ‘+’ and ‘−’ correspond to y ¿ 0 and y ¡ 0, respectively. It follows from (6.18) and (6.19) that jth root of Eq. (6.18) for y ¡ 0 is equal to jth root for y ¿ 0 of opposite sign. That is why we need consider only y ¿ 0. In view of (6.19), the 3rst equation of (6.18) can be conveniently rewritten as vd (x; ∞) Re i S Re +d x (x; ∞) Re 2 2 B − Q0 + (B2 − Q02 ) = B : (6.20) B − 2 2 2 It can be shown that the right-hand side of Eq. (6.20) is small. Therefore the roots of Eq. (6.20) with negative real parts are approximately equal to B21 = −Q0 + WB1 ; where vd (x; ∞) Re B220 = 4 WB1 = −

B22 = B220 + WB2 ;

1+

8iS 16Q02 1+ 2 + 2 vd (x; ∞) Re vd (x; ∞) Re2

+d x (x; ∞) ; 2(iS − vd (x; ∞)Q0 )

WB2 =

(6.21) ;

+d x (x; ∞)B220 : 2 vd (x; ∞)(B220 + Q02 ) + iSB220

(6.22)

34


Equating (6.14)–(6.17) at the point y = y2 (x), we 3nd the following boundary conditions: A1 f01 (y2 ) + A2 f02 (y2 ) = C21 + C22 ;

A1 f11 (y2 ) + A2 f12 (y2 ) = B21 C21 + B22 C22 ;

2 2 C21 + B22 C22 ; A1 f21 (y2 ) + A2 f22 (y2 ) = B21

A˜ 1 3X01 (y2 ) + A˜ 2 3X02 (y2 ) = C˜ 21 + C˜ 22 ;

3 3 A1 f31 (y2 ) + A2 f32 (y2 ) = B21 C21 + B22 C22 ;

A˜ 1 3X11 (y2 ) + A˜ 2 3X12 (y2 ) = B˜ 21 C˜ 21 + B˜ 22 C˜ 22 ;

A˜ 1 3X21 (y2 ) + A˜ 2 3X22 (y2 ) = B˜ 221a C˜ 21 + B˜ 222a C˜ 22 ; A˜ 1 3X31 (y2 ) + A˜ 2 3X32 (y2 ) = B˜ 321 C˜ 21 + B˜ 322 C˜ 22 ; where

(6.23)

9f01 9f02 92 f01 ; f (y ) = ; f (y ) = ; 12 2 21 2 9y y=y2 9y y=y2 9y2 y=y2 92 f02 93 f01 93 f02 f22 (y2 ) = ; f31 (y2 ) = ; f32 (y2 ) = : 9y2 y=y2 9y3 y=y2 9y3 y=y2 93X01 93X02 92 3X01 3X11 (y2 ) = ; 3X12 (y2 ) = ; 3X21 (y2 ) = ; 9y y=y2 9y y=y2 9y2 y=y2 92 3X02 93 3X01 93 3X02 3X22 (y2 ) = ; 3X31 (y2 ) = ; 3X32 (y2 ) = : 9y2 y=y2 9y3 y=y2 9y3 y=y2

f11 (y2 ) =

The eigenvalues of Q0 for the basic boundary-value problem must satisfy the requirement that the determinant f01 (y2 ) f02 (y2 ) 1 1 f11 (y2 ) f12 (y2 ) B21 B22 (6.24) D(Q0 ) = 2 2 f21 (y2 ) f22 (y2 ) B21 B22 3 3 f31 (y2 ) f32 (y2 ) B21 B22 be equal to zero. It can be shown that the eigenvalues for the adjoint boundary-value problem coincide with those for the basic boundary-value problem. The expression for D(Q0 ) can be written as 2 2 D(Q0 ) = q23 (y2 ) − (B21 + B22 )q13 (y2 ) + (B21 + B22 )q12 (y2 ) 2 2 + B21 B22 (q12 (y2 ) + q03 (y2 ) − (B21 + B22 )q02 (y2 )) + B21 B22 q01 (y2 ) ;

(6.25)

where q01 (y) = f01 (y)f12 (y) − f11 (y)f02 (y);

q02 (y) = f01 (y)f22 (y) − f21 (y)f02 (y) ;

q03 (y) = f01 (y)f32 (y) − f31 (y)f02 (y);

q12 (y) = f11 (y)f22 (y) − f21 (y)f12 (y) ;

q13 (y) = f11 (y)f32 (y) − f31 (y)f12 (y);

q23 (y) = f21 (y)f32 (y) − f31 (y)f22 (y) :

(6.26)


35

1.3 10

1.2 1.1 1 0

6

v ph0

Γ0

8 0.5 1

4 3 2 10

0.9 0.8 0.7

2

0.6

5

10 5 3

0.5

0

0 0.5 21

0.4 0

1

(a)

2

3

4

5

6

St

7

0

1

2

(b)

3

4

5

St

Fig. 24. The dependences on the Strouhal number St for b0 = 0:1, q = 3, r0 = 0:5, Re = 25; 000 and diIerent x of: (a) the gain factor 70 and (b) the wave phase velocity vph0 = S=K0 . The value of x is indicated near the corresponding curve in each case.

A direct numerical calculation of D(Q0 ), starting from Eqs. (6.5) and (6.25), gives random values on account of the need to subtract large numbers of the same order. Therefore, instead of Eq. (6.5), we solve the equations for qij (y) which follow from (6.5) and (6.26). They are 9q02 9q12 = q03 (y) + q12 (y); = q13 (y) ; 9y 9y i(S − ud (x; y)Q0 ) Re 9q03 2 = q13 (y) + 2Q0 + q02 (y) 9y 2

9q01 = q02 (y); 9y

Re [vd (x; y)[q03 (y) − Q02 q01 (y)] + +d x (x; y)q01 (y)] ; 2 i(S − ud (x; y)Q0 ) Re 9q13 2 q12 (y) = q23 (y) + 2Q0 + 9y 2 +

+ Q04 q01 (y) +

Re [[iQ02 (S − ud (x; y)Q0 ) − iQ0 +dy (x; y)]q01 (y) + vd (x; y)q13 (y)] ; 2

Re 9q23 = Q04 q02 (y) + [[iQ02 (S − ud (x; y)Q0 ) − iQ0 +dy (x; y)]q02 (y) 9y 2 + vd (x; y)(q23 (y) + Q02 q12 (y)) − +d x (x; y)q12 (y)] :

(6.27)

Solving Eqs. (6.27) with initial conditions q01 (0) = q02 (0) = q03 (0) = q12 (0) = q23 (0) = 0;

2 2 q13 (0) = B11 B12 (B12 − B11 );

(6.28)

and substituting the solution found for y = y2 into (6.25), we calculate D(Q0 ). By varying Q0 until D(Q0 ) becomes equal to zero, we 3nd the eigenvalues of Q0 . The real part of the eigenvalue of Q0 gives the real wave number K0 , whereas its imaginary part gives the wave gain factor 70 . The dependences of the gain factor 70 and wave phase velocity vph0 = S=K0 on the Strouhal number St are given for Re = 25; 000, b0 = 0:1 and a number values of x in Figs. 24a and b.

36


10 8

v ph0

Γ0

1 2

6 4 3

2 0 0

1

2

3

(a)

4

5

6

7

1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4

1 2 3 0

1

(b)

St

2

3

4

5

St

Fig. 25. The dependences on the Strouhal number St of (a) the gain factor 70 and (b) the wave phase velocity vph0 = S=K0 for x = 0 and Re = 25; 000, b0 = 0:1 (curve 1), Re = 100; 000, b0 = 0:05 (curve 2) and Re = 100; 000, b0 = 0:02 (curve 3).

To estimate the in?uence of the Reynolds number and the thickness of the boundary layer at the nozzle exit, we have calculated the eigenvalues of Q0 for x = 0 in two cases: Re = 100; 000, b0 = 0:05 and Re = 100; 000, b0 = 0:02. In the 3rst case the thickness of the boundary layer at the nozzle exit is the same as in Fig. 24, and in the second case it is considerably larger. The results are shown in Fig. 25. We see that, for the same thickness of boundary layer at the nozzle exit the results depend only weakly on the Reynolds number, whereas the thickness of the boundary layer aIects the eigenvalues strongly. To 3nd the eigenfunction and the adjoint eigenfunction corresponding to the eigenvalue Q0 , we use Eq. (6.10) for the calculation of the correction Q1 to the eigenvalue Q0 , and we should in principle calculate the functions f01 (S; x; y), f02 (S; x; y), 3X01 (S; x; y) and 3X02 (S; x; y) and solve the systems of equations (6.23), taking into account that their determinants are equal to zero. However, in the process of a direct numerical solution of Eqs. (6.5) and (6.11) over the region of boundary layer (region III) we face the problem of the strong instability of solutions corresponding to the functions f01 and 3X01 with respect to small rapidly increasing disturbances. This instability becomes more pronounced for larger S. The instability can be illustrated clearly if we pass in Eqs. (6.5) and (6.11) to new variables ˜ x; y) by 9(S; x; y) and 9(S; f0 (S; x; y) = C exp(9(S; x; y));

˜ x; y)) ; 3X0 (S; x; y) = C˜ exp(9(S;

(6.29)

where 9(S; x; y) =

0

y

B(S; x; y) dy;

˜ x; y) = 9(S;

y

0

˜ x; y) dy : B(S;

(6.30)

Substituting (6.29) into Eqs. (6.5) and (6.11) and taking account of (6.30) we obtain the following ˜ x; y): nonlinear equations for B(S; x; y) and B(S; 93 B 92 B 9B +3 + 4B + 2(3B2 − Q02 ) 3 2 9y 9y 9y

9B 9y

2

+ (B2 − Q02 )2


37

2 Re 9B 9B 9B 2 2 2 2 i(S − ud (x; y)Q0 ) − + 3B + B − Q0 + vd (x; y) + B(B − Q0 ) 2 9y 9y2 9y + iQ0 +dy (x; y) + +d x (x; y)B = 0;

(6.31)

2 2˜ ˜ 9B B 9 9 B 93 B˜ 2 2 +3 + 4B˜ 2 + 2(3B˜ − Q0 ) + (B˜ 2 − Q02 )2 3 9y 9y 9y 9y Re 9B˜ 2 2 ˜ i(S − ud (x; y)Q0 ) + B − Q0 − iQ0 (2udy (x; y)B˜ + udyy (x; y)) − 2 9y − vd (x; y)

92 B˜ 9B˜ 9B˜ Q02 2 2 2 ˜ ˜ ˜ ˜ + B(B − Q0 ) − 3vdy (x; y) +B − + 3B 9y2 9y 9y 3

− 3vdyy (x; y)B˜ − vdyyy (x; y) + iQ0 +dy (x; y) − +d x (x; y)B˜ − +d xy (x; y) = 0 :

(6.32)

It is convenient to solve Eqs. (6.31) and (6.32) forward from y = y1 (x) to 1 and backward from y = y2 (x) to 1, and then to sew the solutions found for y = 1. In the 3rst case we should 3nd four partial solutions of these equations with initial conditions B1; 2 (S; x; y1 ) = B˜ 1; 2 (S; x; y1 ) = ±Q0 ;

B3; 4 (S; x; y1 ) = B˜ 3; 4 (S; x; y1 ) = ±B12 :

(6.33)

It is evident that the functions f01 (S; x; y), f02 (S; x; y), 3X01 (S; x; y) and 3X02 (S; x; y), for y1 6 |y| 6 1, are equal to A1 (exp(91 (S; x; y)) − exp(92 (S; x; y))) ; 2 A2 f02 (S; x; y) = (exp(93 (S; x; y)) − exp(94 (S; x; y))) ; 2

f01 (S; x; y) =

3X01 (S; x; y) =

A˜ 1 (exp(9˜ 1 (S; x; y)) − exp(9˜ 2 (S; x; y))) ; 2

3X02 (S; x; y) =

A˜ 2 (exp(9˜ 3 (S; x; y)) − exp(9˜ 4 (S; x; y))) ; 2

(6.34)

(6.35)

where 991; 2 (S; x; y) = B1; 2 (S; x; y); 9y

99˜ 1; 2 (S; x; y) = B˜ 1; 2 (S; x; y) ; 9y

993; 4 (S; x; y) = B3; 4 (S; x; y); 9y

99˜ 3; 4 (S; x; y) = B˜ 3; 4 (S; x; y) 9y

(y1 6 |y| 6 1) :

38

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 1400

800

1200

600

1000

400

f 0i

f 0r

800 600

200 0

400 200

-200

0

-400

-200

-600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 y

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 y

(b)

1400

8

1200

7 6

f0

5

800

4

600

arg

|f| 0

1000

3 2

400

1

200

0

0

(c)

-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 y

0 0.2 0.4 0.6 0.8

(d)

1 1.2 1.4 1.6 1.8 y

2 2.2

Fig. 26. The partial solutions of Eq. (6.31) for (a and b) y1 6 y 6 1 and (c and d) 1 6 y 6 y2 : Re = 25; 000, b0 = 0:1 m, x = 0, S = 22 (Q0 ≈ 44:357517 + 0:597408i). It is found that (a) and (b) all solutions tend to B3 , and (c) and (d) that the 3rst solution tends to the second one.

In the second case we should 3nd two partial solutions of Eqs. (6.31) and (6.32) with initial conditions B1 (S; x; y2 ) = B21 ;

B2 (S; x; y2 ) = B22 ;

B˜ 1 (S; x; y2 ) = B˜ 21 ;

B˜ 2 (S; x; y2 ) = B˜ 22 ;

(6.36)

where B21 , B22 , B˜ 21 and B˜ 22 are de3ned by (6.22). The functions f01 (S; x; y), f02 (S; x; y), 3X01 (S; x; y) and 3X02 (S; x; y), for 1 6 |y| 6 y2 , are equal to f01 (S; x; y) = C1 exp(:1 (S; x; y));

f02 (S; x; y) = C2 exp(:2 (S; x; y)) ;

(6.37)

3X01 (S; x; y) = C˜ 1 exp(:˜ 1 (S; x; y));

3X02 (S; x; y) = C˜ 2 exp(:˜ 2 (S; x; y)) ;

(6.38)

where 9:1; 2 (S; x; y) = B1; 2 (S; x; y); 9y

9:˜ 1; 2 (S; x; y) = B˜ 1; 2 (S; x; y) 9y

(1 6 |y| 6 y2 ) :

Numerical solution of Eqs. (6.31) and (6.32), both forward and backward, has shown that in the 3rst case all partial solutions for B tend to the third partial solution with initial condition B|y=y1 = B12 , whereas in the second case the 3rst partial solution tends to the second one with initial condition B|y=y2 = B22 (see, for example, Fig. 26, where all numerical partial solutions of Eqs. (6.31) and (6.32) with initial conditions (6.33) and (6.36) are shown for S = 22). This means that all partial solutions, except the third one in the 3rst case and the second one in the second case, are ˜ followunstable. That is why we have taken the two partial solutions of the equations for B and B, ing from the Euler equations, as the 3rst and second approximate partial solutions of Eqs. (6.31)


40

30

20 0

~ B1i , B 1i

~ B1r , B 1r

20 10 0 -10

-40

-80

-30 0.9

0.95

(a)

1

1.05

-100 0.9

1.1

40

5

30

0

20

-5 -10

-30 1.05

-40 0.9

1.1

1.05

1.1

0

-20

1

1.1

-10

-20

0.95

(d)

y

1.05

10

-15

0.95

1

y

10

-25 0.9

0.95

(b)

y

~ B4i, B4i

~ B4r , B 4r

-20

-60

-20

(c)

39

1

y

Fig. 27. The partial solutions of Eqs. (6.31) and (6.39) for (a) and (b) y1 6 y 6 1 and (c) and (d) 1 6 y 6 y2 . The parameters are the same as in Fig. 25. It is seen that all solutions are stable.

and (6.32), and, for y1 6 y 6 1, ignored the√fourth partial solution with a large negative real part. The former is valid because |Q0 | ∼ |B21 | Re, and the latter is valid because, for y1 6 y 6 1, exp( B3 (S; x; y) dy)exp( B4 (S; x; y) dy). The equations for B and B˜ following from the Euler equations are 9B 2 2 i(S − ud (x; y)Q0 ) (6.39) + B − Q0 + iQ0 +dy (x; y) + +d x (x; y)B = 0 ; 9y i(S − ud (x; y)Q0 )

9B˜ + B˜ 2 − Q02 9y

− iQ0 (2udy (x; y)B˜ + udyy (x; y))

+ iQ0 +dy (x; y) − +d x (x; y)B˜ − +d xy (x; y) = 0 :

(6.40)

An example of the partial solutions found in this way is given in Fig. 27 for S = 22. Comparing Figs. 27 with 26 we see that, as distinct from the case shown in Fig. 26, all the solutions found are stable. To 3nd the eigenfunctions and adjoint eigenfunctions we have to use expressions (6.34), (6.37) and, respectively, for the adjoint functions, (6.35), (6.38) and the sewing conditions for y = 1. Thus we 3nd the equations for A1 , A2 , C1 and C2 (and, correspondingly, in the case of the adjoint eigenfunctions, for A˜ 1 , A˜ 2 , C˜ 1 and C˜ 2 ): A1 A2 (exp(91 (S; x; 1)) − exp(92 (S; x; 1))) + exp(93 (S; x; 1)) 2 2 =C1 exp(:1 (S; x; 1)) + C2 exp(:2 (S; x; 1)) ;

40


A1 (l) (B (S; x; 1) exp(91 (S; x; 1)) − B2(l) (S; x; 1) exp(92 (S; x; 1))) 2 1 A2 (l) + B (S; x; 1) exp(93 (S; x; 1)) 2 3 =C1 B1(r) (S; x; 1) exp(:1 (S; x; 1)) + C2 B2 1(r) (S; x; 1) exp(:2 (S; x; 1)) ; A1 ((B1(l) (S; x; 1))2 exp(91 (S; x; 1)) − (B2(l) (S; x; 1))2 exp(92 (S; x; 1))) 2 A2 + (B3(l) (S; x; 1))2 exp(93 (S; x; 1)) 2 =C1 (B1(r) (S; x; 1))2 exp(:1 (S; x; 1)) + C2 (B2(r) (S; x; 1))2 exp(:2 (S; x; 1)) ; A1 ((B1(l) (S; x; 1))3 exp(91 (S; x; 1)) − (B2(l) (S; x; 1))3 exp(92 (S; x; 1))) 2 A2 + (B3(l) (S; x; 1))3 exp(93 (S; x; 1)) 2 =C1 (B1(r) (S; x; 1))3 exp(:1 (S; x; 1)) + C2 (B2(r) (S; x; 1))3 exp(:2 (S; x; 1)) ;

(6.41)

A˜ 1 A˜ 2 (exp(9˜ 1 (S; x; 1)) − exp(9˜ 2 (S; x; 1))) + exp(9˜ 3 (S; x; y)) 2 2 =C˜ 1 exp(:˜ 1 (S; x; 1)) + C˜ 2 exp(:˜ 2 (S; x; 1)) ; A˜ 1 ˜ (l) ˜ (B (S; x; 1) exp(9˜ 1 (S; x; 1)) − B˜ (l) 2 (S; x; 1) exp(92 (S; x; 1))) 2 1 +

A˜ 2 ˜ (l) B (S; x; 1) exp(9˜ 3 (S; x; 1)) 2 3 ˜ ˜ ˜ (r) ˜ =C˜ 1 B˜ (r) 1 (S; x; 1) exp(:1 (S; x; 1)) + C 2 B2 1 (S; x; 1) exp(:2 (S; x; 1)) ;

A˜ 1 ˜ (l) 2 ˜ ((B1 (S; x; 1))2 exp(9˜ 1 (S; x; 1)) − (B˜ (l) 2 (S; x; 1)) exp(92 (S; x; 1))) 2 +

A˜ 2 ˜ (l) (B (S; x; 1))2 exp(9˜ 3 (S; x; 1)) 2 3 2 2 ˜ ˜ ˜ (r) ˜ =C˜ 1 (B˜ (r) 1 (S; x; 1)) exp(:1 (S; x; 1)) + C 2 (B2 (S; x; 1)) exp(:2 (S; x; 1)) ;

A˜ 1 ˜ (l) 3 ˜ ((B1 (S; x; 1))3 exp(9˜ 1 (S; x; 1)) − (B˜ (l) 2 (S; x; 1)) exp(92 (S; x; 1))) 2 +

A˜ 2 ˜ (l) (B (S; x; 1))3 exp(9˜ 3 (S; x; 1)) 2 3 3 3 ˜ ˜ ˜ (r) ˜ =C˜ 1 (B˜ (r) 1 (S; x; 1)) exp(:1 (S; x; 1)) + C 2 (B2 (S; x; 1)) exp(:2 (S; x; 1)) :

(6.42)

1400 1200 1000 800 600 400 200 0 -200 -400 -600

600 400 200 0 -200 -400 -600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

(a)

y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

(b)

y 8

1200

6

f0

1400

arg χ , arg

1000

|χ | , |f| 0

41

800

χi , f 0i

χr , f 0r


800 600 400

4 2 0 -2

200 0

-4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

(c)

y

0 0.2 0.4 0.6 0.8

(d)

1 1.2 1.4 1.6 1.8

2 2.2

y

Fig. 28. Plots of the real and imaginary parts and modulus of the eigenfunction f0 , calculated for b0 = 0:1, q = 3, Re = 25; 000, x = 0 and (a) S = 10 (Q0 ≈ 19:952633 + 9:984723i) and (b) S = 18 (Q0 ≈ 37:974133 + 4:244753i).

Because the determinants of the systems of equations (6.41) and (6.42) are equal to zero, these equations allow us to 3nd a = A2 =A1 , c1 = C1 =A1 and c2 = C2 =A1 (and, correspondingly, a˜ = A˜ 2 = A˜ 1 , c˜1 = C˜ 1 = A˜ 1 and c˜2 = C˜ 2 = A˜ 1 ). Examples of the eigenfunctions and adjoint eigenfunctions constructed in this way are illustrated in Figs. 28 and 29 for x = 0, (a) S = 10 and (b) S = 18. We see that the range of the sharp change of the eigenfunctions and adjoint eigenfunctions becomes narrower with increasing S. It is important that the 3rst partial solution of Eq. (6.39) for y1 6 y 6 1, B1(l) (S; x; y), transforms uninterruptedly into the 3rst partial solution of the same equation for 1 6 y 6 y2 , B1(r) (S; x; y) (see Fig. 30). Thus, at a point 9 y = y∗ (S; x) ¿ 1 the real and imaginary parts of B1 (S; x; y) change sign. This change of sign provides an explanation of the formation of vortices within the boundary layer: on diIerent sides of the pivot point the stochastic constituents of the longitudinal velocity are oppositely directed. For S ¡ 3, the form of functions f02 (S; x; y) and 3X02 (S; x; y) means that over the region of the boundary layer (region III) the eigenfunctions f0 (S; x; y) and adjoint eigenfunctions 3X0 (S; x; y) depend strongly on the Reynolds number and diIer markedly from those found from the Euler equations. This is illustrated in Fig. 31, where these functions calculated by the way indicated above and from the Euler equations are compared for S = 1. For S ¿ 8 the eigenfunctions and adjoint eigenfunctions are practically independent of the Reynolds number, and hence may be calculated from the Euler equations. With a knowledge of the eigenfunctions and adjoint eigenfunctions, we can use Eq. (6.10) to calculate the corrections Q1 to the eigenvalues of Q0 . The values of 7 = 70 + 71 =4 and wave 9

This point is called the pivot point.


χ0i

χ0r

1e+16 0 -1e+16 -2e+16 -3e+16 -4e+16 -5e+16 -6e+16 -7e+16 -8e+16 -9e+16

1 1.2 1.4 1.6 y

3e+08 2.5e+08 2e+08 1.5e+08 1e+08 5e+07 0 -5e+07 -1e+08 -1.5e+08 -2e+08 -2.5e+08 0.4 0.6 0.8 1 1.2 1.4 1.6 y

0.4

0.8

1 1.2

1.4

1.6

8e+16 6e+16 4e+16 2e+16 0 -2e+16 -4e+16 -6e+16 0.4 0.6 0.8

7e+08

1.2e+17

6e+08

1e+17

5e+08

1 1.2 1.4 1.6 y

8e+16

4e+08

|χ 0 |

|χ 0|

0.6

y

χ 0i

1e+08 0 -1e+08 -2e+08 -3e+08 -4e+08 -5e+08 -6e+08 -7e+08 0.4 0.6 0.8

χ0r

42

3e+08

6e+16 4e+16

2e+08

2e+16

1e+08 0 0.4 0.6 0.8

(a)

1 1.2 1.4 1.6 y (b)

0 0.4 0.6 0.8

1 y

1.2 1.4 1.6

Fig. 29. Plots of the real and imaginary parts and modulus of the adjoint eigenfunction 3X0 , calculated for b0 = 0:1, q = 3, Re = 25; 000, x = 0 and (a) S = 10 (Q0 ≈ 19:952633 + 9:984723i) and (b) S = 18 (Q0 ≈ 37:974133 + 4:244753i).

30 20

B1i

B1r

10 0 -10 -20 -30 0.85

(a)

0.9

0.95

1

y

1.05

1.1

1.15

40 35 30 25 20 15 10 5 0 -5 -10 0.85

(b)

0.9

0.95

1

1.05

1.1

1.15

y

Fig. 30. The 3rst partial solution of Eq. (6.39) B1 (S; x; y) versus y over the range y1 6 y 6 y2 for x = 0, S = 13.

phase velocity vph = S=(K0 + K1 =4) as functions of St are shown for diIerent x in Figs. 32a and b. Comparing Figs. 32a and 24a we see that for x ¡ 1 the corrections to the eigenvalues of 70 are not small. This is caused by a rather large value of the derivative of 70 with respect to x for small x.


f 0i

f

0r

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 0.8

1

1.2

y

(a)

1

1.2

y

(b)

1.3

1.8

1.2

1.7 1.6

1.1

1.5

χ 0i

1

χ0r

1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.8

43

0.9

1.4 1.3 1.2

0.8

1.1

0.7

1

0.6 0.8

1

y

(c)

0.9 0.8

1.2

1

1.2

y

(d)

Fig. 31. The eigenfunctions and adjoint eigenfunctions versus y over the range y1 6 y 6 y2 for x = 0, S = 1 (bold lines), and the same quantities calculated from the Euler equations (thin lines).

8

1.3 1.2

7

5

1

0.5

v ph

Γ

1.1

0

6 1

4 2

3

3

2

10

0.8 0.7

5

0.6

1

10 5 3 2

0.5

0

0 0.5

1

0.4 0

(a)

0.9

1

2

3

4

5

St

6

7

8

9

0

(b)

1

2

3

4

5

6

St

Fig. 32. The dependences on the Strouhal number St for b0 = 0:1, q = 3, r0 = 0:5, Re = 25; 000 and diIerent x of: (a) 7 = 70 + 71 =4 and (b) the wave phase velocity vph = S=(K0 + K1 =4). The value of x is indicated near the corresponding curve in each case.

The derivatives of 70 and vph0 for diIerent values of x are shown in Fig. 33. As x increases the derivatives of 70 and vph0 , and corrections to the eigenvalues of Q0 become progressively smaller. It can be seen that, as the distance from the nozzle increases, the gain factor decreases for large St and increases slightly for small St. For any given x the gain factor has a maximum at St = Stm , where the greater x is the smaller Stm becomes. It is easily shown that the shift of the gain factor maximum to the low-frequency region is caused by the jet’s divergence. Obviously, this shift of the gain factor maximum results in a shift of the turbulent pulsation power spectrum towards the low-frequency region as the distance from the nozzle increases (see below). It is interesting that, from x ≈ 1, the dependence of Stm on x is of an exponential character.

44

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 0.15 0

0.1 0.05

-4

v 0x

Γ0x

-2

-6

0 -0.05 -0.1

-8

-0.15

-10

-0.2 0

(a)

1

2

3

4

5

St

6

7

0

(b)

1

2

3

4

5

6

7

St

Fig. 33. The dependences of (a) 970 =9x ≡ 70x and (b) 9vph0 =9x ≡ v0x on the Strouhal number St for x = 0, 0.5, 1, 2, 3, 5 and 8.

Another important conclusion that can be drawn from Fig. 32 lies in the fact that the phase velocity of the hydrodynamic waves depends strongly on the Strouhal number, i.e. these waves are rather signi3cantly dispersive. Note also that the resonant character of the dependences of the gain factor on the Strouhal number that we have found indicates that each jet cross-section can be considered as an oscillator whose natural frequency decreases with increasing distance from the nozzle. This fact justi3es consideration of a jet as a chain of coupled resonant ampli3ers, which in turn allows us to understand the analogy between noise-induced pendulum oscillations and the turbulent processes in a jet. Neglecting the correction to the eigenfunction f0 (S; x; y), we can write the generative solutions (t; x; y), u0 (t; x; y) and +0 (t; x; y) as 0 ∞ x 1 f0 (S; x; y) exp iSt − i Q(S; x) d x dS ; 0 (t; x; y) ≈ 2 −∞ 0 ∞ x 9f0 (S; x; y) 1 u0 (t; x; y) ≈ Q(S; x) d x dS ; exp iSt − i 2 −∞ 9y 0 ∞ 2 9 f0 (S; x; y) 1 9Q0 (S; x) 2 f0 (S; x; y) +0 (t; x; y) ≈ − Q (S; x) + i 2 −∞ 9y2 9x x 9f0 (S; x; y) exp iSt − i + iQ(S; x) Q(S; x) d x dS : (6.43) 9x 0 It follows from (6.43) that the vorticity is moderately small outside the boundary layer. Knowing Q(S; x) and the expressions for u0 (t; x; y) and f0 (S; x; y) we can calculate the evolution of the velocity power spectra in the linear approximation. For the sake of simplicity, we do so only for region I. We can expand a random disturbance (t; y) of the longitudinal component of velocity at the nozzle exit into a series in the eigenfunctions of our boundary value problem. Over region I we can approximate cosh(Q(S; 0)y) with eigenvalues of Q as the eigenfunctions for x = 0. Hence, we can set (t; y) = 1 (t) cosh(Q(S; 0)y) + · · · :

(6.44)


45

1

0.9

κ /κ 0

0.8

0.7

0.6

0.5

0.4

0.3 0

5

10

15

20

25

S

Fig. 34. Plot of %(S; 0; 0)=%0 described by (6.47).

The spectral density A1 (S)AX 1 (S) = A2 (S), where A(S) = |A1 |(S), is determined by the spectral density of 1 (t) which is denoted by us as %(S; 0). Because over region I u(S; 0; y) ≈ A1 (S)Q(S; x) cosh(Q(S; 0)y) ;

(6.45)

we 3nd A2 (S) =

%(S; 0) : K 2 (S; 0) + 72 (S; 0)

(6.46)

There is almost no experimental information about %(S; 0), but there is one work [133] giving power spectra of the longitudinal and transverse constituents of velocity pulsations over the range of the Strouhal numbers St 0–8 for a circular jet at diIerent initial turbulence levels. It can be seen from these data that the form of the spectra depends only slightly on the initial turbulence level, that the spectra of the longitudinal and transverse constituents of velocity pulsations are nearly identical, and that the spectral density decreases with increasing St. Since the dependence of the spectral densities presented in [133] on y is close to f(S)|cosh(Q(S; 0)y)|2 , where f(S) is a certain function of S, we can set %(S; 0) ≈ f(S). So, in accordance with data presented in [133], we approximate %(S; 0) by the formula: %0 %(S; 0) = ; (6.47) 1 + b1 S + b2 S2 + b3 S3 where %0 characterizes the level of the disturbances at the nozzle exit, b1 = 0:152, b2 = −0:005 and b3 = 0:000002. The plot of %(S; 0)=%0 described by (6.47) is shown in Fig. 34. Comparison with experimental results for power spectra of velocity pulsations and for the mean longitudinal velocity shows that %0 should be taken as very small. Hereinafter we will set %0 = 8 × 10−28 . It should be noted that, owing to the resonant character of the gain factor, the results are scarcely aIected by the shape of the dependence of %(S; 0).

46

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 36 32

log ( κ / κ 0)

log ( κ/ κ0 )

28 24 20 16 12 8 4 0 0

1

2

3

4

(a)

5

6

7

40 36 32 28 24 20 16 12 8 4 0

8

0

1

2

3

(b)

St

4

5

6

7

8

9

St

Fig. 35. Evolution of %(S; x; y)=%0 in the linear approximation for (a) y = 0 and (b) |y| = 0:7.

St m

St m

10

1

1 1

(a)

10

x

1

(b)

10

x

Fig. 36. The dependences of the Strouhal number Stm on the distance from the nozzle for (a) y = 0 and (b) |y| = 0:7 (the linear (zeroth) approximation). The dependences Stm ≈ 3:2x−0:36 (for y = 0) and Stm ≈ 6:5x−0:68 (for y = 0:7) are shown by solid lines.

It follows from (6.43), (6.45) and (6.46) that the spectral density of the longitudinal velocity pulsations in the linear approximation is x K 2 (S; x) + 72 (S; x) 2 %l (S; x; y) = %(S; 0) 2 : (6.48) 27(S; x ) d x |cosh(Q(S; x)y)| exp K (S; 0) + 72 (S; 0) 0 The evolutions of %l (S; x; y) for y = 0 and |y| = 0:7 are shown in Figs. 35a and b. We see that in the two cases considered the power spectra diIer markedly, especially for small x. In particular, the diIerence shows up as a faster (for |y| = 0:7) decrease of the Strouhal number corresponding to the spectrum maximum (Stm ) with increasing distance from the nozzle. This is more easily seen in Fig. 36, where this decrease is given for both cases considered. As is evident from the 3gures, the experimental dependences can be well-approximated by the curves Stm ≈ 3:2x−0:36 (for y = 0) and Stm ≈ 6:5x−0:68 (for y = 0:7). Unfortunately, experimental data (see Figs. 5 and 6) are available only for the jet axis (corresponding to y = 0) and a line oIset by the radius from the axis (corresponding to y = 1). In the 3rst case our dependence is close to the experimental one, and in the second case it lies between the experimental ones. We emphasize that our results are obtained from the linear theory without taking account of nonlinear phenomena such as the pairing of vortices. These results reinforce our idea that the experimentally observed shift of the power spectrum is explained mainly by the divergence of the jet, not by the pairing of vortices.

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 0.7

0.6

0.6

0.5 0.4

0.4

εu

εu

0.5

0.3

0.3 0.2

0.2

0.1

0.1 0

0 0

(a)

47

1

2

3

4

x

5

6

7

8

0

(b)

1

2

3

4

5

x

Fig. 37. Plots of the mean-root-square values of turbulent velocity pulsations versus x for (a) y = 0 (ju (x; 0)) and (b) y = 0:7 (ju (x; 0:7)).

It should be noted that, as the distance from the nozzle increases, the width of the power spectra decreases signi3cantly. This means that the correlation time increases, i.e. the coherence level increases too. This is a cause of the formation of coherent structures. The mean-root-square value of the turbulent velocity pulsations, which is what is usually is measured experimentally, is equal to 1 ∞ jul (x; y) = %l (S; x; y) dS : (6.49) 0 The plots of jul (x; 0) and jul (x; 0:7) versus y are presented in Fig. 37. It is seen from Fig. 37a that the dependence of jul (x; 0) on x closely resembles the dependence of an order parameter on temperature for a slightly noisy second-order phase transition. This also reinforces our hypothesis that the onset of turbulence is a nonequilibrium noise-induced phase transition of the second order, similar to that for a pendulum with a randomly vibrated suspension axis. It is interesting that for y = 0:7 the root-mean-square value of turbulent velocity pulsations 3rst decreases and then increases with increasing x. This can be explained by the competition between the ampli3cation of the pulsations and the swift decrease of the spectrum width. The condition for validity of all results obtained in this paper is that jul (x; y)1. We see that along the jet axis the results are valid for almost the whole initial part (x ¡ 8), whereas for y = 0:7 they are valid only for x ¡ 5. The change of j(x; y) as x increases is correlated with the change of the mean velocity (see below), but these changes are not fully identical. Let us trace the changes of the group wave velocity at St = Stm , as well as of the wave lengths in the longitudinal and transverse directions (4lon (Stm ; x) = 2 =K(Stm ; x) and 4tr (Stm ; x) = 2 =7(Stm ; x), respectively), with increasing x. The results are presented in Fig. 38. It is seen that along the jet axis, as the distance from the nozzle increases, the group velocity vgr (Stm (x)) 3rst decreases and then increases, whereas for y =0:7 it decreases monotonically. The longitudinal wavelengths increase considerably in both cases, providing evidence for an increase in the scale of the turbulence in the longitudinal direction. Along the jet axis the transverse scale of turbulence increases too, whereas for y = 0:7 it changes nonmonotonically. The increase of the scale of turbulence in the longitudinal direction agrees with the experimental data and reveals itself in the pairing of vortices.

48

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 0.51 0.5

λ lon , λ tr

v gr

0.49 0.48 0.47 0.46 0.45 0.44 0

1

2

3

4

5

6

7

8

x

(a)

2

1

0

1

2

3

4

5

6

7

8

x

(b)

1.2

4.5 4

1.1

2

3.5

λlon , λ tr

1

v gr

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

0.9 0.8 0.7

3 2.5 2 1.5 1

0.6

1

0.5

0.5

0 0

1

2

3

4

5

x

(c)

0

(d)

1

2

3

4

5

x

Fig. 38. The changes of (a, c) the group wave velocity at St = Stm and (b, d) the wave lengths in longitudinal (1) and transverse (2) directions (4lon (Stm ; x) = 2 =K(Stm ; x) and 4tr (Stm ; x) = 2 =7(Stm ; x), respectively), with increasing x: (a, b) for y = 0; (c, d) for y = 0:7.

6.2. The :rst approximation Putting 9A1 9A1 ∼ ∼ j2 ; 9x 9y substituting (6.1) into Eqs. (5.36), (5.37) and equating the terms of order j we obtain the following equations: 9s1 ; r1 − Ws1 = 0; q1 = 9y 9r1 9r1 9r1 9s1 + ud (x; y) + vd (x; y) − +dy (x; y) 9t 9x 9y 9x + +d x (x; y)

2 9s1 − Wr1 = R(t; x; y) ; 9y Re

(6.50)

where R(t; x; y) = and

0

9 0 9+0 9+0 9 0 − ; 9x 9y 9x 9y

and +0 are de3ned by expressions (6.43).

(6.51)


49

We suppose that only waves for which the sign of S is opposite can interact, an assumption that is justi3ed for dispersive waves. It should be noted that the interaction of waves with the same sign of S results in the generation of second harmonic, whereas the interaction of waves for which S is opposite in sign results in the appearance of a constant level. Because the diIerence 9

( S) 0

9x

9+0(S) 9+0(S) 9 0(S) − 9y 9x 9y

is small, the second harmonic is also small. It is therefore suPcient to consider the interaction only of waves for which S is opposite in sign. In so doing, it should be borne in mind that A1 (−S)= AX 1 (S) X and Q(−S) = −Q(S). Thus, we can set A1 (S)A1 (S ) = 2 A2 (S)(S + S), where A(S) = |A1 |(S). In this case R(t; x; y) is independent of t, and we can represent it as ∞ 1 R(x; y) = Rs (S; x; y) dS : (6.52) 2 −∞ A solution of Eqs. (6.50) can be also presented in the form ∞ ∞ 1 1 r1s (S; x; y) dS; s1 (x; y) = s1s (S; x; y) dS ; r1 (x; y) = 2 −∞ 2 −∞ ∞ 1 q1 (x; y) = q1s (S; x; y) dS ; 2 −∞

(6.53) (6.54)

Eqs. (6.50) can be solved analytically only over regions I and II. It follows from (6.43) and our numerical calculations that over region I x Q(S; x) d x ; 0 (S; x; y) ≈ A1 sinh(Q(S; x)y) exp iSt − i 0

+0 (S; x; y) ≈ −iA1 [sinh(Q(S; x)y) + 2Q(S; x)y cosh(Q(S; x)y)] x ×exp iSt − i Q(S; x) d x ;

9Q(S; x) 9x (6.55)

0

and in region II 0 (S; x; y) ≈ A1 [c1 (S; x) exp(B21 (S; x)(y

− y2 (x))) + c2 (S; x) x ×exp(B22 (S; x)(y − y2 (x)))] exp iSt − i Q(S; x) d x ;

(6.56)

0

Q(S; x)+d x (x; ∞) c1 (S; x) exp(B21 (S; x)(y − y2 (x))) iS − Q(S; x)vd x (x; ∞) x 2 + B22 (S; x)c2 (S; x) exp(B22 (S; x)(y − y2 (x))) exp iSt − i Q(S; x) d x ;

+0 (S; x; y) ≈ A1

0

(6.57)

50


where c1 (S; x) and c2 (S; x) are found from Eqs. (6.41), and B21 (S; x) and B22 (S; x) are determined by (6.21) and (6.22). For y − y2 (x)1=|B22 (S; x)| expressions (6.56) may be reduced to x Q(S; x) d x ; 0 (S; x; y) ≈ A1 c1 (S; x) exp(B21 (S; x)(y − y2 (x))) exp iSt − i 0

Q(S; x)+d x (x; ∞) c1 (S; x) exp(B21 (S; x)(y − y2 (x))) iS − Q(S; x)vd x (x; ∞) x Q(S; x) d x : ×exp iSt − i

+0 (S; x; y) ≈ A1

0

(6.58)

6.2.1. Region I It follows from (6.55) that

97(S; x) 9K(S; x) sinh(2K(S; x)y) + Rs (S; x; y) = −2(K (S; x) + 7 (S; x)) 2 9x 9x 97(S; x) 9K(S; x) − 7(S; x) y cosh(2K(S; x)y) ×sin(27(S; x)y) + 2 K(S; x) 9x 9x x 2 7(S; x) d x : (6.59) ×A (S) exp 2 2

2

0

Eqs. (6.50) for r1s , q1s and s1s become r1s − Ws1s = 0;

q1s =

9s1s ; 9y

9r1s 2 − Wr1s = Rs (S; x; y) : 9x Re

(6.60)

Substituting (6.59) into Eqs. (6.60), and ignoring the term proportional to 1=Re, we obtain the following approximate expressions for r1s (S; x; y), s1s (S; x; y) and q1s (S; x; y): x 2 7(S; x) d x : r1s (S; x; y) = r˜1s (S; x; y)A (S) exp 2 0

x 2 s1s (S; x; y) = s˜1s (S; x; y)A (S) exp 2 7(S; x) d x ;

(6.61)

x 7(S; x) d x ; q1s (S; x; y) = q˜1s (S; x; y)A2 (S) exp 2

(6.62)

0

0

where K 2 (S; x) + 72 (S; x) r˜1s (S; x; y) ≈ − 7(S; x)

9K(S; x) 97(S; x) sinh(2K(S; x)y) + 9x 9x

97(S; x) 9K(S; x) − 7(S; x) y cosh(2K(S; x)y) ; ×sin(27(S; x)y) + 2 K(S; x) 9x 9x 2


51

7(S; x) 9K(S; x) 1 97(S; x) 7 + K(S; x) 27(S; x) K 2 (S; x) + 72 (S; x) 9x 9x 9K(S; x) 97(S; x) y cosh(2K(S; x)y) ×sinh(2K(S; x)y) + K(S; x) − 7(S; x) 9x 9x K 2 (S; x) + 72 (S; x) 97(S; x) − y cos(27(S; x)y) ; (6.63) 27(S; x) 9x 1 1 9K(S; x) (372 (S; x) + K 2 (S; x))K(S; x) q˜1s (S; x; y) ≈ − 2 2 27(S; x) K (S; x) + 7 (S; x) 9x 97(S; x) cosh(2K(S; x)y) + (K 2 (S; x) − 72 (S; x))7(S; x) 9x 9K(S; x) 97(S; x) + 2K(S; x)y K(S; x) − 7(S; x) sinh(2K(S; x)y) 9x 9x s˜1s (S; x; y) ≈ −

−

K 2 (S; x) + 72 (S; x) 97(S; x) [cos(27(S; x)y) 27(S; x) 9x

− 27(S; x)y sin(27(S; x)y)]

:

(6.64)

It follows from (6.64) that along the jet axis, where sinh(2K(S; x)y) = sin(27(S; x)y) = 0 and cosh(2K(S; x)y) = cos(27(S; x)y) = 1, we have r1s (S; x; 0) = s1s (S; x; 0) = 0 and 1 1 9K(S; x) (372 (S; x) + K 2 (S; x))K(S; x) q˜1s (S; x; y) ≈ − 2 2 27(S; x) K (S; x) + 7 (S; x) 9x

K 2 (S; x) + 72 (S; x) 97(S; x) 97(S; x) 2 2 − : + (K (S; x) − 7 (S; x))7(S; x) 9x 27(S; x) 9x (6.65) It should be noted that the functions r˜1s , s˜1s and q˜1s are slow functions of x. We emphasize that the function q1 (x; y) determines the dependence on x and y of the additional constant correction to the dynamical constituent of the longitudinal velocity, i.e. the change of the mean velocity due to nonlinear eIects. This change is caused by turbulent pulsations. It is intuitively obvious (and con3rmed by experiment) that the correction found over region I must be negative, i.e. turbulent pulsations must decrease the mean ?ow velocity in region I. Averaging (6.62), substituting (6.46) into q1s (S; x; y) and integrating over S we can 3nd q1 (x; y) for diIerent values of x and y 6 y1 (x). Fig. 39 shows some examples of the dependences of u(x; y) = ud (x; y) + q1 (x; y) on y at 3xed values of x, and on x at 3xed values of y. It is seen from Fig. 39a that the mean velocity pro3le becomes increasingly bell-shaped as x increases. Furthermore (see Fig. 39b), within the initial part (x 6 8) the velocity on the jet axis decreases

52

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 1 0.95

0.85

3

2

0.9 1

0.8 0.75 0.7 0

0.1

0.2

0.3

0.4

0.5

y

(a)

0.6

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

0.7

4

4.5

5

5.5

6

6.5

7

7.5

8

x

(b)

Fig. 39. The dependences of the mean velocity u(x; y) = ud (x; y) + q1 (x; y) taking account of the correction caused by the turbulent pulsations (q1 (x; y)) (a) on y at x = 6 (1), x = 7 (2) and x = 8 (3) and (b) on x at (from right to left) y = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7.

signi3cantly only at its end, and then by no more than 5%. OI-axis, the velocity falls oI much faster. 6.2.2. Region II It follows from (6.58) and (6.21) that 4+d x (x; ∞)|c1 (S; x)|2 K(S; x)S (K 2 (S; x) + 72 (S; x))2 vd2 (x; ∞) + S(S − 27(S; x)vd (x; ∞)) +d x (x; ∞)[(K 2 (S; x) − 72 (S; x))vd (x; ∞) + 7(S; x)S] 2 2 × K (S; x) + 7 (S; x) − 2[K 2 (S; x)vd2 (x; ∞) + (S − 7(S; x)vd (x; ∞))2 ]

x ×exp 2 B21r (S; x)(y − y2 (x)) + 7(S; x) d x A2 (S) ; (6.66)

Rs (S; x; y) = −

0

where

+d x (x; ∞)vd (x; ∞) B21r (S; x) = −K(S; x) 1 − 2 2 2[K (S; x)vd (x; ∞) + (S − 7(S; x)vd (x; ∞))2 ]

:

Eqs. (6.50) for r1s and s1s become r1s − Ws1s = 0 ; vd (x; ∞)

2 9r1s 9s1s + +dx (x; ∞) − Wr1s = Rs (x; y) ; 9y 9y Re

(6.67)

Ignoring the term proportional to 1=Re, we obtain the following approximate expressions for r1s (S; x; y), s1s (S; x; y) and q1s (S; x; y): r1s (S; x; y) ≈

2 (S; x) + 72 (S; x)) 2Rs (S; x; y)(B21r ; 2 2 B21r (S; x)[4(B21r (S; x) + 7 (S; x))vd (x; ∞) + +dx (x; ∞)]


2 1

0.6

yb

ud

0.8

0.4 0.2 0

-2 -1.5

-1 -0.5

0

0.5

1

1.5

2

y

(a)

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

53

2

1

0

1

2

3

4

5

6

7

8

9

x

(b)

Fig. 40. (a) An example of the velocity pro3le taking account of the stochastic constituents (curve 1) and the corresponding pro3le of the dynamical constituent of the velocity (curve 2) for x = 8 (the end of initial part); and (b) the internal (1) and external (2) boundaries of the mixing layer taking account of the stochastic constituents.

s1s (S; x; y) ≈ q1s (S; x; y) ≈

2 2B21r (S; x)[4(B21r (S; x) 2 4(B21r (S; x)

+

Rs (S; x; y) ; + 72 (S; x))vd (x; ∞) + +dx (x; ∞)]

Rs (S; x; y) 2 7 (S; x))vd (x; ∞)

+ +dx (x; ∞)

:

(6.68) (6.69)

Averaging (6.69), substituting (6.46) into q1s (S; x; y) and integrating over S we can 3nd q1 (x; y) for diIerent values of x and y ¿ y2 (x). Extrapolating the values of q1 (x; y) found here and for region I into the region of boundary layer we can estimate the mean longitudinal velocity pro3les width for diIerent values of x, taking account of stochastic constituents. An example of such pro3le for x = 8 (the end of the initial part) is given in Fig. 40a. For comparison, the corresponding pro3le of the dynamical constituent of the velocity is shown in the same 3gure. We see that these pro3les diIer substantially. It should be emphasized that the velocity pro3le that we have found by taking account of the stochastic constituents coincides in form with experimentally measured pro3les. Unfortunately, we cannot calculate exactly the full velocity pro3les, or the change of velocity, for all y because we are restricted to regions I and II and take no account of strong nonlinear eIects. However, we can calculate the width of the internal and external parts of the boundary layer. If we take for the internal boundary the plane where the mean velocity is equal to 0:95U0 , and for the external boundary the plane where the mean velocity is equal to 0:05U0 , then these boundaries are as shown in Fig. 40b. Our results demonstrate that the boundaries of the mixing layer are very far from being the straight lines adduced by many researchers. Up to a certain value of x, these boundaries nearly coincide with the boundaries of the mixing layer for the dynamical constituents. Only for larger x do they strongly move apart. 6.3. The second approximation To derive the equation for the amplitude A1 in the second approximation, we set in expansion (6.1) ∞ x 1 r2 (t; x; y) = C(S) sinh(Q(S; x)y) exp iSt − i Q(S; x) d x dS ; 2 −∞ 0

54


1 s2 (t; x; y) = 2

∞

−∞

D(S) sinh(Q(S; x)y) exp iSt − i

0

x

Q(S; x) d x dS ;

(6.70)

where C(S) ∼ j2 and D(S) ∼ j2 are unknown amplitudes. Like the 3rst approximation, the second approximation can be found analytically only over regions I and II. For simplicity, we restrict our consideration to region I. Equating the terms of order j2 in Eqs. (5.36) and (5.37) for region I, neglecting the derivatives of Q(S; x) with respect to x and taking account of (6.70), we obtain the following system of approximate equations for C and D with determinant equal to zero: 9A1 9A1 − coth(Q(S; x)y) ; C = −2Q(S; x) i 9x 9y 9r1 (x; y) 9r1 (x; y) i(S − Q(S; x))C = −Q(S; x) coth(Q(S; x)y) + i A1 ; (6.71) 9x 9y where r1 (x; y) is de3ned by (6.53) and (6.61). From the condition of compatibility of Eqs. (6.71), we 3nd a truncated equation for the amplitude A1 over region I. It can be written as tanh(Q(S; x)y) where :(S; x; y) =

1 2

9A1 9A1 +i = :(S; x; y)A1 ; 9x 9y

∞

−∞

(6.72)

F(S; s; x; y)A20 (s) exp 2

x

0

7(s; x ) d x ds ;

1 9r˜1s (s; x; y) 27(s; x)r˜1s (s; x; y)) + i tanh(Q(S; x)y) F(S; s; x; y) = 2(Q(S; x) − S) 9y and A0 (S) = |A1 (S; 0; 0)|. Eq. (6.72) can be conveniently rewritten in terms of A1 (S; x; y) z(S; x; y) = ln A1 (S; 0; 0)

(6.73) (6.74)

(6.75)

as tanh(Q(S; x)y)

9z 9z +i = :(S; x; y) : 9x 9y

A solution of Eq. (6.76) can be represented as x ∞ 1 z(S; x; y) = w(S; s; x; y) exp 2 7(s; x ) d x ds : 2 −∞ 0

(6.76)

(6.77)

Substituting (6.77) into Eq. (6.76), taking into account that F(S; s; x; y) and w(S; s; x; y) are slow functions of x, and neglecting the derivative of w with respect to x we obtain the following equation


for w(S; s; x; y): 9w − 2i7(s; x) tanh(Q(S; x)y)w = −iF(S; s; x; y)A20 (s) : 9y

55

(6.78)

The solution of Eq. (6.78) with initial condition w(S; s; x; 0) = 0 is y cosh(Q(S; x)y) −2i7(s;x)=Q(S;x) w(S; s; x; y) = −iA20 (s) F(S; s; x; y ) dy : ) cosh(Q(S; x)y 0

(6.79)

After substituting (6.79) into (6.77) and integrating over s, we 3nd z(S; x; y). Knowledge of z(S; x; y), in view of (6.75), allows us to 3nd A2 (S; x; y) and the mean value of the phase shift. Taking account of (6.75) and using the assumption that |z(S; x; y)|1 we 3nd A2 (S; x; y) ≈ A20 (S)(1 + 2 Re[z(S; x; y)]) ; ’(S; x; y) − ’(S; 0; 0) ≈ Im[z(S; x; y)] ; where i z(S; x; y) = − 2 ×

0

y

∞

−∞

(6.80)

x %(s; 0) exp 2 7(s; x ) d x K 2 (s; 0) + 72 (s; 0) 0

F(S; s; x; y )

cosh(Q(S; x)y) cosh(Q(S; x)y )

−2i7(s;x)=Q(S;x)

dy ds ;

%(S; 0) is de3ned by (6.47). The spectral constituent of the longitudinal velocity pulsations is 9A1 (S; x; y) sinh(Q(S; x)y) u(S; x; y) = A1 (S; x; y)Q(S; x) cosh(Q(S; x)y) + 9y x Q(S; x) d x : ×exp iSt − i 0

(6.81)

(6.82)

Because of the smallness of z(S; x; y) we have A1 (S; x; y) ≈ A0 (S)(1+z(S; x; y)). It follows from here that the spectral density of the longitudinal velocity pulsations, with account taken of nonlinearity, can be represented as 2 tanh(Q(S; x)y) 9z(S; x; y) ; (6.83) %(S; x; y) = %l (S; x; y) 1 + 2z(S; x; y) + Q(S; x) 9y where %l (S; x; y) is determined by (6.48). It follows from (6.81) that for y = 0, in the approximation under consideration, the relative nonlinear correction 2 tanh(Q(S; x)y) 9z(S; x; y) W%(S; x; y) = 2z(S; x; y) + Q(S; x) 9y to the spectral density %l (S; x; y) is absent. For y = 0, however, this correction is essential, and it increases with increasing y and x. Examples of the dependences of W% on the Strouhal number St for y=0:7 and a number values of x and for x=8 and three values of y are given in Figs. 41a and b, respectively. It is seen that for x 6 5 and y = 0:7 the nonlinear correction changes nonmonotonically

56


3

2

2

0.5

0

2

-1

0

1

2

3

(a)

4

5

1

-1

6 7 8

-2 -3

1

4

∆κ

∆κ

1

2 3

-2 6

7

8

-3

9

1

2

3

(b)

St

4

5

6

7

8

9

St

Fig. 41. Examples of the dependences of W% on the Strouhal number St for (a) y = 0:7 and a number values of x, and (b) for x = 8 and three values of y = 0:01 (curve 1), 0.5 (2) and 0.7 (3). Because W% changes strongly as the values of x and y vary, we have plotted not W% but (W%)1=15 .

8e+29

3 7e+29

6e+29

1 5e+29

κ

4 4e+29

3e+29

2 2e+29

1e+29

0 1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

St

Fig. 42. Examples of the velocity pulsations spectral density with taking into account of the nonlinearity: x = 6, (curve 1) y = 0:5 and (curve 2) y = 0:7. For comparison in the same 3gure are given the corresponding dependences found in the linear approximation (curves 3 and 4, respectively).

with increasing x, even changing its sign. Only for x ¿ 5 the changes become monotone, and nearly for all values of St the correction is negative. The latter means that for these St the nonlinearity causes the saturation of turbulent pulsations. We note that the saturation occurs only from a certain value of the Strouhal number. For smaller Strouhal numbers the nonlinear ampli3cation occurs in place of the saturation. Two examples of the velocity pulsations spectral density (for x = 6, and y = 0:5 and 0.7) with taking into account of the nonlinearity are illustrated in Fig. 42. For comparison the corresponding spectral densities calculated in the linear approximation are shown in the same 3gure. It is seen that


57

the nonlinearity results in the signi3cant decrease of the spectral density maximal value, and as a consequence in the decrease of the turbulent pulsations variance. It is seen from Fig. 42 that, for x = 6, the spectral density for y = 0:5 is more than for y = 0:7, even in linear approximation; whereas for small x it increases monotonically with increasing y. We note that 9’(S; x; y)=9x ≡ WK(S; x; y) gives a nonlinear correction to the wave number K(S; x). It is important to note that this correction depends on the transverse coordinate y. It follows from (6.80) and (6.81) that it is x ∞ %(s; 0) WK(S; x; y) ≈ − 7(s; x) exp 2 7(s; x ) d x 2 2 0 −∞ K (s; 0) + 7 (s; 0) −2i7(s;x)=Q(S;x) y cosh(Q(S; x)y) (6.84) F(S; s; x; y ) dy ds : ×Re cosh(Q(S; x)y ) 0 The value of WK(S; x; y) determines a nonlinear correction W4lon to the longitudinal wave length 4lon (S; x) = 2 =K(S; x): 2 WK(S; x; y) : (6.85) W4lon = − K 2 (S; x) Because nearly for all St the values of WK are negative, we can conclude that the nonlinearity causes the faster increase of turbulence scales with increasing x, in comparison with the results of linear consideration. 7. Conclusions The theoretical approach proposed above has enabled us to account for many experimental results, and to demonstrate that a number of widely-accepted interpretations are in fact erroneous. It has led us to a somewhat diIerent and, we believe, more realistic perspective. In particular: (1) Our studies show that the shift of velocity pulsation power spectra to the low-frequency domain is caused mainly by the jet divergence, not pairing of vortices, so that it can therefore be calculated within the linear approximation. (2) The observed phenomenon of vortices pairing can be accounted for in terms of the increase in the longitudinal and transverse turbulent scales, which is caused by jet divergence and not by resonance relations. (3) The transformation of the mean velocity pro3le can be found without the use of the concept of turbulent viscosity. (4) The in?uence of nonlinearity close to the jet symmetry plane (y = 0) is very small within the initial part of the jet, but increases signi3cantly as we approach the boundary layer. (5) The intensity of random disturbances at the nozzle exit necessary for the onset of turbulence may be very small. Our quasi-linear theory is valid only for such small intensities. For larger disturbance intensities, the development of turbulence is from the very outset an essentially nonlinear process. (6) The change of the velocity pulsation variance with distance from the nozzle closely resembles changing order parameter with increasing temperature for a second-order phase transition.

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That is why we guess that the onset of turbulence can be considered as a speci3c noise-induced phase transition similar to that for a pendulum with a randomly vibrated suspension axis. Acknowledgements It is a pleasure to acknowledge useful discussions with G. Kolmakov and helpful advices related to computation of A. Sil’chenko and I. Kaufman. The research was supported in part by the Engineering and Physical Sciences Research Council (UK), and by the Royal Society of London to whom PSL is indebted for a visiting research fellowship at Lancaster during which much of the review was completed. References U [1] G. Hagen, Uber die Bewegung des Wassers in engen zylidrischen RUohren, Pogg. Ann. 46 (1839) 423–442. [2] O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct os sinouos, and the law of resistance in parallel channels, Philos. Trans. Roy. Soc., London 174 (1883) 935–982. Y Y [3] G. Compte-Bellot, Ecoulement Turbulent Entre deux Parois ParallYeles (EditeY e par le service de documentation scienti3que et technique de l’armement, Paris, 1965). [4] A.E. Ginevsky, Ye.V. Vlasov, R.K. Karavosov, Acoustic Control of Turbulence Jets, Fizmatlit, Moscow, 2001, to be published (in Russian) (English Translation: Springer, Heidelberg, 2004). [5] C.C. Lin, The Theory of Hydrodynamic Stability, Cambridge University Press, Cambridge, 1955. [6] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford, 1961. [7] R. Betchov, W.O. Criminale, Stability of Parallel Flows, Academic Press, New York, 1967. [8] M.A. Goldshtik, V.N. Sctern, Hydrodynamic Stability and Turbulence, Nauka, Novosibirsk, 1977 (in Russian). [9] P.D. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981. [10] Yu.S. Kachanov, V.V. Kozlov, V.Ya. Levchenko, The Onset of Turbulence in a Boundary Layer, Novosibirsk, Nauka, 1982 (in Russian). [11] L.D. Landau, E.M. Lifshitz, Hydrodynamics, Nauka, Moscow, 1986 (in Russian). [12] A.S. Monin, A.M. Yaglom, Statistical Gydromechanics, Vols. 1, 2, Gidrometeoizdat, Sankt-Petersburg, 1992 (in Russian). [13] L. Prandtl, FUuhrer durch die StrUomungslehre, F. Vieweg, Braunschweig, 1949. [14] H. Schlichting, Grenzschicht-Theorie, Verlag G. Braun, Karlsruhe, 1965. [15] M.A. Goldshtik, V.N. Shtern, N.I. Yavorsky, Viscous Flows with Paradoxical Properties, Nauka, Novosibirsk, 1989 (in Russian). [16] Yu.L. Klimontovich, Turbulent Motion and the Structures of Chaos, Kluwer Academic Publ., Dordrecht, 1991. [17] P. Holmes, J.L. Lumley, G. Berkooz, Turbulence, Coherent Structures and Symmetry, Cambridge University Press, Cambridge, 1996. [18] R.J. Roache, Computational Fluid Dynamics, Hermosa Publishers Albuquerque, 1976. [19] P. Bradshaw, T. Cebeci, J.H. Whitelaw, Engineering Calculation Methods for Turbulent Flow, Academic Press, New York, 1981. [20] D.C. Wilcox, Turbulence Modelling for CFD, DCW Industries Inc., La Canada, CA, 1998. [21] S.M. Belotserkovsky, A.S. Ginevsky, Simulation of Turbulent Jets and Wakes by Discrete Vortex Technique, Nauka, Moscow, 1995 (in Russian). [22] A.N. Kolmogorov, Local structure of turbulence in incompressible ?uid for very large Reynolds numbers, DAN SSSR 30 (1941) 299–303 (in Russian). [23] A.N. Kolmogorov, The energy dispersion in the case of locally isotropic turbulence, DAN SSSR 32 (1941) 19–21 (in Russian).


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Violations of fundamental symmetries in atoms and tests of uni%cation theories of elementary particles J.S.M. Ginges, V.V. Flambaum∗ School of Physics, University of New South Wales, Sydney 2052, Australia Accepted 9 March 2004 editor J. Eichler

Abstract High-precision measurements of violations of fundamental symmetries in atoms are a very e2ective means of testing the standard model of elementary particles and searching for new physics beyond it. Such studies complement measurements at high energies. We review the recent progress in atomic parity nonconservation and atomic electric dipole moments (time reversal symmetry violation), with a particular focus on the atomic theory required to interpret the measurements. c 2004 Elsevier B.V. All rights reserved. PACS: 32.80.Ys; 11.30.Er; 12.15.Ji; 31.15.Ar

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Manifestations and sources of parity violation in atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The nuclear spin-independent electron–nucleon interaction; the nuclear weak charge . . . . . . . . . . . . . . . . . . . . . 2.2. Nuclear spin-dependent contributions to atomic parity violation; the nuclear anapole moment . . . . . . . . . . . . . 2.3. Z 3 -scaling of parity violation in atoms induced by the nuclear weak charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Measurements and calculations of parity violation in atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Summary of measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Summary of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Method for high-precision atomic structure calculations in heavy alkali-metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. E-mail address: [email protected] (V.V. Flambaum).

c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2004.03.005

65 67 68 68 70 70 71 72 72 73 74 75 75

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J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 4.2. Zeroth-order approximation: relativistic Hartree–Fock method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Correlation corrections and many-body perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. All-orders summation of dominating diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Screening of the electron–electron interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. The hole–particle interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3. Chaining of the self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Other low-order correlation diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Empirical %tting of the energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Asymptotic form of the correlation potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Interaction with external %elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1. Time-dependent Hartree–Fock method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2. E1 transition amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3. Hyper%ne structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4. Structural radiation and normalization of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-precision calculation of parity violation in cesium and extraction of the nuclear weak charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. High-precision calculations of parity violation in cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Mixed-states calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Inclusion of the Breit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3. Neutron distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4. Strong-%eld QED corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5. Tests of accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The vector transition polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. The %nal value for the Cs nuclear weak charge QW and implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Ongoing/future studies of PNC in atoms with a single valence electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atoms with several electrons in un%lled shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Parity nonconservation in thallium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. A method to exclude the error from atomic theory: isotope ratios and the neutron distribution . . . . . . . . . . . . 6.3. Ongoing/future studies of PNC in complex atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The nuclear anapole moment and measurements of P-odd nuclear forces in atomic experiments . . . . . . . . . . . . . . . 7.1. The anapole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Origin of the nuclear anapole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Parity violating e2ects in atoms dependent on the nuclear spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Measurement of nuclear spin-dependent e2ects in cesium and extraction of the nuclear anapole moment . . . 7.4.1. Atomic calculations and extraction of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Extraction of a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. The nuclear anapole moment and parity violating nuclear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. The cesium result and comparison with other experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. Ongoing/future studies of nuclear anapole moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric dipole moments: manifestation of time reversal violation in atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Atomic EDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1. Electronic enhancement mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Enhancement of T -odd e2ects in polar diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Limits on neutron, atomic, and molecular EDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Mechanisms that induce atomic EDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1. The P; T -violating electron–nucleon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2. The electron EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3. P; T -violating nuclear moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P; T -Violating nuclear moments and the atomic EDMs they induce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Electric moments; the Schi2 moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1. The P; T -odd electric %eld distribution in nuclei created by the nuclear Schi2 moment . . . . . . . . . . . .

76 78 79 80 82 83 84 84 85 85 86 87 88 89 90 91 92 96 96 96 97 99 100 101 102 103 104 105 106 106 107 110 112 112 112 113 114 116 117 118 118 119 119 120 122 124 127 127 128 130

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 9.2. Magnetic moments; the magnetic quadrupole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1. The spin hedgehog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. What mechanisms at the nucleon scale induce P; T -odd nuclear moments? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1. The P; T -odd nucleon–nucleon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. The external nucleon EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3. Comparison of the size of nuclear moments induced by the nucleon–nucleon interaction and the nucleon EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Nuclear enhancement mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. Close-level enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Collective enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Octupole deformation; collective Schi2 moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Calculations of atomic EDMs induced by P; T -violating nuclear moments; interpretation of the Hg measurement in terms of hadronic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Current limits on fundamental P; T -violating parameters and prospects for improvement . . . . . . . . . . . . . . . . . . . . . . 10.1. Summary of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Ongoing/future EDM experiments in atoms, solids, and diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The success of the standard electroweak model of elementary particles [1] is extraordinary. 1 It has been tested in physical processes covering a range in momentum transfer exceeding ten orders of magnitude. It correctly predicted the existence of new particles such as the neutral Z boson. However, the standard model fails to provide a deep explanation for the physics that it describes. For example, why are there three generations of fermions? What determines their masses and the masses of gauge bosons? What is the origin of CP violation? The Higgs boson (which gives masses to the particles in the standard model) has not yet been found. The standard model is unable to explain Big Bang baryogenesis which is believed to arise as a consequence of CP violation. It is widely believed that the standard model is a low-energy manifestation of a more complete theory (perhaps one that uni%es the four forces). Many well-motivated extensions to the standard model have been proposed, such as supersymmetric, technicolour, and left-right symmetric models, and these give predictions for physical phenomena that di2er from those of the standard model. Some searches for new physics beyond the standard model are performed at high-energy and medium-energy particle colliders where new processes or particles would be seen directly. However, a very sensitive probe can be carried out at low energies through precision studies of quantities that can be described by the standard model. The new physics is manifested indirectly through a deviation of the measured values from the standard model predictions. The atomic physics tests that are the subject of this review lie in this second category. These tests exploit the fact that low-energy phenomena are especially sensitive to new physics that is manifested in the violations of fundamental symmetries, in particular P (parity) and T (time-reversal), that occur in the weak interaction. The 1

The recent observation of neutrino oscillations calls for a minimal extension, the introduction of neutrino masses and mixing; see, e.g., the review [2].

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J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

deviations from the standard model, or the e2ects themselves, may be very small. To this end, exquisitely precise measurements and calculations are required. More than 20 years ago atomic experiments played an important role in the veri%cation of the standard model. While the %rst evidence for neutral weak currents (existence of the neutral Z boson) was discovered in neutrino scattering [3], the fact that neutral currents violate parity was %rst established in atomic experiments [4] and only later observed in high-energy electron scattering [5]. Now atomic physics plays a major role in the search for possible physics beyond the standard model. Precision atomic and high-energy experiments have di2erent sensitivities to models of new physics and so they provide complementary tests. In fact the energies probed in atomic measurements exceed those currently accessible at high-energy facilities. For example, the most precise measurement of parity nonconservation (PNC) in the cesium atom sets a lower bound on an extra Z boson popular in many extensions of the standard model that is tighter than the bound set directly at the Tevatron (see Section 5). Also, the null measurements of electric dipole moments (EDMs) in atoms (an EDM is a P- and T -violating quantity) place severe restrictions on new sources of CP-violation which arise naturally in models beyond the standard model such as supersymmetry. (Assuming CPT invariance, CP-violation is accompanied by T -violation.) Such limits on new physics have not been set by the detection of CP-violation in the neutral K [6] and B [7] mesons (see, e.g., Ref. [8] for a review of CP violation in these systems). Let us note that while new physics would bring a relatively small correction to a very small signal in atomic parity violation, in atomic EDMs the standard model value is suppressed and is many orders of magnitude below the value expected from new theories. Therefore, detection of an EDM would be unambiguous evidence of new physics. This review is motivated by the great progress that has been made recently in both the measurements and calculations of violations of fundamental symmetries in atoms. This includes the discovery of the nuclear anapole moment (an electromagnetic multipole that violates parity) [9], the measurement of the parity violating electron–nucleon interaction in cesium to 0.35% accuracy [9], the improvement in the accuracy (to 0.5%) of the atomic theory required to interpret the cesium measurement [10], and greatly improved limits on atomic [11] and electron [12] electric dipole moments. The aim of this review is to describe the theory of parity and time-reversal violation in atoms and explain how atomic experiments are used to test the standard model of elementary particles and search for new physics beyond it. 2 We track the recent progress in the %eld. In particular, we clarify the situation in atomic parity violation in cesium: it is now %rmly established that the cesium measurement [9] is in excellent agreement with the standard model; see Section 5. The structure of the review is the following. Broadly, it is divided into two parts. The %rst part, Sections 2–7, is devoted to parity violation in atoms. The second part, Sections 8–10, is concerned with atomic electric dipole moments. In Section 2 the sources of parity violation, and the standard model predictions, are described. In Section 3 a summary of the measurements of parity violation in atoms is given, with particular emphasis on the measurements with cesium. Also the atomic calculations are summarized. In Section 4 we present a detailed description of the methods for high-precision atomic structure calculations 2

There are other tests of fundamental symmetries in atoms not dealt with in this review. These include tests of CPT and Lorentz invariance [13] and tests of the permutation-symmetry postulate and the spin-statistics connection [14].


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applicable to atoms with a single valence electron. The methods are applied to parity violation in cesium in Section 5 and the value for the weak nuclear charge is extracted and compared with the standard model prediction. A discussion of the new physics constraints is also presented. In Section 6 a brief description for the method of atomic structure calculations for atoms with more than one valence electron is given, and the thallium PNC work is discussed. A brief discussion of the prospects for measuring PNC along a chain of isotopes is also presented. Then in Section 7 work on the anapole moment is reviewed. A description of electric dipole moments in atoms is given in Section 8, with a summary of all the measurements and a discussion of the P; T -violating sources at di2erent energy scales. Then in Section 9 a review of P; T -violating nuclear moments is given. In Section 10 a summary of the best limits on P; T -violating parameters can be found. Concluding remarks are presented in Section 11. For a general introduction to atomic P-violation and P; T -violation we refer the reader to the excellent books by Khriplovich [15] and Khriplovich and Lamoreaux [16]. 2. Manifestations and sources of parity violation in atoms Parity nonconservation (PNC) in atoms arises largely due to the exchange of Z 0 -bosons between atomic electrons and the nucleus. The weak electron–nucleus interaction violating parity, but conserving time-reversal, is given by the following product of axial vector (A) and vector (V) currents 3 G [C1N e Q 5 eNQ N + C2N e Q eNQ 5 N ] : (1) hˆ = − √ 2 N Here G = 1:027 × 10−5 =m2p is the Fermi weak constant, N and e are nucleon and electron %eld operators, respectively, and the sum runs over all protons p and neutrons n in the nucleus. The Dirac matrices are de%ned as I 0 0 i 0 −I 0 ≡ = ; 5 = ; i = ; (2) −i 0 0 −I −I 0 and = 2s are the Pauli spin matrices. The coeScients C1N and C2N give di2erent weights to the contributions of protons and neutrons to the parity violating interaction. To lowest order in the electroweak interaction, C1p = 1=2(1 − 4 sin2 W ) ≈ 0:04;

C1n = −1=2 ;

C2p = −C2n = 1=2(1 − 4 sin2 W )gA ≈ 0:05 ;

(3)

where gA ≈ 1:26. The Weinberg angle W is a free parameter; experimentally it is sin2 W ≈ 0:23. The suppression of the coeScients C1p and C2N due to the small factor (1 − 4 sin2 ) makes |C1n | about 10 times larger than C1p and |C2N |. There is a contribution to atomic parity violation arising due to Z 0 exchange between electrons. However, this e2ect is negligibly small for heavy atoms [17–19]. It is suppressed by a factor 3

We use units ˝ = c = 1 throughout unless otherwise stated.

68


(1 − 4 sin2 )K=(QW R(Z)) compared to the dominant electron–nucleon parity violating interaction, where QW is the nuclear weak charge [see below, Eq. (6)], K is a numerical factor that decreases with Z and R(Z) is a relativistic factor that increases with Z [18]. For 133 Cs 6S1=2 − 7S1=2 , K ≈ 2 and R(Z) = 2:8 and so the suppression factor is ≈ 0:04% of the dominant amplitude [18]. This number was con%rmed in [19]. We will consider this interaction no further. 2.1. The nuclear spin-independent electron–nucleon interaction; the nuclear weak charge Approximating the nucleons as nonrelativistic, the time-like component of the interaction (Ae ; VN ) is given by the nuclear spin-independent Hamiltonian (see, e.g., [15]) G hˆW = − √ 5 [ZC1p !p (r) + NC1n !n (r)] ; (4) 2 Z and N are the number of protons and neutrons. This is an e2ective single-electron operator. The proton and neutron densities are normalized to unity, !n; p d 3 r = 1. Assuming that these densities coincide, !p = !n = !, this interaction reduces to G (5) hˆW = − √ QW !(r)5 : 2 2 The nuclear weak charge QW is very close to the neutron number. To lowest order in the electroweak interaction, it is QW = −N + Z(1 − 4 sin2 W ) ≈ −N :

(6)

This value for QW is modi%ed by radiative corrections. The prediction of the standard electroweak model for the value of the nuclear weak charge in cesium is [20] SM 133 (55 Cs) = −73:10 ± 0:03 : QW

(7)

The nuclear weak charge is protected from strong-interaction e2ects by conservation of the nuclear vector current. The clean extraction of the weak couplings of the quarks from atomic measurements makes this a powerful method of testing the standard model and searching for new physics beyond it. The nuclear spin-independent e2ects arising from the nuclear weak charge give the largest contribution to parity violation in heavy atoms compared to other mechanisms. However, note that the weak interaction (5) does not always “work”. This interaction can only mix states with the same electron angular momentum (it is a scalar). Nuclear spin-dependent mechanisms (see below), which produce much smaller e2ects in atoms, can change electron angular momentum and so can contribute exclusively to certain transitions in atoms and dominate parity violation in molecules. 2.2. Nuclear spin-dependent contributions to atomic parity violation; the nuclear anapole moment Using the nonrelativistic approximation for the nucleons, the nuclear spin-dependent interaction due to neutral weak currents is G hˆNC = − √ · C2p (8) p + C2n n !(r) ; 2 p n


69

where #i = 0 i . This term arises from the space-like component of the (Ve ; AN ) coupling. Averaging this interaction over the nuclear state with angular momentum I in the single-particle approximation gives G K − 1=2 · I!(r) ; hÎNC = − √ 2 2 I (I + 1)

(9)

where K = (I + 1=2)(−1)I +1=2−l , l is the orbital angular momentum of the unpaired nucleon, and 2 = −C2 . There are two reasons for the suppression of this contribution to parity violating e2ects in atoms. First, unlike the spin-independent e2ects [Eq. (5)], the nucleons do not contribute coherently; in the simple nuclear shell model only the unpaired nucleon which carries nuclear spin I makes a contribution. Second, the factor C2 ˙ (1 − 4 sin2 W ) is small in the standard model. There is another contribution to nuclear spin-dependent PNC in atoms arising from neutral currents: the “usual” weak interaction due to the nuclear weak charge, hˆW , perturbed by the hyper%ne interaction [21]. In the single-particle approximation this interaction can be written as [21,22] ·I G !(r) ; hÎQ = √ Q I 2

(10)

#N 1 = 2:5 × 10−4 A2=3 N ; Q = − QW 3 m p RN

(11)

with

RN = r0 A1=3 is the nuclear radius, r0 = 1:2 fm, 4 A = N + Z is the mass number, # = 1=137 is the %ne structure constant, and N is the magnetic moment of the nucleus in nuclear magnetons. For 133 Cs, N = 2:58 and Q = 0:017. However, the neutral currents are not the dominant source of parity violating spin-dependent e2ects in heavy atoms. It is the nuclear anapole moment a that gives the largest e2ects [23]. This moment arises due to parity violation inside the nucleus, and manifests itself in atoms through the usual electromagnetic interaction with atomic electrons. The Hamiltonian describing the interaction between the nuclear anapole moment and an electron is 5 G K · I!(r) : hâ = √ a 2 I (I + 1)

(12)

The anapole moment a increases with atomic number, a ˙ A2=3 . This is the reason it leads to larger parity violating e2ects in heavy atoms compared to other nuclear spin-dependent mechanisms. In heavy atoms a ∼ #A2=3 ∼ 0:1 − 1 [23,24]. (Note that the interaction (10), (11) also increases as A2=3 , however the numerical coeScient is very small.) The spin-dependent contributions [Eqs. (9), (10) and (12)] have the same form and produce the same e2ects in atoms. We will continue our discussion of the nuclear anapole moment and nuclear spin-dependent e2ects in atoms in Section 7. 4 Throughout the review we take the number (r0 = 1:1 or 1:2 fm) used in the work being cited in order to quote numerical results. 5 In fact, the distribution of the anapole magnetic vector potential is di2erent from the nuclear density. However, the corrections produced by this di2erence are small; see Section 7.

70


2.3. Z 3 -scaling of parity violation in atoms induced by the nuclear weak charge In 1974 the Bouchiats showed that parity violating e2ects in atoms increase with the nuclear charge Z faster than Z 3 [17,25]. This result was the incentive for studies of parity violation in heavy atoms. Let us brie?y point out where the factor of Z 3 originates. Taking the nonrelativistic limit of the electron wave functions and considering the nucleus to be point-like, the Hamiltonian (5) reduces to G hˆW = √ ( · p%3 (r) + %3 (r) · p)QW ; (13) 4 2m where m, , p are the electron mass, spin, and momentum. The weak Hamiltonian hˆW mixes electron states of opposite parity and the same angular momentum (it is a scalar). It is a local operator, so we need only consider the mixing of s and p1=2 states. The matrix element p1=2 |hˆW |s , with nonrelativistic single-particle s and p1=2 electron states, is proportional to Z 2 QW . One factor of Z here comes from the probability for the valence electron to be at the nucleus, and the other from the operator p which, near the nucleus (unscreened by atomic electrons), is proportional to Z. The nuclear weak charge |QW | ≈ N ∼ Z. (See [17,25,15] for more details.) It should be remembered thatrelativistic e2ects are important, since Dirac wave functions diverge at r = 0, j ˙ r −1 , = (j + 1=2)2 − Z 2 #2 . Taking into account the relativistic nature of the wave functions brings in a relativistic factor R(Z) which increases with the nuclear charge Z. The factor R ≈ 10 when Z = 80. As a consequence, the parity nonconserving e2ects in atoms increase as p1=2 |hˆW |s ˙ R(Z)Z 2 QW ;

(14)

that is, faster than Z 3 . 3. Measurements and calculations of parity violation in atoms An account of the dramatic story of the search for parity violation in atoms can be found in the book [15]. Below we will brie?y discuss how parity violation is manifested in atoms, which experiments have yielded nonzero signals, what quantity is measured, and what is required to interpret the measurements. Parity violation in atoms produces a spin helix, and this helix interacts di2erently with rightand left-polarized light (see, e.g., Ref. [15]). The polarization plane of linearly polarized light will therefore be rotated in passing through an atomic vapour. The weak interaction mixes states of opposite parity (parity violation), e.g., |p + )|s , where the mixing coeScient ) is pure imaginary. Therefore, a magnetic dipole (M 1) transition 6 in atoms will have a component originating from an electric dipole (E1) transition between states of the same 6 Atomic PNC studies are not limited, of course, to M 1 transitions, although all unambiguous signals to date have been obtained with them (see below). An experiment on an E2 transition (6S1=2 − 5D3=2 ) with singly ionized barium is being considered at Seattle [26]; the Berkeley dysprosium experiment [27] involves quantum beats on an E1 transition. See Ref. [28].


71

nominal parity, EPNC , e.g. p1=2 − p3=2 . The rotation angle per absorption length in such a transition is proportional to the ratio Im(EPNC )=M 1. While it may appear that it is more rewarding to study M 1 transitions that are highly forbidden, where there is a larger rotation angle, the ordinary M 1 transitions are in fact more convenient for experimental investigation since the angle per unit length ≈ ImEPNC M 1 (see, e.g., [15]). In measurements of parity violation in highly forbidden M 1 transitions, an electric %eld j is applied to open up the forbidden transition. The M 1 transition then contains a Stark-induced E1 component EStark which the parity violating amplitude interferes with. In such experiments the ratio Im(EPNC )= is measured, where is the vector transition polarizability, EStark ∼ j. Atomic many-body theory is required to calculate the parity nonconserving E1 transition amplitude EPNC . This is expressed in terms of the fundamental P-odd parameters like the nuclear weak charge QW . Interpretation of the measurements in terms of the P-odd parameters also requires a determination of M 1 or . 3.1. Summary of measurements Zel’dovich was the %rst to propose optical rotation experiments in atoms [29]. Unfortunately, he only considered hydrogen where PNC e2ects are small. Optical rotation experiments in Tl, Pb, and Bi were proposed by Khriplovich [30], Sandars [31], and Sorede and Fortson [32]. These proposals followed those by the Bouchiats to measure PNC in highly forbidden transitions in Cs and Tl [25,17]. The %rst signal of parity violation in atoms was seen in 1978 at Novosibirsk in an optical rotation experiment with bismuth [4]. Now atomic PNC has been measured in bismuth, lead, thallium, and cesium. PNC e2ects were measured by optical rotation in the following atoms and transitions: in 209 Bi in the transition 6s2 6p3 4 S3=2 − 6s2 6p3 2 D5=2 by the Novosibirsk [4], Moscow [33], and Oxford [34,35] groups and in the transition 6s2 6p3 4 S3=2 − 6s2 6p3 2 D3=2 by the Seattle [36] and Oxford [37,38] groups; in 6s2 6p2 3 P0 − 6s2 6p2 3 P1 in 208 Pb at Seattle [39,40] and Oxford [41]; and in the transition 6s2 6p 2 P1=2 − 6s2 6p 2 P3=2 in natural Tl (70:5% 205 Tl and 29:5% 203 Tl) at Oxford [42,43] and Seattle [44]. The highest accuracy that has been reached in each case is: 9% for 209 Bi 4 S3=2 −2 D5=2 [35], 2% for 209 Bi 4 S3=2 −2 D3=2 [38], 1% for 208 Pb [40], and 1% for Tl [44]. The Stark-PNC interference method was used to measure PNC in the highly forbidden M 1 transitions: 6s 2 S1=2 −7s 2 S1=2 in 133 Cs at Paris [45–48] and Boulder [49,50,9] and 6s2 6p 2 P1=2 −6s2 7p 2 P1=2 in 203;205 Tl at Berkeley [51,52]. In the most precise Tl Stark-PNC experiment [52] an accuracy of 20% was reached. In 1997, PNC in Cs was measured with an accuracy of 0.35% [9]—an accuracy unprecedented in measurements of PNC in atoms. Results of atomic PNC measurements accurate to sub-5% are listed in Table 1. Several PNC experiments in rare-earth atoms have been prompted by the possibility of enhancement of the PNC e2ects due to the presence of anomalously close levels of opposite parity [53]. Another attractive feature of rare earth atoms is their abundance of stable isotopes. Taking ratios of measurements of PNC in di2erent isotopes of the same element removes from the interpretation the dependence on atomic theory [53]; see Section 6. Null measurements of PNC have been reported for M 1 transitions in the ground state con%guration 4f6 6s2 of samarium at Oxford [54,55] and for the 4f9 5d2 6s J = 10 − 4f10 5d6s J = 10 transition in dysprosium at Berkeley [27]. The upper limit for dysprosium was smaller than expected by theory.

72


Table 1 Measurements of PNC in atoms with precision better than 5% Atom

Transition

Group

Year

Ref.

Measurement −Im(EPNC =M 1) (10−8 )

209

Bi Pb

4

208 205

Tl

6P1=2 − 6P3=2

133

Cs

6S1=2 − 7S1=2

3

S3=2 − 2 D3=2 P0 − 3 P1

Oxford Seattle Oxford Oxford Seattle Boulder Boulder

1991 1993 1996 1995 1995 1988 1997

[38] [40] [41] [43] [44] [50] [9]

10.12(20) 9.86(12) 9.80(33) 15.68(45) 14.68(17)

−Im(EPNC =) (mV/cm)

1.576(34) 1.5935(56)

Results of optical rotation experiments are given in terms of Im(EPNC =M 1); Stark-PNC experiments are given in terms of Im(EPNC =).

For a recent review of measurements of atomic PNC, we refer the reader to [28]; for a review of the early measurements, see, e.g., [57]. For comprehensive reviews, please see the book [15] and the more recent review [58]. 3.2. Summary of calculations The interpretation of the single-isotope PNC measurements is limited by atomic structure calculations. The theoretical uncertainty for thallium is at the level of 2.5–3% for the transition 6P1=2 −6P3=2 [59,60], and is worse for the transition 6P1=2 − 7P1=2 at 6% [59] and for lead (8%) [61] and bismuth (11% for the 876 nm transition 4 S3=2 − 2 D3=2 [61,62] and 15% for the 648 nm transition 4 S3=2 − 2 D5=2 [63]). Cesium is the simplest atom of interest in PNC experiments, it has one electron above compact, closed shells. The precision of the atomic calculations for Cs is 0.5% [10] (see also calculations accurate to better than 1%, [62,19,64]). For references to earlier calculations for the above atoms and transitions, see, e.g., the book [15]. In Table 2 we present the values of the most precise calculations for the PNC amplitudes corresponding to those atoms and transitions in which high-precision measurements (¡ 5% error) have been performed (Table 1). Using these calculations one may conclude that all parity violation experiments are in excellent agreement with the standard model. Note that the calculations were performed before the accurate measurements. It is interesting to note that the actual accuracy of the many-body calculations in all atoms (Cs, Tl, Pb, Bi) has been found to be better than claimed in the original theoretical papers! 3.3. Cesium Because of the extraordinary precision that has been achieved in measurements of cesium, and the clean interpretation of the measurements (compared to other heavy atoms), in this review we concentrate mainly on parity violation in cesium. The high precision of the nuclear weak charge extracted from cesium has made this system important in low-energy tests of the standard model


73

Table 2 Most precise calculations of PNC amplitudes EPNC for atoms and transitions listed in Table 1. Units: −10−11 ieaB (−QW =N ) Atom

Transition

EPNC a

Ref.

209

Bi Pb 205 Tl

4

208

3

133

6S − 7S

26(3) 28(2) 27.0(8) 27.2(7) 0.904(5)

[61,62] [61] [59] [60] [10]

Cs

S3=2 − 2 D3=2 P0 − 3 P1 6P1=2 − 6P3=2

a The values for the PNC amplitudes for Cs [10] and for Tl 6P1=2 − 6P3=2 [60] include corrections beyond the other calculations. In particular, for Cs the contributions of the Breit interaction and vacuum polarization due to the strong nuclear Coulomb %eld are included. For Tl the Breit interaction is also included. The remaining corrections for Cs and Tl are discussed in detail in Sections 5, 6, respectively. These corrections would be inside the error bars for the other atoms and transitions in the table.

Table 3 Summary of experimental results for PNC in cesium 6S − 7S, −Im(EPNC )=; units: mV/cm Group

Year

Ref.

Value

Paris Boulder Boulder Boulder Paris

1982, 1984 1985 1988 1997 2003

[45–47] [49] [50] [9] [48]

1.52(18) 1.65(13) 1.576(34) 1.5935(56) 1.752(147)

and has made it one of the most sensitive probes of new physics. Measurements of parity violation in cesium have also opened up a new window from which parity violation within the nucleus (the nuclear anapole moment; see Section 7) can be studied. Below we list the measurements and calculations for cesium that have been performed over the years, culminating in a 0.35% measurement and 0.5% calculation. 3.3.1. Measurements Measurements of parity violation in the highly forbidden 6S − 7S transition in Cs were %rst suggested and considered in detail in the landmark works of the Bouchiats [17,25]. Measurements have been performed independently by the Paris group [45–48] and the Boulder group [49,50,9]. The results of the Cs PNC experiments are summarized in Table 3. The Paris result in the %rst row is the average [47] of their (revised) results for the measurements of PNC in the transitions 6SF=4 −7SF=4 [45] and 6SF=3 −7SF=4 [46]. (The nuclear angular momentum of 133 Cs I = 7=2 and the electron angular momentum J = 1=2, so the total angular momentum of the atom is F = 3; 4). The Paris group have very recently performed a new measurement of PNC in Cs (last row) using a novel approach, chiral optical gain [48]. Each of the Boulder results [49,50,9] cited in the table is an average of PNC in the hyper%ne transitions 6SF=4 − 7SF=3 and 6SF=3 − 7SF=4 . The accuracy of the latest result is 0.35%, several times more precise than the best measurements of parity violation in other atoms.

74


The PNC nuclear spin-independent component, arising from the nuclear weak charge, makes the same contribution to all hyper%ne transitions. So averaging the PNC amplitudes over the hyper%ne transitions gives the contribution from the nuclear weak charge. PNC in atoms dependent on the nuclear spin was detected for the %rst (and only) time in Ref. [9] where it appeared as a di2erence in the PNC amplitude in di2erent hyper%ne transitions. The dominant mechanism for nuclear spin-dependent e2ects in atoms, the nuclear anapole moment, is the subject of Section 7. 3.3.2. Calculations Numerous calculations of the Cs 6S − 7S EPNC These calculations are summarized in Table 4. The performed more than 10 years ago represented a theory and parity violation in atoms. At the time,

amplitude have been performed over the years. many-body calculations [62,19], accurate to 1%, signi%cant step forward for atomic many-body these calculations were unmatched by the PNC

Table 4 Summary of calculations of the PNC E1 amplitude for the cesium 6S − 7S transition; units are −10−11 ieaB (−QW =N ) Authors

Year

Bouchiat, Bouchiat Loving, Sandars Neu2er, Commins Kuchiev, Sheinerman, Yahontov Das Bouchiat, Piketty, Pignon Dzuba, Flambaum, Silvestrov, Sushkov SchZafer, MZuller, Greiner, Johnson Ma[ rtensson-Pendrill Plummer, Grant SchZafer, MZuller, Greiner Johnson, Guo, Idrees, Sapirstein Johnson, Guo, Idrees, Sapirstein Bouchiat, Piketty Dzuba, Flambaum, Silvestrov, Sushkov Johnson, Blundell, Liu, Sapirstein Parpia, Perger, Das Dzuba, Flambaum, Sushkov Hartley, Sandars Hartley, Lindroth, Ma[ rtensson-Pendrill Blundell, Johnson, Sapirstein Safronova, Johnson Kozlov, Porsev, Tupitsyn Dzuba, Flambaum, Ginges

1974, 1975 1977 1981 1981 1983 1984, 1984 1985 1985 1985 1985, 1985, 1986 1987 1988 1988 1989 1990 1990 1990, 2000 2001 2002

a

1975

1985

1986 1986

1992

Ref.

Value

[17,25]a [65]a [66]a [67]a [68]b [69]a [70,71]b [72]b [73]b [74]b [75]b [76]c [76]b [77]a [59]b [78]b [79]c [62]b [80]c [81]b [19]b [82]b [64]bd [10]bd

1.33 1.15 1.00 0.75 1.06 0.97(10) 0.88(3) 0.74 0.886 0.64 0.92 0.754, 0.876, 0.856 0.890 0.935(20)(30) 0.90(2) 0.95(5) 0.879 0.908(9) 0.904(18) 0.933(37) 0.905(9) 0.909(11) 0.901(9) 0.904(5)

Semi-empirical calculations. Ab initio many-body calculations. c Combined many-body and semi-empirical calculations. d The di2erence between the values of [64,10] and previous ones is due to the inclusion of the Breit interaction in [64] and the Breit and strong %eld vacuum polarization in [10]. b


75

measurements which were accurate to 2%. The method of calculation used in Ref. [62] is the subject of Sections 4 and 5. The method used in Ref. [19] is based on the popular coupled-cluster method, and we refer the interested reader to this work for details. In the last 10 years a series of new measurements have been performed for quantities used to test the accuracy of the atomic calculations [62,19], such as electric dipole transition amplitudes (see [83]). The new measurements are in agreement with the calculations, resolving a previous discrepancy between theory and experiment. This inspired Bennett and Wieman [83] to claim that the atomic theory is accurate to 0.4% rather than 1% claimed by theorists. Since then, contributions to the PNC amplitude have been found (a correction due to inclusion of the Breit interaction and more recently the strong-%eld radiative corrections) that enter above the 0.4% level but below 1% (see Section 5). A re-calculation of the work [62], with some further improvements, was performed recently, with a full analysis of the accuracy of the PNC amplitude. This work, Ref. [10], represents the most accurate (0.5%) calculation to date. It is described in detail in Section 5. The result of [10] di2ers from [62,19] by only ∼ 0:1% if Breit, vacuum polarization, and neutron distribution corrections are excluded. One may interpret this as grounds for asserting that the many-body calculations [62,19,64,10] have an accuracy of 0.5% in agreement with the conclusion of [83]. 4. Method for high-precision atomic structure calculations in heavy alkali-metals In this section we describe methods that can be used to obtain high accuracy in calculations involving many-electron atoms with a single valence electron. These are the methods that have been used to obtain the most precise calculation of parity nonconservation in Cs. They were originally developed in works [7,84,85,59] and applied to the calculation of PNC in Cs in Ref. [62]. In [62] it was claimed that the atomic theory is accurate to 1%. A complete re-calculation of PNC in Cs using this method, with a new analysis of the accuracy, indicates that the error is as small as 0.5% [10]. (We refer the reader to Section 5, where this question of accuracy is discussed in general; please also see Section 5 for an in-depth discussion of PNC in Cs.) In this section the method is applied to energies, electric dipole transition amplitudes, and hyper%ne structure. A comparison of the calculated and experimental values gives an indication of the quality of the many-body wave functions. Note that the above quantities are sensitive to the wave functions at di2erent distances from the nucleus. Hyper%ne structure, energies, and electric dipole transition amplitudes are dominated by the contribution of the wave functions at small, intermediate, and large distances from the nucleus. We concentrate on calculations for Cs relevant to the 6S − 7S PNC E1 amplitude (see Eq. (56) and Section 5.1.5). A brief overview of the method is presented in Section 4.1. For those not interested in the technical details of the atomic structure calculations, Sections 4.2–4.8 may be omitted without loss of continuity. 4.1. Overview The calculations begin in the relativistic Hartree–Fock (RHF) approximation. The N − 1 selfconsistent RHF orbitals of the core are found (N is the total number of electrons in the atom), and

76


the external electron orbitals are found in the potential of the core electrons (the Vˆ N −1 potential). We get RHF wave functions, energies, and Green’s functions in this way. Correlation corrections to the external electron orbitals are included in second (lowest) order in the residual interaction (Vêxact − Vˆ N −1 ), where Vêxact is the exact Coulomb interaction between the atomic electrons. The correlations are included into the external electron orbitals by adding the correlation potential (the self-energy operator) to the RHF potential when solving for the external electron. Using the Feynman diagram technique, important higher-order diagrams are included into the self-energy in all orders: screening of the electron–electron interaction and the hole–particle interaction. The self-energy is then iterated using the correlation potential method. Interactions of the atomic electrons with external %elds are calculated using the time-dependent Hartree–Fock (TDHF) method; this method is equivalent to the random-phase approximation (RPA) with exchange. Using this approach we can take into account the polarization of the atomic core by external %elds to all orders. Then the major correlation corrections are included as corrections to electron orbitals (Brueckner orbitals). Small correlation corrections (structural radiation, normalization) are taken into account using many-body perturbation theory. 4.2. Zeroth-order approximation: relativistic Hartree–Fock method The full Hamiltonian we wish to solve is the many-electron Dirac equation 7 Hˆ =

N

[i · pi + ( − 1)m − Ze2 =ri ] +

i=1

i¡j

e2 : |ri − rj |

(15)

Here m and p are the electron mass and momentum, and are Dirac matrices, Ze is the nuclear charge and N is the number of electrons in the atom (N = 55 for cesium). This equation cannot be solved exactly, so some approximation scheme must be used. This is done by excluding the complicated Coulomb term and adding instead some averaged potential in which the electrons move. The Coulomb term, minus the averaged potential, can be added back perturbatively. It is well known that choosing the electrons to move in the self-consistent Hartree–Fock potential Vˆ N −1 , in the zeroth-order approximation, simpli%es the calculations of higher-order terms (we will come to this in the next section). The single-particle relativistic Hartree–Fock (RHF) Hamiltonian is hˆ0 = · p + ( − 1)m − Ze2 =r + Vˆ N −1 ; Hˆ 0 =

i

hˆ(i) 0 , where the Hartree–Fock potential

Vˆ N −1 = Vˆdir + Vêxch ;

7

In Section 5.1.2 we discuss the inclusion of the Breit interaction into the Hamiltonian.

(16)


(a)

77

(b)

Fig. 1. Hartree–Fock (a) direct and (b) exchange diagrams for energies, %rst-order in the Coulomb interaction. The solid and dashed lines are the electron and Coulomb lines, respectively.

is the sum of the direct and nonlocal exchange potentials created by the (N − 1) core electrons n, N −1 † n (r1 ) n (r1 ) 3 2 ˆ d r1 (r) (17) Vdir (r) = e |r − r1 | n=1 Vêxch (r) = −e2

N −1 n=1

†

n (r1 )

(r1 ) 3 d r1 |r − r1 |

n (r)

:

(18)

The SchrZodinger equation hˆ0

i

= ji

i

;

(19)

where i , ji are single-particle wave functions and energies, is solved self-consistently for the N − 1 core electrons. The Hartree–Fock potential is then kept “frozen” and the RHF equation (16), (19) is solved for the states of the external electron. The Hamiltonian hˆ0 thus generates a complete orthogonal set of single-particle orbitals for the core and valence electrons [86]. Hartree–Fock diagrams for energies are presented in Fig. 1. The single-particle electron orbitals have the form fnjl (r)3jlm 1 ; (20) njlm (r) = r −i#( · n)gnjl (r)3jlm where # is the %ne structure constant, = 2s is the electron spin, n = r=r, fnjl (r) and gnjl (r) are radial functions, and 3jlm is a spherical spinor (see, e.g., [15]). Because we are performing calculations for heavy atoms, and we are interested in interactions that take place in the vicinity of the nucleus (the weak and hyper%ne interactions), the %nite size of the nucleus must be taken into account. We use the standard formula for the charge distribution Z!(r) in the nucleus !0 !(r) = ; (21) 1 + exp[(r − c)=a] where !0 is the normalization constant found from the condition !(r) d 3 r = 1, t = a(4 ln 3) is the skin-thickness, and c is the half-density radius. For 133 Cs we take t = 2:5 fm and c = 5:6710 fm (r 2 1=2 = 4:804 fm) [87]. Energy levels of cesium states relevant to the 6S − 7S E1 PNC transition are presented in Table 5. It is seen that the RHF energies agree with experiment to 10%.

78


Table 5 Removal energies for Cs in units cm−1 State

RHF

5ˆ (2)

5ˆ

Experimenta

6S 7S 6P1=2 7P1=2

27954 12112 18790 9223

32415 13070 20539 9731

31492 12893 20280 9663

31407 12871 20228 9641

a

Taken from [88].

In order to obtain more realistic wave functions, we need to take into account the e2ect of correlations between the external electron and the core. We describe the techniques used to calculate these correlations in the following sections. 4.3. Correlation corrections and many-body perturbation theory The subject of this section is the inclusion of electron–electron correlations into the single-particle electron orbitals using many-body perturbation theory. We will see that high accuracy can be reached in the calculations by using the Feynman diagram technique as a means of including dominating classes of diagrams in all orders. The correlation corrections can be most accurately calculated in the case of alkali-metal atoms (for example, cesium). This is because the external electron has very little overlap with the electrons of the tightly bound core, enabling the use of perturbation theory in the calculation of the residual interaction of the external electron with the core. The exact Hamiltonian of an atom [Eq. (15)] can be divided into two parts: the %rst part is the sum of the single-particle Hamiltonians, and the second part represents the residual Coulomb interaction Hˆ =

N i=1

Uˆ =

i¡j

hˆ0 (ri ) + Uˆ ;

(22) N

e2 − Vˆ N −1 (ri ) : |ri − rj | i=1

(23)

Correlation corrections to the single-particle orbitals are included perturbatively in the residual interaction Uˆ . By calculating the wave functions in the Hartree–Fock potential Vˆ N −1 for the zeroth-order approximation, the perturbation corrections are simpli%ed. The %rst-order corrections (in the residual Coulomb interaction Uˆ ) to the ionization energy vanish (the two terms in Eq. (23) cancel each other), since the correlation corrections %rst-order in the Coulomb interaction are nothing but the %rstorder Hartree–Fock ones (Fig. 1). The lowest-order corrections therefore correspond to those arising in second-order perturbation theory, Uˆ (2) . These corrections are determined by the four Goldstone diagrams in Fig. 2 [86]. They can be calculated by direct summation over intermediate states [86] or by the “correlation potential” method [71]. This latter method gives higher accuracy and, along with the Feynman diagram technique to be discussed in the following section, enables the inclusion

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 α α

β

α

α

79 α

α

β n

n n

γ

β γ

β

α

m

n

m α

(a)

(b)

(c)

(d)

Fig. 2. Second-order correlation corrections to energy of the valence electron. Dashed line is the Coulomb interaction; loop is polarization of the atomic core, corresponding to virtual creation of an excited electron and a hole. The state of the external electron is denoted by #; n, m are core states; and , are states outside the core.

of higher-order e2ects: electron–electron screening, the hole–particle interaction, and the nonlinear contributions of the correlation potential. The correlation potential method corresponds to adding a nonlocal correlation potential 5ˆ to the potential Vˆ N −1 in the RHF equation (16) and then solving for the states of the external electron. The correlation potential is de%ned such that its average value coincides with the correlation correction to the energy, ˆ # %j# = # |5| ˆ 5 # = 5(r1 ; r2 ; j# ) # (r1 ) d 3 r1 :

(24) (25)

It is easy to write the correlation potential explicitly. For example, a part of the operator 5(r1 ; r2 ; j# ) corresponding to Fig. 2(a) is given by † † −1 −1 † n (r4 )r24 (r4 ) (r2 ) (r3 ) (r1 )r13 n (r3 ) 2(a) 4 3 3 d r3 d r 4 : (26) 5 (r1 ; r2 ; j# ) = e j# + j n − j − j n;;

Note that 5ˆ is a single-electron and energy-dependent operator. By solving the RHF equation for the ˆ we obtain “Brueckner” orbitals and energies. 8 states of the external electron in the %eld Vˆ N −1 + 5, The largest correlation corrections are included in the Brueckner orbitals. See Table 5 for Brueckner energies of the lower states of cesium calculated in the secondorder correlation potential. 5ˆ (2) ≡ Uˆ (2) denotes the “pure” second-order correlation potential (without screening, etc.). It is seen that the inclusion of these corrections improves the energies signi%cantly, from the level of 10% deviation from experiment for the RHF approximation to the level of 1%. 4.4. All-orders summation of dominating diagrams We saw in the previous section that when we take into account second-order correlation corrections, the accuracy for energies is improved signi%cantly beyond that for energies calculated in the 8

Note that there is a slight distinction in the de%nition of these Brueckner orbitals and those de%ned in, e.g., [89].

80


RHF approximation. However, the corrections are overestimated. This overestimation is largely due to the neglect of screening in the electron–electron interaction. In this section we describe the calculations of three series of higher-order diagrams: screening of the electron–electron interaction and the hole–particle interaction, which are inserted into the ˆ and iterations of 5. ˆ With the inclusion of these diagrams the accuracy for correlation potential 5; energies is improved to the level of 0.1% (see Table 5). The screening of the electron–electron interaction is a collective phenomenon and is similar to Debye screening in a plasma; the corresponding chain of diagrams is enhanced by a factor approximately equal to the number of electrons in the external closed subshell (the 5p electrons in cesium) [84]. The importance of this e2ect can be understood by looking at a not dissimilar example in which screening e2ects are important, for instance, the screening of an external electric %eld in an atom. According to the Schi2 theorem [90], a homogeneous electric %eld is screened by atomic electrons (and at the nucleus it is zero). (See [91] where a calculation of an external electric %eld inside the atom has been performed.) The hole–particle interaction is enhanced by the large zero-multipolarity diagonal matrix elements of the Coulomb interaction [85]. The importance of this e2ect can be seen by noticing that the existence of the discrete spectrum excitations in noble gas atoms is due only to this interaction (see, e.g., [92]). The nonlinear e2ects of the correlation potential are calculated by iterating the self-energy operator. These e2ects are enhanced by the small denominator, which is the energy for the excitation of an external electron (in comparison with the excitation energy of a core electron) [85]. All other diagrams of perturbation theory are proportional to powers of the small parameter Qnd =\jint ∼ 10−2 , where Qnd is a nondiagonal Coulomb integral and \jint is a large energy denominator corresponding to the excitation of an electron from the core (due to an interaction in an internal electron line of the perturbation diagrams) [85]. 4.4.1. Screening of the electron–electron interaction The main correction to the correlation potential comes from the inclusion of the screening of the Coulomb %eld by the core electrons. Some examples of the lowest-order screening corrections are presented in Fig. 3. When screening diagrams in the lowest (third) order of perturbation theory are taken into account, a correction is obtained of opposite sign and almost the same absolute value as the corresponding second-order diagram [84]. Due to these strong cancellations there is a need to sum the whole chain of screening diagrams. However, this task causes diSculties in standard perturbation theory as the screening diagrams in the correlation correction cannot be represented by a simple geometric progression due to the overlap of the energy denominators of di2erent loops

(a)

(b)

(c)

Fig. 3. Lowest order screening corrections to the diagram in Fig. 2(a).


ε+ω

ε

81

ε

ω1 (a)

ω2

(b)

Fig. 4. Second-order correlation corrections to energy in the Feynman diagram technique.

ε+ω r

r

1

2

ε Fig. 5. Polarization operator.

[such an overlap indicates a large number of excited electrons in the intermediate states; see, e.g., Figs. 3(b) and (c)]. This summation problem is solved by using the Feynman diagram technique. The correlation corrections to the energy in the Feynman diagram technique are presented in Fig. 4. The Feynman Green’s function is of the form G(r1 ; r2 ; j) =

†

n (r1 ) n (r2 )

j − jn − i%

n

+

†

(r1 ) (r2 )

j − j + i%

;

%→0 ;

(27)

where n is an occupied core electron state, is a state outside the core. While the simplest way of calculating the Green’s function is by direct summation over the discrete and continuous spectrum, there is another method in which higher numerical accuracy can be achieved. As is known, the radial Green’s function G0 for the equation without the nonlocal exchange interaction Vexch can be expressed in terms of the solutions 70 and 7∞ of the SchrZodinger or Dirac equation that are regular at r → 0 and r → ∞, respectively: G0 (r1 ; r2 ) ˙ 70 (r¡ )7∞ (r¿ ), r¡ = min(r1 ; r2 ), r¿ = max(r1 ; r2 ). ˆ The exchange interaction is taken into account by solving the matrix equation Gˆ = Gˆ 0 + Gˆ 0 Vêxch G. The polarization operator (Fig. 5) is given by ∞ dj ˆ 1 ; r2 ; !) = G(r1 ; r2 ; ! + j)G(r2 ; r1 ; j) : 9(r (28) −∞ 2; This integration is carried out analytically, giving † ˆ 1 ; r2 ; !) = i 9(r n (r1 )[G(r1 ; r2 ; jn + !) + G(r1 ; r2 ; jn − !)] n (r2 ) :

(29)

n

Using formulae (27) and (29), it is easy to perform analytical integration over ! in the calculation of the diagrams in Fig. 4. After integration, diagram 4(a) transforms to 2(a) and (c) and diagram 4(b) transforms to 2(b) and (d).

82

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 +

+

+ ....

Fig. 6. Screening diagram chain for e2ective polarization operator.

Fig. 7. Insertion of the hole–particle interaction into the second order correlation correction.

To include electron–electron screening to all orders in the Coulomb interaction in the Feynman diagram technique, the polarization operator is chained (Fig. 6) before integration over ! is carried out. The screened polarization operator is −1 ˆ ˆ ;(!) ˆ = 9(!)[1 + iQˆ 9(!)] :

(30)

The integration over ! is performed numerically. The integration contour is rotated 90◦ from the real axis to the complex ! plane parallel to the imaginary axis—this aids the numerical convergence by keeping the poles far from the integration contour. The all-order electron–electron screening reduces the second-order correlation corrections to the energies of S and P states of 133 Cs by 40%. 4.4.2. The hole–particle interaction In Fig. 7 we include the hole–particle interaction into the polarization loop of the second-order correlation correction. The hole–particle interaction accounts for the alteration of the core potential due to the excitation of the electron from the core to the virtual intermediate state. This electron now moves in the potential created by the N − 2 electrons, and no longer contributes to the Hartree–Fock potential. Denoting Vˆ 0 as the zero multipolarity direct potential of the outgoing electron, the potential which describes the excited and core states simultaneously is [85] ˆ Vˆ 0 (1 − P) ˆ ; Vˆ = Vˆ N −1 − (1 − P)

(31)

where Pˆ is the projection operator on the core orbitals, Pˆ =

N −1

|n n| :

(32)

n=1

The projection operator Pˆ is introduced into the potential to make the excited states orthogonal to the core states. It is easily seen that for the occupied orbitals Vˆ = Vˆ N −1 , while for the excited orbitals Vˆ = Vˆ N −1 − Vˆ 0 . Strictly one should also make subtractions for higher multipolarities and for the exchange interaction as well, however these contributions are relatively small and are therefore safe to ignore [85]. To obtain high accuracy, the hole–particle interaction in the polarization operator needs to be taken into account in all orders (Fig. 8). This is achieved by calculating the Green’s function in the potential (31) and then using it in the expression for the polarization operator (29). The screened


=

+

+

83

+ ....

Fig. 8. Hole–particle interaction in the polarization operator.

=

+

=

+

+

+ ....

Fig. 9. Renormalization of the Coulomb line due to the hole–particle interaction and screening.

Σ

=

+

Fig. 10. The electron self-energy operator with screening and hole–particle interaction included.

Σ

+

Σ

Σ

+

Σ

Σ

Σ

+

...

Fig. 11. Chaining of the self-energy operator.

polarization operator, with hole–particle interaction included, is found by using the Green’s function in Eq. (30). The Coulomb interaction, with screening and the hole–particle interaction included in all orders, is calculated from the matrix equation [85] Q˜ = Qˆ − iQˆ ;ˆQˆ :

(33)

This is depicted diagrammatically in Fig. 9. The series of diagrams representing the screening and hole–particle interaction can now be included into the correlation potential. This is done by introducing the renormalized Coulomb interaction (Fig. 9) and the polarization operator (Fig. 8) into the second-order diagrams according to Fig. 10. The screened second-order correlation corrections to the energies of S and P states of cesium are increased by 30% when the hole–particle interaction is taken into account in all orders. 4.4.3. Chaining of the self-energy The accuracy of the calculations can be further improved by taking into account the nonlinear contributions of the correlation potential 5ˆ (Fig. 11). The chaining of the correlation potential

84


Fig. 12. Third-order diagrams of the interaction of a hole and particle from the loop with an external electron.

Fig. 13. Correlation corrections to occupied orbitals of closed shells.

(Fig. 10) to all orders is calculated by adding 5ˆ to the Hartree–Fock potential, Vˆ N −1 , and solving the equation (hˆ0 + 5ˆ − j) = 0

(34)

iteratively for the states of the external electron. The inclusion of 5ˆ into the SchrZodinger equation is what we call the “correlation potential method” and the resulting orbitals and energies “Brueckner” orbitals and “Brueckner” energies (see Section 4.3). Iterations of the correlation potential 5ˆ increase the contributions of 5ˆ (with screening and hole–particle interaction) to the energies of S and P states of cesium by about 10%. The %nal results for the energies are listed in Table 5. The inclusion of the three series of higher-order diagrams improves the accuracy of the calculations of the energies to the level of 0.1%. 4.5. Other low-order correlation diagrams Third-order diagrams for the interaction of a hole and particle in the polarization loop with an external electron are depicted in Fig. 12. These are not taken into account in the method described above. However, these diagrams are of opposite sign and cancel each other almost exactly [85]: the small and almost constant potential of a distant external electron practically does not in?uence the wave functions of the core and excited electrons in the loop; it shifts the energies of the core and excited electrons by the same amount. This cancellation was proved in the work [93] by direct calculation. Also, correlation corrections to the external electron energy arising from the inclusion of the self-energy into orbitals belonging to closed electron shells, depicted in Fig. 13, are small and can be safely omitted [71]. 4.6. Empirical ˜ = > + %>. This gives rise to E1 transitions between states of the same nominal parity. The parity violating 6S − 7S E1 transition amplitude in Cs is Hˆ E1 |6S = %(7S)|Hˆ E1 |6S + 7S|Hˆ E1 |%(6S) : EPNC = 7S|

(54)

Calculations of PNC E1 amplitudes can be performed using the following approaches: from a mixed-states approach, in which there is a small opposite-parity admixture in each state [Eq. (54)]; or from a sum-over-states approach, in which the amplitude [Eq. (54)] is broken down into contributions arising from opposite-parity admixtures and a direct summation over the intermediate states is performed [Eq. (55)]. In the sum-over-states approach, the Cs 6S − 7S PNC E1 transition amplitude is written in terms of a sum over intermediate, many-particle states NP1=2

7S|Hˆ E1 |NP1=2 NP1=2 |Hˆ W |6S 7S|Hˆ W |NP1=2 NP1=2 |Hˆ E1 |6S : (55) EPNC = + E6S − ENP1=2 E7S − ENP1=2 N There are three dominating contributions to this sum: EPNC =

7S|Hˆ E1 |6P1=2 6P1=2 |Hˆ W |6S 7S|Hˆ W |6P1=2 6P1=2 |Hˆ E1 |6S + E6S − E6P1=2 E7S − E6P1=2 +

7S|Hˆ E1 |7P1=2 7P1=2 |Hˆ W |6S + ··· E6S − E7P1=2

= 1:908 − 1:493 − 1:352 + · · · = −0:937 + · · · ;

(56)

the units are 10−11 ieaB (−QW =N ). The numbers are from the work [19] where the sum-over-states method was used; here we just demonstrate that these terms dominate. An advantage of the sum-overstates approach is that experimental values for the energies and E1 transition amplitudes can be explicitly included into the sum. This was the procedure for some of the early calculations of PNC in Cs (see, e.g., [77]). 10

It is seen from Eqs. (3), (4), by inserting the coeScients C1N , that the density !(r) is essentially the (poorly understood) neutron density in the nucleus. In the calculations, !(r) will be taken equal to the charge density, Eq. (21), and then in Section 5.1.3 we will consider the e2ect on the PNC E1 amplitude as a result of correcting for !(r).

92


If one neglects con%guration mixing, the sum can be represented in terms of single-particle states; in this case, the sum also runs over core states (corresponding to many-particle states with a single core excitation),

7s|hÊ1 |np1=2 np1=2 |hˆW |6s 7s|hˆW |np1=2 np1=2 |hÊ1 |6s EPNC = : (57) + j6s − jnp1=2 j7s − jnp1=2 n In Refs. [62,19,64,10] PNC calculations were performed in the mixed-states approach, and in Ref. [19] a calculation was carried out in the sum-over-states approach also. Here we refer to the most precise calculations, 6 1% accuracy. 5.1.1. Mixed-states calculation In the TDHF method (Section 4.8.1, Eq. (37)), a single-electron wave function in external weak and E1 %elds is =

0

+ % + X e−i!t + Y ei!t + %X e−i!t + %Y ei!t ;

(58)

where 0 is the unperturbed state, % is the correction due to the weak interaction acting alone, X and Y are corrections due to the photon %eld acting alone, and %X and %Y are corrections due to both %elds acting simultaneously. These corrections are found by solving self-consistently the system of the TDHF equations for the core states (hˆ0 − j)% = −(hˆW + %VˆW )

;

(59)

(hˆ0 − j − !)X = −(hÊ1 + %VÊ1 )

;

(60)

† (hˆ0 − j + !)Y = −(hˆ†E1 + %VÊ1 )

;

(61)

(hˆ0 − j − !)%X = −%VÊ1 % − %VˆW X − %VÊ1W

;

(62)

† † (hˆ0 − j + !)%Y = −%VÊ1 % − %VˆW Y − %VÊ1W

;

(63)

where %VˆW and %VÊ1 are corrections to the core potential due to the weak and E1 interactions, respectively, and %VÊ1W is the correction to the core potential due to the simultaneous action of the weak %eld and the electric %eld of the photon. The TDHF contribution to EPNC between the states 6S and 7S is given by TDHF = EPNC

ˆ

7s |hE1

+ %VÊ1 |%

6s

+

ˆ + %VˆW |X6s +

7s |hW

ˆ

7s |%VE1W | 6s

:

(64)

The corrections % 6s and X6s are found by solving Eqs. (59), (60) in the %eld of the frozen core (of course, amplitude (64) can instead be expressed in terms of corrections to 7s ). Now we need to include the correlation corrections to the PNC E1 amplitude. In the previous sections (Sections 4.3, 4.4, 4.8.4) we have discussed two types of corrections: the dominant Brueckner-type corrections, represented by diagrams in which the external %eld appears in the external electron line (see Fig. 17); and structural radiation, in which the external %eld acts on an internal electron line. In the case of PNC E1 amplitudes, in order to distinguish between structural radiation diagrams with di2erent %elds, we refer to diagrams with the weak interaction attached to


Σ

Σ

93

Σ

Σ

Fig. 17. Brueckner-type correlation corrections to the PNC E1 transition amplitude in %rst order in the weak interaction; the crosses denote the weak interaction and the dashed lines denote the electromagnetic interaction.

δΣ

δΣ

Σ

Σ

δΣ

δΣ

(a)

(b)

(c)

Fig. 18. External %eld inside the correlation potential. In diagrams (a) the weak interaction is inside the correlation potential (%5 denotes the change in 5 due to the weak interaction); this is known as the weak correlation potential. Diagrams (b,c) represent the structural radiation (photon %eld inside the correlation potential). In diagram (b) the weak interaction occurs in the external lines; in diagram (c) the weak interaction is included in the electromagnetic vertex.

the internal electron line as “weak correlation potential” diagrams. Structural radiation and the weak correlation potential diagrams are presented in Fig. 18. We will consider %rst the dominating Brueckner-type corrections to the E1 PNC amplitude. ˆ Remember that the correlation potential is energy-dependent, 5ˆ = 5(j). This means that the 5ˆ operators for the 6s and 7s states are di2erent. We should consider the proper energy-dependence at least in %rst-order in 5ˆ (higher-order corrections are small and the proper energy-dependence is not important for them). The %rst-order in 5ˆ correction to EPNC is presented diagrammatically in Fig. 17. We can write this as

ˆ

7s |5s (j7s )|%X6s

+ %

ˆ

7s |5p (j7s )|X6s

+ %Y7s |5ˆ s (j6s )|

6s

+ Y7s |5ˆ p (j6s )|%

6s

:

(65)

The nonlinear in 5ˆ contribution to the Brueckner-type correction is found using the correlation potential method (Section 4.3): the all-orders in 5ˆ contribution is calculated and from this the %rst-order contribution, found in the same method, is subtracted. The all-orders term is calculated

94


using external electron orbitals, and corrections to these orbitals induced by the weak interaction ˆ The PNC E1 amplitude is then calculated, and the photon %eld, found in the potential Vˆ N −1 + 5. using these new orbitals, in the same way as in the usual time-dependent Hartree–Fock method. The all-orders contribution to EPNC is ˆ − E (Vˆ N −1 ) : E all-orders = E (Vˆ N −1 + 5) (66) PNC

PNC

PNC

The %rst-order in 5ˆ contribution is found by placing a small coeScient a before the correlation ˆ When a1, the linear in 5ˆ contribution to EPNC dominates. Its extrapolation to potential, 5ˆ → a5. a = 1 gives the %rst-order in 5ˆ contribution. So the nonlinear in 5ˆ contribution to EPNC is [116] non-lin ˆ − EPNC (Vˆ N −1 )] − 1 [EPNC (Vˆ N −1 + a5) ˆ − EPNC (Vˆ N −1 )] : = [EPNC (Vˆ N −1 + 5) (67) EPNC a To complete the calculation of corrections second-order in the residual Coulomb interaction the weak correlation potential, structural radiation, and normalization contributions to the PNC amplitude must be included. The weak correlation potential is calculated by direct summation over intermediate states. See Section 4.8.4 for the approximate form for structural radiation in length form and for the form for the normalization of the many-body states. Due to parity violation there is an opposite-parity correction to the orbitals # ≡ # and ≡ , #˜ = # + %# and ˜ = + %, and to the correlation ˆ 5˜ = 5ˆ + %5. ˆ potential 5, Structural radiation is then given by ˜ ˜ ˜ 95 + 95 D|# ˜ str = − 1 |D ˜ : (68) M 2 9j 9j There are two contributions to structural radiation for the PNC E1 amplitude: one in which the electromagnetic vertex is parity conserving, the weak interaction included in the external lines: ˆ ˆ ˜ 95 + 95 D|# ˜ Fig: 18(b) = − 1 |D M ˜ (69) 2 9j 9j (see diagram (b) Fig. 18); and the other in which the weak interaction is included in the electromagnetic vertex (we call this structural radiation and not weak correlation potential): ˜ ˜ ˜ Fig: 18(c) = − 1 |D 95 + 95 D|# (70) M 2 9j 9j (see diagram (c) of Fig. 18). Note that in each case the amplitude %rst-order in the weak interaction is considered. The normalization contribution is ˆ ˆ 9 5 9 5 1 ˜ # ˜ norm = |D| ˜ | | + #| |# : (71) M 2 9j 9j The results of the calculation [10] for the 6S −7S PNC amplitude are presented in Table 8. Taking into account all corrections discussed in this section, the following value is obtained for the 6S − 7S PNC amplitude in cesium EPNC = −0:9078 × 10−11 ieaB (−QW =N ) :

(72)


95

Table 8 Contributions to the 6S − 7S EPNC amplitude for Cs in units −10−11 ieaB (−QW =N ) (5ˆ corresponds to the (un%tted) “dressed” self-energy operator) TDHF Brueckner-type correlations 7s |5ˆ s (j7s )|%X6s % 7s |5ˆ p (j7s )|X6s %Y7s |5ˆ s (j6s )| 6s Y7s |5ˆ p (j6s )|% 6s Nonlinear in 5ˆ correction Weak correlation potential Structural radiation Normalization Subtotal Breit Neutron distribution correction QED radiative corrections Vacuum polarization (Uehling) Self-energy and vertex Total

0.8898 0.0773 0.1799 −0.0810 −0.1369 −0.0214 0.0038 0.0029 −0.0066 0.9078 −0.0055 −0.0018 0.0036 −0.0072 0.8969

This corresponds to “Subtotal” of Table 8. This is in agreement with the 1989 result [62]. Notice the stability of the PNC amplitude. The time-dependent Hartree–Fock value gives a contribution to the total amplitude of about 98%. The point is that there is a strong cancellation of the correlation corrections. The mixed-states approach has also been performed in [19,64] to determine the PNC amplitude in cesium. However, in these works the screening of the electron–electron interaction was included in a simpli%ed way. In [19] empirical screening factors were placed before the second-order correlation corrections 5ˆ (2) to %t the experimental values of energies. Kozlov et al. [64] introduced screening factors based on average screening factors calculated for the Coulomb integrals between valence electron states. The results obtained by these groups (without the Breit interaction, i.e., corresponding to the Subtotal of Table 8) are 0.904 [19] and 0.905 [64]. 11 As a check, a pure second-order (i.e., using 5ˆ (2) ) calculation with energy-%tting was also performed in [10] (in the same way as [19]), and the result 0.904 was reproduced. Contributions of the Breit interaction, the neutron distribution, and radiative corrections to EPNC are considered in the following sections.

11

The numbers di2er from those presented in Table 4 due to the Breit interaction. In [19] a value for Breit of −0:2% of the PNC amplitude was included (this value was underestimated), while in [64] the magnetic (Gaunt) part of the Breit interaction was included and calculated to be −0:4%. See Section 5.1.2.

96


5.1.2. Inclusion of the Breit interaction The Breit interaction is a two-particle operator 2

e Hˆ Breit = − 2

i · j + (i · nij )(j · nij ) i¡j

|ri − rj |

;

(73)

are Dirac matrices, nij = (ri − rj )=|ri − rj |. It gives magnetic (Gaunt) and retardation corrections to the Coulomb interaction. A few years ago it was thought that the correction to EPNC arising due to inclusion of the Breit interaction in the Hamiltonian (15) is small (safely smaller than 1%). In the work [62] the Breit interaction was neglected, and in [19] it was only partially calculated. (Remember that these works claimed an accuracy of 1%.) The huge improvement in the experimental precision of the cesium PNC measurement in 1997 [9] and the claim of Bennett and Wieman in 1999 [83] that the theoretical accuracy is 0.4% prompted theorists to revisit their calculations. Naturally this also involves a consideration of previously neglected contributions which, while at the 1% level could be neglected, are signi%cant at the 0.4% level. Derevianko [110] calculated the contribution of the Breit interaction to EPNC and found that it is larger than had been expected. Its contribution to EPNC is −0:6%. This result has been con%rmed by subsequent calculations [117,64,10]. 5.1.3. Neutron distribution The weak Hamiltonian Eq. (5) was used to obtain the result Eq. (72) with !(r) taken to be the charge density, parametrized according to Eq. (21). However, as we mentioned in a footnote at the beginning of Section 5.1, the weak interaction is sensitive to the distribution of neutrons in the nucleus. Here we look at the e2ect of correcting for the neutron distribution. For the neutron density the two-parameter Fermi model (21) is used. The result of Ref. [118] was used in [10] for the di2erence \rnp = 0:13(4) fm in the root-mean-square radii of the neutrons rn2 1=2 and protons rp2 1=2 . Three cases which correspond to the same value of rn2 were considered: (i) cn = cp , an ¿ ap ; (ii) cn ¿ cp , an ¿ ap ; and (iii) cn ¿ cp , an = ap (using the relation rn2 ≈ 3 2 c + 75 ;2 a2n ). It is found that EPNC shifts from −0:18% to −0:21% when moving from the extreme 5 n cn = cp to the extreme an = ap . Therefore, EPNC changes by about −0:2% due to consideration of the neutron distribution. This is in agreement with Derevianko’s estimate, −0:19(8)% [119]. 5.1.4. Strong-<eld QED corrections It was noted in Ref. [111] that corrections to the PNC amplitude due to vacuum polarization by the strong Coulomb %eld of the nucleus could be comparable in size to the Breit correction. This has been con%rmed by calculations, the strong-%eld radiative corrections associated with the Uehling potential (vacuum polarization) increase EPNC by 0:4% [113,112,10]. In Ref. [120] it was pointed out that the self-energy correction can give a larger contribution to 133 Cs PNC with opposite sign (∼ −0:65%). The self-energy and vertex corrections were %rst calculated in [114] and found to be −0:73(20)% for 133 Cs. The relation between the PNC correction and radiative corrections to %nite nuclear size energy shifts was used in this work. This result was con%rmed in direct analytical calculations using Z# expansion [121,115,122] and by all-orders in Z# numerical calculations of the PNC matrix element of the 2S1=2 − 2P1=2 transition in hydrogenic ions performed in [123].


97

Note that corrections occur at very small distances (r . 1=m) where the nuclear Coulomb %eld is not screened and the electron energy is negligible. Therefore, the relative radiative corrections to weak matrix elements in neutral atoms like Cs are approximately the same as for the 2S1=2 − 2P1=2 transition in hydrogenic ions. There is good agreement between the di2erent calculations for all values of Z; see [124] and the review [125]. For the strong-%eld self-energy and vertex contribution to PNC in 133 Cs we will quote the value −0:8%, which is the average value of [114,122] corresponding also to the value obtained in [123]. Above we discussed the radiative corrections to the weak matrix elements. However, the sum-overstates expression for the PNC amplitude contains also energy denominators and E1 electromagnetic amplitudes. It was shown in [120,10] that for Cs the corrections to the energies (−0:3%) and E1 matrix elements (+0:3%) cancel. The contributions of strong-%eld radiative corrections to EPNC of cesium are listed in Table 8. 5.1.5. Tests of accuracy There are two main methods used to estimate the accuracy of the PNC amplitude EPNC : (i) root-mean-square (rms) deviation of the calculated energy intervals, E1 amplitudes, and hyper%ne structure constants from the accurate experimental values; (ii) in?uence of %tting of energies and hyper%ne structure constants on the PNC amplitude. The PNC amplitude can be expressed as a sum over intermediate states. Notice that there are three dominating contributions to the 6S − 7S PNC amplitude in Cs; see Eq. (56). Each term in the sum is a product of E1 transition amplitudes, weak matrix elements, and energy denominators. Therefore, this amplitude is sensitive to the electron wave functions at all distances. (The weak matrix elements, energies, and E1 amplitudes are sensitive to the wave functions at small, intermediate, and large distances from the nucleus, respectively.) While mixed-states calculations of PNC amplitudes do not involve a direct summation over intermediate states, it is instructive to analyse the accuracy of the weak matrix elements, energy intervals, and E1 transition amplitudes which contribute to Eq. (56) calculated using the same method as that used to calculate EPNC . The accuracy of these quantities is determined by comparing the calculated values with experiment. Note that we cannot directly compare weak matrix elements with experiment. However, like the weak matrix elements, hyper%ne structure is determined by the electron wave functions in the vicinity of the nucleus, and this is known very accurately. In Section 4 we presented calculations of the energies, E1 transition amplitudes, and hyper%ne structure constants relevant to Cs 6S − 7S EPNC . The calculated removal energies are presented in Table 5. The Hartree–Fock values deviate from experiment by 10%. Including the second-order correlation corrections 5ˆ (2) reduces the error to ∼ 1%. When screening and the hole–particle interaction are included into 5ˆ (2) in all orders, the energies improve, 0:2 − 0:3%. The rms deviation between the calculated and experimental energy intervals E6S − E6P1=2 , E7S − E6P1=2 , and E6S − E7P1=2 is 0.3%. We mentioned at the end of Section 4.1 that the experimental values for energies can be %tted ˆ The stability of the amplitude EPNC exactly by placing a coeScient before the correlation potential 5. (as well as the E1 amplitudes and hfs constants) with %tting gives us an indication of the size of omitted contributions. Note that the accuracy for the energies is already very high and the remaining discrepancy with experiment is of the same order of magnitude as the Breit and radiative corrections.

98


Therefore, generally speaking, we should not expect that %tting of the energy will always improve the results for amplitudes and hyper%ne structure. In fact, as we will see below, some values do improve while others do not. The overall accuracy, however, remains at the same level. ˆ Below we present results for EPNC obtained in three di2erent approximations: with un%tted 5, (2) 12 ˆ ˆ and with 5 and 5 %tted with coeScients to reproduce experimental removal energies. First, we analyse the E1 transition amplitudes and hfs constants calculated in these approximations. The relevant E1 transition amplitudes (radial integrals) are presented in Table 6. These are calculated with the energy-%tted “bare” correlation potential 5ˆ (2) and the (un%tted and %tted) “dressed” ˆ Structural radiation and normalization contributions are also included. The rms devipotential 5. ations of the calculated E1 amplitudes from experiment are the following: without energy %tting, the rms deviation is 0.1%; %tting the energy gives a rms deviation of 0.2% for 5ˆ (2) and 0.3% for ˆ Note, these correspond to the deviations between the calculations and the central the complete 5. points of the measurements. The errors associated with the measurements are in fact comparable to this di2erence. So it is unclear if the theory is limited to this precision or is in fact much better. Regardless, the uncertainty in the theoretical accuracy remains the same. The hyper%ne structure constants calculated in di2erent approximations are presented in Table 7. Corrections due to the Breit interaction, structural radiation, and normalization are included. The rms deviation of the calculated hfs values from experiment using the un%tted 5ˆ is 0.5%. With %tting, the rms deviation in the pure second-order approximation is 0.3%; with higher orders it is 0.4%. The point is to estimate the accuracy of the s − p1=2 weak matrix elements. It seems reasonable then to use the square-root formula, hfs(s)hfs(p1=2 ). Notice that by using this approach the deviation is smaller. Without energy %tting, the rms deviation is 0.5%. With %tting, the rms deviation in the ˆ it is 0.3%. second-order calculation (5ˆ (2) ) is 0:2% and in the full calculation (5) From the above consideration it is seen that the rms deviation for the relevant parameters is 0:5% or better. Note that from this analysis the error for a sum-over-states calculation of EPNC would be larger than this, as the errors for the energies, hfs constants, and E1 amplitudes contribute to each of the three terms in Eq. (56). However, in the mixed-states approach, the errors do not add in this way. We now consider calculations of the PNC amplitude performed in [10] in di2erent approximations ˆ and with energy-%tted 5ˆ (2) and 5). ˆ The spread of the results can be used to estimate (with un%tted 5, the error. The results are listed in Table 9. It can be seen that the PNC amplitude is very stable. The PNC amplitude is much more stable than hyper%ne structure. This can be explained by the much smaller correlation corrections to EPNC (∼ 2% for EPNC and ∼ 30% for hfs; compare Table 8 with Table 9 Values for EPNC in di2erent approximations; units −10−11 ieaB (−QW =N )

EPNC

5ˆ (2) With %tting

5ˆ

5ˆ With %tting

0.901

0.904

0.903

Note that %tting 5ˆ (2) is an empirical method to estimate screening corrections (which were accurately calculated in ˆ Agreement between results with %tted 5ˆ (2) and ab initio 5ˆ shows that the %tting procedure is a reasonable way to 5). estimate omitted diagrams. 12


99

Table 7). One can say that this small value of the correlation correction is a result of cancellation of di2erent terms in (65) but each term is not small (see Table 8). However, this cancellation has a regular behaviour. The stability of EPNC may be compared to the stability of the usual electromagnetic amplitudes where the error is very small (even without %tting). 13 In [10] the %tting of hyper%ne structure was also considered, using di2erent coeScients before ˆ The %rst-order in 5ˆ correlation correction (65) changes by about 10%. It was found that each 5. the PNC amplitude changes by about 0.4%. It is also instructive to look at the spread of EPNC obtained in di2erent schemes. The result of the work [10] (the number we present here) is in excellent agreement with the earlier result [62]. The only other calculation of EPNC in Cs that is as complete as [62,10] is that of Blundell et al. [19]. Their result in the all-orders sum-over-states approach is 0.909 (without Breit) and is very close to the value of 0.908 (corresponding to “Subtotal” of Table 8). A note on the sum-over-states procedure. The authors of Ref. [19] include single, double, and selected triple excitations into their wave functions. Note, however, that even if wave functions of 6S, 7S, and intermediate NP states are calculated exactly (i.e., with all con%guration mixing included) there are still some missed contributions in this approach. Consider, e.g., the intermediate state 6P ≡ 5p6 6p. It contains an admixture of states 5p5 ns6d: 6P = 5p6 6p + #5p5 ns6d + · · · . This mixed state is included into the sum (55). However, the sum (55) must include all many-body states of opposite 5 ns6d = 5p5 ns6d − #5p6 6p + · · · should also be included into parity. This means that the state 5p] the sum. Such contributions to EPNC have never been estimated directly within the sum-over-states approach. However, they are included into the mixed-states calculations [62,19,64,10]. It is important to note that the omitted higher-order many-body corrections are di2erent in the sum-over-states [19] and mixed-states [62,10] calculations. This may be considered as an argument that the omitted many-body corrections in both calculations are small. Of course, here it is assumed that the omitted many-body corrections to both values (which, in principle, are completely di2erent) do not “conspire” to give exactly the same magnitude. A comparison of calculations of EPNC in second-order with %tting of the energies is also useful in determining the accuracy of the calculations of EPNC . (Remember that this value is in agreement with results of similar calculations performed in [19,64]; see Section 5.1.1.) One can see that replacing the all-order 5ˆ by its very rough second-order (with %tting) approximation changes EPNC by less than 0.4% only. On the other hand, if the higher orders are included accurately, the di2erence between the two very di2erent approaches is 0.1% only. The maximum deviation obtained in the above analysis is 0.5%. This is the error claimed in the EPNC calculation [10]. 5.2. The vector transition polarizability The determination of the nuclear weak charge from the Stark–PNC interference measurements also requires knowledge of the vector transition polarizability . This can be found in a number

13

Note that di2erent methods also give di2erent signs of the errors for hfs. This is one more argument that the true value of EPNC is somewhere in the interval between the results of di2erent calculations in Table 9.

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of ways: (i) from a direct calculation of . can be expressed as a sum over intermediate states and experimental E1 transition amplitudes and energies can be used [17] (see also [19,126]). However, this calculation is unstable due to strong cancellations of di2erent terms in the sum (see Ref. [126]). These cancellations are explained by the fact that is proportional to the spin–orbit interaction, therefore for zero spin–orbit interaction the sum for must be zero; (ii) from the measurement of the ratio of the o2-diagonal hyper%ne amplitude to the vector transition polarizability, Mhfs = [127]. is then extracted from the ratio using a theoretical determination of Mhfs ; (iii) from the measurement of the ratio of the scalar to vector polarizabilities, #=. # can be calculated accurately using experimental values for E1 transition amplitudes and energies in the sum-over-states approach (the calculation of # is much more stable than that of [126]). There are currently two very precise determinations of . One was obtained from the analysis [128] (calculation of Mhfs ) of the measurement [83] of the ratio Mhfs =, = 26:957(51)a3B , and another is from an analysis [10] (semi-empirical calculation of #; see [126] for details, where a similar calculation was performed) of the measurement [129] of the ratio #= using the most accurate experimental data for E1 transition amplitudes including the recent measurements of Ref. [130], = 27:15(11)a3B . An average of these values gives = 26:99(5)a3B :

(74)

5.3. The s0) =

T

M

+

T

M

Fig. 7. Illustration of the sources of a discontinuity in the amplitude A from two baryon intermediate states. The dashed horizontal lines indicate the presence of a two baryon unitarity cut (diagrams with initial state interaction are not shown explicitly (cf. Fig. 5)).

N T -matrix. 10 Eq. (12) follows by simply using the de6nition of the elastic phase shift: 7T (m2 ) = ei3 sin 3, where 7 = p 4 denotes the phase-space density here expressed in terms of the reduced mass of the N system 4 = (MN M )=(MN + M ). We will discuss at the end of this section how to control the possible inNuence of the K interactions. The solution of (10) in the physical region can then be written as (see [38] and references therein) 2

A(s; t; m2 ) = eu(m +i0) /(s; t; m2 ) ; where

m˜ 2

dm 2 D(s; t; m 2 ) −u(m 2 ) e m 2 − m 2 −∞ and, in the absence of bound states, 1 ∞ 3(m 2 ) 2 dm : u(z) = m20 m 2 − z 1 /(s; t; m ) = 2

(13)

(14)

(15)

The m2 dependence of / is dominated by the m2 dependence of the production operator. The momentum transfer in the production operator, however, is controlled by the initial momentum and therefore one should expect the m2 dependence of / to be weak as long as the corresponding relative momentum of the -nucleon pair is small. Thus, in a large momentum transfer reaction, the m2 dependence of the production amplitude A is governed by the elastic scattering phase shifts of the dominant two particle reaction in the 6nal state! The relation between the phase shifts and the m2 dependence of the amplitude as can be extracted from an invariant mass distribution is 9xed by analyticity and unitarity! 10

The appearance of the complex conjugation here follows from the direct evaluation of the discontinuity of the T -matrix by employing the unitarity of the S matrix: from S † S = 1 it follows, that 1 disc(T ) := (T (m2 + i0) − T (m2 − i0)) = −7|T (m2 )|2 : 2i

C. Hanhart / Physics Reports 397 (2004) 155 – 256

173

So far we are in line with the reasoning of the previous section, however the exponential factor in Eq. (13) is in general not simply the elastic scattering amplitude. For illustration we investigate the form of A(m2 ) for a 6nal state interaction that is fully described by the 6rst two terms in the e@ective range expansion (cf. Eq. (7)), p ctg(3(m2 )) = 1=a + ( 12 )rp 2 , where p is the relative momentum of the 6nal state particles under consideration in their center-of-mass system. Then A can be given in closed form as [37] (p 2 + ,2 )r=2 /(s; t; m2 ) ; (16) 2 1=a + (r=2)p − ip where , = (1=r)(1 + 1 + 2r=a). Note: The case of an attractive interaction without a bound state in this convention corresponds to a positive value of a and a positive value of r. In the limit a → ∞, as is almost realized in NN scattering, the energy dependence of A(m2 ) is given by 1=(1−iap ) as long as p 1=r. This, however, exactly agrees with the energy dependence for NN on-shell scattering. This prediction was experimentally con6rmed by the near-threshold measurements of the reaction pp → pp0 [43,44]. However, for interactions where the e@ective range is of the order of the scattering length, the numerator of Eq. (16) plays a role and thus the full production amplitude is no longer given by the on-shell elastic scattering times terms whose energy dependence is independent of the scattering parameters. It is often argued that since the full production amplitude is given by a term with a plane wave 6nal state as well as one with the strong interactions, that an appropriate parameterization is given by a Watson term (proportional to the elastic scattering of the outgoing particles) plus a constant term, and their relative strength should be taken as a free parameter. The considerations in the previous paragraphs show, however, that this is not the case: Eq. (13) (and thus also the special form given in Eq. (16)) describes the m2 dependence of the full invariant mass spectrum. One more comment is in order: In the previous section we stressed that, although the shape of the invariant mass spectra as well as the energy dependence of the total cross section are governed by the on-shell interactions in the subsystems, the overall normalization is not. This statement, based on e@ective 6eld theory arguments, is illustrated in Appendix D. On the other hand, Eq. (13) gives a closed expression for the invariant mass spectrum for arbitrary values of m2 . On the 6rst glance this look like a contradiction. However, it should be stressed that Eq. (13) does in general not allow one to relate the asymptotic form of the full production amplitude with information on the scattering parameters of the strongly interacting subsystems in the 6nal state to the amplitude in the close-to-threshold regime. Besides the trivial observation that Eq. (13) holds for each amplitude individually and that far away from the threshold there should be many partial waves contributing, there is no reason to believe that the function / is constant over a wide energy range. Therefore, we want to emphasize that the m2 dependence of Eq. (13) is controlled by the scattering phase shifts of the most strongly interacting subsystem only in a very limited range of invariant masses. In fact, the leading singularity that contributes to an m2 dependence of / is that of the t-channel meson exchange in the production operator. The m2 dependence that originates from this singularity can be estimated through a Taylor expansion of the momentum transfer and thus turns out to be governed by (p =p)2 , where p denotes the relative momentum of the 6nal state particle pair of interest. For p we may use that of the initial state particles at the production threshold, all to be taken in the over all center-of-mass system. As a consequence, for values of m2 that do not signi6cantly deviate A(s; t; m2 ) =

174


from m20 (or equivalently p p) the assumption of / being constant is justi6ed. However / can show a signi6cant m2 dependence over a large range of invariant masses. Let us now return to our main goal: namely to derive from (13) a formula that allows extraction of the elastic scattering phase shifts from an invariant mass spectrum. In this course we will derive a dispersion relation in terms of the function |A(m2 )|2 . Here we will use a method similar to the one used in [55,56]. However, in contrast to the formulas derived in these references, we will present a integral representation for the elastic scattering phase shifts from production data that involves a 6nite integration range only. For this we 6rst observe that Eq. (15) holds for purely elastic scattering only and thus is of very limited practical use. On the other hand, a signi6cant contribution stems from large values of m 2 which depends only weakly on m2 in the near-threshold region, and therefore can be absorbed into the function /. Let 2 3(m 2 ) 1 mmax 2 2 ˜ dm /(m2max ; m2 ) ; (17) A(m ) = exp m20 m 2 − m2 − i0 ˜ 2max ; m2 ) = /(m2 )/m2 (m2 ), with where /(m max ∞ 2) 3(m 1 2 dm : /m2max (m2 ) = exp m2max m 2 − m2 − i0

(18)

The quantity m2max is to be chosen by physical arguments in such a way that both /(m2 ), and /m2max (m2 ) vary slowly on the interval (m20 ; m2max ). Obviously, it needs to be suOciently large that the structure in the amplitude we are interested in can be resolved. 11 On the other hand, it should be as small as possible, since this will keep the inNuence of the inelastic channels small. In order to estimate the minimal value of mmax we see from Eq. (16) that the values of p that enter in the integral given in Eq. (22) should be at least large enough that the scattering length term plays a signi6cant role. Thus we require pmax ∼ 1=atyp . In case of the pn interaction (the spin triplet pn scattering length is 5:4 fm) this would correspond to a value of pmax of only 10 MeV corresponding to a value for jmax = mmax − m0 —the maximum excess energy that should occur within the integral in Eq. (17) (as well as Eq. (22))—as small as 2 MeV. On the other hand, if we want to study the hyperon–nucleon interaction, we want to ‘measure’ scattering lengths of the order of a few fermi we choose atyp = 1 fm, leading to a value of 40 MeV for jmax . Note that 40 MeV is still signi6cantly smaller than the thresholds for the closest inelastic channels (that for the KN 6nal state is at 75 MeV, that for the KN channel at 140 MeV). The integral in (17) contains an unphysical singularity of the type log(m2max − m2 ), which is ˜ 2max ; m2 ), but this does not a@ect the region near threshold. canceled by the one in /(m Notice next that the function 2 ˜ 2max ; m2 )} 1 1 mmax 3(m 2 ) log{A(m2 )= /(m 2 = dm (19) 2 2 − i0 2 2 2 2 2 2 2 2 2 m − m (m − m0 )(mmax − m ) (m − m0 )(mmax − m ) m0 11

One immediately observes that the integral given in Eq. (22) goes to zero for mmax → m0 . At the same time the theoretical uncertainty 3ammax , de6ned in Eq. (24), tends to the value of the scattering length.


175

mmax

mo

mo

(a)

(b) 2

Fig. 8. The integration contours in the complex m plane to be used in the original treatment by Geshkenbein [55,56] (a) and in the derivation of Eq. (18) (b). The thick lines indicate the branch cut singularities.

has no singularities in the complex plane except the cut from m20 to m2max (cf. Fig. 8b) and its value below the cut equals the negative of the complex conjugate from above the cut. Hence, m2max ˜ 2max ; m 2 )|2 1 log |A(m 2 )= /(m m2max − m2 3(m2 ) 2 P = − dm : (20) 2 2 2 2 2 2 mmax − m m − m0 m0 m 2 − m2 (m 2 − m2 ) 0

It is an important point to stress that P

m2max

m20

1 m 2 − m20 (m 2 − m2 )

m2max − m2 2 dm = 0 2 2 mmax − m

(21)

as long as m2 is in the interval between m20 and m2max . Therefore, if the function / only weakly depends on m2 , as it should in large momentum transfer reactions, it can well be dropped in the above equation. In addition, up to kinematical factors, |A|2 agrees with the cross section for the production of a p pair of invariant mass m. We can therefore replace it in Eq. (20) by the cross section where, because of Eq. (21), all constant prefactors can be dropped. Thus we get

2 mmax 1 M + Mp m2max − m2 2 P dm aS = lim M Mp m2 →m20 2 m2max − m 2 m20 2 d &S 1 1 : (22) × log p dm 2 dt m 2 − m20 (m 2 − m2 ) This is the desired formula: namely, the scattering length is expressed in terms of an observable. Note that Eq. (22) is applicable only if it is just a single YN partial wave that contributes to the cross section &S . In Section 4.3.1 we will discuss how the use of polarization in the initial state can be used to project out a particular spin state in the 6nal state.

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Up to the neglect of the kaon–baryon interactions, Eq. (20) is exact. Thus, 3a(th) —the theoretical uncertainty of the scattering length extracted using Eq. (22)—is given by the integral

1 M + Mp (th) 3a = − lim M Mp m2 →m20 2 m2max ˜ 2max ; m 2 )|2 log |/(m m2max − m2 2 dm : (23) ×P 2 2 2 m − m 2 2 m0 max m − m2 (m − m2 ) 0

˜ 2max ; m2 )|2 = log |/(m2 )|2 + log |/m2 (m2 )|2 we may write 3a(th) = 3a(lhc) + 3ammax , Since log |/(m max where the former, determined by /(m2 ), is controlled by the left-hand cuts, and the latter, determined by /m2max (m2 ), by the large-energy behavior of the N scattering phase shifts. The closest left-hand singularity is that introduced by the production operator, which is governed by the momentum transfer. Up to an irrelevant overall constant, we may therefore estimate the variation of / ∼ 1 + 3(p =p)2 , where we assume 3 to be of the order of 1. Evaluation of integral (23) then gives =p2 ) ∼ 0:05 fm ; 3a(lhc) ∼ 3(pmax 2 where we used pmax = 24jmax with jmax ∼ 30 MeV and the threshold value for p ∼ 900 MeV. On the other hand, using the de6nition of /m2max (m2 ) given in Eq. (18) one easily derives 2 2 ∞ 3(y) dy mmax 6 |3max | ; (24) |3a | = 2 (3=2) pmax 0 (1 + y ) pmax

where y2 = (m − mmax )=jmax . Thus, in order to estimate 3ammax we need to make an assumption about the maximum value of the elastic N phase for m2 ¿ m2max . Recall that we implicitly assume the inelastic channel not to play a role. This assumption was con6rmed in Ref. [32] within a model calculation. The denominator in the integral appearing in Eq. (24) strongly suppresses large values of y. Since for none of the existing N models does 3max exceed 0:4 rad, we estimate 3ammax ∼ 0:2 fm : When using the phase shifts as given by the models directly in the integral, the value for 3ammax is signi6cantly smaller, since for all models the phase changes sign at energies above m2max . Combining the two error estimates, we conclude 3a(th) . 0:3 fm : In the considerations of the theoretical uncertainty of the parameters extracted so far we did not talk about the possible e@ect of the kaon–baryon interaction. Unfortunately, it cannot be quanti6ed a priori; however, it can be controlled experimentally. As will be discussed in Section 4, a signi6cant meson–baryon interaction, due, for example, to the presence of a resonance, will show up as a band in the Dalitz plot. If this band overlaps with the region of 6nal state interactions we are interested in, interference e@ects might heavily distort the signal [57]. By choosing a di@erent beam energy, the FSI region and the resonance band move away from each other. Thus there should be an energy regime where the FSI can be studied undistorted. It is therefore necessary for a controlled extraction of FSI parameters to do a Dalitz plot analysis at the same time. An additional possible cross-check


177

of the inNuence of a meson–baryon interaction would be to do the full analysis at two di@erent beam energies separated in energy by at least the typical hadronic width (about 100–200 MeV). If the parameters extracted agree with each other, there was no substantial inNuence from other subsystems. The possible gain in experimental accuracy for the extraction of the N scattering lengths is illustrated and discussed in detail in Section 7.4. To our knowledge so far the corresponding formalism including the Coulomb interaction is not yet derived. 3. The initial state interaction As mentioned earlier, the collision energy of the two nucleons in the initial state is rather large, especially when we study the production of a heavy meson. However, as long as the excess energy is small, only very few partial waves contribute in the 6nal state and thus the conservation of total angular momentum as well as parity and the Pauli principle allow for only a small number of partial waves in the initial state. In addition, only small total angular momenta are relevant. In such a situation the standard methods developed by Glauber, which include the initial state interaction via an exponential suppression factor [58], cannot be applied. The role of the initial state interaction relevant to meson production reactions close to the threshold was discussed for the 6rst time in Ref. [40]. It should be clear a priori that for quantitative predictions the ISI has to play an important role: to allow a production reaction to proceed the nucleons in the initial state have to approach each other very closely—actually signi6cantly closer that the range of the NN interaction. Thus, especially for the production o@ heavy mesons, a large number of elastic and inelastic NN reactions can happen before the two-nucleon pair comes close enough together to allow for the meson production. Thus the e@ective initial current gets reduced signi6cantly. Technically, the inclusion of the initial state interaction in a calculation for a meson production process requires a convolution of the production operator with the half o@-shell T -matrix for the scattering of the incoming particles (cf. Figs. 5c and d). Contrary to the 6nal state interaction, for the high initial energies the energy dependence of the NN interaction is rather weak. One might therefore expect that the real part of the convolution integral is small and that the e@ect of the initial state interaction is dominated by the two-nucleon unitarity cut [40]. This contribution, however, can be expressed completely in terms of on-shell NN scattering parameters: 2 |:L |2 = 12 ei3L (p) ( L (p)ei3L (p) + e−i3L (p) ) = − L (p) sin2 (3L (p)) + 14 [1 + L (p)]2 6 14 [1 + L (p)]2 ;

(25)

where p denotes the relative momentum of the two nucleons in the initial state with the total energy E, and 3L ( L ) denote the phase shift (inelasticity) in the relevant partial wave L. Each partial wave amplitude should be multiplied by :L , de6ned in Eq. (25), in order to account for the dominant piece of the ISI. This method was used e.g. in Ref. [59]. Using typical values for phase shifts and inelasticities at NN energies that correspond to the thresholds of the production of heavier mesons, Eq. (25) leads to a reduction factor of the order of 3, thus clearly indicating that a consideration of the ISI is required for quantitative predictions. One should keep in mind, however, that Eq. (25) can

178


only give a rough estimate of the e@ect of the initial state interaction and should whenever possible be replaced by a full calculation. This issue is discussed in detail in Ref. [34]. Unfortunately, the applicability of Eq. (25) is limited to energies where scattering parameters for the low NN partial waves are available. Due to the intensive program of the EDDA collaboration for elastic proton–proton scattering [60] and the subsequent partial wave analysis documented in the SAID database [61], scattering parameters in the isospin—one channel are available up to energies that correspond to the ; production threshold. The situation is a lot worse in the isoscalar channel. Here pn scattering data would be required. At present, those are available only up to the production threshold. It is very fortunate that there is a proposal in preparation to measure spin observables in the pn system in this unexplored energy range at COSY [62]. 4. Observables In this section the various observables experimentally accessible are discussed. After some general remarks about the three-body kinematics in the 6nal state, in the next subsection we will discuss unpolarized observables. In the following subsection we will then focus on polarized observables. Even if x denotes just a single meson, the reaction NN → B1 B2 x is subject to a 6ve-dimensional phase: three particles in the 6nal state introduce 3×3=9 degrees of freedom, but the four momentum conservation reduces this number to 5. As we will restrict ourselves to the near-threshold regime, the 6nal state can be treated non-relativistically. The natural coordinate system is therefore given by the Jacobi–coordinates in the overall center-of-mass system (cf. Ref. [42]), where 6rst the relative momentum of one pair of particles is constructed and then the momentum of the third particle is calculated as the relative momentum of the third particle with respect to the two-body system. Obviously, there are three equivalent sets of variables possible. In the center-of-mass this choice reads ˜ j − Mj p ˜i Mi p p ˜ ij = ; Mi + M j ˜qk =

pk − Mk (˜ pi + p ˜ j) (Mi + Mj )˜ =p ˜k ; Mi + M j + M k

(26)

where we labeled the three 6nal state particles as ijk; qk = pk holds in the over all center-of-mass system only. For simplicity in the following, we will drop the subscripts when confusion is excluded. For reactions of the type NN → B1 B2 x it is common to work with the relative momentum of the two-nucleon system and to treat the particle x separately. From the theoretical point of view this choice is most convenient, for one is working already with the relative momentum of the dominant 6nal state interaction and the assignment of partial waves as used in Section 1.6 is straightforward. Note that for any given relative energy of the outgoing two-nucleon system j = p 2 =MN the modulus of the meson momentum |˜q | is 6xed by energy conservation; we thus may characterize the phase space by the 5 tuple = = {j; .p ; .q } ;

(27)

where .k = (cos(>k ); ;k ) denotes the angular part of vector ˜k. The coordinate system is illustrated in Fig. 9. The center-of-mass nucleon momentum in the initial state will be denoted by p ˜ . Throughout

C. Hanhart / Physics Reports 397 (2004) 155 – 256 B1 p'−MB1/(MB1+MB2) q'

p

x q'

179

B2 −p'− MB2/(MB1+MB2) q'

−p

Fig. 9. Illustration of the choice of variables in the over all center-of-mass system.

this report we choose the beam axis along the z-axis. Explicit expressions for the vectors appearing are given in Appendix B.1. Due to the high dimensionality of the phase space it is not possible to present the full complexity of the data in a single plot. At the end of Section 4.2 we will discuss a possible way of presenting the data through integrations subject to particular constraints. A di@erent choice would be simply to present highly di@erential observables as is done for bremsstrahlung, 12 or at least to use the two dimensional representation of the Dalitz plot to show some correlations. 4.1. Unpolarized observables As long as the initial state is unpolarized, the system has azimuthal rotation symmetry, reducing the number of degrees of freedom from 5 to 4. In the case of a three-particle decay (a 1 → 3 reaction) the physics does not depend on the initial direction. The same is true for reactions where the initial state does not de6ne a direction, like in the experiment series for pp G annihilation at rest carried out by the Crystal Barrel collaboration at LEAR (see Ref. [63] and references therein). Therefore, the number of degrees of freedom is further reduced from four to two. 13 Those are best displayed in the so-called Dalitz plot (see [64] and references therein) that shows a two dimensional representation in the plane of the various invariant masses m2ik = (pi + pk )2 . In reactions with two particles in the initial state and three particles in the 6nal state (2 → 3 reactions), however, the initial momentum de6nes an axis and therefore a single fully di@erential plot is no longer possible. Especially in the example given below, it will become clear how the appearance of the additional momentum changes drastically the situation. To account for the higher complexity of the 2 → 3 reactions, if enough statistics were collected in a particular experiment, one can present di@erential Dalitz plots—a new plot for each di@erent orientation of the reaction plane [65]. However, in what follows, we will call Dalitz plot the representation of a full data set/calculation in the plane of invariant masses, ignoring the initial direction. Obviously this means throwing out some correlations, for in general any integration reduces the amount of information in an observable. We will brieNy review the properties of a Dalitz plot, closely following Ref. [64]. In Fig. 10 a schematic picture of it is shown. In this plot also di@erent 12 It should be noted that in the early bremsstrahlung experiments the method of integration was not possible, because the detectors used had very limited angular acceptance. 13 This is quite obvious, since one can look at a three particle decay as the crossed channel reaction of two-body scattering which is well known to be characterized by two variables in the unpolarized case.

180


m232 3

1

2

1 3

F(12) 2 1

F(31)

3

2

v1=v3 p2 max.

R

v1=v2 p3 max.

1

3

2

F(23) 1

2

3

I 1

3

v2=v3

2

p1 max.

m122 Fig. 10. Sketch of a Dalitz plot for a reaction with a three-body 6nal state (particles are labeled as 1, 2 and 3). Regions of possibly strong 6nal state interactions in the (ij) system are labeled by F(ij) ; a resonance in the (12)—system would show up like the band labeled as R. The region of a possible interference of the two is labeled as I .

regimes are speci6ed: those of potentially strong 6nal state interaction for the subsystem (ij) are labeled F(ij) . They should occur when particle i and j move along very closely. On the other hand, resonances will show up as bands in the Dalitz plot. Labeled as R in the 6gure, the e@ect of a resonance in the (12) system is shown. The total area of the Dalitz plot scales with the phase-space volume. Therefore, as the excess energy decreases, resonance and 6nal state interaction signals or the regions of di@erent 6nal state interactions might start to overlap, leading to interference phenomena (cf. the hatched area I in Fig. 10). It was demonstrated recently [57] that those can rather strongly distort resonance and 6nal state interaction signals. At least for known 6nal state properties, these patterns might help to better pin down resonance parameters [57]. On the other hand, as the excess energy increases the di@erent structures move away from each other and one should then be able to study them individually. This observation is of great relevance if one wants to extract parameters of a particular 6nal state interaction from a production reaction (cf. discussion in Section 2). In case of particle decays of spinless or unpolarized particles, the Dalitz plot not only contains the information about the occurrence of a resonance, but also its quantum numbers can be extracted by projecting the events in the resonance band (labeled as R in Fig. 10) on the appropriate axis (for the example of the 6gure this is the 23 axis). In the case of 2 → 3 reactions, however, this projection is not necessarily conclusive. To explain this statement we have to have a closer look at the angles of the system. First of all there is a set of angles, the so-called helicity angles, that can be constructed from the 6nal momenta only. One example is cos(>p q ) =

p ˜ · ˜q |˜ p | |˜q |

(28)


181

were p and q where de6ned in Eq. (26). These angles that can be extracted from the Dalitz plot directly. In addition there are those angles that are related to the initial momentum—the Jackson angles. One example is cos(>pp ) =

˜ p ˜ · p : |˜ p | |˜ p|

(29)

It should be stressed that it is not in the distribution of the helicity angles but that of the Jackson angles that the subsystems reNect their quantum numbers [64]. 14 Therefore a pure Dalitz plot analysis is insuOcient for production reactions and the distributions for the Jackson angles have to be studied as well. This will be illustrated in an example in the following subsection. Note that in the presence of spin there are even additional axes in the problem. This will be discussed in detail in Section 4.2. 4.1.1. Example: analysis of pp → dKG 0 K + close-to-threshold Recently, a 6rst measurement of the reaction pp → dKG 0 K + close to the production threshold was reported [66] at an excess energy Q = 46 MeV. The data, as well as the corresponding theoretical analysis, based on the assumption that only the lowest partial waves contribute, will now be used to illustrate the statements of the previous section. It will become clear, especially, that the information encoded in the distributions of the Jackson angles and the helicity angles is rather di@erent. As can be seen in Table 2, a 6nal state that contains s-waves only is not allowed in this reaction. For later convenience 15 we work with the relative momentum of the kaon system (˜ pKK ˜ ; G ≡ p cf. Eqs. (26)) and the deuteron momentum with respect to this system (˜qd ≡ ˜q ). Given our assumptions, that only the lowest partial waves contribute, the amplitudes that contribute to the production reaction are either linear in q or linear in p . In Ref. [67] the full amplitude was constructed (cf. also discussion in Section 7.5), however it should be clear that the spin averaged square of the matrix element can be written as ˜ · p) G 2 = C q q 2 + C p p 2 + C q (q˜ · p) ˆ 2 + C1p (p ˆ 2 |M| 0 0 1

˜ · q˜ ) + C3 (p ˜ · p)( ˆ q˜ · p) ˆ ; + C2 (p

(30)

˜ or linear in q˜ . Here pˆ = p since all terms in the amplitude are either linear in p ˜ =|˜ p| denotes the beam direction. Since the two protons in the initial state are identical, any observable has to be symmetric under the transformation p ˜ = −˜ p. This is why pˆ appears in even powers only. Figs. 11 and 12 show the data as well as a 6t based on Eq. (30). The parameters extracted are ˜ · p)=p given in Table 3. The 6rst two panels of Fig. 11 contain the distributions of the angles (p ˆ ˜ · q˜ )=(q p ). The and (q˜ · p)=q ˆ . The last panel contains the distribution of the helicity angle (p solid line corresponds to a complete 6t to the data including both p-waves in the KG 0 K + system as well as those in the d(KG 0 K + ) system. For the long dashed line the KG 0 K + p-waves were set to G s-waves) were set zero, whereas for the dotted line the d(KG 0 K + ) p-waves (corresponding to KK 14

As is stressed in Ref. [64], only under special conditions, namely for peripheral production as it occurs in high-energy experiments, the information in the helicity angles and the Jackson angles agrees. Close to threshold, however, meson production is not at all peripheral. 15 In Section 7.5 it will be argued, that the reaction pp → dKG 0 K + can be used to study scalar resonance a+ 0 (980). Thus we are especially interested in the partial waves of the kaon system, that should show a strong 6nal state interaction.


dσ/dΩ [nb/sr]

182 5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

0

0

0.5 |cos(θpp')|

1

0

0

0.5 |cos(θpq')|

1

0

-1

0 -0.5 0.5 cos(θp'q')

1

dσ/dM [µb/GeV]

Fig. 11. Angular distributions for the reaction pp → dKG 0 K + measured at Q = 46 MeV [66]. The solid line shows the G s-waves as well as p-waves. To obtain the dashed (dotted) line, the parameters result of the overall 6t including both KK for the p-wave (s-wave) were set to zero (see text). The small error shows the statistical uncertainty only, whereas the large ones contain both the systematic as well as the statistical uncertainty added linearly.

1.5

1.5

1

1

0.5

0.5

0 0.98

1 1.02 MKK [GeV]

1.04

0

2.38

2.4 MdK [GeV]

2.42

Fig. 12. Various mass distributions for the reaction pp → dKG 0 K + (Line code as in Fig. 11). The small error bars show the statistical uncertainty only, whereas the large ones contain both the systematic as well as the statistical uncertainty (cf. Ref. [66]).

to zero. Thus, the 6rst two panels truly reNect the partial wave content of the particular subsystems individually, whereas the helicity angle (which can also be extracted from the Dalitz plot) shows a Nat distribution only if both subsystems are in a p-wave simultaneously. Therefore, the helicity angle can well be isotropic although one of the subsystems is in a high partial wave. Note that this statement is true even if all particles were spinless. The only thing that would change is that C0q


183

Table 3 Results for the C parameters from a 6t to the experimental data C0p

C0q

0 ± 0:1

C1p

1 ± 0:03

1:26 ± 0:08

C1q

C2 + 13 C3

−0:6 ± 0:1

−0:36 ± 0:17

The parameters are given in units of C0q .

and C0p would vanish (cf. Section 4.3.2). In Fig. 12 two Dalitz plot projections (invariant mass q p spectra) are shown. The 6rst one (d&=dmKK G ) is needed to disentangle C0 and C0 . The second one does not give any additional information. What is now the information contained in the two-dimensional Dalitz plot? Since it does not contain any information about the initial state, the parts of the squared amplitude that can be extracted from the Dalitz plot are easily derived from Eq. (30) by integrating over the beam direction [64], giving 1 q 2 1 p 2 q p 2 G d.p |M| = C0 + C1 q + C0 + C1 p 3 3 1 ˜ · q˜ ) : + C2 + C3 (p (31) 3

G s-wave strength (C0q + ( 1 )C1q ), the Therefore from the Dalitz plot one can extract the total KK 3 G p-wave strength (C0p + ( 1 )C1p ), as well as the strength of the interference of the two total KK 3 (C2 + ( 13 )C3 ). Note that in this particular example, all the coeOcients given in Eq. (31) (and even the C0 and C1 individually) can as well be extracted from the angular distributions given in Fig. 11 G invariant mass distribution (left panel of Fig. 12) directly; the Dalitz plot here does not and the KK provide any additional information. Obviously, as we move further away from the threshold, the complexity of the amplitude increases and the Dalitz plot contains information not revealed in the projections. To summarize, in order to allow for a complete analysis of the production data, in addition to the Dalitz plot the angular distributions of the 6nal momenta on the beam momentum need to be analyzed as well. The latter distributions are the ones that give most direct access to the partial wave content of the subsystems. 4.2. Spin-dependent observables Polarization observables for 2 → 2 reactions are discussed in great detail in Ref. [68]. In our case, however, we have one more particle in the 6nal state and therefore there are more degrees of freedom available. Here we will not only derive the expressions for the observables in terms of spherical tensors but also relate these to the partial wave amplitudes of the production matrix elements. In this section we closely follow Refs. [47,69].

184


In terms of the so-called Cartesian polarization observables, the spin-dependent cross section can be written as  &(=; ˜P b ; ˜P t ; ˜Pf ) = &0 (=) 1 + ((Pb )i Ai0 (=) + (Pf )i D0i (=)) +

ij

+

i

((Pb )i (Pt )j Aij (=) + (Pb )i (Pf )j Dij (=))  (Pb )i (Pt )j (Pf )k Aij; k0 (=) : : : ;

(32)

ijk

where &0 (=) is the unpolarized di@erential cross section, the labels i; j and k can be either x; y or z, and Pb , Pt and Pf denote the polarization vector of beam, target and the 6rst one of the 6nal state particles, respectively. All kinematic variables are collected in =, de6ned in Eq. (27). The observables shown explicitly in Eq. (32) include the beam analysing powers Ai0 , the corresponding quantities for the 6nal state polarization D0i , the spin correlation coeOcients Aij , and the spin transfer coeOcients Dij . In this context it is important to note that baryons that decay weakly, as the hyperons do, have a self analyzing decay. In other words, the angular pattern of the decay particles depends on the polarization of the hyperon, therefore, the hyperon polarization in the 6nal state can be measured without an additional polarimeter (see e.g. Ref. [70]). All those observables that can be de6ned by just exchanging ˜P b and ˜P t , such as the target analyzing power A0i , are not shown explicitly. From Eq. (32) it follows that for example 1 (33) &0 Aij; k0 = Tr(&k(f) M&i(b) &j(t) M† ) ; (2sb + 1)(2st + 1) where the &i(b) (&j(t) ) are the Pauli matrices acting in the spin space of beam and target, respectively. The production matrix element is denoted by M. In addition, sb (st ) denote the total spin of the beam (target) particles. It is straightforward to relate the polarization observables to the partial wave amplitudes that can be easily extracted from any model. For this purpose it is convenient to use spherical tensors de6ned through 1 (b) (t) † 1 )† (f2 )† Tr[%(f Tkk13qq13;k; k24qq24 = (34) k3 q3 %k4 q4 M%k1 q1 %k2 q2 M ] ; (2sb + 1)(2st + 1) where the %kq denote the spherical representation of the spin matrices 1 (35) %10 = &z ; %1±1 = ∓ √ (&x ± i&y ); %00 = 1 : 2 To relate the observables to the spherical tensors, the easiest method is to use the de6nitions of Eqs. (35) inside the various equations (34). For the observables for which the 6nal polarization remains undetected, the relations between the various T and the corresponding observables are shown in Table 4. In Table 5 a few of the observables that contain the 6nal state polarization are listed. Triple polarization observables are not listed explicitly, but it is straightforward to derive also the relevant expressions for these, such as √ 3 1100 Axx; x + Ayy; x − (Axy; y − Ayx; y ) = 2 Re(T111 −1 ) :


185

Table 4 Relations between spherical tensors and some observables that do not contain the 6nal state polarization following Ref. [69] Cartesian observable Di@erential cross section &0 Beam analyzing powers &0 Ax0 &0 Ay0 &0 Az0 Target analyzing powers &0 A0x &0 A0y &0 A0z Spin correlation parameters &0 Azz & 0 A & 0 AC &0 Axz &0 Azx &0 Ayz &0 Azy &0 [Axy + Ayx ] & 0 A[

Tk1 q1 k2 q2

Q i = q1 + q 2

T0000 (s; j)

0

*

√ −√2 Re(T1100 (s; j)) − 2 Im(T1100 (s; j)) T1000 (s; j)

1 1 0

*

√ −√2 Re(T0011 (s; j)) − 2 Im(T0011 (s; j)) T0010 (s; j)

1 1 0

(∗)

T1010 (s; j) −2 Re(T111−1 (s; j)) 2 Re(T √ 1111 (s; j)) −√2 Re(T1110 (s; j)) −√2 Re(T1011 (s; j)) −√2 Im(T1110 (s; j)) − 2 Re(T1011 (s; j)) 2 Im(T1111 (s; j)) 2 Im(T111−1 (s; j))

0 0 2 1 1 1 1 2 0

* * * (∗) *

*

(∗)

*

To simplify notation, the indices specifying the 6nal state polarization are dropped. The symbol * indicates a possible set of independent observables. Note: For pp-induced reactions more observables become equivalent, as described in the text. Those are marked by (∗). The linear combinations of spin correlation observables appearing in the table are de6ned in Eqs. (39).

After some algebra given explicitly in Appendix C, one 6nds 1 Q TD (p; ˆ q) ˆ = BL˜l;˜ : (q; ˆ p)A ˆ LD˜l;˜ : ; 4

(36)

˜ L˜l:

where D = {k1 q1 ; k2 q2 ; k3 q3 ; k4 q4 } and Q = q1 + q2 − q3 − q4 . All the angular dependence is contained in 1 ˜ l |:Q YL4 ˜ L ; l4 L4 BLQ˜l;˜ : (q; ˆ p) ˆ = ˆ l4 ˆ (37) ˜ L (p)Y ˜ l (q) 4 4 ;4 L

and ALD˜l;˜ : =

,;,G

l

CL,;˜l;˜,;G:D M , (M ,G)† :

(38)

Here , and ,G are multi-indices for all the quantum numbers necessary to characterize a particular partial wave matrix element and C denotes a coupling coeOcient that can be expressed in terms of Clebsch–Gordan coeOcients. Its explicit form is given in Eq. (C.9). From Eq. (36) one can derive the angular dependences of all observables.

186


Table 5 Relations between spherical tensors and some observables that do contain the 6nal state polarization Cartesian observable

Tkk13qq13kk24qq24

Q = q 1 + q 2 − q3 − q 4

Induced polarization &0 D0x &0 D0y &0 D0z Spin transfer coeOcients &0 Dzz & 0 D & 0 DC &0 Dxz &0 Dzx &0 Dyz &0 Dzy &0 [Dxy + Dyx ] & 0 DE

√ 1−100 (s; j)) √2 Re(T0000 1−100 2 Im(T0000 (s; j)) 1000 T0000 (s; j)

1 1 0

*

1000 T1000 (s; j) 1100 2 Re(T1100 (s; j)) 1−100 −2 Re(T (s; j)) 1100 √ 1000 − 2 Re(T (s; j)) 1100 √ 1−100 2 Re(T (s; j)) 1000 √ 1000 − 2 Im(T (s; j)) 1100 √ 1−100 2 Re(T1000 (s; j)) 1−100 −2 Im(T1100 (s; j)) 1100 2 Im(T1100 (s; j))

0 0 2 1 1 1 1 2 0

* * * * *

*

*

The symbol * indicates a possible set of independent observables. The linear combinations of spin correlation observables appearing in the table are de6ned in Eqs. (40).

In what follows, it is convenient to de6ne the following quantities: A = Axx + Ayy ;

AC = Axx − Ayy ;

and

AE = Axy − Ayx

(39)

and, analogously, D = Dxx + Dyy ;

DC = Dxx − Dyy ;

and

DE = Dxy − Dyx :

(40)

Using the conservation of parity and the explicit expression for the C coeOcient given in Eq. (C.9) CL,;˜l;˜,;G:D = (−)(k1 +k2 +k3 +k4 ) CL,;˜Gl;˜,;:D :

(41)

Since the parameter C is real, the analyzing powers are proportional to the imaginary part of M, M,∗G , whereas the di@erential cross section as well as the spin correlation parameters depend on the real part of M, M,∗G . Thus, it is either the real part or the imaginary part of B that contributes to the angular structure. As we will see in the following subsection, this observation allows for a straightforward identi6cation of the possible azimuthal dependences of each observable. Another obvious consequence of Eq. (41) is, that those observables for which ki is odd have to be small when only a single partial wave dominates. Thus, at the threshold, analyzing powers will vanish. The structure of Eq. (36) is general—no assumption regarding the number of contributing partial waves was necessary. However, if we want to make statements about the expected angular dependence of observables, the number of partial waves needs to be restricted. For example, if we allow for at most p-waves for the NN system as well as the particle x with respect to the NN system, then the largest value of L˜ and l˜ that can occur is 2, which strongly limits the possible >-dependences that can occur in the angular function B de6ned in Eq. (37).


187

If the spin of the particles is not detected there is no interference between di@erent spin states in the 6nal state. This leads to a severe selection rule in case of the reaction pp → ppX : since the 6nal state is necessarily in a T = 1 state the Pauli principle demands that di@erent spin be accompanied by di@erent angular momentum. Therefore, for the a reaction with a pp 6nal state the partial waves can be grouped into two sets, namely {Ss; Sd; Ds; Dp} and {Pp; Ps}, where only the members of the individual sets interfere with each other. 4.2.1. Equivalent observables All the angular dependence of the observables is contained in the function B de6ned in Eq. (37). In this subsection we will discuss some properties of B and relate these to properties of particular observables. The functional form of B enables one immediately to read o@ the allowed azimuthal dependences for each observable as well as to identify equivalent observables. To see this we rewrite Eq. (37) as BLQ˜l;˜ : (q; ˆ p) ˆ = fL˜l;˜ :; Q; n (>p ; >q ) exp{i[(Q − n);p + n;q ]} ; (42) n

which directly translates into the following ;-dependences for the spherical tensors (cf. Eq. (36)): ˆ q) ˆ = TD (p;

N

gD; n (>p ; >q ) exp{i[(Q − n);p + n;q ]} :

(43)

n=−N

Note that N is given by the highest partial waves that contribute to the reaction considered: (44) − L˜ max 6 (Q − N ) 6 L˜ max and N 6 l˜max ; where L˜ max (l˜max ) is given by twice the maximum baryon–baryon (meson) angular momentum. These limits are inferred by the Clebsch–Gordan coeOcient appearing in the de6nition of B in Eq. (37). Eq. (43) directly relates the real and the imaginary parts of the spherical tensors: Im(TD (>p ; ;p + =(2Q); >q ; ;q + =(2Q))) = Re(TD (>p ; ;p ; >q ; ;q )) :

(45)

Thus, two observables are equivalent if they are given by the real and imaginary parts of the same spherical tensor with Q = 0. In Table 4 the relations of the various observables to the spherical tensors are given. Thus, using Eq. (45) we can identify the following set of pairwise equivalent observables: Ay0 ≡ Ax0 ;

A0y ≡ A0x ;

Axx − Ayy ≡ Axy + Ayx ;

(46)

and analogously for observables for which the 6nal state polarization is measured as well. Notice that there is no connection between Axx + Ayy and Axy − Ayx , for these have Q = 0 and therefore there is no transformation, such as the one given in Eq. (45), that relates real and imaginary parts of the spherical tensors. For identical particles in the initial state, as in pp-induced reactions, all observables should be equivalent under the exchange of beam and target. This further reduces the number of independent observables, for now the beam analyzing powers are equivalent to the target analyzing powers and Axz is equivalent to Azx . In Tables 4 and 5 a possible set of independent observables

188


is marked by a ∗. Those of these that are not independent for identical particles in the initial state are labeled as (∗). From the discussion in the previous section it follows (cf. Eq. (41)), that all those observables with an even (odd) value of k1 + k2 + k3 + k4 lead to a real (imaginary) value for gD; n , de6ned in Eq. (43). As a consequence, for all coeOcients appearing in the expansion of the observables, the ;-dependence is 6xed (cf. Table 4); for example, the terms that contribute to &0 , &0 Azz , and &0 A behave as cos(n(;q − ;p )). As was stressed above, the phase space for the production reactions is of high dimension. To allow for a proper presentation of the data as well as of calculations, one either needs high-dimensional plots (see discussion in the previous section) or the dimensionality needs to be reduced to one-dimensional quantities, 16 while, however, still preserving the full complexity of the data. As can be seen in Eq. (43), each polarization observable is described by 2N + 1 functions gD; n (>p ; >q ), where the number of relevant terms is given by the number of partial waves. In order to allow disentanglement of these functions, in Ref. [47] it was proposed to integrate each observable over both azimuthal angles under a particular constraint, / = m;p + n;q = c :

(47)

This integration projects on those terms that depend on / or do not show any azimuthal dependence at all [47]. To further reduce the dimensionality of the data, either the relative proton angle or the meson angle can be integrated to leave one with a large number of observables that depend on one parameter only. Those are then labeled as A/ij (>k ), where k is either p or q. In Figs. 13 and 14 some observables reported in Ref. [47] are shown for the energy with highest statistics, namely = 0:83. In the 6gures the data are compared to the model predictions of Ref. [71]. The solid lines are the results of the full model whereas for the dashed lines the contribution from the Delta isobar was switched o@. In Section 6 this model will be discussed in more detail. In the case of Ref. [47] the complete set of polarization observables for the reaction p ˜p ˜ → pp0 is given. Since the particles in the initial state are identical there are 7 independent observables, all functions of 5 independent parameters (cf. Table 4). In the analysis of the data presented in the same reference it was assumed that only partial waves up to p-waves in both the NN as well as the (NN ) system were relevant. Thus the various integrations described in the previous paragraph lead to 32 independent integrated observables that depend only on a single parameter. On the other hand, if the assumption about the maximum angular momenta holds, only 12 partial waves have to be considered in the partial wave analysis. Since the amplitudes are complex and two phases are not observable (cf. discussion at the end of Section 4.2), a total of 22 degrees of freedom needs to be 6xed from the data. Thus a complete partial wave decomposition for the reaction p ˜p ˜ → pp0 seems feasible.

16

For the experimental side this is the far more demanding procedure, for the angular dependence of eOciency as well as acceptance needs to be known very well over the full angular range for those variables that are integrated in order not to introduce false interferences.

C. Hanhart / Physics Reports 397 (2004) 155 – 256 2

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Fig. 13. Some polarization observables reported in Ref. [47] for the reaction p ˜p ˜ → pp0 at = 0:83 as a function of the pion angle compared to predictions of the model of Ref. [71]. The solid lines show the results for the full model whereas contributions from the Delta where omitted for the dashed lines.

4.3. General structure of the amplitudes In this section we give the recipe for constructing the most general transition amplitude for reactions of the type NN → B1 B2 x, where we focus on spin 12 baryons in the 6nal state. A generalization to other reactions is straightforward. For further applications we refer to a recent review [72]. For simplicity, let us restrict ourselves to those reactions in which there is only one meson produced. The system is then characterized by three vectors, p ˜ ; ˜q ; and p ˜ ; denoting the relative momentum of the two nucleons in the initial state, the meson momentum, and the relative momentum of the two nucleons in the 6nal state, respectively—in the over all center-of-mass system. In addition, as long as x denotes a scalar or pseudo-scalar meson, we 6nd 6 axial vectors, namely those that can be constructed from the above: i(˜ p×p ˜ );

i(˜ p × ˜q );

and

i(˜ p × ˜q )

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C. Hanhart / Physics Reports 397 (2004) 155 – 256 1.5

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Fig. 14. Some polarization observables reported in Ref. [47] for the reaction p ˜p ˜ → pp0 at = 0:83 as a function of cos(>p ) compared to predictions of the model of Ref. [71]. The solid lines show the results for the full model whereas contributions from the Delta were omitted for the dashed lines.

and those that contain the 6nal or initial spin of the two-nucleon system ˜S; ˜S ; and i(˜S × ˜S ) ; where ˜S = H1T &y˜&H2 and ˜S = H3†˜&&y (H4† )T . Here Hi denotes the Pauli spinors for the incoming (1,2) and outgoing (3,4) nucleons and ˜& denotes the usual Pauli spin matrices. If x is a vector particle, an additional axial vector, namely the polarization vector of the vector meson ˜j∗ occurs. In addition, if instead of a two-nucleon state in the continuum a deuteron occurs in the 6nal state, its polarization direction will be characterized by the same ˜j∗ . Since the energy available for the 6nal state is small (we focus on the close to threshold regime), we restrict ourselves to a non relativistic treatment of the outgoing particles. This largely simpli6es the formalism since a common quantization axis can be used for the whole system. In order to construct the most general transition amplitude that satis6es parity conservation, we have to combine the vectors and axial vectors given above so that the 6nal expression form a scalar


191

or pseudo-scalar for reactions where the produced meson has positive or negative intrinsic parity, respectively. The most general form of the transition matrix element may be written as ˜ · (˜SI ) + iÃ · (˜S I) + (Si Sj )Bij ; M = H (II ) + iQ

(48)

where I = (H1T &y H2 ) and I = (H3† &y (H4T )† ). In addition, the amplitudes have to satisfy the Pauli principle as well as invariance under time reversal. This imposes constraints on the terms that are allowed to appear in the various coeOcients. The 9 amplitudes that contribute to B may be further decomposed according to the total spin to which ˜S and ˜S may be coupled Bij = bs 3ij + bvk jijk + btij ; where the superscripts indicate if the combined spin of the initial and the 6nal state are coupled to 0 (s), 1 (v) or 2 (t), where btij is to be a symmetric, trace free tensor. Once the amplitudes are identi6ed the evaluation of the various observables is straightforward. In this case the polarization comes in through Hi Hi† = 12 (1 + ˜P i · ˜&) ; where ˜P i denotes the polarization direction of particle i. Using the formulas given in Appendix B.2 one easily derives (summation over equal indices is implied): ˜ 2 + |Ã|2 + |Bmn |2 ; 4&0 = |H |2 + |Q|

(49)

∗ 4A0i &0 = +ijijk (Qj∗ Qk + Bjl Bkl ) + 2 Im(Bil∗ Al − Qi∗ H ) ;

(50)

∗ 4D0i &0 = −ijijk (A∗j Ak + Blj Blk ) − 2 Im(Bli∗ Ql − A∗i H ) ;

(51)

4Aij &0 = 3ij (−|H |2 + |Q|2 − |A|2 + |Bmn |2 ) ∗ + 2 Re(jlij (Ql∗ H − A∗m Blm ) − Qi∗ Qj − Bim Bjm ) ;

∗ 4Dij &0 = 2 Re Qi∗ Aj + jilm Ql∗ Bmj + jjml Bil A∗m + 12 jilm jjnk Bln Bmk + Bij∗ H ;

(52) (53)

∗ 4Aij; k0 &0 = Im(3ij (2H ∗ Ak − 2Ql∗ Blk − j,Kk (A∗, AK − Bl, BlK )) ∗ + 2jlij (Ql∗ Ak − jnmk A∗n Blm + HBlk ) ∗ ∗ + 2(Qi∗ Bjk + Qj∗ Bik ) − jmnk (Bim Bjn + Bjm Bin )) :

(54)

For illustration we also give here the explicit expressions for A and AC de6ned in the previous section: &0 A = 12 (−|H |2 + |Qz |2 − |Ã|2 + |Bzn |2 ) ; ∗ ∗ Bxm − Bym Bym ) : &0 AC = − 12 (Qx∗ Qx − Qy∗ Qy + Bxm

As was stressed in the previous section, the method of spherical tensors is very well suited to identifying equivalent observables. This is signi6cantly more diOcult in the amplitude approach.

192


Table 6 List of a possible set of the independent amplitude structures that contribute to the reaction pp → pp + (pseudo-scalar) Amplitudes

Structures

H ˜ Q Ã Bij

(˜ p×p ˜ )˜q (˜ p·p ˜ ) p ˜ p ˜ jijk pk (˜ p · ˜q ) 3ij (˜ p×p ˜ )˜q pi pj (˜ p×p ˜ )˜q

Lowest pw p·p ˜ ) p ˜ (˜ p ˜ (˜ p·p ˜ ) jijk pk (˜ p · ˜q ) (˜ p×p ˜ )i qj (˜ p×p ˜ )i pj (˜ p · ˜q )

˜q (˜ p · ˜q ) ˜q (˜ p · ˜q ) jijk qk (˜ p · p ˜) (˜ p × ˜q )i pj (˜ p × ˜q )i pj (˜ p·p ˜ )

Ds Ss, Ds, Sd Ps, Pd Pp

In the last column shows the lowest partial waves for the 6nal state that contribute to the given amplitude (using the notation of Section 1.6). To keep the expressions simple we omitted to give the symmetric, trace free expressions for the terms listed in the last two lines.

However the amplitude method becomes extremely powerful if—due to physical arguments or appropriate kinematical cuts—one of the subsystems can be assumed in an s-wave, for then the number of available vectors is reduced signi6cantly and rather general arguments become possible (cf. Section 4.3.1). Since any amplitude can be made successively more complex by multiplying it by an arbitrary scalar, in most of the cases an ordering scheme is demanded in order to make the approach useful. In the near-threshold regime this is given by the power of 6nal momenta occurring—in analogy with the partial wave expansion. Actually, the amplitude approach presented here and the partial wave expansion presented in the previous subsection are completely equivalent. However, in the near-threshold regime the amplitude method is more transparent. As one goes away from the threshold the number of partial waves contributing as well as the number of the corresponding terms in the amplitude expansion increases rapidly. As a consequence the construction of the most general transition amplitude is rather involved. The partial wave expansion, on the other hand, can be easily extended to an arbitrary number of partial waves. As follows directly from Eq. (48), in the general case the matrix element M is described by 16 complex valued scalar functions. One can show, e.g. by explicit construction, that for general kinematics of the reaction NN → NNx all 16 amplitudes are independent. A possible choice is given explicitly in Table 6 for the reaction pp → pp + (pseudo-scalar). However, for particular reaction channels or an appropriate kinematical situation their number sometimes reduces drastically. For example in collinear kinematics (where p ˜ , ˜q , and p ˜ are all parallel) the number of amplitudes that fully describes the reactions pp → pp + (pseudo-scalar) is equal to 3 (this special case is discussed in Ref. [76]). This can be directly read from Table 6, for under collinear conditions all cross products vanish and all structures of one group that are given by the di@erent vectors of the ˜ → ,˜ system collapse to one structure (Q p, Ã → Kp, Bij → jijk pk ). Another interesting example is that of elastic pp scattering. The Pauli principle demands (cf. Section 1.6) that odd (even) parity states are in a spin triplet (singlet). Time reversal invariance requires the amplitude to be invariant under the interchange of the 6nal and the initial state. When parity conservation is considered in addition, one also 6nds that the total spin is conserved. Thus, only H and Bij will contribute to elastic pp scattering. In addition, from the two vectors available in the system (˜ p and p ˜ ) one can construct only 4 structures that contribute to the latter, namely


193

3ij (˜ p ·˜ p )b1 , jijk (˜ p ×˜ p )k b2 , (pi pj +pj pi −( 23 )3ij )(˜ p ·˜ p )b3 , and (pi pj +pj pi −( 23 )˜ p2 3ij )(˜ p ·˜ p )b4 . 17 Thus, pp scattering is completely characterized by 5 scalar functions. As a further example and to illustrate how the formalism simpli6es in the vicinity of the production threshold, we will now discuss in detail the production of pions in NN collisions. Throughout this report, however, the formalism will be applied to various reactions. In our example there are three reaction channels experimentally accessible, namely pp → pp0 , pp → pn+ , and pn → pp− , that can be expressed in terms of the three independent transition amplitudes ATi Tf (as long as we assume isospin to be conserved), where Ti (Tf ) denote the total isospin of the initial (6nal) NN system [27] (cf. Section 1.5). As in Section 1.6, we will restrict ourselves to those 6nal states that contain at most one p-wave. We may then write for the amplitudes of A11 , H 11 = 0 ; ˜ 11 = a1 pˆ ; Q

Ã11 = a2 p ˜ )pˆ − 13 p ˜ + a3 (pˆ · p ˜ ; Bij11 = 0 ;

(55)

for the amplitudes of A10 , H 10 = 0 ; ˜ 10 = 0 ; Q

Ã10 = b2˜q + b3 (˜q · p) ˆ pˆ − 13 ˜q ; ˜B10 ˆk ; ij = jijk b1 p

(56)

for the amplitudes of A01 , H 01 = 0 ;

˜ 01 = c1˜q + c2 p( ˆ pˆ · ˜q ) − 13 ˜q ; Q Ã01 = 0 ; Bij01 = jnmk p ; ˜ k c3 3in 3jm + c4 3jm pˆ i pˆ n − 13 3in + c5 3in pˆ j pˆ m − 13 3jm

(57)

where pˆ denotes the initial NN momentum, normalized to 1, and p ˜ and ˜q denote the 6nal nucleon and pion relative momentum, respectively. The coeOcients given are directly proportional to the corresponding partial wave amplitudes as listed in Table 1; e.g., a3 is proportional to the transition amplitude 1 D2 → 3 P2 s. When constructing amplitude structures for higher partial waves care has to be taken not to list dependent structures. In order to remove dependent structures the reduction formula, Eq. (B.7), proved useful. In addition, one should take care that the number of coeOcients appearing exactly matches the number of partial waves allowed. For example, the partial waves that contribute to Bij01 are 3 S1 → 3 P1 s, 3 D1 → 3 P1 s and 3 D2 → 3 P2 s. 17

Note, this choice of structures in not unique; we could also have used (˜ p×p ˜ )i (˜ p×p ˜ )j as a replacement of any t other structure in bij .

194


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60

40

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45

90 135 θtot (deg)

180

0

45

90 135 θtot (deg)

180

Fig. 15. Sensitivity of the analyzing power as well as the di@erential cross section for the reaction pn → pp− to the sign of the 3 P0 → 1 S0 s amplitude a1 . The lines corresponds to the model of Ref. [73]: the solid line is the model prediction whereas for the dashed line the sign of a1 was reversed. The experimental data are from Refs. [74,75] at TLab = 353 MeV ( = 0:65).

The large number of zeros appearing in the above list of amplitude structures reNects the strong selection rules discussed in Section 1.6. As an example we will calculate the beam analyzing power and the di@erential cross section for the reaction p ˜ n → pp− . These observables were measured at TRIUMF [74,75] and later at PSI [77,78]—here, however, with a polarized neutron beam (see Fig. 15). In accordance with the TRIUMF experiment, where the relative NN energy in the 6nal state was restricted to at most 7 MeV, we assume that the outgoing NN system is in a relative S-wave. This largely simpli6es the expressions. We then 6nd &0 = 14 |a1 |2 + 12 q Re a∗1 c1 + 23 c2 cos(>) ; 2 &0 Ay = 14 q Im(c1∗ c2 ) sin(2>) − 12 q Im a∗1 c1 − 13 c2 sin(>) cos(;) ; (58) ˆ y = −q sin(>) cos(;) and (˜q · p) ˆ = q cos(>) where we used the de6nitions (˜q × p) (cf. Appendix B.1). Thus, the forward–backward asymmetry in &0 as well as the shift of the zero in Ay directly measure the relative phase of the 3 P0 → 1 S0 s transition in A11 (a1 ) in the pion p-wave transitions of A01 (c1 and c2 ), as was 6rst pointed out in Ref. [79]. This issue will be discussed below (cf. Section 6). Note: As before we neglected here pion d-waves, since they are kinematically suppressed close to the threshold (cf. Eq. (9)). 4.3.1. Example I: polarization observables for a baryon pair in the 1 S0 9nal state As an example of the eOciency of the amplitude method, in this subsection we will present an analysis of the angular pattern of some polarization observables for the reaction pp → B1 B2 x under the constraint that the outgoing two baryon state (B1 B2 ) is in the 1 S0 partial wave and x is a pseudo-scalar, as is relevant for the reaction pp → pK . The dependence of the observables on the meson emission angle is largely constrained under these circumstances, as was stressed in Ref. [80]. In contrast to the discussion in Section 1.6, here we will not assume B1 and B2 to be


195

identical particles. The results of this subsection will show how to disentangle in a model independent way the two spin components of the hyperon–nucleon interaction [32]. The analysis starts with identifying the tensors that are to be considered in the matrix element of Eq. (48). 18 For this we go through a chain of arguments similar to those in Section 1.6. Given ˜ can be non-zero. In addition, the that we restrict ourselves to a spin-zero 6nal state, only H and Q quantum numbers of the 6nal state are 6xed by lx , the angular momentum of the pseudo-scalar with respect to the two baryon system, since J = lx

and

tot = (−)(lx +1)

(59)

for the total angular momentum and the parity, respectively. Conservation of parity and angular momentum therefore gives (−)L = (−)lx +1 = (−)J +1 → S = 1 ;

(60)

and consequently we get H = 0. In addition, for odd values of lx we see from the former equation that L must be even. In pp systems, however, even values of L correspond to S = 0 states, not allowed in our case. Therefore lx must be even. We may thus make the following ansatz: ˜ = ,˜ Q p + K˜q (˜q · p ˜) ;

(61)

p·p ˜ ). All other coeOcients appearing in Eq. (48) where , and K are even functions of p, p and (˜ vanish. This has serious consequences for the angular dependences of the various observables. For example, the expression for the analyzing power collapses to i 1 A0i &0 = jijk (Qj∗ Qk ) = Im(K∗ ,)(˜q · p ˜ )(˜q × p ˜ )i : (62) 4 2 Therefore, independently of the partial wave of the pseudo-scalar emitted, for a two-baryon pair in the 1 S0 state the analyzing power A0y vanishes if the pseudo-scalar is emitted either in the xy-plane or in the zx-plane. In Ref. [32] this observation was used to disentangle the di@erent spin states of the N interaction. 4.3.2. Example II: amplitude analysis for pp → dKG 0 K + close-to-threshold In Section 4.1.1 we discussed in some detail the data of Ref. [66] for the reaction pp → dKG 0 K + based on rather general arguments on the cross section level. In this subsection we will present the corresponding production amplitude based on the amplitude method presented above. This study will allow us at the same time to extract information on the relative importance of the a+ 0 and the

(1405) in the reaction dynamics. G system or the deuteron with respect to the KK G As in Section 4.1.1, we assume that either the KK system is in a p-wave, whereas the other system is in an s-wave, calling for an amplitude linear in p or q , respectively. We use the same notation as in Section 4.1.1. Therefore the 6nal state has odd parity and thus also the amplitude needs to be odd in the initial momentum p ˜ . An odd parity ˜ isovector NN state has to be S = 1 and thus has to be linear in S, de6ned in Section 4.3. In addition, the deuteron in the 6nal state demands that each term is linear in the deuteron polarization vector j. 18

Note, here we could as well refer to Table 6 to come to the same conclusion as in this section, for the 1 S0 state is allowed for the pp system. However, the argument given is quite general and instructive.

196


We therefore get for the full transition amplitude, in slight variation to Eq. (48) due to the presence of the deuteron in the 6nal state, M = Bij˜S i˜j∗j ;

(63)

where ˜j appears as complex conjugate, since the deuteron is in the 6nal state, and ˆ Bij = aSp pˆ i˜qj + bSp (pˆ · ˜q )3ij + cSp˜qi pˆ j + dSp pˆ i pˆ j (˜q · p) + aPs pˆ i p ˜ j + bPs (pˆ · p ˜ )3ij + cPs p ˜ i pˆ j + dPs pˆ i pˆ j (˜q · p) ˆ ;

(64)

G system and small where capital letters in the amplitude label indicate the partial wave of the KK G system. letters that of the deuteron with respect to the KK Once the individual terms in the amplitude are identi6ed, it is straightforward to express the Ci de6ned in Section 4.1.1 in terms of them. We 6nd, for example,

C0q = 12 (|aSp |2 + |cSp |2 ) ;

C1q = |bSp |2 + 12 |bSp + dSp |2 + Re[a∗Sp cSp + (aSp + cSp )∗ (bSp + dSp )] ; ∗ C2 = a∗Sp aPs + cSp cPs :

A 6t to the experimental data revealed that, within the experimental uncertainty, C0p is compatible with zero. Thus, given the previous formulas, both aPs and cPs have to vanish individually. G s-waves in Ref. [66] it was argued that the reaction pp → Based on the strongly populated KK dKG 0 K + is governed by the production of the a+ 0 . In Ref. [81], however, it was argued that the strong G Kd interaction caused by the proximity of the (1405) resonance should play an important role as G s-wave. We now want to calculate the contribution of this well. This FSI should enhance the Kd G p-wave based on the amplitudes given in Table 3. This will illustrate partial wave relative to the Kd a further strength of the amplitude method, for within this scheme changing the coordinate system G system and in the Kd G system is trivial. The coordinate system suited to study resonances in the KK are illustrated in the left and right panels of Fig. 16. All we need to do now is express the vectors ˜ and ˜P . We 6nd that appear in Eq. (30) in terms of Q ˜ ˜q = ˜P − ,Q

and

˜ + ˜P ) ; p ˜ = 12 ((2 − ,)Q

where , = md =(md + mKG ). Obviously, the squared amplitude expressed in the new coordinates reveals the same structure as Eq. (30): G = BQ Q 2 + BP P 2 + BQ (Q ˜ · p) |M| ˆ 2 + B1p (P˜ · p) ˆ 2 0 0 1

˜ ) + B3 (P˜ · p)( ˜ · p) ˆ Q ˆ ; + B2 (P˜ · Q

(65)

where the coeOcients appearing can be expressed in terms of the C coeOcients of Eq. (30) so that, for example,

B0Q =

B1Q =

(2 − ,)2 q ,(2 − ,) 1 C0 + ,2 C0k − C2 ; 4 2 2 (2 − ,)2 q ,(2 − ,) 1 C1 + ,2 C1k − C3 : 4 2 2

(66)

C. Hanhart / Physics Reports 397 (2004) 155 – 256 Ko

K+

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p'

Q'

q'

p p

197

p p

(a)

p

p

(b)

Fig. 16. (a)–(b) Illustration of the coordinate system used in the analysis for the reaction pp → dKG 0 K + .

With these expressions at hand it is easy to verify, that K KG s-waves contribute to 83% to the total G s-waves contribute to 54% only [82]. Here we used the total s-wave cross section, whereas Kd q G G (B0Q + ( 1 )B1Q ), as a measure strength for KK, (C0 + ( 13 )C1q ), and the total s-wave strength for Kd, 3 of the strength of the partial waves. G s-wave, it appears that the (1405) does As we do not see a signi6cant population of the Kd not play an essential role in the reaction dynamics of pp → dKG 0 K + close-to-threshold in contrast to the a+ 0. 4.4. Spin cross sections As early as 1963, Bilenky and Ryndin showed [83], that from the spin correlation coeOcients that can be extracted from measurements with polarized beam and target, the cross section can be separated into pieces that stem from di@erent initial spin states. Their results were recently re-derived [84] and the formalism was generalized to the di@erential level in Ref. [85]. With these so-called spin cross sections it can easily be demonstrated how the use of spin observables enable one to 6lter out particular aspects of a reaction. We begin this subsection with a derivation of the spin cross sections using the amplitude method of the previous subsection and then use the reaction p ˜p ˜ → pn+ as an illustrative example. Since we have the amplitude decomposition of the individual observables given in Eqs. (49)–(54), one easily 6nds &0 (1 − Axx − Ayy − Azz ) = |H |2 + |A|2 = : 1 &0 ;

(67)

&0 (1 + Axx + Ayy − Azz ) = |Qz |2 + |Bzn |2 = : 3 &0 ;

(68)

&0 (1 + Azz ) = 12 (|Qx |2 + |Qy |2 + |Bxn |2 + |Byn |2 )= : 3 &1 ;

(69)

198


Table 7 List of the lowest partial waves in the 6nal state that contribute to the individual spin cross sections 2S+1

&m

Possible 6nal states for pp → bb x Slx

Plx

1

3

3

3

1

3

&0 &0 3 &1

S1 p S0 s, 1 S0 d, 3 S1 d 3 S1 s, 1 S0 d, 3 S1 d

Pj s, 1 P1 s Pj p, 1 P1 p 3 Pj p, 1 P1 p

Capital letters denote the baryon–baryon partial waves whereas small letters that of the meson with respect to the baryon–baryon system.

where the assignment of the various spin cross sections (2S+1) &MS , with S (MS ) the total spin (projection of the total spin on the beam axis) of the initial state can be easily con6rmed from the de6nition of the amplitudes in Eq. (48). As was shown in Section 1.6, in case of two-nucleon initial or 6nal states restrictive selection rules apply. For example, for pp 6nal states the isospin of the 6nal NN system is 1 and therefore states with even (odd) angular momentum have total spin 0 (1). From Table 7 it thus follows, that the pp S-wave in connection with a meson s-wave contributes only to 3 &0 . This example of how spin observables can be used to 6lter out particular 6nal states was used previously in Section 2 (cf. discussion to Fig. 6). For illustrative purposes we show in Fig. 17 the spectra for the various cross sections for the reaction p ˜p ˜ → pn+ , as a function of the relative energy of the nucleon pair in the 6nal state. The curves correspond to the model of Ref. [71] that very well describes the available data in the + production channel. The model is described in detail in Section 6. In the upper left panel the unpolarized cross section is shown. It is dominated by the Sp 6nal state (the dominant transition is 1 D2 → 3 S1 p), and from this spectrum alone it would be a hard task to extract information on 6nal states other than the 3 S1 NN state. This can be clearly seen by the similarity of the shapes of the dot–dashed line and the solid line. The spin cross sections, however, allow separation of the spin singlet from the spin triplet initial states. Naturally 1 &0 (lower left panel of Fig. 17) is now saturated by the Sp 6nal state, but in 3 &0 and 3 &1 other structures appear: the former is now dominated by the transition 3 P0 → 1 S0 s and the latter by 3 P1 → 3 S1 s. 4.5. Status of experiment In this presentation we will be rather brief on details about current as well as planned experiments, as this subject was already covered in recent reviews [1,2]. Here we only wish to give a brief list of observables and reactions that are measured already or are planned to be measured in nucleon–nucleon and nucleon–nucleus-induced reactions. In the case of pion production, measurements with vector- and tensor-polarized deuteron and vector-polarized proton targets and polarized proton beams have been carried out at IUCF [134]. Because of good 4 detection of photons and charged particles, CELSIUS, at least in the near future, is well equipped for studies involving mesons. For the production of heavier mesons,


199

dσ/dε [µb/MeV]

8 σtot

6

3

0.6

σ1

0.5 0.4

4

0.3 0.2

2

0.1 0

0 0

5

10 15 ε [MeV]

20

25

0

5

10

15

20

25

ε [MeV]

7 3

1σ 0

6

0.5

5

σ0

0.4

4

0.3

3 0.2

2

0.1

1 0

0 0

5

10 15 ε [MeV]

20

25

0

5

10 15 ε [MeV]

20

25

Fig. 17. Demonstration of the selectivity of the spin cross sections. Shown are the spectra of the reaction p ˜p ˜ → pn+ as a function of j for an excess energy of 25 MeV. In each panel the solid line shows the full result for the corresponding cross section, the dot–dashed, long-dashed, dashed, and dotted lines show the Sp, Ss, Sd, and Pp contribution, respectively. The curves are from the model of Ref. [71].

ppπo

pnπ+

dπ+ ppη'

pKΛ ΝΝπ

Ν∆

ppω

ppη

ppφ ppao

Ecms

IUCF

pKΣ

ppfo

CELSIUS COSY

Fig. 18. The lowest meson production thresholds for single meson production in proton–proton collisions together with the corresponding energy ranges of the modern cooler synchrotrons.

COSY, due to its higher beam energies and intensities, but due most importantly to the possibility to use polarization, is in a position to dominate the 6eld during the years to come. The energy range of the di@erent cooler synchrotrons is illustrated in Fig. 18.

200


Table 8 List of observables measured for various NN → NNx channels for excess energies up to Q = 40 MeV Channel

&tot

d&=d.

d&=dm

Aoi

Aij

pp → pp0 pp → pn+ pn → pp− pp → d+

[43,44,86–89] [91–93] [77] [95–104]

[90,88,87] [92] [77,74] [95,103,104]

[87–89] [77] —

[89,47] [92] [78,75] [105–107]

[47] [94]

pp → pp pn → pn pn → d

[109–117] [33] [120,114]

[113,112,118]

[113,112]

[119]

pp → pp

[117,29,121]

pp → pK + pp → pK + 0

[122,123,46] [124]

pp → pp! pn → d!

[125] [126]

pp → pp;

[127]

pp → ppf0 =a0

[128]

pp → pp+ − pp → pn+ 0 pp → pp0 0

[129–131] [131] [131]

[129–131] [131] [131]

[129–131] [131] [131]

pp → ppK + K − pp → dK + KG 0

[132] [133]

[133]

[133]

[108]

—

[46]

— [127]

In Table 8 a list is given for the various NN -induced production reactions measured in recent years at SACLAY, TRIUMF, PSI, COSY, IUCF, and CELSIUS in the near-threshold regime (Q ¡ 40 MeV). The corresponding references are listed as well. In the subsections to come we will discuss some examples of the kind of physics that can be studied with the various observables in the many reaction channels. 5. Symmetries and their violation As was already mentioned in several places in this article, symmetries strongly restrict the allowed pattern for various observables. This leads to observable consequences. Naturally, it is therefore also straightforward to investigate the breaking of these symmetries by looking at a violation of these symmetry predictions. Probably, the most prominent example of an experiment for a storage ring is that proposed by the TRI collaboration to be performed at the COSY accelerator. The goal is to do a null experiment


201

in order to put an upper limit on the strength of T -odd P-even interactions via measuring Ay; xz in polarized proton–deuteron scattering, which should vanish if time reversal invariance holds [135]. For details we refer to Ref. [136]. 5.1. Investigation of charge symmetry breaking (CSB) If the masses of the up and down quark were equal, the QCD Lagrangian were invariant under the exchange of the two quark Navors. In reality, these masses are not equal and also the presence of electro magnetic e@ects leads to small but measurable charge symmetry breaking e@ects. Note, the mass di@erence of a few MeV [137] is small compared to the typical hadronic scale of 1 GeV. Quantifying CSB e@ects therefore allows to extract information on the light quark mass di@erences from hadronic observables. As should be clear from the previous paragraph, CSB is closely linked to the isospin symmetry. However, does isospin symmetry demand an independence of the interaction under an arbitrary rotation in isospin space, a system is charge symmetric, if the interaction does not change under a 180◦ rotation in isospin space. Therefore isospin symmetry or charge independence is the stronger symmetry than charge symmetry. For an introduction into the subject we refer to Ref. [5]. The advantage of reactions with only nucleons or nuclei in the initial state is, that one can prepare initial states with well-de6ned isospin. This is in contrast to photon-induced reactions, since a photon has both isoscalar and isovector components. Therefore, in the case of meson photo- or electro-production, CSB signals can only be observed as deviations from some expected signal. (As an example of this reasoning see Ref. [138].) In the case of hadron-induced reactions on the other hand, experiments can be prepared that give a non-vanishing result only in the presence of CSB. This makes the unambiguous identi6cation of the e@ect signi6cantly easier. One complication that occurs if a CSB e@ect is to be extracted from the comparison of two cross sections with di@erent charges in the 6nal state is that of the proper choice of energy variable. Obviously, the quantity that changes most quickly close to the threshold is the phase space, so that it appears natural to compare two reaction channels that have the same phase-space volume. However, due to the di@erences in the particle masses, this calls for di@erent initial energies. In the case of a resonant production mechanism this might lead to e@ects of the same order as the e@ect of interest. In Ref. [139] this is discussed in detail for the reactions pp → d+ and pn → d0 . In this report we will concentrate on the implications of CSB on observables in NN and dd collisions. For details of the mechanisms of CSB we refer to Ref. [5]. In the corresponding class of experiments the deuteron as well as the alpha particle play an exceptional role since as isoscalars they can act as isospin 6lters. The most transparent example of a CSB reaction is dd → ,x ; where x is some arbitrary isovector. The reaction with x = o was recently measured at IUCF for the 6rst time close to the threshold [140]. The initial state as well as the , are pure isoscalars. Thus the 6nal state as an isovector cannot be reached as long as isospin is conserved. In addition pn reactions can be used for producing clean signals of CSB. More generally, whenever a pn pair has a well-de6ned isospin they behave as identical particles. Speci6cally, the di@erential cross section needs to be forward–backward symmetric because nothing should change if beam and

202


x

a0 a0 π

(a)

x

f0 η

π

(b)

Fig. 19. Illustration of di@erent sources of charge symmetry breaking: diagram (a) shows CSB in the production operator through − mixing and diagram (b) shows CSB in the propagation of the scalars. Thin solid lines denote nucleons, thick solid lines scalar mesons and dashed ones pseudo-scalar mesons. The X indicates the occurrence of a CSB matrix element.

target are interchanged. Any deviation from this symmetry is an unambiguous signal of CSB [5]. An experiment performed at TRIUMF recently claimed for the 6rst time a non-vanishing forward– backward asymmetry in pn → d0 [6]. Note that not every forward–backward asymmetry in pn reactions stems from isospin violation. A counterexample was given at the end of Section 4.3, where the di@erential cross section for pn → pp− is discussed in detail. There, in contrast to the previous example, T = 0 and 1 initial states interfere. In case of pion production in nucleon–nucleon collisions it is possible to de6ne a convergent e@ective 6eld theory (see Section 6). Within this theory it is possible to relate e@ects of CSB in these reactions directly to the up–down quark mass di@erence [5,141]. Preliminary studies show, that the relative importance of di@erent CSB mechanisms in the reaction pn → d0 and dd → ,0 are very di@erent [142], and thus it should be possible to extract valuable information on the leading CSB operators from a combined analysis of the two reactions. In the arguments given all that was used was the isovector character of the meson produced. Thus, the same experimental signals will be seen also in pn → d(0 ) [143] and dd → ,(0 ) [144]. In Ref. [67] also the analyzing power was identi6ed as a useful quantity for the extraction of the a0 −f0 mixing matrix element (see also discussion in Ref. [145]). The (0 )s-wave is interesting especially G threshold, since it should give insight into the nature of the light scalar mesons close to the KK a0 (980) and f0 (980) (cf. Section 7.5). Note that the channel is the dominant decay channel of the a0 , which is an isovector–scalar particle. It should be stressed that the charge symmetry breaking signal in case of the scalar mesons is signi6cantly easier to interpret in comparison to the case of pion production. The reason is that the two scalar resonances of interest overlap and therefore the e@ect of CSB as it occurs in the propagation of the scalar mesons is enhanced compared to mixing in the production operator [82]. To make this statement more quantitative we compare the impact of f0 − a0 mixing in the propagation of the scalar mesons (Fig. 19b) to that of − mixing in the production operator (Fig. 19a). We regard the latter as a typical CSB e@ect and thus as a reasonable order-of-magnitude estimate for CSB in the production operator. Observe that the relevant dimensionless quantity for this comparison is the mixing matrix element times a propagator


203

(cf. Fig. 19). In the production operator the momentum transfer—at least close to the production threshold—is given by t = −MN mR , where mR denotes the invariant mass of the meson system produced (or equivalently the mass of the resonance) and MN denotes the nucleon mass. Thus, the appearance of the propagator introduces a factor of about 1=t into the amplitude, since tm2 . On the other hand, the resonance propagator is given by 1=(mR MR ), as long as we concentrate on invariant masses of the outgoing meson system close to the resonance position. Here MR denotes the width of the scalar resonance. Thus we 6nd using MR =50 MeV [146] that the CSB in the production operator is kinematically suppressed by a factor of more than MR =MN ∼ 1=20 as compared to CSB in the propagation of the scalars. In addition the mixing matrix element is enhanced in the case of f0 − a0 mixing (cf. Section 7.5) and therefore it should be possible to extract the f0 − a0 mixing matrix element from NN - and dd-induced reactions.

6. The reaction NN → NN The production of pions in nucleon–nucleon collisions has a rather special role. First of all, it is the lowest hadronic inelasticity for the nucleon–nucleon interaction and thus an important test of our understanding of the phenomenology of the NN interaction. Secondly, since pions are the Goldstone bosons of chiral symmetry, it is possible to study this reaction using chiral perturbation theory. This provides the opportunity to improve the phenomenological approaches via matching to the chiral expansion as well as to constrain the chiral contact terms via resonance saturation. Last but not least, a large number of (un)polarized data is available (cf. Table 8). After a brief history, we continue this chapter with a discussion of a particular phenomenological model for pion production near the threshold, followed by a presentation of recent results from chiral perturbation theory. 6.1. Some history In Section 1.2 it was argued, that for the near-threshold regime the distorted wave born approximation is appropriate, and here we will concentrate on those models that work within this scheme. 19 Pioneering work on pion production was done by Woodruf [147] as well as by Koltun and Reitan [31] in the 1960s. The diagrams included are shown in Figs. 2a and b where, in these early approaches, the N → N transition amplitudes (denoted by T in diagram 2b) were parameterized by the scattering lengths. When the 6rst data on the reaction pp → pp0 close to the threshold were published [43,44], it came as a big surprise that the model of Koltun and Reitan [31] underestimated the data by a factor of 5–10. This is in vast contrast to the reaction pp → pn+ reported in Ref. [91], where the discrepancy was less than a factor of 2. On the other hand, it was shown that the energy dependence of the total cross section can be understood from that of the NN FSI once the Coulomb interaction is properly included [45] (cf. Section 2). 19

In the next section a further argument in favor of the distorted wave born approximation will be given.

204


Niskanen investigated whether the inclusion of the Delta isobar, as well as keeping the rather strong on-shell energy dependence of the N interaction, could help to improve the theoretical results for neutral pion production [148]. Although these improvements lead to some enhancement, the cross section was still missed by more than a factor of 3. The 6rst publication that reported a quantitative understanding of the pp → pp0 data was that by Lee and Riska [149] and later con6rmed by Horowitz et al. [150], where it was demonstrated that short range mechanisms as depicted in Fig. 2c, can give a sizable contribution. However, shortly after this discovery Hern\andez and Oset demonstrated, using various parameterizations for the N → N transition amplitude and qualitatively reproducing earlier work by Hachenberg and Pirner [151] that the strong o@-shell dependence of that amplitude can also be suOcient to remove the discrepancy between the Koltun and Reitan model and the data. Gedalin et al. [152] came to the same conclusion within a relativistic one boson exchange model. In Ref. [153] the N amplitude needed as input for the evaluation of diagram 2b was extracted from a microscopic model. Also there a signi6cant although smaller contribution from the pion rescattering was found. Thus, in this model still some additional short range mechanism is needed. In the succeeding years many theoretical works presented calculations for the pp → pp0 cross section. In Refs. [154,155] covariant one boson exchange models were used in combination with an approximate treatment of the nucleon–nucleon interaction. Both models turned out to be dominated by heavy meson exchanges and thus give further support to the picture proposed in Ref. [149]. However, in Ref. [156] the way that the anti nucleons were treated in Refs. [149,150,154] was heavily criticized: the authors argued that the anti nucleon contributions get signi6cantly suppressed once they are included non-perturbatively. It is interesting to note, that also in Bremsstrahlung the contribution from anti-nucleons in a non-perturbative treatment is signi6cantly reduced compared to a perturbative inclusion [157,158]. Additional short range contributions were also suggested, namely the D−! meson exchange current [159], resonance contributions [160] and loops that contain resonances [161], all those, however, turned out to be smaller compared to the heavy meson exchanges and the o@-shell pion rescattering, respectively. At that time the hope was that chiral perturbation theory might resolve the true ratio of rescattering and short range contributions. It came as a big surprise, however, that the 6rst results for the reaction pp → pp0 [162,163] found a rescattering contribution that interfered destructively with the direct contribution (diagram a in Fig. 2), making the discrepancy with the data even more severe. In addition, the same isoscalar rescattering amplitude also worsened the discrepancy in the + channel [164,165]. Some authors interpreted this 6nding as a proof for the failure of chiral perturbation theory in these large momentum transfer reactions [155,166]. Only recently was it demonstrated, that it is possible to appropriately modify the chiral expansion in order to make it capable of analysing meson production in nucleon–nucleon collisions. We will report on those studies in Section 6.3 that in the future will certainly prove useful for improving the phenomenological approaches (cf. Section 6.5). Before we close this section a few remarks on the o@-shell N amplitude are necessary. For this purpose we write the relevant piece of the N interaction T -matrix in the following form TN = −t (+)

1 † 1 † N · N + t (−) N % · ( × )N ˙ ; 2 2m

(70)


205

where t (+) (t (−) ) denote the isoscalar (isovector) component. Note that it is only the former that can contribute to the reaction pp → pp0 , 20 for the isospin structure of the latter changes the total isospin of the two-nucleon system. As long as we neglect the distortions due to the 6nal and initial state interactions, what is relevant for the discussion in this paragraph is the half o@-shell N amplitude. We thus may write t =t(s; k 2 ), where s denotes the invariant energy of the N system and k 2 is the square of the four momentum of the incoming pion. At the threshold for elastic N scattering (s = (m + MN )2 , k 2 = m2 ) we may write m 4 (+) 2 1 1+ t (s0 ; m ) := (a1 + 2a3 ) = (−0:05 ± 0:01)m− ; 3 MN m 4 (− ) 2 1 1+ t (s0 ; m ) := (a1 − a3 ) = (1:32 ± 0:02)m− (71) ; 3 MN where the a2I denote the scattering lengths in the corresponding isospin channels I . The corresponding values were extracted from data on − d atoms in Ref. [167]. Note that the dominance of the isovector interaction is a consequence of the chiral symmetry: the leading isoscalar rescattering is suppressed by a factor m =MN compared to the leading isovector contribution—the so-called Weinberg–Tomozawa term [168,169]. However, it is still remarkable, that also the higher order chiral corrections are small leaving a value consistent with zero for the isoscalar scattering length. A detailed study showed, that this smallness is a consequence of a very eOcient cancellation of several individually large terms that are accompanied with di@erent kinematical factors [170]. To be concrete: to order O(p2 ) one 6nds 2 gA2 (+) 2 t = 2 −2m c1 + q0 k0 c2 − + (q k)c3 ; (72) f 8MN where k and q denote the four momentum of the initial and 6nal pion, respectively. The values for the various ci are given in Table 9. For on-shell scattering at the threshold (q = k = (m ; ˜0)) 1 one gets t (+) (s0 ; m2 ) = −0:24m− using the values of Ref. [171]. 21 Please note, that the linear combination of the ci appearing above turns out to be an order of magnitude smaller than the individual values. As a consequence, the on-shell isoscalar amplitude shows a rather strong energy dependence above threshold. It should not then come as a surprise, that the transition amplitude corresponding to Eq. (70), when evaluated in the kinematics relevant for pion production in NN collisions, within chiral perturbation theory turns out to be rather large numerically [162,163]. For the non-covariant expression given in Eq. (72) this translates into q = (m ; ˜0) and k = (m =2; ˜k), leading 1 to t (+) ((m + MN )2 ; −MN m ) = 0:5m− . This is why the 6rst calculations using chiral perturbation theory found a big e@ect from pion rescattering—unfortunately increasing the discrepancy with the data. In Ref. [173] the tree level chiral perturbation theory calculations were repeated using a di@erent prescription for the energy of the exchange pion. 22 The authors found agreement with the data, but with a sign of the full amplitude opposite to the one of the direct term. 20

This is only true if we do not include the C isobar explicitly, as will be discussed in the next section. Note, this value is inconsistent with the empirical value given in Eq. (71). To come to a consistent value one has to go to one loop order as discussed in Ref. [172]. 22 This prescription was later criticized in Ref. [174]. 21

206


One year earlier it was shown, that within phenomenological approaches the isoscalar transition amplitude evaluated in o@-shell kinematics is also signi6cantly di@erent from its on-shell value. For example, in the Jûlich meson exchange model it is the contribution from the iterated D t-channel exchange and the & exchange that are individually large but basically cancel in threshold kinematics in the isoscalar channel [153]. This cancellation gets weaker away from the threshold point. This rescattering contribution, however, turned out to interfere constructively with the direct term. Since chiral perturbation theory as the e@ective 6eld theory for low energy strong interactions is believed to be the appropriate tool to study pion reactions close-to-threshold, it seemed at this stage as if there were a severe problem with the phenomenology. However, as was shown in Section 4.3, there are observables that are sensitive to the sign of the s-wave pp → pp0 amplitude—relative to a p-wave amplitude that is believed to be under control—and the experimental results [74,75,77,78] agree with the sign as given by the phenomenological model (cf. Fig. 15) . Does this mean that chiral perturbation theory is wrong or not applicable? No. As we will discuss in the subsequent sections, it was demonstrated recently that the chiral counting scheme needs to be modi6ed in the case of large momentum transfer reactions. No complete calculation has been carried out up to now, but intermediate results look promising for a consistent picture to emerge in the years to come. The insights gained so far from the e@ective 6eld theory studies call also for a modi6cation of the phenomenological treatment. This will be discussed in detail in Section 6.5. 6.2. Phenomenological approaches As stressed in the previous section, the number of phenomenological models for pion production is large. For de6niteness in this section we will focus on one particular model, namely that presented in Refs. [71,73,153], mainly because it incorporates most of the mechanisms proposed in the literature for pion production in nucleon–nucleon collisions, its ingredients are consistent with the data on N scattering, and it is the only model so far whose results have been compared to the polarization data recently measured at IUCF [47,94,108]. Results of the model for the various total cross sections are shown in Fig. 20. The model is the 6rst attempt to treat consistently the NN as well as the N interaction for meson production reactions close to the threshold: both were taken from microscopic models (described in Refs. [175,176] for the NN and the N interaction, respectively). These were constructed from the same e@ective Lagrangians consistent with the symmetries of the strong interaction and are solutions of a Lippman–Schwinger equation based on time-ordered perturbation theory. Although not all parameters and approximations used in the two systems are the same, this model should still be viewed as a benchmark calculation for pion production in NN collisions within the distorted wave Born approximation. We will start this section with a description of the various ingredients of the model and then present some results. 6.2.1. The NN interaction A typical example of a so-called realistic model for NN scattering is the Bonn potential [26]. The model used for the NN distortions in the 6nal and initial states is based on this model, where a pseudo-potential is constructed based on the t-channel exchanges of all established mesons below one GeV in mass between both nucleons and Delta isobars.


207

101 pp→pnπ+

σtot [mb]

100

pn→ppπ−

10-1 10-2 10-3 10-4 pp→dπ+

pp→ppπ0

10-1 10-2 10-3 10-4 0.1

1

0.1

η = qπ(max)/mπ

1

Fig. 20. Comparison of the model predictions of Ref. [73] to the data. The references for the experimental data can be found in Table 8. The solid lines show the results of the full model; the dashed line shows the results without the C contributions.

The interaction amongst the various dynamical 6elds in the model derived from the following Lagrange densities: fNN G (73) 5 4 · 94 ; LNN = m fNND G LNND = gNND G 4 · 4 + (74) &4O · (94 O − 9O 4 ) ; 4MN LNN! = gNN! G 4 !4 LNN& = gNN& G &

;

(75)

;

LNNa0 = gNNa0 G · a0

(76) ;

fN S G ˜ 4 T · 9 4 + h:c: ; m fN SD G ˜ · (94 O − 9O 4 ) LN SD = i 5 4 T m

(77)

LN S =

(78) O

+ h:c :

(79)

Note: The particle called 3 in the original Bonn publication [26] is nowadays called a0 . The ˜ as well as the 6elds are de6ned in Ref. [26]. Note that there is no tensor coupling for operator T the !NN vertex given for the 6t to the elastic NN scattering data did not need any such coupling.

208


60

δ [degrees]

40

1S

20

20

0

0

-20

-20

-40

-40 3P

0

3

40

0

S1

3P

-10

0

1

-20 -20 -30 -40

-40 3P

1

2

D2

15 20 10 10

5 0

0 0

200

400

ELAB [MeV]

600

0

200

400

600

ELAB [MeV]

Fig. 21. The NN phase shifts for the model of Ref. [175] in the energy range relevant for pion production. The pion production threshold is at ELAB = 286 MeV. The experimental data are from Refs. [177] (triangles) and [61] (circles).

There is a di@erence between the nucleon–nucleon model used [175] and the Bonn potential [26]. The original Bonn model has an energy-dependent interaction, for it keeps the full meson retardation in the intermediate state. This, however, leads to technical problems, once the model is to be evaluated above the pion production threshold due to the occurrence of three-body singularities. In Ref. [24] those singularities were handled by solving the dynamical equations in the complex plane. Unfortunately, this method is not useful for the application in a distorted wave Born approximation. Instead, we used a model based on the so-called folded diagram formalism developed in Ref. [178]. This formalism, worked out to in6nite order, is fully equivalent to time-ordered perturbation theory. When truncated at low order, however, it leads to energy independent potentials that can formally be evaluated even above the pion production threshold. In addition, the model of Ref. [175] is constructed as a coupled-channel model including the NN as well as the NC and CC channels. This enables us to treat the C isobar on equal footing with the nucleons. The resulting phase shifts are shown in Fig. 21. Note that the model parameters were adjusted to the phase shifts below the pion production threshold only, which is located at ELAB = 286 MeV.

C. Hanhart / Physics Reports 397 (2004) 155 – 256 π

N

N

N (a)

π

N

∆

N

π

π

N

π

N

π

∆

π

N (c)

(b)

N

209

N

π

(d) π

N

V

Tππ

J=0,1 π

N (e)

Fig. 22. (a)–(e) Contributions to the potential of the model of Ref. [179].

Fig. 21 thus clearly illustrates that using this model for the NN interaction is indeed justi6ed and we may conclude that—at least up to laboratory energies of 600 MeV—the NN phenomenology is well understood. 6.2.2. The N model The N interaction that enters in the pion rescattering diagrams can be taken from a meson exchange model as well [179]. This allows a consistent treatment of the meson and nucleon dynamics. It should be stressed that this is the precondition for comparing the results of the phenomenological model to those of chiral perturbation theory, as we will do below. In addition, since we also want to include the rescattering diagram in partial waves higher than the s-wave, after a 6t to the N data the pole contributions (nucleon and Delta) need to be removed from the amplitudes in order to avoid double counting with the direct production. This is possible only within a microscopic model. The main features of the N model of Ref. [179] are that it is based on an e@ective Lagrangian consistent with chiral symmetry to leading order and that the t-channel exchanges in the isovector (D) and isoscalar (&) channel are constructed from dispersion integrals. 23 For details on how the t-channel exchanges are included we refer to Ref. [179]. The diagrams that enter the potential are displayed in Fig. 22. This potential is then unitarized with a relativistic Lippmann–Schwinger equation—in complete analogy to the nucleon–nucleon interaction. Within this model, at tree level the isoscalar and isovector N interaction are given by the corresponding t-channel exchanges. In the latter case the unitarization does not have a big inNuence in the near-threshold regime, and thus also the isovector scattering length is governed by the tree 23

There are ambiguities in how to extrapolate the results of the dispersion integrals to o@-shell kinematics. This issue is discussed in detail in Ref. [180].

210

C. Hanhart / Physics Reports 397 (2004) 155 – 256 0.0

20.0 S11 15.0

-5.0

10.0

-10.0

5.0

-15.0

0.0

-20.0

δ [degrees]

1.0

P11

-1.0

S31

P31

-2.0 0.0 -3.0 -1.0

-4.0

-2.0

-5.0 P13

80.0

-1.0

P33

60.0 -2.0 40.0 -3.0 -4.0 1050

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Ecms [MeV]

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0.0 1050

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Fig. 23. The N phase shifts for the model of Ref. [179] in the energy range relevant for pion production. The experimental data are from Refs. [183,184]. Note the di@erent scales of the various panels.

level D-exchange. The famous KFSR relation [181,182], that relates the couplings of the D-meson to pions and nucleons to the coupling strength of the Weinberg–Tomozawa term, is a consequence of this. On the other hand, for the isoscalar N interaction the e@ects of the unitarization are large and lead to an almost complete cancellation of the isoscalar potential with the iterated D-exchange in the near-threshold regime. As one moves away from the threshold value this cancellation gets weaker leading to the strong variation of the o@-shell isoscalar N T -matrix mentioned at the end of Section 6.1. In Fig. 23 the results for the model of Ref. [179] are compared to the data of Refs. [183,184]. As one can see the model describes the data well, especially in the most relevant partial waves: S11 ; S31 and P33 . 6.2.3. Additional short-range contributions and model parameters As was stressed above, a large class of additional mechanisms was suggested in the literature to contribute signi6cantly to pion production in nucleon–nucleon collisions. Since they are all of rather

C. Hanhart / Physics Reports 397 (2004) 155 – 256 pp→dπ+

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short range and mainly inNuence the production of s-wave pions, in this work only a single diagram was included (heavy meson exchange through the !—Fig. 2c) to parameterize these various e@ects. Consequently, the strength of this contribution was adjusted to reproduce the total cross section of the reaction pp → pp0 close to the production threshold. The short-range contributions turn out to contribute about 20% to the amplitude. After this is done, all parameters of the model are 6xed. 6.2.4. Results The results of the model presented have already appeared several times in this report (Figs. 13, 14, 15, 17, 20 and 24)—mainly for illustrative purposes—and they are discussed in detail in Refs. [73,71]. Overall the model is rather successful in describing the data, given that only one parameter was adjusted to the total cross section for low-energy neutral pion production (see Section 6.2.3). One important 6nding is that the sign of the s-wave neutral pion production seems to be in accord with experiment, as is illustrated in Fig. 15 (cf. corresponding discussion in Section 4.3), in contrast to the early calculations using chiral perturbation theory. As we will see in Section 6.3, today we know that those early calculations using e@ective 6eld theory were incomplete. The most striking di@erences appear, however, for double polarization observables in the neutral pion channel, as shown in Figs. 13 and 14. As a general pattern the amplitudes seem to be of

212


the right order of magnitude, but show a wrong interference pattern. To actually allow a detailed comparison of the model and data, a partial wave decomposition of both is necessary. Work in this direction is under way. It is striking that for charged pion production the pattern is very di@erent, for here almost all observables are described satisfactorily. In Fig. 24 the results of the model for charged and neutral pion production are compared to the data for a few observables. In contrast to the neutral channel, the charged pion production is completely dominated by two transitions, namely 3 P1 → 3 S1 s, which is dominated by the isovector pion rescattering, and 1 D2 → 3 S1 p, which governs the cross section especially in the regime of the Delta resonance. The prominence of the Delta resonance is a consequence of the strong transition 1 D2 (NN ) → 5 S2 (NC) that even shows up as a bump in the NN phase shifts (see Fig. 21). This e@ect was 6rst observed long ago and is well known (see, e.g., discussion in Ref. [185]). As a consequence, the NN → NC transition potential should be rather well constrained by the NN scattering data and is not the case for all the many transitions relevant in case of the neutral pion production, where the Sp 6nal state is not allowed due to selection rules (see Section 1.6). One might therefore hope to learn more about the SN interaction from the pion production data. One can also ask how well we know the production operator. Fortunately, at the pion production threshold it is still possible to analyze meson production in NN collisions within e@ective 6eld theory. This analysis will give deeper insight into the production dynamics, as will be explained in the following section. The two approaches are then compared in Section 6.5. 6.3. Chiral perturbation theory The phenomenological approaches, such as the one described in the previous section, lack a systematic expansion. Thus it is neither possible to estimate the associated model uncertainties nor to systematically improve the models. On the other hand, for various meson production reactions the phenomenological approaches proved to be quite successful. One might therefore hope that e@ective 6eld theories will give insights into why the phenomenology works, as was 6rst stressed by Weinberg [186]. A 6rst attempt to construct—model independently—the transition amplitude NN → NN was carried out almost 40 years ago [187–189]. The authors tried to relate what is known about nucleon– nucleon scattering to the production amplitude via the low-energy theorem of Adler and Dothan [190], which is a generalization of the famous Low theorem for conserved currents [191] to partially conserved currents via the PCAC relation (see, e.g., Ref. [192]). However, it soon was realized that the extrapolation from the chiral limit (m = 0) to the physical point can change the hierarchy of di@erent diagrams. The reason for the non-applicability of soft radiation theorems to meson production close to the threshold is easy to see: a necessary condition for the soft radiation theorem to be applicable is that the energy emitted is signi6cantly smaller than the typical energy scale, that characterizes variations of the nuclear wavefunction. In the near-threshold regime, however, the scale of variation is set by the inverse of the NN scattering lengths—thus a soft radiation theorem of the type of Low or that of Adler and Dothan could only be applicable to meson production if m 1=(MN a2 ). In reality, however, the pion mass exceeds the energy scale introduced by inverse NN scattering length by more than two orders of magnitude. The range of applicability of soft radiation theorems is discussed in Ref. [193] in a di@erent context.


213

More recent analyses, however, show that in the case of pion production it is indeed possible to de6ne a convergent e@ective 6eld theory that allows a systematic study of the structure of the production operator. As we will show, most of the diagrams included in the Koltun and Reitan model [31] are indeed the leading operators in pion production. In addition this study will show • • • •

that the use of the distorted wave born approximation is justi6ed, why neutral pion production is the more problematic case, the importance of loop contributions, that there is a close connection between pion production in nucleon–nucleon collisions and the three nucleon problem.

The problem with strong interaction phenomena is their non-perturbative nature with respect to the coupling constants. To construct a controlled expansion it is necessary, to identify a small expansion parameter. In general this is only possible for a limited energy range. The conditio sine qua non-for constructing an e@ective 6eld theory for any system is the separation of scales characteristic for the system. Once the—in this context light—scales are identi6ed, one treats them dynamically, while all the dynamics that are controlled by the heavy scales are absorbed in contact interactions. As long as the relevant external momenta and energies are such that structures of the size of the inverse of the heavy scale cannot be resolved, this procedure should always work. Weinberg [194] as well as Gasser and Leutwyler [195] have shown that this general idea works even when loops need to be included. In case of low-energy pion physics it is the chiral symmetry that provides both preconditions for the construction of an e@ective 6eld theory, in that it forces not only the mass of the pion m , as the Goldstone boson of the chiral symmetry breaking, to be low, but also the interactions to be weak, for the pion needs to be free of interactions in the chiral limit for vanishing momenta. The corresponding e@ective 6eld theory is called chiral perturbation theory (HPT ) and was successfully applied to meson–meson [196] scattering. Treating baryons as heavy allows straightforward extension of the scheme to meson–baryon [197] as well as baryon–baryon [198–201] systems. In all these references the expansion parameter used was p= H ∼ p=(4f ) ∼ p=MN , where H denotes the chiral symmetry breaking scale, f the pion decay constant and MN the nucleon mass. Recently, it was shown that also the Delta isobar can be included consistently in the e@ective 6eld theory [202]. The authors treated the new scale, namely the Delta nucleon mass splitting C = MC − MN , to be of the order of p that is taken to be of the order of m . An additional new scale √ occurs for meson production in nucleon–nucleon collisions, namely the initial momentum pi ∼ m MN . Note that, although larger than the pion mass, this momentum is still smaller than the chiral symmetry breaking scale and thus the expansion should still converge, but slowly. A priori there are now two options to construct an e@ective 6eld theory for pion production. The 6rst option, called Weinberg scheme in what follows, treats all light scales to be of order of m . Thus, in this case, there is one expansion parameter, namely HW = m =MN . The other option is to expand in two scales simultaneously, namely m and pi . In this case the expansion parameter is m ∼ 0:4 : H= MN

214


This scheme was advocated in Refs. [163,203] and applied in Ref. [204]. The two additional scales, namely C and m , are identi6ed with C pi m p2 ∼ = H and ∼ i2 = H2 ; (80)

H

H

H

H where the former assignment was made due to the numerical similarity of the two numbers 24 (C = 2:1m and pi = 2:6m ). Only explicit calculations can reveal which one is the more appropriate approach. Within the Weinberg counting scheme, tree level calculations were performed for s-wave pion production in the reactions pp → pp0 [162,163,173] as well as pp → pn+ [164,165]. In addition, complete calculations to next-to-next-to-leading order (NNLO), where in the Weinberg scheme for the 6rst time loops appear, are available for pp → pp0 [205,206]. The authors found that some of the NNLO contributions exceeded signi6cantly the next-to-leading (NLO) terms leading them to the conclusion that the chiral expansion converges only slowly, if at all. This point was further stressed in Refs. [207,155], however, it was shown recently that as soon as the scale induced by the initial momentum is taken into account properly (expansion in H and not in HW ), the series indeed converges [203,204]. For illustration, in this section we compare the order assignment of the Weinberg scheme to that of the modi6ed scheme. In Appendix E the relevant counting rules of the new scheme are presented as well as justi6ed via application to a representative example. Especially, it is not clear a priori what scale to assign to the zeroth component l0 of the four dimensional integration volume d4 l as it occurs in covariant loops. After all, the typical energy of the system is given by m , but the momentum by pi . As is shown in the appendix by matching a covariant analysis to one carried out in time-ordered perturbation theory, in loops l0 ∼ pi . This assignment was also con6rmed in explicit calculations [204]. The starting point is an appropriate Lagrangian density, constructed to be consistent with the symmetries of the underlying more fundamental theory (in this case QCD) and ordered according to a particular counting scheme. Omitting terms that do not contribute to the order we will be considering here, we therefore have for the leading order Lagrangian [208,199,197] 1 1 1 1 2 2 2 2 L(0) = 94 94 − m2 2 + m ( · 9 ) − 4 2 2 2f2 4 1 † · ( × ) ˙ N + N i90 − 4f2 gA † 1 ˜ ˜ + N · ˜& · ∇ + ( · ∇) N 2f 2f2 hA ˜ [N † (T · ˜S · ∇)0 (81) C + h:c:] + · · · : 2f The expressions for interactions with more than two pions depend on the interpolating 6eld used. The choice made here was the so called sigma gauge—cf. Appendix A of Ref. [197], where also the corresponding vertex functions are given explicitly. 25 + 0C† [i90 − C]0C +

24 25

Note that C stays 6nite in the chiral limit, whereas both pi as well as m vanish. As usual, all observables are independent of the choice of the pion 6eld.


215

For the next-to-leading order Lagrangian we get L(1) =

1 ˜ 2 N + 0† ∇ ˜ 2 0C ] [N † ∇ C 2mN 1 ˜ · ∇N ˜ + h:c:) (iN † · ( × ∇) 8MN f2 1 g2 ˜ 2 − 2c1 m2 2 + 2 N † c2 + c3 − A ˙2 − c3 (∇) f 8mN 1 1 c4 + jijk jabc &k %c 9i a 9j b N − 2 4mN +

−

gA ˜ + h:c:] − hA [iN † T · ˙˜S · ∇0 ˜ C + h:c:] [iN † · ˜ ˙& · ∇N 4mN f 2mN f

−

d2 d1 † ˜ N ( · ˜& · ∇)N N †N − jijk jabc 9i a N † &j %b N N † &k %c N + · · · ; f 2f

(82)

where f denotes the pion decay constant in the chiral limit, gA is the axial-vector coupling of the nucleon, hA is the SN coupling, and ˜S and T are the transition spin and isospin matrices, normalized such that Si Sj† = 13 (23ij − ijijk &k ) ;

(83)

Ti Tj† = 13 (23ij − ijijk %k ) :

(84)

These de6nitions are in line with the ones introduced in Section 6.2.1. The dots symbolize that what is shown are only those terms that are relevant for the calculations presented. As demanded by the heavy baryon formalism, the baryon 6elds N and 0C are the velocity-projected pieces of the relativistic 6elds appearing in the interactions discussed in Section 6.2.1; e.g. N = 12 (1 + v=) , where v4 denotes the nucleon 4-velocity. The terms in the Lagrangians given are ordered according to the conventional counting (p m ). A reordering on the basis of the new scheme does not seem appropriate, for what order is to be assigned to the energies and momenta occurring depends on the topology of a particular diagram (see also Appendix E). The constants ci can be extracted from a 6t to elastic N scattering. This was done in a series of papers with successively improved methods [171,172,209–212]. However, here we will focus on the values extracted in Refs. [171,172] for it is those that were used in the calculations for pion production in nucleon–nucleon collisions. In the former work the ci were extracted at tree level and in the latter to one loop. The corresponding values are given in Table 9. In both papers the Delta isobar was not considered as explicit degree of freedom. In an e@ective 6eld theory the low-energy constants appearing depend on the dynamical content of the theory. Thus, in a theory without explicit Deltas, their e@ect is absorbed in the low-energy constants [172]. This is illustrated graphically in Fig. 25. Thus we need to subtract the Delta contribution from the values given in the 6rst columns

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Table 9 The various low-energy constants ci in units of GeV−1 i

citree

ciloop

citree (C=)

ciloop (C=)

1 2 3 4

−0.64 1.78 −3.90 2.25

−0.93 3.34 −5.29 3.63

−0.64 0.92 −1.20 0.9

−0.93 0.64 −2.59 2.28

The 6rst two columns give the values from the original references (left column: tree level calculation of Ref. [171]; right column: one loop calculation of Ref. [172]), whereas the last two columns give those values reduced by the Delta contribution as described in the text. It is those numbers that where used in the calculations presented here.

with ∆

without ∆

Fig. 25. Illustration of resonance saturation.

of Table 9. Analytical results for those contributions are given in Refs. [213,172]: 26 c2C = −c3C = 2c4C =

h2A = 2:7 GeV−1 : 9(MC − MN )

(85)

Thus, the only undetermined parameters in the interaction Lagrangian are d1 and d2 . Since they are the strength parameters of 4-nucleon contact interactions that do not contain any derivatives of the nucleon 6elds, they only contribute to those amplitudes that have an NN S-wave to NN S-wave plus pion p-wave transition. This automatically excludes a transition from an isospin triplet to an isospin triplet state, for this would demand to go from 1 S0 to 1 S0 accompanied by a p-wave pion, which is forbidden by conservation of total angular momentum. Thus, only two transitions are possible: T = 0 → T = 1, as it can be studied in pn → pp− , and T = 1 → T = 0, as it can be studied in pp → pn+ . Even more importantly, in both channels the di appear with the same linear combination d which is completely 6xed by the corresponding isospin factors: d=

1 3 (d1 + 2d2 ) ∼ 2 ; 3 f MN

(86)

where we have introduced the dimensionless parameter 3. In order for the counting scheme to work, 3 = O(1) has to hold. As we will show below, this order of magnitude is indeed consistent with the data. Please note, the four-nucleon–pion contact term with p-wave discussed here was to some extend also investigated in Ref. [214], however, within a di@erent expansion scheme. Thus direct comparison is not possible. 26

√ To match the results of the two references the large NC value has to be used for hA = 3gA = 2 2:7.


(a)

217

(b)

Fig. 26. (a)–(b) Illustration of the role of the 4N contact term in NN → NN and three nucleon scattering. Solid lines denote nucleons, dashed lines denote pions.

The parameter d is very interesting, for it is at the same time the leading short-range–long-range contribution 27 to the three nucleon force, as illustrated in Fig. 26. Naturally, it can also be 6xed from pd scattering data directly. This was done in Ref. [215]. We come back to this point below. The e@ect of the large scale on the vertices is best illustrated with an example. In L(0) the so-called Weinberg–Tomozawa (WT) term 1=(4f2 )N † · ( × )N ˙ appears. The corresponding recoil ˜ · ∇N ˜ + h:c:) appears in L(1) . Therefore the vertex function correction 1=(8MN f2 )(iN † · ( × ∇) derived from the WT term is proportional to (q0 +k0 )=f2 , where q0 (k0 ) denote the zeroth component of the 4-vectors for the outgoing (incoming) pion. If the WT term appears as the N vertex in the rescattering contribution (cf. Fig. 2b), in threshold kinematics k0 = m =2 and q0 = m . Also in threshold kinematics, where the three momentum transfer equals the initial momentum, we 6nd for the contribution from the recoil term p2 =(2MN f2 ). Obviously, since p2 = MN m , the contribution from the WT term and from its recoil term are of the same order. This changes if the WT term appears inside a loop, for then the scale for k0 is also set by p—in this case the recoil term is suppressed by one chiral order compared to the WT term itself. The assignments made are con6rmed by explicit calculation [204] as well as by a toy model study [174]. Since we know now the interactions of pions and nucleons we can investigate the relevance of sub-leading loops, where we mean loops that can be constructed from a low order diagram by inserting an additional pion line. Obviously, this procedure introduces at least one NN vertex ∼ p=f , a pion propagator ∼ 1=p2 , an integral measure p4 =(4)2 and either an additional NN vertex together with two-nucleon propagators, or one additional nucleon propagator together with a factor 1=f . To get the leading piece of the loops in the latter case, the integration over the energy variable has to pick the large scale and thus we are to count the nucleon propagator as 1=p, leading to an overall suppression factor for this additional loop of p2 =MN2 . In the former case, however, a topology is possible that contains a two-nucleon cut. This unitarity cut leads to an enhancement of that intermediate state, for it pulls a large scale (MN ) into the numerator. Based on this observation Weinberg established a counting scheme for NN scattering that strongly di@ers from that in 27

In this context, pion exchanges are called long ranged, whereas any exchange of heavier mesons—absorbed in the contact terms—are called short ranged.

218


as well as N scattering [186]. The corresponding factor introduced by this loop is p=MN (and in addition typically comes with a factor of ). As was stressed by Weinberg, this suppression is compensated by the size of the corresponding low energy constants of the two-nucleon interaction and therefore all diagrams that can be cut by crossing a two-nucleon line only are called reducible and the initial as well as 6nal state NN interaction is summed to all orders. This is what is called distorted wave Born approximation. There is one exception to this rule: namely when looking at the two-nucleon intermediate state close to the pion production vertex in diagram a of Fig. 2. Either the incoming or the outgoing nucleon needs to be o@-shell and thus this intermediate state does not allow for a two-nucleon unitarity cut. Therefore those two-nucleon intermediate states are classi6ed as irreducible. There is a special class of pion contributions not yet discussed, namely those that contain radiative pions (on-shell pions in intermediate states). We restrict ourselves to a kinematic regime close to the production threshold. As soon as an intermediate pion goes on-shell, the typical momentum in the corresponding loop automatically needs to be of order of the outgoing momenta. This leads to an e@ective suppression of radiative pions; e.g., for s-wave pions the e@ects of pion retardation become relevant at N5 LO. Note that each loop necessarily contributes at many orders simultaneously. The reason for this is that the two scales inherent to the pion production problem can be combined to a dimensionless number smaller than one: m =pi = H. However, what can always be done on very general grounds is to assign the minimal order at which a diagram can start to contribute and this is suOcient for an eOcient use of the e@ective 6eld theory. The next step is the consistent inclusion of the nuclear wavefunctions. However, the NN potentials constructed consistently with chiral perturbation theory [198–201] are not applicable at the pion production threshold. Therefore, we use the so-called hybrid approach originally introduced by Weinberg [216], where we convolute the production operator, constructed within chiral perturbation theory, with a phenomenological NN − NC wavefunction. We use the CCF model described in the previous section [175]. Let us start with a closer look at the production of p-wave pions, for those turned out easier to handle than s-wave pions. The reason for this pattern lies in the nature of pions as Goldstone bosons of the chiral symmetry: since in the chiral limit for vanishing momenta the interaction of pions with matter has to vanish, the coupling of pions naturally occurs in the company of either a derivative or an even power of the pion mass. 28 As a consequence, the leading piece of the NN vertex is of p-wave type, whereas the corresponding s-wave piece is suppressed as ! =MN , where ! denotes the pion energy. Note that also in the case of neutral pion photoproduction the s-wave amplitude is dominated by pion loops, whereas the p-wave amplitude is dominated by tree level diagrams [218,197]. The corresponding diagrams are shown in Table 10 up to N3 LO. So far in the literature calculations have been carried out only up to N2 LO [203]. An important test of the approach is to show its convergence. For that we need an observable to which s-wave pions do not contribute. Such an observable is given by the spin cross section 3 &1 recently measured at IUCF for the reaction 28

Only even powers of the pion mass are allowed to occur in the interaction, since, due to the Gell–Mann–Oakes– Renner relation [217], m2 ˙ mq , where mq is the current quark mass and in the interaction no terms non-analytic in the quark masses are allowed.


219

Table 10 Comparison of the corresponding chiral order in the Weinberg scheme (p ∼ m ) and the new counting scheme √ (p ∼ m MN ) for several nucleonic contributions for p-wave pion production p-Wave diagrams (nucleons only)

. .

. .

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p∼

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LO

LO

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LO

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N2 LO

pq=MN2

N2 LO

N3 LO

m M N

.

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.

√

Scale

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. c)

.

.

. .

.

d)

g)

f)

e)

. . . .

. . .

.

. . h)

.. . . ..

. i)

j)

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pp → pp0 [47] (cf. Table 7 in Section 4.2). The parameter-free prediction of chiral perturbation theory compared to the data is shown in Fig. 27. As one can see, the total amplitude is clearly dominated by the leading order suggesting a convenient rate of convergence for the series. In addition, the prediction agrees with the data. Thus we are now prepared to extract the parameter d from data on the reaction pp → pn+ [219]. As it was argued above, only the amplitude corresponding to the transition 1 S0 → 3 S1 p, called a0 , is inNuenced by the corresponding contact interactions. The results of the chiral perturbation theory calculations are shown in Fig. 28. The 6gure shows four curves for di@erent values of the parameter 3 de6ned in Eq. (86), namely the result for 3 = 0 (dot–dashed line), for 3 = −0:2—the authors of Ref. [220] claim this value to yield an important contribution to Ay in Nd scattering at energies

220


40

3

σ1 [µb]

60

20

0 0.5

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0.7

η

0.8

0.9

1

Fig. 27. Comparison of the predictions from e@ective 6eld theory at LO (dashed line) and NLO (with the N parameters from a NLO (dotted line) and an NNLO (solid line) analysis) with the 3 &1 cross section for pp → pp0 [47]. (Reproduced from Phys. Rev. Lett. with permission from AIP.)

1

a0 [µb1/2]

0.5

0

-0.5

-1 0.1

0.3

0.5

0.7

η

Fig. 28. a0 of pp → np+ in chiral perturbation theory. The di@erent lines correspond to values of the parameter related to the three-nucleon force: 3 = 1 (long-dashed line). 3 = 0 (dot–dashed line), 3 = −0:2 (solid line), and 3 = −1 (short-dashed line). Data are from Ref. [219].

of a few MeV 29 —as well as the results we get when 3 is varied within its natural range 3 = +1 and 3 = −1 shown as the long-dashed and the short-dashed curve, respectively. Thus we 6nd that the results for a0 are indeed rather sensitive to the strength of the contact interaction. This might be surprising at 6rst glance, since we are talking about a sub-leading operator, however it turned 29

The calculation of Ref. [220] su@ers from numerical problems.


221

out that the leading (diagrams (a) and (b) in Table 10) and sub-leading contributions (the same diagrams with a C intermediate state) largely cancel. Thus, to draw solid conclusions the p-wave calculations should be improved by one chiral order, the corresponding diagrams of which are shown in Tables 10g–j. Please note that the rescattering contribution involving c4 that occurs at the same order as the d contribution turned out to be sensitive to the regulator used for the evaluation of the convolution integral with the nuclear wavefunction. This cuto@ dependence can be absorbed in d, which in turn is now cuto@-dependent. Therefore, in order to compare the results for d from the pion production reaction to those extracted from the three-nucleon system [215] a consistent calculation that is not possible at present has to be performed. Note that the d parameter as 6xed in Ref. [215] (there it is called E) also turned out to be sensitive to the regulator. On the long run, however, a consistent description of pion production and three-nucleon scattering should be a rather stringent test of chiral e@ective 6eld theories at low and intermediate energies. As should be clear from the discussion above, to yield values for the low-energy constant 3 that are compatible both calculations pd scattering as well as -production have to be performed using the same dynamical 6elds—at present the C-isobar is not considered as explicit degree of freedom in Ref. [215], but plays a numerically important role in the extraction of 3 from NN → NN. In Section 4.3 it was shown that the di@erential cross section as well as the analysing power for the reaction pn → pp− is sensitive to an interference term of the s-wave pion production amplitude of the A11 amplitude (3 P0 → 1 S0 s) and the p amplitudes of A01 : 3 S1 → 1 S0 p and 3 D1 → 1 S0 p. Obviously, the 4-nucleon contact interaction contributes to the former. Thus, once a proper chiral perturbation theory calculation is available for the s-wave pion production—whose status will be discussed in the subsequent paragraphs—the reaction pn → pp− close to the production threshold might well be the most sensitive reaction to extract the parameter d. Let us now turn to the s-wave contributions. The leading diagrams containing nucleons only are shown in Table 11; those that contain the C are shown in Table 12. Again, the Weinberg scheme and the new scheme are compared. The list is complete up to NNLO in both schemes. Please note, however, that one class of diagrams (k and l) that is of NLO in the Weinberg counting in the new scheme is pushed to N4 LO! Thus in the new counting scheme—contrary to the Weinberg scheme—the leading pieces of some loops appear one order lower than the tree level isoscalar rescattering amplitudes. As can also be read from the table, the corresponding order is m =MN . If we consider, in addition, that due to the odd parity of the pion the initial state has to be a p-wave, the loops themselves have to scale as √ m . Since no counter term non-analytic in m is allowed, the loops have to be 6nite or to cancel exactly. This requirement is an important consistency check of the new counting scheme. The details of the loops calculations in threshold kinematics can be found in Ref. [204]. Here we only give the results. Note, we only evaluate the leading order pieces of the integrals corresponding to the diagrams of Table 11. For example, in the integrals we drop terms of order m compared to l0 . After these simpli6cations, straightforward evaluation gives for the production amplitude from the loops with nucleons only

˜k MN m i 3 An = 3 gA ˜&1 · –n ; (87) F 128F2 |˜k|

222


Table 11 Comparison of the corresponding chiral order in the Weinberg scheme (p ∼ m ) and the new counting scheme √ (p ∼ m MN ) for several nucleonic contributions for s-wave pion production s-Wave diagrams (nucleons only)

.

p ∼ m

p∼

m =p

LO

LO

p=M

NLO

LO

p2 =MN2

N2 LO

NLO

pm =MN2

N2 LO

N2 LO

m2 =(pMN )

NLO

N2 LO

m3 =pMN2

N2 LO

N4 LO

m M N

. a)

. .

.

.

. b)

c)

. . . .

.

.

. .

.

g)

.

f)

e)

.

.

. . . . . .

. . . .

d)

. . .

. i)

h)

. j)

. .

√

Scale

. . . k)

. . .

. . . l)

.

Subleading vertices are marked as . Not shown explicitly are the recoil corrections for low-order diagrams. For example, recoil corrections to diagram (b) appear at order pm =MN2 .


223

Table 12 Comparison of the corresponding chiral order in the Weinberg scheme (p ∼ m ) and the new counting scheme √ (p ∼ m MN ) for the leading and next-to-leading C contributions for s-wave pion production s-Wave diagrams (Deltas only)

. .

. .

b)

. . . .

. .

.

c)

. .

C ∼ m

C∼

pm =SMN

NLO

NLO

p3 =(SMN2 )

N2 LO

NLO

√

m MN

.

a)

.

Scale

. .

.

.

.

. .

.

. .

.

d)

. . . .

. . . .

f)

e)

Subleading vertices are marked as . Not shown explicitly are the recoil corrections for low-order diagrams.

where n denotes the diagram (labels as in the 6gure). For the isospin functions we 6nd –d = −%c1 ;

–e = − 14 (%c1 + %c2 );

–f = 32 %c1 :

(88)

Those can be easily evaluated in the di@erent isospin channels. We 6nd for –(TT3 ) = TT3 |–d + –e + –f |11 (it is suOcient here to look at the pp initial state only) –(11) = −1 −

1 2

+

3 2

= 0;

–(00) = −1 + 0 +

3 2

= − 12 :

(89)

The 6rst observation is that the NLO contributions are of the order of magnitude expected by the power counting, since M N m = 0:8H2 ; 128F2 where we used F =93 MeV. The power counting proposed in Refs. [163,203] thus is indeed capable of treating properly the large scale inherent to the NN → NN reaction. We checked that our results for the individual diagrams agree with the leading non-vanishing pieces from the calculations of Ref. [205]. 30 In addition, for almost all diagrams given in Table 11, 30

In this reference the same choice for the pion 6eld is made and thus a comparison of individual diagrams makes sense. Note that there is a sign error in the formula for diagram f in Ref. [205]. We are grateful to Myhrer for helping us to resolve this discrepancy.

224


Table 13 Comparison of the results of the analytic calculation of Ref. [205] with the expectations based on the two counting schemes as discussed in the text Diagram

d

e

g

jR

k

l

Ref. [205] p∼√ m p ∼ m MN

1.0 1.0 1.0

−1.0 1.0 1.0

0.1 1.0 0.4

0.4 1.0 0.4

0.03 1.0 0.06

0.02 1.0 0.06

The diagrams are labeled as in Fig. 11 (the label jR shows that here the recoil term of diagram j is calculated; it appears at NNLO in the Weinberg scheme as well as in the new counting scheme).

Ref. [205] gives explicit numbers for the amplitudes in threshold kinematics. It is intriguing to compare those to what one expects from the di@erent counting schemes. This is done in Table 13, where the 6rst line speci6es the particular diagram according to Table 11 and the second gives the result of the analytical calculation of Ref. [205]. 31 (normalized to the 6rst column). In the following two lines those numbers are compared to the expectations based on the counting schemes—6rst showing those for the Weinberg scheme and then those for the new counting scheme. As can clearly be seen, the latter does an impressive job of predicting properly the hierarchy of diagrams. Thus, at least when ISI and FSI are neglected, the counting scheme proposed is capable of dealing with these large momentum transfer reactions. It is striking that the sum of loops in the case of the neutral pion production vanishes (–(11) = 0 in Eq. (89)). In addition, for neutral pion production there is no meson exchange current at leading order and the nucleonic current (diagrams b and c in Table 11) gets suppressed by the poor overlap of the initial and 6nal state wavefunctions (see discussion in Section 1.4)—an e@ect not captured by the counting—and interferes destructively with the direct production o@ the Delta (diagrams a and b in Table 12). Thus the 6rst signi6cant contributions to the neutral pion production appear at NNLO. This is the reason why many authors found many di@erent mechanisms, all of similar importance and capable of removing the discrepancy between the Koltun and Reitan result and the data, simply because there is a large number of diagrams at NNLO. The situation is very di@erent for the charged pions. Here there is a meson exchange current at leading order and there are non-vanishing loop contributions. We therefore expect charged pion production to be signi6cantly better under control than neutral pion production and this is indeed what we found in the phenomenological model described in the previous section. Next, let us have a look at the loops that contain Delta isobars (cf. Fig. 12). In Ref. [204] it was shown that the individual loops diverge already at leading order, because the Delta–Nucleon mass di@erence introduces a new scale. Therefore, as was argued above, the sum of the diagrams has to cancel, for at NLO there is no counter term. It is an important check of the counting scheme that this does indeed happen. We take this as a strong indication that expanding in H is consistent with the chiral expansion. The observation that we have now established a counting scheme for reactions of the type NN → NN also has implications for the understanding of other reactions. One example is the analysis elastic d scattering, which is commonly used to extract the isoscalar N scattering length 31

In Ref. [205] the full result for the particular loops are given. Thus, any loop contains higher order contributions.


225

that is diOcult to get at otherwise [167]. As mentioned above, due to chiral symmetry constraints the isoscalar scattering length does not get a contribution at leading order and thus, for an accurate extraction of this important quantity from data on d scattering or on d bound states, a controlled calculation of the few-body corrections is compulsory. One of these corrections are the so-called dispersive corrections (see Ref. [221] and references therein): loop contributions to the elastic scattering that have intermediate two-nucleon states. Obviously, the imaginary part of those loops gives the essential contribution to the imaginary part of the d scattering length, 32 however there is also a real part to these loops that needs to be calculated within a scheme consistent with that used for the calculation of the other contributions. Within chiral perturbation theory that has not been done up to now. Given the progress reported here, such a calculation is now feasible. The same technique can then also be used to calculate the corresponding corrections for 3 He scattering, recently calculated in chiral perturbation theory for the 6rst time [223]. 6.4. On the signi9cance of o8-shell e8ects A few years a ago there existed a strong program at several hadron facilities to measure bremsstrahlung in NN collisions with the goal to identify the true o@-shell behavior of the NN interaction. Indeed it was found that the predictions for several highly di@erential observables are signi6cantly di@erent when di@erent NN potentials are employed. On the other hand, a change in the o@-shell behavior of any T -matrix can be realized on the level of the e@ective interaction by a 6eld rede6nition and it is known since long that S-matrix elements do not change under those transformations (as long as one works within a well de6ned 6eld theory) [224,225]. Thus, for any given set of 6elds the o@-shell amplitudes might well have a signi6cant impact on the values of various observables, but it is an intrinsic feature of quantum 6eld theory that they cannot be separated from the short range interactions constructed within the same model space. In a very pedagogical way those results were presented later in Refs. [226,227]. How can one understand this seemingly contradictory situation: on the one hand o@-shell amplitudes enter the evaluation of matrix elements and in some cases inNuence signi6cantly the result (cf. discussion in previous sections), on the other hand they are claimed not to have any physical signi6cance? Following Ref. [226] to start the discussion, let us consider some general half-o@ shell N T -matrix. In a covariant form the list of its arguments contains the standard Mandelstam variables s, t as well as q2 —the four momentum squared of the o@-shell particle(s). For our discussion let this be the incoming pion and for simplicity omit all spin indices. We now de6ne TR (s; t; q2 ) via T (s; t; q2 ) = T˜ (s; t; q2 ) + (q2 − m2 )TR (s; t; q2 ) ; where T˜ (s; t; q2 ) is arbitrary up to the condition that it has to agree with the on-shell amplitude for q2 = m2 . Obviously, this is always possible and we can assume TR to be smooth around the on–shell point. Then, when introduced into the NN → NN transition amplitude, we get A=T

32

q2

1 1 WNN = T˜ 2 WNN + TR WNN ; 2 − m q − m2

About one third of the imaginary part was found to be related to the reaction d → NN [222].

(90)

226


T

T

Fig. 29. Visualisation of Eq. (90).

where WNN denotes the NN vertex function. This example highlights two important facts: (i) a change in the o@-shell dependence of a particular amplitude can always be compensated via an appropriate additional contact term. The only quantities that are physically accessible are S-matrix elements and those are by de6nition on-shell; and (ii) the contribution stemming from the o@-shell N T -matrix is a short range contribution. Therefore the distinction between short range and o@-shell rescattering is arti6cial (see Fig. 29). 6.5. Lessons and outlook The previous section especially made clear that it is rather diOcult to construct a model that gives quantitatively satisfactory results for the reaction pp → pp0 . On the other hand, it turned out that for the reaction pp → pn+ current models as well as the e@ective 6eld theory approach do well. In the previous sections this di@erence was traced back to a suppression of meson exchange currents in the neutral pion production. What is the lesson to be learned from this? First of all, any model for meson production close to the threshold should contain the most prominent meson exchange currents; as long as these are not too strongly suppressed, one can expect them to dominate the total production cross section close to the threshold. However if there is no dominant meson exchange current, then there is a large number of sub-leading operators that compete with each other and make a quantitative understanding of the cross section diOcult. The optimistic conclusion to be drawn from these observations is that (also for heavier mesons) the leading meson exchange currents should give a reasonable description of the data, while the contributions from irreducible loops largely cancel. Obviously, in heavy meson production there is no reason anymore to consider pions only as the exchange particles. In this sense it is + production that is the typical case, whereas the 0 is exceptional due to the particular constraints from chiral symmetry in that channel. Note that also in the case of pion photoproduction close to the threshold the 0 plays a special role, in that the s-wave amplitude is dominated by loops [218]. Let us look in somewhat more detail at the production operator for neutral pion production. Within the model described above, the most prominent diagram for neutral pion production close to the production threshold is pion rescattering via the isoscalar pion–nucleon T -matrix that, for the kinematics given, is dominated by a one-sigma exchange (diagram b of Fig. 2, where the T -matrix is replaced by the isoscalar potential given by diagram e of Fig. 22). Within the e@ective 6eld theory the isoscalar potential is built up perturbatively. This is illustrated in Fig. 30. As was shown above, the leading piece of the one-sigma exchange gets canceled by other loops that cannot be interpreted as a rescattering diagram (the sum of diagrams d–f of Table 11 vanishes) and are therefore


+

+

+

227

+

...

Fig. 30. The one sigma exchange as it is perturbatively built up in the e@ective 6eld theory, starting from the left with the lowest order diagram (NLO). The chiral order increases by one power in H between each diagram from the left to the right.

not included in the phenomenological approach. This is an indication that in order to improve the phenomenological approach, at least in case of neutral pion production, pion loops should be considered as well. 33 There is one more important conclusion to be drawn from the insights reported in the two previous sections: We can now identify what was missing in the straightforward extension of the Bonn potential reported in Ref. [24]. This is especially relevant, since we will see that the failure to describe the low-energy pion production data in that approach does not point at a missing degree of freedom in the nucleon–nucleon phenomenology, but at an incomplete treatment of the cut structure. Diagrammatically, the dressed NN vertex function is given by

Γ

TNP

where the solid dot indicates the so-called bare vertex (in this picture it is to be understood as a parameterization of both the extended structure of the nucleon as well as meson dynamics not included in the non-pole N T -matrix T NP , such as D [229] or & [230] correlations) as well as the dressing due to the N interactions parameterized in T NP . The latter structure, however, contains a N cut not considered in Ref. [24]. It is straightforward to map the di@erent three-body cuts in the one pion exchange potential to those that must occur in the pion production operator in order to allow for a consistent description of scattering and production. This is indicated in Fig. 31: the additional cut due to the pion dynamics in the vertex function is related to the pion rescattering diagram in the production operator. Therefore, at least close to threshold in Ref. [24], the most relevant mechanism that leads to inelasticities was missed. As we saw, however, as we move away from the near-threshold region the contribution from the Delta isobar becomes more and more signi6cant. Consequently, the results of Ref. [24] look signi6cantly better at higher energies.

33

Note that within the e@ective 6eld theory approach the convergence of the series shown in Fig. 30 should also be checked, as pointed out in a di@erent context in Ref. [228].

228


TNP

Γ Γ

Γ Γ

Fig. 31. Leading cut structure of the one pion exchange potential in nucleon–nucleon scattering.

7. Production of heavier mesons We start this section with some general remarks and then discuss some special examples. The phenomenology will be discussed only very brieNy; for details on this we refer to the original references or the recent reviews [1,2]. The list of reactions discussed in detail is by no means complete. For example, we will not discuss here production, for this was already discussed in great detail in Ref. [2]. Neither will the production of vector mesons be discussed in detail. Here we refer to the recent contributions to the literature (Ref. [59] and references therein). As can be seen from the headlines already, special emphasis will now be put on the physics that can be extracted from the particular reactions. 7.1. Generalities As the mass of the produced meson increases, the initial energy needs to increase as well and consequently also the typical momentum transfer at threshold. This has various implications: • the NN interaction needed for a proper evaluation of the initial state interaction is less under control theoretically; for the T = 0 channel there is not even a partial wave analysis available above the -production threshold, due to a lack of data; • a larger momentum transfer makes it more diOcult to construct the production operator; at least one should expect that the exchanges of heavier mesons become even more relevant compared to the pion case;


229

• it is not known, how to formulate a convergent e@ective 6eld theory; 34 • less is known about the interactions of the subsystems; • the treatment of three-body singularities due to the exchange of light particles requires more care. In the upcoming section we will not discuss in detail the role of the three-body singularities. Only recently was a method developed that in the future will allow inclusion of those singularities in calculations within the distorted wave born approximation [231]. This method was already applied in a toy model calculation [232], where its usefulness was demonstrated. In calculations these singularities can occur only if the initial state interaction is included. For the production of mesons heavier than the , however, no reliable model is available at present. Up to now no model for the production of heavy mesons takes three-body singularities into account, and thus their role is yet unclear. 7.2. Remarks on the production operator for heavy meson production As in Section 6 we will only look at analyses, that work within the distorted wave born approximation. For the two-baryon states nothing changes, other than that the nucleon–nucleon interaction needed for the initial state interaction gets less reliable from the phenomenological point of view as we go up in energy. The construction of the production operator, however, is now even more demanding, for with increasing momentum transfer heavier exchange mesons can play a signi6cant role. Therefore, in order to get anything useful out of a model calculation, as many reaction channels as possible should be studied simultaneously. Only in this way can the phenomenological model parameters can be 6xed and useful information be extracted. In this context also the simultaneous analysis of pp- and pn-induced reactions plays an important role, for di@erent isospin structures in the production operator will lead to very di@erent relative strengths of the two channels (cf. discussion in Section 1.5). Then, once a basic model is constructed that is consistent with most of the data, deviations in particular reaction channels can be studied. In Ref. [34] the model described in Section 6.2 was extended to production in nucleon–nucleon collisions, where the single channel N T -matrix used in the pion production calculations was replaced by the multi channel meson–baryon model of Ref. [233] to properly account for the exchange of heavy mesons. So far only s-waves were considered in the calculation. Results for the various total cross sections using two di@erent models for the 6nal state NN interaction are shown in Fig. 32. The studies of Ref. [34] indicate, that a complete calculation for NN → NN will put additional constraints on the relative phases of the various meson–baryon → N transition amplitudes. Unfortunately, up to now the is the heaviest meson that can be investigated using this kind of microscopic approach, for there is no reliable model for the NN interaction for energies signi6cantly above the production threshold. The strategy that therefore needs to be followed is to study many reaction channels consistently. This was done, for example, in a series of papers by Kaiser and others for , , [155], ! [234] as well as strangeness production [235] and by Nakayama and coworkers for ! [59,236,237], 34

In principle, one might expect an approach like that presented in Section 6.3 to work for the production of all Goldstone bosons. However, the next threshold after pion production is that for eta production and it already corresponds to an initial momentum of pi = 770 MeV. Thus the expansion parameter would be m =MN = 0:8.

230


pn→dη 10 σ [µb]

pn→pnη

pp→ppη

1

0.1 0

10

20 30 Q [MeV]

40

50

Fig. 32. Results for the total cross sections of the reactions pp → pp , pn → pn , and pn → d from Ref. [34] employing di@erent NN models for the 6nal state interaction. The solid lines represent the results with the CCF NN model [175] whereas the dashed–dotted lines were obtained for the Bonn B model [25]. Data are from Refs. [33,109–112,120].

; [237,238], [239,240] and [241] production. Both groups construct the production operator as relativistic meson exchange currents, where the parameters are constrained by other data such as decay ratios. The striking di@erence between the two approaches is that the former studies s-waves only, does not consider e@ects of the initial state interaction and treats the 6nal state interaction in an approximate fashion. The latter group includes the ISI through the procedure of Ref. [40] (cf. Section 3), treats the FSI microscopically and includes higher partial waves as well. It is important to stress that, where compatible, the two approaches give qualitatively similar results, as stressed in Ref. [234]. Recently, it was observed that the onset of higher partial waves can strongly constrain the production operator [240]. In the same reference it was demonstrated that to unambiguously disentangle effects from FSI or higher partial waves, polarization observables are necessary. Thus one should expect that once a large amount of polarized data is available on the production of heavy mesons, the production operators and thus the relevant short-range physics for the production of a particular meson, can be largely 6xed. As an example, this particular issue is discussed in detail in the next section. 7.3. The reaction NN → NN or properties of the S11 (1535) The meson is a close relative of the pion, since it is also a member of the nonet of the lightest pseudo-scalar mesons. It is an isoscalar with a mass of 547:3 MeV—the di@erence in mass from the pion can be understood from its content of hidden strangeness through the Gell–Mann–Okubo mass formula [242,243]. As for the pion, the couples strongly to a resonance, but a resonance with di@erent quantum numbers: is it the positive parity C(1232) for the pion that in the C rest frame leads to p wave pion production, the production is dominated by the negative parity S11 (1535), 35 which leads 35

The quantum numbers are chosen in accordance with the partial wave in which the resonance would appear in N scattering. Therefore positive parity resonances appear in odd partial waves.


231

to s wave production in its rest frame. As a consequence near-threshold production is completely dominated by the S11 (1535) resonance. This statement can well be reversed: studying production in various reaction channels close to the production threshold allows selective study of the properties of the S11 (1535) resonance in various environments. In this context it is interesting to note that the nature of this particular resonance is under discussion: within the so-called chiral unitary approach the resonance turns out to be dynamically generated [244,245], while on the other hand, detailed studies within the meson exchange approach [246], as described in Section 6, as well as quark models (see Ref. [247] and references therein) call for a genuine quark resonance. One expects that a molecule behaves di@erently in the presence of other baryons than a three-quark state due to the naturally enhanced aOnity to meson baryon states of the former. In this sense data on production in few-baryon systems should provide valuable information about this lowest negative parity excitation of the nucleon. The reaction NN → NN was studied intensively both theoretically [34,155,239,240,248–259] as well as experimentally (cf. Table 8) in recent years. The outstanding feature of production in NN collisions is the strong e@ect of the N FSI, most visible in the reaction pn → d , where in a range of 10 MeV the amplitude grows by about an order of magnitude [120]. This phenomenon was analyzed by means of Faddeev calculations, showing this enhancement to be consistent with a real part of the N scattering length of 0:4 fm [248]. It is interesting that the same value of the N scattering length was recently extracted from an analyses of the reaction d → pn [260] and that this value is consistent with that stemming from the microscopic model for meson–baryon scattering of Ref. [233]. First three-body calculations for the reaction with a two-nucleon pair in the continuum show only a minor impact of the N interaction on the invariant mass spectra [261]. Most model calculations for the production operator share the property that the production operator was calculated within the meson exchange picture. 36 This was then convoluted with the NN FSI, treated either microscopically or in some approximate fashion. In most analyses it was found that the dominant production mechanism is via the S11 (1535), with the exception of Ref. [249], where the reaction pp → pp turned out to be dominated by isoscalar meson exchanges, in analogy with the heavy meson exchanges in the reaction pp → pp0 . The largest qualitative di@erences between the various models is the relative importance of the exchanged mesons. In Refs. [155,252–255] the D-exchange turned out to be the dominant process, whereas it played a minor role in Refs. [34,256,249]. In Ref. [259] the D-exchange was not even considered, but exchange turned out to be signi6cant there. It remains to be seen, however, if models that are dominated by isoscalar exchanges, Refs. [259,249], are capable of accounting for the large ratio of pn- to pp-induced eta production, as discussed in Section 1.5. As stressed above, in this section we do not want to focus on the details of the dynamics of the production operator, but instead on the physical aspect of what we can learn about the S11 resonance from studying eta meson production in NN collisions. Thus, in the remaining part of this section we will restrict ourselves to a model-independent analysis of a particular set of data based on the amplitude method introduced in Section 4.3. Here we closely follow the reasoning of Ref. [240].

36

The only exception to the list given above is Ref. [257], where the ratio of pp- to pn-induced production is explained by the instanton force. The relation between this result and the hadronic approach is unclear [257].

232


dσ/dmpp [arbitrary units]

0.3

0.2

0.1

0 0

5

10

15

mpp [MeV]

Fig. 33. Invariant mass spectrum for the two-proton system for the reaction pp → pp at Q = 15 MeV. The dashed line shows the Ss contribution, the dot–dashed line the Ps distribution and the solid line the incoherent sum of both. The data are taken from Ref. [113].

To be de6nite we will now concentrate on the measurement of angular distributions and invariant mass spectra for pp → pp at Q=15 MeV [113]. The experiment shows that the angular distribution of both the two-proton pair and of the are Nat, suggesting that only s-waves are present. On the other hand, the invariant mass distribution d&=dmpp deviates signi6cantly from what is predicted based on the presence of the strong pp FSI only (cf. discussion of Section 2). This is illustrated in Fig. 33, where the distribution for pp S-wave, s-wave (Ss) is shown as the dashed line. Since the N interaction is known to be strong, its presence appears as the natural explanation for the deviation of the dashed line from the data. The structure, however, is even more pronounced at Q = 42 MeV (cf. Ref. [240]) in contrast to what should be expected for the N interaction. On the other hand, the discrepancy between the data and the dashed curve can be well accounted for by a Ps con6guration (given by phase-space times a factor of p 2 , cf. Section 2.1). This is also illustrated in Fig. 33, where the pure Ps con6guration is shown as the dashed–dotted line and the total result—the incoherent sum of Ps and Ss cross section—as the solid line. How is this compatible with the angular distribution of the two-nucleon system being Nat? This question can be very easily addressed in the amplitude method described in Section 4.3. As long as we restrict ourselves to s-waves and at most P waves in the NN system, the only non-vanishing ˜ containing the amplitude for the Ss 6nal state, and Ã, containing the two terms in Eq. (48) are Q, possible amplitudes for the Ps 6nal state. The explicit expressions for the amplitudes were given previously in Eqs. (55). In this particular case we 6nd for example (cf. Eqs. (49)–(54)) ˜ 2 + |Ã|2 ) ; &0 = 14 (|Q| A0i &0 = 0 ; ˜ 2 − |Ã|2 ) ; Axx &0 = 14 (|Q|


233

where ˜ 2 = |a1 |2 ; |Q| 2 |Ã|2 = p (|a2 − (1=3)a3 |2 + x2 {|a3 |2 − (2=3) Re(a∗3 a2 )})

p · p). ˆ Here the amplitudes a1 , a2 and a3 correspond to the transitions 3 P0 → 1 S0 s, and p x = (˜ 1 3 S0 → P0 s, and 1 D2 → 3 P2 s, respectively. From these equations we directly read that • if a3 is negligible, the pp di@erential cross section d&=d x is Nat; • the observable &0 (1 − Axx ) directly measures the NN P-wave admixture; • any non-zero value for the analysing power is an indication for higher partial waves for the . It is easy to see that for a3 = 0 the di@erential cross section has to be Nat, since then the only partial wave that contributes to the NN P-waves is 1 S0 → 3 P0 s and a J = 0 initial state does not contain any information about the beam direction. Certainly, the N 6nal state interaction has to be present in this reaction channel as well and it is important to understand its role in combination with the two-nucleon continuum state. It should be clear, however, that in order to understand quantitatively the role of the N interaction, it is important to pin down the NN P-wave contribution 6rst. This is why a measurement of Axx for pp → pp is so important. What does it imply if the conjecture of a presence of NN P-waves at rather low excess energies were true? In Ref. [240] it is demonstrated that the need to populate predominantly the 1 S0 → 3 P0 s instead of the 1 D2 → 3 P2 s very strongly constrains the NN → S11 N transition potential. This indicates that a detailed study of NN → NN should reveal information about the S11 in a baryonic environment that might prove valuable in addressing the question of the nature of the lowest negative-parity nucleon resonance. It should be mentioned that in Ref. [262] an alternative explanation for the shape of the invariant mass spectrum was given, namely an energy dependence was introduced to the production operator. Based on the arguments given in the previous chapters, we do not believe that this is a natural explanation. However, as we just outlined, with Axx an observable exists that allows to unambiguously distinguish between the two possible explanations. The experiment is possible at COSY [263]. 7.4. Associated strangeness production or the hyperon–nucleon interaction from production reactions In Section 5 the small breaking of SU (2) isospin symmetry present in the strong interaction was discussed. If we include strange particles in the analysis we can also study the breaking of Navor SU (3). It is well-known that the light mesons and baryons can be arranged according to the irreducible representations of the group SU (3). The mass splittings within the multiplet can be well accounted for by the number of strange quarks in some baryon or meson. However, not much is known about the dynamics of systems that contain strangeness. Many phenomenological models for, e.g., hyperon–nucleon scattering [268–271] use the Navor SU (3) to 6x the meson baryon–meson couplings. The remaining parameters, like the cut-o@ parameters, are then 6t to the data. As we will discuss below, so far the existing data base for hyperon–nucleon scattering is insuOcient to judge, if this procedure is appropriate.

234


80

d2σ/dΩ/dmΛp (µb/str/MeV)

500

σ (mb)

400

300

200

100

0

60

40

20

0 0

100

200

pLab (MeV)

300

2060

2080

2100

2120

mΛp (MeV)

Fig. 34. Comparison of the quality of available data for the reactions p elastic scattering (data are from [264–266]) and pp → K + p at TLab = 2:3 GeV [267]. In both panels the red curve corresponds to a best 6t to the data. In the left panel the dashed lines represent the spread in the energy behavior allowed by the data, according to the analysis of Ref. [266]; analogous curves in the right panel would lie almost on top of the solid line and are thus not shown explicitly.

As was stressed previously, e@ective 6eld theories provide the bridge between the hadronic world and QCD. In connection with systems that contain strangeness there are still many open questions. Up to now it is not clear if the kaon is more appropriately treated as heavy or as light particle. In addition, in order to establish the counting rules it is important to know the value of the SU (3) chiral condensate. For a review this very active 6eld of research as well as the relevant references we refer to Ref. [272]. To further improve our understanding of the dynamics of systems that contain strangeness, better data are needed. The insights to be gained are relevant not only for few-body physics, but also for the formation of hypernuclei [273], and might even be relevant to the structure of neutron stars (for a recent discussion on the role of hyperons in the evolution of neutron stars see Ref. [274]). Naturally, the hyperon–nucleon scattering lengths are the quantities of interest in this context. In the left panel of Fig. 34 we show the world data set for elastic N scattering. 37 In Ref. [266] a Likelihood analysis based on the elastic scattering data was performed in order to extract the low-energy N scattering parameters. The resulting contour levels are shown in Fig. 35, clearly demonstrating that the available elastic hyperon–nucleon scattering data do not signi6cantly constrain the scattering lengths: the data allow for values of (−1; 2:3) as well as (6,1) (all in fermi) for (as ; at ), respectively. Later models were used to extrapolate the data. However, also in this way the scattering lengths could not be pinned down accurately. For example, in Ref. [269] one can 6nd six di@erent models that equally well describe the available data but whose (S-wave) scattering lengths range from 0.7 to 2:6 fm in the singlet channel and from 1.7 to 2:15 fm in the triplet channel. 37

The data were taken inclusively (only the kaon was measured in the 6nal state), however, for the small invariant masses shown only the K channel is open.


235

3

2

at (fm)

x

1

0

0

4

8 as (fm)

12

16

Fig. 35. Values allowed for the spin singlet and spin triplet scattering length by the N elastic scattering data according to Ref. [266]. The dark shaded area denotes the 1& range for the parameters and the light shaded area the 2& range. The cross shows the best 6t value (as = 1:8 fm and at = 1:6 fm).

Production reactions are therefore a promising alternative. In the literature the reactions K − d → n [9], d → K + n (Ref. [53] and references therein) and pp → pK + [122] were suggested. Therefore the method described in Section 2.2, that applies to the latter two reactions, is an important step towards a model-independent extraction of the hyperon–nucleon scattering lengths. A natural question that arises is the quality of data needed e.g. for the reaction pp → pK + in order to signi6cantly improve our knowledge about the hyperon–nucleon scattering lengths. In Ref. [32] it is demonstrated that data of the quality of the Saclay experiment for pp → K + X [267], shown in right panel of Fig. 34 38 that had a mass resolution of 2 MeV, allows for an extraction of a scattering length with an experimental uncertainty of only 0:2 fm. Note, however, that the actual value of the scattering length extracted with Eq. (22) from those data is not meaningful, since the analysis presented can be applied only if just a single partial wave contributes to the invariant mass spectrum. The data set shown, however, represents the incoherent sum of the 3 S1 and the 1 S0 hyperon–nucleon 6nal state. The two spin states can be separated using polarization measurements. In Section 4.3.1 as well as in Ref. [32] it was shown, that for the spin singlet 6nal state the angular distributions of various polarization observables are largely constrained. This is suOcient to extract the spin dependence of the -nucleon interaction from the reaction pp → pK + . One more issue is important to stress: to make sure that the structure seen in the invariant mass spectrum truly stems from the 6nal state interaction of interest, a Dalitz plot analysis is necessary, for resonances can well distort the spectra. This is discussed in detail in Section 4.1 as well as at the end of Section 2.2.

38

Shown is only the low m2 tail of the data taken at TLab = 2:3 GeV and an angle of 10◦ .

236

C. Hanhart / Physics Reports 397 (2004) 155 – 256 K0

K+ Λ=

f0

f0

a0

a0

 K−

K0

Fig. 36. Graphical illustration of the leading contribution to the f0 − a0 mixing matrix element de6ned in Eq. (91).

7.5. Production of scalar mesons or properties of the lightest scalar The lightest scalar resonances a0 (980) and f0 (980) are two well-established states seen in various reactions [146], but their internal structure is still under discussion. Analyses can be found in the literature identifying these structures with conventional qqG states (see Ref. [275] and references therein), compact qq-qGqG states [276,277] or loosely bound K KG molecules [278,279]. In Ref. [280] a close connection between a possible molecule character of the light scalar mesons and chiral symmetry was stressed. It has even been suggested that at masses below 1:0 GeV a complete nonet of 4-quark states might exist [281]. Resolution of the nature of the light scalar resonances is one of the most pressing questions of current hadron physics. First of all we need to understand the multiplet structure of the light scalars in order to identify possible glueball candidates. In addition, it was pointed out recently that a0 (980), f0 (980) as well as the newly discovered Ds [282] might well be close relatives [283]. Thus, resolving the nature of the light scalar mesons allows one simultaneously to draw conclusions about the charmed sector and will also shed light on both the con6ning mechanism in light–light as well as light–heavy systems. Although predicted long ago to be large [284], the phenomenon of a0 − f0 mixing has not yet been established experimentally. In Ref. [284] it was demonstrated that the leading piece of the f0 − a0 mixing amplitude can be written as 39 2 √ pK 0 − pK2 + ; (91)

= f0 |T |a0 = igf0 K KG ga0 K KG s(pK 0 − pK + ) + O s where pK denotes the modulus of the relative momentum of the kaon pair and the e@ective coupling 2 constants are de6ned through MxK KG = gxK p . Obviously, this leading contribution is just that of KG K the unitarity cut of the diagrams shown in Fig. 36 and is therefore model-independent. Note that in Eq. (91) electromagnetic e@ects were neglected, because they are expected to be small [284]. The contribution shown in Eq. (91) is unusually enhanced between the K + K − and the KG 0 K 0 thresholds, a regime of only 8 MeV width. Here it scales as (this formula is for illustration only—the Coulomb interaction contributes with similar strength to the kaon mass di@erence [285]) m2K 0 − m2K + mu − m d ∼ ; 2 2 mˆ + ms mK + + m K 0 where mu , md and ms denote the current quark mass of the up, down and strange quark, respectively, 40 and m=(m ˆ which scale as (mu −md )=(m+m ˆ u +md )=2. This is in contrast to common CSB e@ects s ), 39

Here we deviate from the original notation of Achasov et al. in order to introduce dimensionless coupling constants in line with the standard Flatt\e parameterization. 40 Here we denote as common CSB e@ects those that occur at the Lagrangian level.


237

1

|Λ| [arb. units]

0.8 0.6 0.4 0.2

0 970

980

990

1000 s [MeV]

1010

1020

Fig. 37. Modulus of the leading piece of the mixing amplitude de6ned in Eq. (91). The two kinks occur at the K + K − (at 987:35 MeV) and the KG 0 K 0 (995:34 MeV) threshold, respectively.

since they have to be analytic in the quark masses. It is easy to see that away from the kaon thresholds √

returns to a value of natural size. This s dependence of is depicted in Fig. 37. G scattering the mixing of a0 and f0 was studied in Within a microscopic model for and KK Ref. [286]. Within this model both resonances are of dynamical origin. The only mixing mechanism considered was the meson mass di@erences. Within this model the predictions of Ref. [284] were con6rmed. As was demonstrated by Weinberg for the case of the deuteron [287], the e@ective couplings of resonances to the continuum states contain valuable information about the nature of the particles. In the case of the deuteron this analysis demonstrated that the e@ective coupling of the deuteron to the pn continuum, as can be derived from the scattering length and the e@ective range, shows that the deuteron is a purely composite system. Recently, it was demonstrated that, under certain conditions that apply in the case of a0 and f0 , the analysis can be extended to unstable scalar states as well [288]. Accurate data on the e@ective couplings of the scalars to kaons should therefore provide valuable information about their nature. As was argued in Section 5, the occurrence of scalar mixing dominates the CSB e@ects in production reactions. Therefore quantifying the mixing matrix element might be one of the cleanest ways to measure the e@ective decay constants of a0 and f0 . In Section 5 it was demonstrated that from studies of production in NN and dd collisions one should be able to extract the a0 − f0 mixing amplitude. Those arguments were supported in Section 4.3.2, where G s-waves. We should it was shown, that the reaction pp → dKG 0 K + is indeed dominated by KK therefore expect a signi6cant signal for the mixing as well. In the years to come we can thus expect the experimental information about the scalar mesons to be drastically improved. 8. Meson production on light nuclei In this section we wish to make a few comments concerning meson production on light nuclei. Note, dd-induced meson production was discussed to some extent in Section 5. A more extensive discussion can be found in Ref. [2].

238

C. Hanhart / Physics Reports 397 (2004) 155 – 256 3He

3He

N

x

N

x

π

d (a)

N

N

x

N (b)

d

N

d

(c)

Fig. 38. Possible diagrams to contribute to the reaction Nd → 3Nx. In diagram (a) and (b) the three nucleons in the 6nal state from a bound state, whereas diagram (c) shows the dominant diagram in quasi-free kinematics. Here it is assumed that only a particular nucleon pair (to be identi6ed through proper choice of kinematics) interacts in the 6nal state.

8.1. Generalities Almost all general statements made for meson production in NN collisions apply equally well for meson production involving light nuclei. Naturally, now the selection rules are di@erent (see discussion at the end of Section 1.6) and the possible breakup of the nuclei introduces additional thresholds that must be considered in theoretical analyses. A comparison of meson production in two-nucleon collisions and in few-nucleon systems should improve our understanding of few-nucleon dynamics. As a result of a systematic study of, for example, both NN - and pd-induced reactions, a deeper insight into the importance of three-body forces should be gained. For example, a recent microscopic calculation using purely two-body input to calculate the reaction +3 He → ppp, found that the data [289,290] call for three-body correlations [291], con6rming earlier studies [292]. It should be stressed, however, that this 6eld is still in its infancy and a large amount theoretical e@ort is urgently called for. 8.2. 2 → 2 reactions The reaction channel studied best up to date is pd → 3 He x. Data exist mainly from SATURNE for x = [293], x = ! [294], x = [295,296], as well as x = and ; [297]. Most of the experiments were done using a polarized deuteron beam. In addition the reactions dd → ,0 [140] 41 and dd → , [298] were measured. The reaction pd → 3 He 0 could be understood quantitatively from diagram (a) of Fig. 38 [299]. However, it turned out that diagram (a) alone largely under predicted the data [300,301] for the reaction pd → 3 He , whereas diagram (b) contributes suOciently strongly to allow description of the data [300,302,303]. In all these approaches the individual amplitudes were taken from data directly; so far no microscopic calculation exists. The relevance of the pion exchange mechanism was explained in Ref. [304] as what the authors called a kinematic miracle: in the near-threshold 41

Note that this reaction is charge symmetry breaking and was discussed in Section 5.


239

regime the kinematics for eta production almost exactly matches that for pp → d+ followed by + n → p, with the intermediate pion on-shell. The striking feature of reactions with an -nucleus 6nal state is the pronounced energy dependence that was already described in Section 7.3 for the reaction pn → d —often interpreted as a signal of an existing (quasi)bound state in the -nucleus system. Indeed, since the quark structure of the G is proportional to the number operator, one should expect the -nucleus interaction to ∼ uu G + dd get stronger with increasing number of light quark Navors present in the interaction region. For a more detailed discussion of this issue we refer to Ref. [2]. 8.3. Quasi-free production A deuteron is a loosely bound state of a proton and a neutron. For properly chosen kinematics, the deuteron can therefore be viewed as an e@ective neutron beam/target as alternative to neutron beams (see Ref. [2] and references therein). The corresponding diagram is shown in Fig. 38(c). Obviously, the existence of a three-nucleon bound states already shows that there must be a kinematic regime, where the interaction of all three nucleons in the 6nal state is signi6cant, namely, when all three have small relative momenta. On the other hand, if the spectator nucleon escapes completely una@ected, its momentum distribution should be given by just half the deuteron momentum, convoluted with the deuteron wavefunction times the phase-space factors. Thus, we should expect a momentum distribution for the spectator that shows in addition to a pronounced peak from the quasi-free production a long tail stemming from rescattering in the 6nal state. Experimental data exist for the proton-spectator momentum distribution for the reaction pd G → 3+ 2− p [305]. In Fig. 39 these data are shown together with the results of a calculation [306] that considers both the quasi-free piece as well as a rescattering piece. The data clearly show the quasi-elastic peak. Thus, when considering spectator momenta that are of about 100 MeV or less, to a good approximation, the reaction should be quasi-free. Experimentally, this conjecture can be checked by comparing data for pp induced reactions with those stemming from a pd initial state with a neutron spectator. Those comparisons were carried out at TRIUMF for pion production [74,75] as well as at CELSIUS for production [33,120], showing that the quasi-elastic assumption is a valid approximation. In Section 5 it was argued that a forward–backward asymmetry in the reaction pn → d(0 )s-wave is a good system from which to extract the f0 − a0 mixing matrix element. If investigated at COSY, this reaction can only be studied with a deuteron as e@ective neutron target. Since we are after a CSB e@ect, the expected asymmetry is of the order of a few percent. Thus we need to ensure a priori that the spectator does not introduce an asymmetry of this order through its strong interaction with, say, the deuteron. Theoretically, it would be very demanding to control such a small e@ect in a four-particle system. Fortunately, we can test experimentally to what extent the spectator introduces a forward–backward asymmetry by studying, for example, the reaction pd → + dn. If the reaction were purely quasi-elastic, the angular distribution of the + d system in its rest frame would be forward–backward symmetric, for it would be stemming from a pp initial state. However, any interaction of the spectator with either the pion or the deuteron, should immediately introduce some forward–backward asymmetry. Experimental investigations of this very sensitive test of the spectator approach are currently under way [307].

240


600

events/10 MeV/c

500

400

300

200

100

0

0

0.2

0.4 Ps, GeV/c

0.6

Fig. 39. Momentum distribution of proton-spectators in the reaction pd G → 3+ 2− p. The data are taken from Ref. [305] while the solid line is based on a calculation that includes the quasi-free production as well as meson rescattering. The 6gure is taken from [306], where also details about the corresponding calculation can be found.

9. Summary The physics of meson production in nucleon–nucleon collisions is very rich. The various observables that are nowadays accessible experimentally are, for example, inNuenced by baryon resonances and 6nal state interactions as well as their interference. It is therefore inevitable to use polarized observables to disentangle the many di@erent physics aspects. From the authors personal point of view, the most important issues for the 6eld are: • that NN - and dd induced reactions are very well suited for studies of charge-symmetry breaking [5]. Especially, investigation of the reactions pn → d(0 )s-wave and dd → ,(0 )s-wave should shed light on the nature of the scalar mesons. For the experiments planned in this context we refer to Ref. [308]; • that suOciently strong 6nal state interactions can be extracted from production reactions with large momentum transfer, as, for example, pp → pK + . The condition for an accurate extraction is a measurement with high resolution, as should be possible with the HIRES experiment at COSY [309]; • that an e@ective 6eld theory was developed for large momentum transfer reactions such as NN → NN. Once pushed to suOciently high orders, those studies will not only provide us with a better understanding of the phenomenology of meson production in NN collisions but also allow us to identify the charge symmetry breaking operators that lead to the cross sections reported in Refs. [6,140]; • that resonances can be studied in NN -induced reactions. Those studies are complementary to photon-induced reactions, since the resonances can be excited by additional mechanisms, which


241

can be selected using, for example, spin 0 particles as spin 6lter [4]. In addition, also the properties of resonances in the presence of another baryon can be systematically investigated. There are very exciting times to come in the near future, for not only does the improvement of experimental apparatus in recent years permit the acquisition of a great deal of high-precision data but also because or theoretical understanding has now also improved to a point that important physics questions can be addressed systematically.

Acknowledgements This article would not have been possible without the strong support, the lively discussions as well as the fruitful collaborations of the author with V. Baru, M. Bûscher, J. Durso, Ch. Elster, A. Gasparyan, J. Haidenbauer, B. Holstein, V. Kleber, O. Krehl, N. Kaiser, S. Krewald, B. Kubis, A.E. Kudryavtsev, U.-G. Mei_ner, K. Nakayama, J. Niskanen, A. Sibirtsev, and J. Speth. Thanks to all of you! I am especially grateful for the numerous editorial remarks by J. Durso.

Appendix A. Kinematical variables In this report we only deal with reactions, in which the energies of the 6nal states are so low that a non-relativistic treatment of the baryons is justi6ed. This greatly simpli6es the kinematics. Thus, in the center-of-mass system we may write for the reaction NN → B1 B2 x, ˜ 2 + MN2 Etot = 2 p 2 2 2 2 = M1 + p1 + M2 + p2 + m2x + q 2 p 2 q 2 + + ≈ M1 + M2 + 2412 2(M1 + M2 )

m2x + q 2 ;

(A.1)

where p denotes the momentum of the initial nucleons, p the relative momentum of the 6nal nucleons and q the center-of-mass momentum of the meson of mass mx . The reduced mass of the outgoing two baryon system is denoted by 412 = M1 M2 =(M1 + M2 ). The kinematical variable traditionally used for the total energy of a meson production reaction is , the maximum meson momentum in units of the meson mass. Then one gets from Eq. (A.1): Etot ( ) ≈ M1 + M2 + 2

m2x + mx 1 + 2 : 2(M1 + M2 )

(A.2)

Another often used variable for the energy of a meson production reaction is Q=

√

s − (M1 + M2 + mx ) :

(A.3)

242


Table 14 The values for TLab , Q and for the di@erent channels for pion production

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30 1.40 1.50

pp → d+

pp → pp0

pp → pn+

TLab

Q

TLab

Q

TLab

Q

287.6 288.0 289.2 291.2 293.9 297.5 301.8 306.8 312.5 318.9 325.9 333.5 341.7 350.5 359.8 369.5 379.8 390.5 401.6 413.1 425.0 449.8 476.0 503.4 531.9 561.4

0.00 0.19 0.75 1.68 2.97 4.62 6.61 8.94 11.58 14.53 17.77 21.29 25.06 29.09 33.34 37.81 42.49 47.36 52.40 57.62 63.00 74.19 85.91 98.10 110.71 123.69

279.6 280.0 281.2 283.1 285.8 289.2 293.3 298.1 303.6 309.8 316.5 323.9 331.8 340.2 349.1 358.5 368.4 378.6 389.3 400.4 411.8 435.7 460.9 487.1 514.5 542.8

0.00 0.18 0.72 1.62 2.87 4.46 6.38 8.62 11.17 14.02 17.14 20.53 24.18 28.05 32.16 36.47 40.98 45.67 50.54 55.57 60.75 71.54 82.83 94.57 106.72 119.23

292.3 292.7 293.9 295.9 298.7 302.3 306.6 311.6 317.3 323.7 330.7 338.3 346.6 355.3 364.6 374.4 384.6 395.3 406.5 418.0 429.9 454.8 481.0 508.4 536.9 566.4

0.00 0.19 0.75 1.68 2.97 4.62 6.61 8.94 11.58 14.53 17.77 21.29 25.06 29.08 33.34 37.81 42.48 47.35 52.40 57.62 62.99 74.18 85.90 98.09 110.69 123.68

It is straightforward to express in terms of Q: (s − Mf2 − m2x )2 − 4(mx Mf )2 1 = 2mx s 4 1 Q 1−3 : 24Q 1 + mx 44 Mf + m x

(A.4)

where Mf = M1 + M2 and the reduced mass of the full system is given by 4 = Mf mx =(Mf + mx ). For the latter approximation we used that in the close-to-threshold regime Q(Mf + mx ). Traditionally, is used for the pion production only, whereas Q is used for the production of all heavier mesons. To simplify the comparison of results for those di@erent reaction channels, we present in Table 14 the various values for , Q as well as TLab for the di@erent pion production channels.


243

For completeness we also give the relation to the Laboratory variables TLab =

s − 4M 2 ; 2M

2 2 = TLab + 2MTLab : pLab

(A.5) (A.6)

Appendix B. Collection of useful formulas B.1. De9nition of coordinate system In this appendix we give the explicit expressions for the vectors relevant for the description of a general 2 → 3 reaction. We work in the coordinate system, where the z-axis is given by the beam momentum p ˜ . These formulas are particularly useful for Section 4.3. Thus we have       0 sin(>p ) cos(;p ) sin(>q ) cos(;q )           p ˜ = p ˜ = p  0; p  sin(>p ) sin(;p )  ; ˜q = q  sin(>q ) sin(;q )  : 1 cos(>p ) cos(>q ) From these one easily derives   −sin(>p ) sin(;p )   ) cos(;p )  ; sin(> i(˜ p×p ˜ ) = ipp  p   0   cos(>q ) sin(>p ) sin(;p ) − cos(>p ) sin(>q ) sin(;q )    i(˜ p × ˜q ) = ip q   −cos(>q ) sin(>p ) cos(;p ) + cos(>p ) sin(>q ) cos(;q )  ; −sin(>p ) sin(>q ) sin(;q − ;p )

(B.1)

as well as the analogous expression for i(˜ p × ˜q ). B.2. Spin traces In this appendix some relations are given, that are useful to evaluate expression that arise from in the amplitude method appearing described in Section 4.3. &y Hi∗ HiT &y = 12 &y (1 + ˜P i · ˜&T )&y = 12 (1 − ˜P i · ˜&) :

(B.2)

The sum over the spins of the external particles leads to traces in spin space, such as tr(&i ) = 0 ;

(B.3)

tr(&i &j ) = 23ij ;

(B.4)

tr(&i &K &j ) = 2ijiKj ;

(B.5)

244


tr(&, &i &K &j ) = 2(3i, 3jK + 3j, 3iK − 3ij 3,K ) :

(B.6)

To identify dependent structures the following reduction formula is useful: ã 2˜b · (˜c × ˜d) = (ã · ˜b)ã · (˜c × ˜d) + (ã · ˜d)ã · (˜b × ˜c) + (ã · ˜c)ã · (˜d × ˜b) :

(B.7)

To recall the sign or, better, the order of the vectors appearing, observe that on the right-hand side the vectors other than ã are rotated in cyclic order. To prove Eq. (B.7) we use 3ij jklm − 3im jklj = jkl (3ij 3m − 3im 3j ) = jkl ji, jjm, = jjm, (3k, 3li − 3ki 3l, ) = 3il jjmk − 3ki jjml : Appendix C. Partial wave expansion In this appendix we give the explicit relations between the partial wave amplitudes for reactions of the type NN → NNx and the spherical tensors de6ned in Eq. (34), where x is a scalar particle. The relations between the spherical tensors and the various observables is given in Tables 4 and 5. In terms of the partial wave amplitudes, we can write for two spin- 12 particles in the initial state 1 S MS ; p ˜ ;˜q |M |SMS ; p ˜ SG MG S ; p ˜ |M † |SG MG S ; p ˜ ;˜q Tkk13qq13;k; k24qq24 = 16 1 )† (f2 )† ×SMS |%k(b) %(t) |SG MG S SG MG S |%(f k3 q3 %k4 q4 |S MS 1 q1 k 2 q2 1 (2LG + 1)(2L + 1) = 4 (2JG + 1)(2J + 1)

×S MS ; L ML |j Mj j Mj ; l ml |JMJ SMS ; L0|JMJ

G JG MG J ×S MS ; LG MG L |jG MG j jG MG j ; lG mG l |JG MG J SG MG S ; L0|

1 )† (f2 )† ×SMS |%k(b) %(t) |SG MG S SG MG S |%(f k3 q3 %k4 q4 |S MS 1 q1 k 2 q2

×Yl ml (qˆ )YL ML (pˆ )YlG mG (qˆ )∗ YLG MG L (pˆ )∗ M , (s; j)M ,G(s; j)† : l

(C.1)

In order to proceed the following identities are useful [310]: (2l1 + 1)(2l2 + 1) ˆ l2 m2 (p) ˆ = Yl1 m1 (p)Y (2l + 1)4 lm

ˆ ×l1 m1 ; l2 m2 |lm l1 0; l2 0|l0 Ylm (p) √ % $ &|%kq |& = (−)q 2k + 1 12 &; k (−q)| 12 & :

(C.2) (C.3)


245

The latter, for instance, allows evaluation of the matrix element of the spin operators: %(t) |SG MG S SMS |%k(b) 1 q1 k 2 q2 G G = SMS |%k(b) |m1 m2 m1 m2 |%(t) k2 q2 |S M S 1 q1 $ %$ % = SMS | 12 (MS − m2 ); 12 m2 SG MG S | 12 m1 ; 12 (MG S − m1 ) & '& ' 1 G × 1 (MS − m2 )|%k(b)q |m1 m2 |%(t) | ( M − m ) S 1 k q 2

=

1 1

2 2

2

(−)q1 +q2 (2k1 + 1)(2k2 + 1) $ %$ % × SMS | 12 (MS − m2 ); 12 m2 SG MG S | 12 m1 ; 12 (MG S − m1 ) $ %$ % × 12 (MS − m2 ); k1 (−q1 )| 12 m1 12 m2 ; k2 (−q2 )| 12 (MG S − m1 ) :

(C.4)

It is convenient to couple the remaining spherical harmonics to a common angular momentum and to de6ne 1 YL˜M˜ L (p)Y ˆ l˜m˜ l (q)= ˆ : L˜ M˜ L ; l˜m˜ l |:Q BLQ˜l;˜ : (q; ˆ p) ˆ ; (C.5) 4 :

where we used the fact that the sum of the projections turns out to be equal to q1 + q2 = Q; B is then 1 ˜ l |:Q YL4 ˜ L ; l4 ˆ p) ˆ = ˆ l4 ˆ (C.6) BLQ˜l;˜ : (q; L4 ˜ L (p)Y ˜ l (q) 4 4 ;4 L

l

and normalized such that ˆ p) ˆ = 3:0 3L0 d`p d`q BLQ˜l;˜ : (q; ˜ 3l0 ˜ 3Q0 ˜ :

(C.7)

Some properties of B are derived in the next section. After putting together the individual pieces we arrive at the 6nal result: 1 Q ˆ q) ˆ = BL˜l;˜ : (q; ˆ p)A ˆ LD˜l;˜ : ; TD (p; 4 ˜ L˜l:

where D = {k1 q1 ; k2 q2 ; k3 q3 ; k4 q4 }, and ,; ,;G D CL˜l;˜ : M , (M ,G)† ALD˜l;˜ : = ,;,G

with CL,;˜l;˜,;G:D =

1 (−)MS +MS 7S MS ; L ML |j Mj j Mj ; l ml |JMJ SMS ; L0|JMJ 4

G JG MG J ×SG MG S ; LG MG L |jG MG j jG MG j ; lG mG l |JG MG J SG MG S ; L0|

˜ L˜ M˜ ; l˜m|:Q ˜ ×L ML ; LG − MG L |L˜ M˜ l ml ; lG − mG l |l˜m

(C.8)

246


%$ $ % G ˜ ˜ ×L 0; LG 0|L0 l 0; l 0|l0 SMS | 12 (MS − m2 ); 12 m2 SG MG S | 12 m1 ; 12 (MG S − m1 ) $ %$ % × 12 (m1 + q1 ); k1 (−q1 )| 12 m1 12 m2 ; k2 (−q2 )| 12 (MG S − m1 ) $ %$ % × 12 (m1 − q3 ); k3 q3 | 12 m1 12 m2 ; k4 ; q4 | 12 (MG S − m1 ) ;

(C.9)

where the sum runs over {MS ; MS ; ML ; MG L ; m1 ; m1 } and (2l + 1)(2lG + 1)(2L + 1)(2LG + 1)(2LG + 1)(2L + 1)(2k1 + 1)(2k2 + 1) 7= : (2L˜ + 1)(2l˜ + 1)(2J + 1)(2JG + 1) Appendix D. On the non-factorization of a strong nal state interaction In this appendix we will demonstrate the need for a consistent treatment of both the NN scattering and production amplitudes in order to obtain quantitative predictions of meson-production reactions. Let us assume a separable NN potential V (p ; k) = ,g(p )g(k) ;

(D.1)

where , is a coupling constant and g(p) an arbitrary real function of p. With this potential the T -matrix scattering equation can be readily solved to yield T (p ; k) =

V (p ; k) 1 − R(p ) + i7(p )V (p ; p )

with

R(p ) ≡ mP

0

∞

d k

k 2 V (k ; k ) p 2 − k 2

(D.2)

(D.3)

and 7(p) = p4 denotes the phase-space density here expressed in terms of the reduced mass of the outgoing two-nucleon pair 4 = mN =2. Note that for an arbitrary function g(k), such as g(k) ≡ 1 as discussed below, R(p ) may be divergent. In this case R is to be understood as properly regularized. The principal value integral R(p ) given above is therefore a model-dependent quantity, for it depends on the regularization scheme used. The condition that the on-shell NN scattering amplitude should satisfy Eq. (6) relates this to the on-shell potential, V (p ; p ): R(p ) = 1 + 7(p ) cot(3(p ))V (p ; p ) ;

(D.4)

where it is assumed that (p ) = 1. This shows that, for a given potential, the regularization should be such that Eq. (D.4) be satis6ed in order to reproduce Eq. (6). Indeed, in conventional calculations based on meson exchange models, where one introduces form factors to regularize the principal value integral, the cuto@ parameters in these form factors are adjusted to reproduce the NN scattering phase shifts through Eq. (6). Conversely, for a given regularization scheme, the NN potential should be adjusted such as to obey Eq. (D.4). This is the procedure used in e@ective 6eld theories [311], where the coupling constants in the NN potential are dependent on the regularization.


247

We also assume that the production amplitude M is given by a separable form, M (E; k) = Kg(k)h(p) ;

(D.5)

where K is a coupling constant and h(p) an arbitrary function of the relative momentum p of the two nucleons in the initial state. With this we can express the total transition amplitude as 1 i3(p ) M (E; p ) e : (D.6) sin(3(p )) A(E; p ) = − 7(p ) V (p ; p ) Eq. (D.6) is the desired formula for our discussion. It allows us to study the relationship between the NN potential and the production amplitude M (E; p ) explicitly as di@erent regularization schemes are used. For this purpose let us study the simplest case of a contact NN potential (setting the function g=1) in the limit p → 0. If we regularize the integrals by means of the power divergent subtraction (PDS) scheme [311] we get a4 R=− ; 1 − a4 where 4 denotes the regularization scale. Substituting this result into Eq. (D.4), we obtain 1 2a ,= m 1 − a4 for the NN coupling strength. Note that for 4 = 0 the PDS scheme reduces to that of minimal subtraction [311]. Since the total production amplitude A should not depend on the regularization scale we immediately read o@ Eq. (D.6) that K ˙ (1 − a4)−1 : Therefore the model clearly exhibits the point made in Section 2: Namely, the necessity of calculating both the production amplitude and the FSI consistently in order to allow for quantitative predictions. Appendix E. Chiral counting for pedestrians In this appendix we demonstrate how to estimate the size of a particular loop integral. This is a necessary step in identifying the chiral order of a diagram. It should be clear, however, that the same methods can be used to estimate the size of any integral. However, the importance of the chiral symmetry is that it ensures the existence of an ordering scheme that suppresses higher loops. The necessary input are the expressions for the vertices and propagators at any given order. For the chiral perturbation theory those can be found in Ref. [197]. In addition we need an estimate for the measure of the integral. Once each piece of a diagram is expressed in terms of the typical momenta/energies, one gets an estimate of the value of the particular diagram. The procedure works within both time-ordered perturbation theory and covariant theory. Obviously, for each irreducible diagram both methods have to give the same answer. If a diagram has a pure two-nucleon intermediate state, as is the case for the direct production, the covariant counting can only give the leading order piece of the counting within TOPT. In this appendix we study only diagrams that are three-particle irreducible; i.e., the topology of the diagram does not allow an intermediate two-nucleon state to go on-shell. The reducible diagrams

248


require a di@erent treatment and are discussed in detail in the main text. There is another group of diagrams mentioned in the main text that is not covered by the counting rules presented, namely radiation pions. Those occur if a pion in an intermediate state goes on-shell. It is argued in the main text that these are suppressed, because the pion, in order to go on-shell, is only allowed to carry momenta of the order of the external momenta, and thus the momentum scale within the loop is of order of the pion mass and not of the order of the initial momentum. Therefore we do not consider radiation pions any further. E.1. Counting within TOPT As mentioned above, if we want to assign a chiral order to a diagram, all we need to do is to replace each piece in the complete expression for the evaluation of the matrix element by its value when all momenta are of their typical size. In case of meson production in nucleon–nucleon collisions this typical momentum is given by the initial momentum pi . Time-ordered perturbation theory contains only three-dimensional integrals and thus we do not need to 6x the energy scale in the integral. The counting rules are • the energy of virtual pions is interpreted as O(pi ) 42 and thus; ◦ every time slice that contains a virtual pion is interpreted as 1=pi (see Fig. 40), ◦ for each virtual pion line put an additional 1=pi (from the vertex factors), • interpret the momenta in the vertices as pi , • every time slice that contains no virtual pion is interpreted as 1=m ; most of these diagrams, however, are reducible (cf. main text); • the integral measure is taken as pi3 =(4)2 . Here we used that pi2 =MN m pi , in accordance with Eqs. (80), and thus nucleons can be treated as static in the propagators if there is an additional pion present. However if there is a time slice that contains two nucleons only, the corresponding propagator needs to be identi6ed with the inverse of the typical nucleon energy 1=m and the static approximation is very bad [155]. In the diagram of Fig. 40 three NN vertices, each pi =f , appear as well as the NN Weinberg–Tomozawa vertex pi =f2 . In addition the three time slices give a factor 1=pi3 and we also need to include a factor 1=pi2 , since there are two virtual pions. We therefore 6nd 3 pi 1 5 pi3 m pi 1 M TOPT : f f2 pi (4)2 f3 MN √ Here we used 4f MN and pi MN m . E.2. Counting within the covariant scheme Naturally, as TOPT and the covariant scheme are equivalent, the chiral order that is to be assigned to some diagram needs to be the same in both schemes. The reason why we demonstrate both is that 42

There is one exception to this rule: if a time derivative acts on a pion on a vertex, where all other particles are on-shell, then energy conservation 6xes the energy.


249

Fig. 40. A typical loop that contributes to pion production in nucleon–nucleon collisions. Solid lines denote nucleons and dashed lines pions. The solid dots show the points of interactions. The horizontal dashed lines indicate equal time slices, as needed for the evaluation of the diagram in time-ordered perturbation theory.

here we are faced with a problem in which the typical energy scale m and the typical momentum scale pi are di@erent. In the covariant approach a four-dimensional integral measure enters and naturally the question arises whether m or pi is appropriate for the zeroth component of this measure p0 . For example, in Ref. [163] it was argued, that one should choose m , although this choice is by no means obvious from the structure of the integrals. However, given the experience we now have in dealing with loops in time-ordered perturbation theory, where these ambiguities do not occur, the answer is simple: we just have to assign that scale to p0 that will reproduce the same order for any diagram as in the counting within TOPT [204]. Once the choice for p0 is 6xed, the much easier to use covariant counting can be used to estimate the size of any loop integral. Thus have the following rules: • the energy of virtual pions is interpreted as O(pi ); 43 • each pion propagator is taken as O(1=pi2 ), • each nucleon propagator that cannot occur in a two-nucleon cut is taken as O(1=p0 ) (the leading contribution of a nucleon propagator that can occur in a two-nucleon cut is O(1=m ); most of these diagrams, however, are reducible (cf. main text)), • interpret the momenta in the vertices as pi , • the integral measure is taken as p0 pi3 =(4)2 (when the diagram allows for a two-nucleon cut the measure reads (m pi3 )=(4)2 ). 43

With the same exception as in the time-ordered situation (cf. corresponding footnote).

250


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Four sorts of meson D.V. Bugg∗ Queen Mary, University of London, London E1 4NS, UK Accepted 8 March 2004 editor: J.V. Allaby

Abstract An extensive spectrum of light non-strange qq/ states up to a mass of 2400 MeV has emerged from Crystal Barrel and PS172 data on pp / → Resonance → A + B in 17 4nal states. These data are reviewed with detailed comments on the status of each resonance. For I = 0, C = +1, the spectrum is complete and very secure. Six ‘extra’ states are identi4ed. Four of them have I = 0, C = +1 and spin-parities predicted for glueballs. Their mass ratios agree closely with predictions from lattice QCD calculations. However, branching ratios for decays are not nite potential strength while the anticlassical limit corresponds to the limit of vanishing strength of the inverse-square potential. For nonhomogeneous potentials with two or more energy or length scales, the de>nition of the semiclassical limit is not always straightforward and unambiguous. Consider, for example the Woods-Saxon potential step, VWS (r) = −

V0 ; 1 + exp(r=)

(13)

which is characterized by its depth V0 = ˝2 (K0 )2 =(2M) and the di?useness parameter . A quantum particle approaching such a step with a total positive energy E = ˝2 k 2 =(2M) is partially reAected (see Section 5), and the reAection probability PR is known [43,46] to be sinh[(q − k)] 2 PR = ; (14) sinh[(q + k)] where q = k 2 + (K0 )2 is the asymptotic wave number on the down side of the step, r → −∞. In the Schr4odinger equation (1) we can consider the formal limit ˝ → 0 while keeping the energy E and potential strength V0 >xed. This corresponds to taking the limit k → ∞ and K0 → ∞, and the reAection probability (14) becomes, PR

k;K0 →∞

∼

exp(−4k) ;

(15)

which is now independent of q. If on the other hand, we respect the fact that ˝ is >xed and study the high-energy (k → ∞) limit of Eq. (1), then the parameter K0 de>ning the height of the step

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

remains constant and the formula for the reAection probability becomes, 2 (K0 )2 k →∞ PR ∼ exp(−4k) : k

367

(16)

We see that the high-energy limit for PR di?ers from the “formal” semiclassical limit (15) by a factor inversely proportional to the energy. This is another subtle example demonstrating that the high energy limit need not coincide with the semiclassical limit Sc =˝ → ∞. 2.2. The WKB approximation For vanishing or constant potential, the local classical momentum (2) is constant, and the Schr4odinger equation (1) has plane wave solutions, i (r) ˙ exp ± pr : (17) ˝ When the potential is not constant, p is a function of r, and looking at Eq. (17) suggests a more general ansatz for the wave function, i S(r) : (18) (r) = exp ˝ Inserting (18) in the Schr4odinger equation (1) gives a di?erential equation for the function S(r), S (r)2 − i˝S (r) = p(r)2 : In order to obtain an approximate solution for S(r), and thus for the wave function S(r) in a formal series in ˝, which is regarded as a small parameter, 2 ˝ ˝ S2 (r) + · · · : S(r) = S0 (r) + S1 (r) + i i

(19) (r), we expand (20)

In the spirit of conventional semiclassical theory, the expansion (20) assumes that all other relevant quantities of the dimension of an action are large compared to ˝, as discussed in the preceding section. Inserting Eq. (20) in Eq. (19) gives a di?erential equation for the Si (r), p2 (r) − (S0 )2 + i˝(S0 + 2S0 S1 ) + ˝2 (S1 + 2S0 S2 + (S1 )2 ) : : : = 0 :

(21)

Eq. (21) can only be ful>lled if all terms of O(˝n ) vanish independently. Starting with n = 0 we get (22) S0 = ±p(r) ⇒ S0 = ± p(r) dr : In the classically allowed regions, the local classical momentum (2) is real, and we shall always assume p(r) to refer to the positive square root of 2M[E − V (r)]; except for a possible sign, S0 is the classical action Sc , Eq. (3). The terms of >rst order in ˝ in Eq. (21) give, S1 = −

S0 p (r) ; = − 2S0 2p(r)

(23)

368


and, after integration, 1 S1 = − ln p(r) : 2

(24)

After inserting these results in the ansatz (18), we obtain the general form of the >rst-order WKB wave function, r C1 i (r) = p(r ) dr exp WKB ˝ r0 p(r) C2 i r + ; (25) exp − p(r ) dr ˝ r0 p(r) with arbitrary complex coeScients C1 and C2 . In classically allowed regions, Eq. (25) represents a superposition of a rightward travelling (>rst term) and a leftward travelling (second term) wave. The lower integration point r0 is a “point of reference” which determines the phase of each term; this point of reference must always be speci>ed when de>ning a WKB wave function. An alternative choice of WKB waves is given by the sine or cosine of the WKB integrals, e.g. r 1 1 : (26) p(r ) dr − cos WKB (r) ˙ ˝ r0 2 p(r) Two linearly independent WKB wave functions can be obtained by choosing two di?erent phases in (26), but they must not di?er by an integral multiple of 2. In classically forbidden regions, V (r) ¿ E, the local classical momentum (2) becomes purely imaginary, but WKB wave functions can still provide a feasible approximation of the exact solution of the Schr4odinger equation. Two independent WKB wave functions are now given by 1 r 1 ; (27) exp ± (r) ˙ |p(r )| dr WKB ˝ r0 |p(r)| which are exponentially increasing or decreasing functions of r. The terms of higher order in ˝ in Eq. (21) allow a systematic derivation of the terms Sn in the expansion (20). It is convenient [27] to introduce the functions n (r), n = −1; 0; 1; 2; : : :, via r dr n−1 (r ); Sn = n−1 : (28) Sn (r) = r0

The n are then given by −1 = S0 = ±p;

0 = S1 = −

p ; 2p

and the recursion relation   n 1  n+1 = − n + j n− j  2−1 j=0

(29)

for n ¿ 0 :

(30)


In particular, 1 1 = S2 = − ( + 02 ) = ± 2−1 0

p 3 p 2 − 4p2 8 p3

369

:

(31)

The series de>ned by Eq. (20) is asymptotic and does not converge. The inclusion of higher and higher terms eventually gives less accurate results. There are special potentials where the series terminates. For example, for potentials V (r) ˙

1 ; (ar + b)4

(32)

with real constants a and b, all n (r), n ¿ 1 vanish identically at zero energy, so the WKB wave functions (25) or (27) are exact solutions of the Schr4odinger equation for E = 0. For V (r) = (ar 2 + br +c)−2 and E =0, we >nd 2 ≡ 0, but we are left with the calculation of the remaining integrals in Eq. (28). 2.3. Accuracy of WKB wave functions as a local property of the Schrodinger equation The WKB approximation is, of course, expected to work well near the semiclassical limit ˝ → 0, see Section 2.1. However, since the expansion (20) depends on the coordinate r, the accuracy of a truncated expansion including a given number of terms must be expected to also depend on r. A frequently formulated condition for the accuracy of the WKB approximation is based on the requirement, that the second term in the left-hand side of Eq. (19) should be small compared to the >rst term, |i˝S (r)||S (r)2 | ; inserting S (r) ≈ p(r) according to Eq. (22) gives p (r) 1 d 1 ; ˝ 2 = (r) p (r) 2 dr

(33)

(34)

which corresponds to the requirement that the local de Broglie wavelength, (r) = 2˝=p(r) should be slowly varying. Note, however, that the leading contribution to S on the left-hand side of Eq. (33), namely S0 , is already included via the >rst-order terms (23), (24) in the >rst-order WKB wave functions (25)–(27), so it does not make sense to require this term to be small as a condition for the accuracy of the wave functions. Indeed, the frequently accepted condition (34) is not necessary for the >rst-order WKB wave functions to be accurate. A striking counter-example is the potential (32) at energy E = 0. Although >rst-order WKB wave functions are exact solutions of the Schr4odinger equation, the derivative of the de Broglie wavelength goes to in>nity for r → ∞. As a criterion for the accuracy of the WKB wave functions, it makes more sense to require smallness of the >rst term of the series (20) which is not considered in the de>nition of the wave functions, i.e., the term (31). This idea is supported by considering the modi>ed Schr4odinger equation for which the WKB wave function is an exact solution. By calculating the second derivative of the

370


WKB wave function (25) [or (26) or (27)] it is easy to see that Schr4odinger-type equation p(r)2 3 p 2 p − WKB (r) − WKB (r) = 0 : WKB (r) + ˝2 4 p2 2p

WKB

is an exact solution of the (35)

It should be a good approximation to the solution of the original Schr4odinger equation if the third term on the left-hand side of (35) is small compared to the second term, i.e. the absolute value of the function 3 (p )2 p 2 Q(r) = ˝ (36) − 4 p4 2p3 should be much smaller than unity, |Q(r)|1 :

(37)

This corresponds to requiring |1 =−1 | to be small, where 1 is given in Eq. (31); 1 is the derivative of S2 and (−˝2 )S2 is the >rst term in the series (20) not to be considered in the de>nition of the (>rst-order) WKB wave functions. The function Q(r) de>ned in Eq. (36) can be positive or negative, but it is always real, even in a classically forbidden region where the local classical momentum p(r) is purely imaginary. The condition (37) is a local condition, because Q is a function of r. This function has been called “badlands function” [55,56], because it is large in regions of coordinate space where the WKB approximation is bad. On the other hand, large (positive or negative) values of Q(r) mean that quantum e?ects are important in this region of coordinate space. So a more positive name for the function (36) is the “quantality function”. In regions of high quantality, the condition (37) is violated, the WKB approximation is poor and quantum e?ects are important. Now it is also clear that the simple condition Eq. (34) is not in general suScient for the WKB wave function to be an accurate approximation of an exact solution of the Schr4odinger equation. Consider a potential oscillating rapidly with small amplitude and a moderate total energy such that the local de Broglie wavelength 2˝=p(r) behaves as 1 + sin(qr)=q3=2 . In the limit of large q values, the simple condition (34) is ful>lled suggesting good applicability of the WKB approximation. However, √ the term involving the second derivative in Eq. (36) gives a contribution proportional to q sin qr, which results in a diverging quantality for large q. By analogous arguments it follows that small but nonvanishing values of |Q(r)| are not always suScient for the WKB approximation to be accurate; it is in principle possible that p which contributes to the next-order correction 2 according to Eq. (30) is large even though |Q(r)| is small. Nevertheless, smallness of |Q(r)| is clearly a more reliable indication of the accuracy of the WKB approximation than the simple condition Eq. (34). Outside the quantal region, the WKB wave function is often an excellent approximation to the exact solution of the Schr4odinger equation, even if the global condition for semiclassical approximations, i.e. small ˝, Eq. (5), is not ful>lled. In such situations, it may be possible to >nd accurate solutions of the Schr4odinger equation in the quantal regions—by analytical or numerical means—and then construct globally accurate wave functions by appropriately matching the exact (or highly accurate) wave functions from the quantal region to WKB wave functions in the semiclassical, “WKB regions” where the condition (37) is well ful>lled. This simple philosophy underlies the various applications discussed in the subsequent sections of this review.


371

2.4. Examples A simple set of examples is provided by step-wise constant potentials, Vsteps (r) = Vi ;

r ∈ Li ;

(38)

where {Li } represents a covering of coordinate space with a set of intervals, Li = (ri ; ri+1 ). At total energy E, the motion in each interval Li is classically allowed if E ¿ Vi and forbidden if E ¡ Vi . WKB wave functions are exact within each interval. In the classically allowed intervals there are two independent solutions proportional to exp(±iki r) with wave number ki = 2M(E − Vi )=˝, in the classically forbidden intervals there are two independent solutions proportional to exp(±%i r) with inverse penetration depth %i = 2M(Vi − E)=˝. The quantality function (36) is zero everywhere, except at the borders ri of the intervals, and a global exact solution of the Schr4odinger equation can be obtained by matching superpositions of the two independent solutions in each interval such that the wave function and its >rst derivative are continuous. This method gives exact results regardless of any consideration of the semiclassical limit. In fact, it is just how one would go about solving the Schr4odinger equation without any knowledge of or reference to WKB wave functions. At each step ri , the >rst derivative of the wave function should be continuous, but the second derivative should not, because in the Schr4odinger equation (1), the discontinuity in the potential must be compensated by a corresponding discontinuity in . For a single discontinuity separating two regions 1 and 2 where V (r) is constant, we have a sharp-step potential. A quantum particle approaching the step on the upper level is partially reAected, and the reAection probability is √ √ 2 E − V1 − E − V2 q−k 2 √ PR = = √ (39) q+k E − V1 + E − V2 where q and k are the wave numbers on the down side and the up side of the step. Eq. (39) is a standard textbook result; it also follows from the reAection probability for the Woods-Saxon potential (13) in the limit of vanishing di?useness, → 0. The formula (39) does not contain ˝, i.e. it is not a?ected by taking the formal semiclassical limit, ˝ → 0. One way of understanding this is to realize, that the characteristic length of the potential, de>ned as the distance over which it changes appreciably, is zero for the sharp step. This is always small compared to quantum mechanical lengths such as wavelengths or penetration depths, regardless how close we may be to the semiclassical limit. (See also footnote in Section 25 of Ref. [43]). In this context it is interesting to discuss the case that the potential V is itself continuous, but has a step-like discontinuity in one of its derivatives. Assume that V (i) is continuous for 0 6 i ¡ n and that the nth derivative V (n) (r) is discontinuous at a point r0 . The next derivative, V (n+1) (r), then has a delta-function singularity at r0 , and so do the (n + 1)st derivative of p(r) and the function (r), cf. Eqs. (29), (31). The function S Sn+1 n+1 (r) has a step-like discontinuity at r0 , and this enters in order ˝n+1 in the expansion (20). The ansatz (18) thus contains a step-like discontinuity of order ˝n at the point r0 , which is incompatible with the requirement, that (r) be a continuous and (at least) twice di?erentiable function. The continuity of the wave function (18) is repaired by adding a second term with an amplitude of order O(˝n ). In the classically allowed regime, this leads to classically forbidden reAection with a reAection amplitude of order O(˝n ). The case n = 0

372


as for the sharp step potential leads to a reAection amplitude of order ˝0 , i.e., independent of ˝. For a continuous potential with a kink, i.e. with a step-like discontinuity in the >rst derivative, the amplitude for classically forbidden reAection vanishes as ˝ in the semiclassical limit. For analytical potentials, which are continuously di?erentiable to all orders, the amplitude for classically forbidden reAection generally vanishes exponentially in the semiclassical limit, e.g. as exp(−C=˝), see Section 5, Eqs. (253), (254). For homogeneous potentials (7), the semiclassical and anticlassical limits can be reached by appropriate variations of the energy and/or the potential strength, see Section 2.1. Negative degrees d corresponding to inverse-power potentials, V& (r) = ±

C& ; r&

&¿0 ;

(40)

occur in the description of various physical phenomena. For example: & = 1 for Coulomb potentials, & = 2 for centrifugal or monopole-dipole potentials, & = 3 for the van der Waals potential between a neutral polarizable particle and a surface, & = 4 for the interaction between a neutral and a charged particle, &=6 for the van der Waals potential between two neutral particles and &=7 for the retarded van der Waals potential between two neutral particles [57]. At zero energy, the local classical momentum (2) in the repulsive or attractive potential (40) is proportional to r −&=2 , and the quantality function (36) has a very simple form, & & −2 & 1− r : (41) Q(r) ˙ 4 4 As discussed for the potential (32), Q(r) ≡ 0 for the special case & = 4. More importantly, Q(r) is seen to vanish for small r when & ¿ 2 and for large r when & ¡ 2. For Coulombic potentials and near-threshold energies, WKB wave functions become increasingly accurate for r → ∞. For >nite (positive or negative) energies, the inAuence of the potential becomes less important for r → ∞ and the local classical momentum tends to a constant. WKB wave functions become increasingly accurate for r → ∞ for any power & ¿ 0. For positive energies E = ˝2 k 2 =(2M), the error in the WKB wave function decreases asymptotically as 1=(kr)&+1 when its phase is correctly matched [58]. Towards small r values, a >nite energy in the Schr4odinger equation becomes increasingly irrelevant as r → 0, so the result (41) holds in this case as well. For & ¿ 2, WKB wave functions become increasingly accurate for r → 0. For attractive potentials, this singular behaviour cannot continue all the way to the origin, but there may be a region of small but nonvanishing r values where the WKB approximation is highly accurate. Repulsive inverse-power potentials can be meaningfully used all the way down to r = 0, and, for & ¿ 2, the WKB wave functions accurately describe the behaviour of the wave function near the origin. Expressing the strength C& of the potential (40) through a length & , C& = ˝2 (& )&−2 =(2M), we have (&−2)=2 1 & r →0 &=4 : (42) ; '= (r) ˙ r exp −2' r &−2 For 0 ¡ & ¡ 2, the quantality function diverges for r → 0, and the WKB approximation fails near the origin. For the special case & = 2, the quantality function tends towards a constant as r → 0. For a repulsive inverse-square potential, C2 = ˝2 (=(2M), ( ¿ 0, the WKB wave function WKB and


the exact wave function r →0 1 +√( ; WKB (r) ˙ r 2

373

behave a little di?erently near the origin, r →0

1 √

(r) ˙ r 2 +

(+1=4

:

(43)

Inverse-square potentials will be discussed in more detail later on; see in particular Eq. (64) in Section 3.3 and the whole of Section 4.4. 3. Beyond the semiclassical limit As was shown in the preceding section the accuracy of the WKB approximation is a local property of the Schr4odinger equation in coordinate space rather than depending on the global validity of the conditions of the semiclassical limit. By bridging the gaps due to regions of high quantality, appropriate WKB wave functions may be constructed and used to derive accurate quantum results, even under conditions which are globally far from the semiclassical limit. In this section we discuss concrete applications to three di?erent physical situations: scattering by singular potentials, quantization in potential wells and tunnelling through potential barriers. 3.1. Connection formulas at classical turning points At a classical turning point, rt , of a one-dimensional system the local classical momentum (2) vanishes, p(rt ) = 0, and the quantality function Q(r) [Eq. (36)] is singular. The exact wave function may nevertheless be well represented by a superposition of exponentially increasing and decreasing WKB waves (27) on the classically forbidden side of rt , and/or by oscillating WKB waves (25), (26), on the classically allowed side. For a consistent description of both regions, we have to say how the WKB waves on either side of rt are to be connected. In the most general case, the connection formulas can be written as r N 2 1 1 r ↔ − (44) cos exp − p(r ) dr p(r ) dr 2 ; ˝ rt ˝ rt p(r) |p(r)| r U 1 NU 1 r 1 ↔ (45) cos exp p(r ) dr − p(r ) dr : ˝ rt 2 ˝ rt p(r) |p(r)| The two parameters and N in Eq. (44) can be determined by comparing the exact solution corresponding to an exponentially decreasing wave on the classically forbidden side with the oscillating WKB waves on the allowed side. Since the decaying wave on the forbidden side is unique except for a constant factor, the parameters and N are well de>ned. In contrast, the asymptotic behaviour of the exponentially increasing solution in Eq. (45) masks any admixtures of the decaying wave; since such an admixture signi>cantly a?ects phase and amplitude of the wave function on the allowed side, the two parameters U and NU are not well de>ned. In conventional semiclassical theory [43–47], the connection formulas are derived assuming that the potential can be approximated by a linear function of r in the vicinity of rt , and that this region extends “suSciently far” to either side of rt . “SuSciently far” means that the exact solutions of the Schr4odinger equation in the linear potential, which can be expressed in terms of Airy functions [59],

374


are valid until the asymptotic forms of the respective Airy functions can be matched to the WKB waves on both sides. For the well de>ned parameters and N this conventional matching leads to conv = ; N conv = 1 : (46) 2 The continuity equation relates the four parameters , N , U and NU [60]. To see this consider a superposition of Eqs. (44) and (45) with arbitrary complex coeScients, = A × (44) + B × (45), and calculate the current density, j = (˝=M)I( ∗ ). For de>niteness we assume the classically allowed (forbidden) region to lie to the left (right) of rt . Inserting the left-hand sides of Eqs. (44) and (45) in the superposition gives the current density on the classically allowed side, 2 − U ∗ I(A B) sin ; (47) jallowed = M 2 whereas inserting the right-hand sides gives the current density on the classically forbidden side, 2 jforbidden = (48) I(A∗ B)N NU : M The conservation of the current density, jallowed = jforbidden , requires U − : (49) N NU = sin 2 The ill-de>ned parameters U and NU are irrelevant for a totally reAecting potential. Here the wave function must vanish asymptotically in the classically forbidden region, so the coeScient of the exponentially growing solution (45) must vanish. In the more general case, the uncertainty in the de>nition of the barred quantities can be removed by introducing a second condition by convention, e.g. by assuming a >xed phase di?erence of =2 between the oscillating waves in Eqs. (44) and (45). Such a choice is used in the de>nition of the irregular continuum wave function in scattering theory [49,61]. From Eq. (49) it would follow that NU = 1=N . However, other choices are possible as long as the left-hand sides of Eqs. (44) and (45) remain linearly independent. 3.2. The re?ection phase In the case of total reAection at a classical turning point the current density is zero and it is possible to represent the quantum wave by a real function. The expressions (44) represent the exact wave function in the semiclassical regions on either side of the turning point. The amplitude parameter N can be used to calculate the particle density | |2 in the classically forbidden region [62], but the choice of N is not so important in the classically allowed region if the normalization of the wave function is either irrelevant or deducible from other considerations. The more important quantity in many cases is the phase , which inAuences the phase of the WKB wave in the whole of the classically allowed region. Rewriting the left-hand side of Eq. (44) as r ei=2 i i r () −i + e (50) (r) = p(r ) dr exp p(r ) dr exp − WKB ˝ ˝ p(r) rt

rt


375

reveals that is the phase loss su?ered by the WKB wave due to reAection, the “reAection phase” [63,64]. With the conventional choice (46) of the reAection phase and the amplitude factor, the WKB wave function in the classically allowed region is r 2 1 conv : (51) p(r ) dr − cos WKB (r) = ˝ rt 4 p(r) For this approximation to be valid, the Airy function must match the exact wave function both in the classically allowed as well as in the classically forbidden region. The potential must be close to linear in a region extending at least a few times the wavelength of the Airy function on the classically allowed side and a few times the penetration depth on the forbidden side. This condition is sometimes hard to meet, even though the general condition (37) for accuracy of the WKB approximation may be well ful>lled on both sides of the classical turning point rt , e.g. when the kinetic energy of the particle is small and thus the wavelength large even far away from rt . Sometimes the slow variation of the potential compared to the wavelength and penetration depth coincides with the linearity requirement. However, there are important examples where these conditions are not ful>lled at the same time. The separation of the linearity requirement and the resulting >xation (46) of the reAection phase from the application of the WKB approximation greatly widens the range of applicability of WKB waves to a variety of new situations. A very simple example is the sharp potential step, V = V0 ,(r), at energies below the step, 0 ¡ E ¡ V0 . On the classically forbidden side of the step, r ¿ 0, the WKB wave function is pro portional to exp(−%|r|) where % = 2M(V0 − E)=˝ is the inverse penetration depth, and this is an exact solution of the Schr4odinger equation. On the classically allowed side, r ¡ 0, the WKB wave √ function is proportional to cos(k|r| − =2) where k = 2ME=˝ is the wave number, and this too is an exact solution of the Schr4odinger equation. The problem is solved exactly in the whole of coordinate space, if the reAection phase and amplitude factor N are chosen as %k : (52) = 2 arctan(%=k); N = 2 2 % + k2 √ Note that assumes the semiclassical value =2 only for %=k whereas N =1 only for %=(2± 3)k. The sharp step potential is particularly easy to handle, because the quantal region reduces to a single point; WKB wave functions are exact everywhere, except at the discontinuity in the potential. For a billard system in more than one dimension, i.e. a particle moving in an area (or volume) with a sharp boundary, the generalization from an in>nitely repulsive boundary to a potential of >nite height simply means replacing the Dirichlet boundary condition appropriate for the in>nite step to the conditions (52) for the component normal to the bounding surface. This is one of the few examples where quantum e?ects beyond the semiclassical limit have been successfully included in the semiclassical description of multidimensional systems [24,25,65]. 3.3. Scattering by a repulsive singular potential Repulsive singular potentials have a large application >eld in particle scattering. They are excellent examples to demonstrate the usefulness of the concept of the reAection phase. Consider a repulsive

376


homogeneous potential, C& ˝2 (& )&−2 ˝2 v& (r) : V&rep (r) = & = = r 2M r& 2M The quantality function (36) for the potential (53) at energy E = ˝2 k 2 =(2M) is

& 1 & v& (r) 2 − 1 v Q(r) = (& + 1)k + (r) : & 4 r 2 [k 2 − v& (r)]3 4

(53)

(54)

From Eq. (54) we observe that Q(r) = O(r &−2 ) for small r. As already mentioned in Section 2.4, the WKB approximation becomes increasingly accurate for r → 0 as long as & ¿ 2. This holds in particular for small energies—including vanishing energy. In the special case & = 4 and k = 0 the quantality function vanishes for all r, Q(r) ≡ 0, and WKB wave functions are exact solutions of the Schr4odinger equation, see also Eq. (32). For the large-r limit one has to distinguish the cases E = 0 and E ¿ 0. While the Q(r) diverges as r &−2 (& ¿ 2; & = 4) for zero energy, it approaches zero for large r as r −&−2 for E ¿ 0. The classical turning point is given by rt = & (& k)−2=& :

(55) & −2

as r → 0, so the WKB For 0 ¡ & ¡ 2, the quantality function Q(r) diverges proportional to r approximation deteriorates towards the origin. Decaying WKB waves cannot be expected to be good approximations of the exact wave function on the classically forbidden side of rt for potentials which are less singular than 1=r 2 . On the allowed side of rt however, Q(r) vanishes faster than 1=r 2 for any positive value of & (and E ¿ 0), so the WKB approximation becomes increasingly accurate for r → ∞. Hence an oscillating WKB wave function such as on the left-hand side of Eq. (44), r 1 2 () ; (56) cos p(r ) dr − WKB (r) = ˝ rt 2 p(r) is a valid representation of the exact wave function on the classically allowed side of the turning point, regardless of whether the corresponding decaying WKB wave functions are good approximations for r → 0, as is the case for & ¿ 2, or poor approximations, as is the case for 0 ¡ & ¡ 2. In any case, the reAection phase is to be chosen such that the phase of the WKB wave (56) asymptotically matches the phase of the regular exact wave function which vanishes at r = 0. Inverse-square potentials, corresponding to & = 2, represent a special case, ˝2 ( V( (r) = : (57) 2M r 2 Potentials of this form are of considerable physical importance since they appear in the interaction of a monopole charge with a dipole and as centrifugal potential in radial Schr4odinger equation in two or more dimensions. The centrifugal potential depends on the angular momentum quantum number lD and is of the form (57) with ( = l3 (l3 + 1);

l3 = 0; 1; 2; : : : ;

for three dimensions, and 1 ( = (l2 )2 − ; l2 = 0; ±1; ±2; : : : ; 4 for two dimensions.

(58) (59)


377

The Schr4odinger equation with an inverse-square potential (57) contains no length or energy scale. Its regular solution is essentially a simple Bessel function of the product kr, 1 r J0 (kr); where 0 = ( + : (60) (r) = 2 4 √ The classical turning point is given by krt = (, and the quantality function is (2 ( 3 5 + : (61) 2 3 2 4 ((kr) − () 2 ((kr) − ()2 Away from the classical turning point, the absolute value of Q(r) is small for large ( corresponding to the semiclassical limit, and large for small ( corresponding to the anticlassical limit, see Section 2.1. For large values of kr, the asymptotic form [59] of the Bessel function implies that the wave function (60) behaves as 1 kr →∞ 1 ; (62) (r) ˙ √ cos kr − (+ − 2 4 4 k Q(r) =

whereas for the WKB wave function (56), √ kr →∞ 1 () : (63) (− WKB (r) ˙ √ cos kr − 2 2 k In conventional semiclassical theory, the reAection phase in the WKB wave function (63) is assumed to be =2 and the resulting discrepancy of the phases in Eqs. (63) and (62) is repaired with the help of the Langer modi>cation [66]. This consists of modifying the potential used to calculate the WKB wave functions according to 1 ˝2 1 ( → ( + ; V( (r) → V( (r) + : (64) 4 2M 4r 2 This also leads to a correct behaviour of the WKB wave function in the classically forbidden region: 0 ˙ (kr) for kr → 0, 0 = ( + 1=4, cf. Eq. (43) in Section 2.4. However, the classical turning point is shifted. Comparison of Eq. (62) with Eq. (63) suggests an alternative approach. The phases of exact and WKB waves can be made to match asymptotically by choosing the reAection phase in the following way [63,64], 1 √ = + (+ − ( : (65) 2 4 Examination of higher order asymptotic (kr → ∞) terms shows that the error in the WKB wave function (63) with the reAection phase (65) is proportional to (kr)−3 . This is better by two orders in 1=(kr) than the conventional semiclassical treatment, which is based on a reAection phase =2 and the Langer modi>cation (64); in the conventional treatment, the error in the WKB wave function only falls o? as 1=(kr) asymptotically [63,64]. For the inverse-square potential (57), the semiclassical limit is independent of energy and is reached for ( → ∞, see Section 2.1 and Eq. (61), whereas the anticlassical limit corresponds to ( → 0. As one would expect, the reAection phase (65) approaches =2 in the semiclassical limit;

378


in the anticlassical limit it approaches the value , as for a free particle reAected by a hard wall [Eq. (52) in the limit %=k → ∞]. The general repulsive potential (53) proportional to 1=r & with & = 2 is characterized by a strength parameter with the physical dimension of a length, & . The properties of the Schr4odinger equation at energy E = ˝2 k 2 =(2M) depend not on energy and potential strength independently, but only on the product k& . Proximity to the semiclassical limit may be expressed quantitatively by comparing the classical turning point (55) with the quantum mechanical length corresponding to the inverse wave number, k −1 . We call the ratio of these two quantities the reduced classical turning point, a = krt = (k& )1−2=& :

(66)

In the semiclassical limit, the classical turning point is large compared to k −1 , i.e. a → ∞, which corresponds to k → ∞ when & ¿ 2, but to k → 0 when 0 ¡ & ¡ 2, in accordance with Section 2.1. Conversely, the anticlassical limit is given by a → 0. The inverse-square potential (57) >ts into this √ scheme if we identify krt = ( as the reduced classical turning point a. In terms of the classically de>ned quantities, total energy E, particle mass M and strength parameter C& of the potential (53) or (57), the reduced classical turning point (66) is, a=

√ 1 1−1 pas rt E 2 & (C& )1=& 2M = : ˝ ˝

(67)

According to Eq. (67), the reduced classical turning point a is just the classical action obtained by multiplying the asymptotic (r → ∞) classical momentum pas = ˝k and the classical turning point rt , measured in units of Planck’s constant ˝. The reAection phase in the WKB wave function (56) is directly related to the phase shift 1, which determines the asymptotics of the regular solution of the Schr4odinger equation [43,61], reg (r)

r →∞

˙ sin(kr + 1) :

(68)

Comparing this to the WKB wave function (56) gives an explicit expression for 1 in terms of , r 1 p(r ) dr − kr − 1 = + lim : (69) 2 r →∞ ˝ rt 2 For potentials falling o? faster than 1=r asymptotically, the square bracket above remains >nite as r → ∞ and 1 is a well de>ned constant depending only on the wave number k. For & ¿ 2, the anticlassical limit of the Schr4odinger equation corresponds to the threshold, k → 0, and the phase loss of the WKB wave due to reAection by the singular potential (53) can be derived from the exact solution of the Schr4odinger equation for zero energy, √ & 1=(2') 1 : (70) ; '= rK±' 2' reg (r) ˙ r &−2 This is the regular solution, reg (0)=0, which is unique to within a constant factor. Since the energy enters the Schr4odinger equation with a term of order E=O(k 2 ), the zero-energy solution (70) remains valid for small energies to order below O(k 2 ). For large values of r, the argument of the Bessel


379

function K±' in (70) is small, and the leading behaviour of the wave function is, reg (r)

r →∞

˙ '2'

r 2(1 − ') − : 2(1 + ') &

(71)

The asymptotic behaviour of the regular solution of the Schr4odinger equation is given by Eq. (68) and, for potentials falling o? more rapidly than 1=r 3 , the low-energy behaviour of the phase shift is, k →0

1 ∼ n − a0 k ;

(72)

with the scattering length a0 (see Ref. [43] and Section 4.1). So for large r and small k we have, reg (r)

˙ k(r − a0 ) :

(73)

Comparing Eqs. (73) and (71) gives an explicit expression for the scattering lengths of repulsive inverse power-law potentials (53) with & ¿ 3, a0 = '2'

2(1 − ') & ; 2(1 + ')

'=

1 : &−2

(74)

The asymptotic (r → ∞) expression for the action integral in the WKB wave function (56) is, √ 1 r 2(1 − &1 ) a & −1 r →∞ : (75) p(r ) dr ∼ kr − a + O ˝ rt 2 2( 32 − &1 ) kr For & ¿ 3 the phase of the WKB wave function has the correct near-threshold behaviour, see Eqs. (72) and (74), when the reAection phase is given by [58], k →0

∼ −

2 √ 2(1 − &1 ) 2(1 − ') k& ; 3 1 (k& )1− & + 2'2' 2(1 + ') 2( 2 − & )

(& ¿ 3) :

(76)

The second term on the right-hand side of Eq. (76) is linear in a and cancels the corresponding contribution from the WKB action integral (75); the third term (proportional to a1+2' ) yields the contribution −ka0 with the scattering length a0 in Eq. (74). Near the semiclassical limit a → ∞, a semiclassical expansion for the phase shifts [67,68] can be used [58] to derive the leading contributions to the reAection phase for the repulsive homogeneous potentials (53), √ (& + 1)2( &1 ) 1 a→∞ + : (77) ∼ +O 2 a 12&2( 12 + &1 ) a3 For >nite values of the reduced classical turning point a, between the semiclassical (77) and anticlassical (76) limits, the reAection phases can be obtained by solving the Schr4odinger equation numerically and comparing the phase of the solution with the phase of the WKB wave function (56) as r goes to in>nity. The results are shown in Fig. 1 as functions of a for integer powers & from 2 to 7. In all cases we observe a monotonous decline from the value in the anticlassical limit a = 0 towards the semiclassical expectation =2 for large a. The smooth dependence of the reAection phases in Fig. 1 on a and & suggests that a simple algebraic formula based on the exact result (65) for & = 2 might be e?ective. Indeed, if we replace

380


Fig. 1. Exact reAection phases for reAection by a repulsive homogeneous potential (53) as function of the reduced classical turning point (66). From [58].

the reduced classical turning point a ≡ aS =

a ; S(&)

√

( in Eq. (65) with a scaled reduced classical turning point,

2 & + 1 2( &1 ) ; S(&) = √ 3 & 2( 12 + &1 )

then the appropriate generalization of Eq. (65), namely 1 (aS )2 + − aS ; = + 2 4

(78)

(79)

reproduces the numerically calculated exact reAection phases with an error never exceeding 0:006 for all powers & shown in Fig. 1 and all energies. The scaling factor S(&) in Eq. (78) was chosen such that the term proportional to 1=a in the large-a expansion (77) is given correctly by the formula a→0 (79). Near the anticlassical limit, the approximate formula Eq. (79) corresponds ∼ − a=S(&), and the next-to-leading term −a=S(&) does not agree exactly with the result (76). The accurate algebraic approximation (79) of the reAection phases yields a convenient and accurate approximation for the scattering phase shifts via Eq. (69), namely √ 2(1 − &1 ) 1 1= −a (80) (aS )2 + − aS : − 4 2 2( 32 − &1 ) 2 4 This is illustrated in Fig. 2, where the phase shifts are plotted as functions of k& . The case & = 1:4 is included in order to show that the formula (79) also gives good results in the regime 1 ¡ & ¡ 2, where the semiclassical limit a → ∞ corresponds to the low-energy limit k → 0. For Coulomb potentials, & = 1, the phase shifts have to be de>ned with respect to appropriately distorted waves rather than plane waves (68), but the concept of the reAection phase and Eq. (79) are still useful [58,69].


381

Fig. 2. Phase shifts 1 for scattering by a repulsive homogeneous potential (53) proportional to 1=r & , which are functions of k& . The thin solid lines are the exact results and the thin dashed lines are the conventional WKB results, which are obtained by inserting the value =2 for the reAection phase in the expression (69). The thick dashed lines show the results obtained with Eq. (80). The curves for & = 8; & = 4 and & = 1:4 are shifted by ; 2 and 4 respectively. From [58].

3.4. Quantization in a potential well Consider a binding potential in one dimension with two classical turning points, rl to the left and rr to the right. We assume that there is a region between rl and rr , where WKB wave functions are accurate approximations to the exact solution of the Schr4odinger equation. Then each turning point can be assigned a reAection phase, l and r for the left and right turning point, respectively, and the bound state wave function can be written either as r 1 l 1 (81) p(r ) dr − cos l (r) ˙ ˝ rl 2 p(r) or as

rr 1 r 1 p(r ) dr − cos r (r) ˙ ˝ r 2 p(r)

(82)

at any point r in the WKB region. For the wave function to be continuous around r, the cosines in Eqs. (81) and (82) must agree, at least to within a sign, so the sum of the arguments, which does

382


not depend on r, must be an integral multiple of . This yields the rather general quantization rule l r 1 rr (E) + : (83) p(r) dr = n + ˝ rl (E) 2 2 We retrieve the conventional WKB quantization rule,

1 rr (E) 0M ; 0M = 2 ; p(r) dr = n + ˝ rl (E) 4

(84)

by inserting =2 for both l and r in (83) as prescribed by Eq. (46). The parameter 0M is the Maslov index which counts how often a phase loss of =2 occurs due to reAection at a classical turning point during one period of the classical motion. Allowing for reAection phases which are not integral multiples of =2 amounts to allowing nonintegral Maslov indices [64]. The conventional quantization rule (84) works well near the semiclassical limit, but it breaks down near the anticlassical limit of the Schr4odinger equation. The modi>ed quantization rule (83) represents a substantial generalization of the conventional WKB rule, in that it avoids the restrictive assumptions underlying the conventional choice of reAection phases at the turning points; it only requires the WKB approximation to be accurate in some, perhaps quite small region of coordinate space between the turning points. 3.4.1. Example: Circle billard As a simple example consider the circle billard [24–26], a particle moving freely in an area bounded by a circle of radius R. The quantum mechanical wave functions of this √ separable twodimensional problem can be written in polar coordinates as (r; ’) = eil2 ’ l2 (r)= r, and the radial wave function l2 (r) obeys the one-dimensional Schr4odinger equation with the centrifugal potential (57), (59); l2 = 0; ±1; ±2; : : : is the angular momentum quantum number. The √ exact solutions for 2 2 (r) at energy E=˝ k =(2M) are essentially Bessel functions [59], (r) ˙ rJ|l2 | (kr), and bound l2 l2 states for a given angular momentum l2 exist when the wave function vanishes at the boundary: J|l2 | (kn R) = 0. The >rst, second, third, etc. zeros of the Bessel function de>ne bound states with radial quantum number n = 0; n = 1; n = 2, etc., and angular momentum quantum number l2 . The circle billard has served as a popular model system for testing semiclassical periodic orbit theories, and in leading order, standard periodic orbit theories essentially yield the energy eigenvalues obtained via conventional WKB quantization of the separable system. Main [26] has recently used the circle-billard as an example to demonstrate the applicability of an extension of standard periodic orbit theories to include terms of higher order in ˝. In conventional WKB quantization (84) of the radial motion, the Maslov index is taken to be 0M = 3 in order to account for the hard-wall reAection with reAection phase at r = R; furthermore, the centrifugal potential is taken to be (l2 )2 ˝2 =(2Mr 2 ) in accordance with the Langer modi>cation 2 2 (64). The resulting energy eigenvalues En;WKB |l2 | [in units of ˝ =(2MR )] are tabulated in Table 1 for angular momentum quantum numbers |l2 |=0 and 1 together with the exact eigenvalues En;exact |l2 | , which are just the squares of the zeros of the corresponding Bessel functions. For the modi>ed quantization rule (83) the reAection phase at the outer classical turning point r = R is also taken as , corresponding to hard-wall reAection, but the reAection phase at the inner classical turning point has, for l2 = 0, the energy independent value =2 + (|l2 | −

(l2 )2 − 14 ) according to Eq. (65); also the centrifugal


383

Table 1 Energy eigenvalues En; |l2 | [in units of ˝2 =(2MR2 )] for angular momentum quantum numbers l2 = 0 and |l2 | = 1 in the circle billard. The superscript “WKB” refers to conventional WKB quantization of the radial degree of freedom, “mqr” refers to the modi>ed quantization rule (83), “exact” labels the exact results and “(1)” the results obtained by Main [26] in higher-order semiclassical periodic orbit theory n

En;WKB 0

En;mqr 0

En;exact 0

En;(1)0

En;WKB 1

En;mqr 1

En;exact 1

En;(1)1

0 1 2 3 4 5 6 7 8

5.551652 30.22566 74.63888 138.7913 222.6829 326.3138 449.6839 592.7931 755.6416

5.798090 30.47498 74.88861 139.0412 222.9329 326.5637 449.9338 593.0431 755.8916

5.783186 30.47126 74.88701 139.0403 222.9323 326.5634 449.9335 593.0429 755.8914

5.804669 30.47647 74.88960 139.0418 222.9329 326.5656

14.39777 48.95804 103.2445 177.2678 271.0297 384.5306 517.7705 670.7494 843.4676

14.65833 49.21105 103.4959 177.5187 271.2803 384.7809 518.0207 670.9997 843.7178

14.68197 49.21846 103.4995 177.5208 271.2817 384.7819 518.0214 671.0002 843.7182

14.70160 49.22259 103.5015 177.5135 271.2822

potential (57), (59) is left intact—it is not subjected to the Langer modi>cation. A naive application of the modi>ed quantization rule (83) does not work for l2 = 0, i.e. for s-waves in two dimensions, because the centrifugal potential is actually attractive without the Langer modi>cation, and the WKB action integral diverges when taken from r = 0. This can be overcome by shifting the inner integration limit to a small positive value and adjusting the reAection phase accordingly as described in Ref. [70]; see also Section 4.4. The eigenvalues obtained with the modi>ed quantization rule (83) are listed as En;mqr |l2 | in Table 1.

Also included in Table 1 are the energies En;(1)|l2 | obtained via the extension of semiclassical periodic-orbit quantization to higher order in ˝ according to Main [26]. The performance of the various approximations is illustrated in Figs. 3 and 4 showing the errors approx |En; |l2 | − En;exact |l2 | | on a logarithmic scale. The energies obtained in conventional WKB quantization including the Langer modi>cation of the potential are, both for l2 = 0 and |l2 | = 1, too small by an error which is very close to 0:25˝2 =(2MR2 ) and virtually independent of n. Using the modi>ed quantization rule (83)—without the Langer modi>cation—reduces the error by one to three orders of magnitude and yields the same sort of accuracy as the higher-order periodic orbit quantization according to Ref. [26]. In contrast to the higher-order periodic orbit theory, however, the application of the modi>ed quantization rule is very simple indeed—just as simple as applying the conventional WKB quantization rule. For large angular momentum quantum numbers |l2 |, conventional WKB quantization becomes increasingly accurate and the improvement due to the modi>ed quantization rule is less dramatic. For small values of l2 , the modi>ed quantization rule is an extremely simple and powerful tool for obtaining highly accurate energy eigenvalues beyond the conventional WKB approximation. 3.4.2. Example: Potential wells with long-ranged attractive tails In this subsection we consider a deep potential well with a long ranged attractive tail, as occurs, e.g., in the interaction of atoms and molecules with each other and with surfaces. We focus our

384


l2=0

0

-1

conv. WKB

log10(error)

mod. quan. rule higher-order p.o. theory

-2

-3

-4 0

1

2

3

4 n

5

6

7

8

Fig. 3. Logarithmic plot of the error |En;approx − En;exact 0 | of various approximations of the energy eigenvalues En; |l2 | of the 0 circle billard for angular momentum number l2 = 0. The >lled triangles are the errors obtained via conventional WKB quantization including the Langer modi>cation of the potential and the empty triangles are the errors of the eigenvalues obtained by Main [26] via higher-order periodic orbit theory. The >lled squares show the errors obtained via the modi>ed quantization rule (83), adapted for the weakly attractive centrifugal potential as described in Ref. [70].

attention on homogeneous tails, C& ˝2 (& )&−2 = − ; &¿2 : (85) r& 2M r& The repulsive homogeneous potentials (53) studied in the previous section can, in principle, be the whole potential in the Schr4odinger equation, but the attractive potentials (85) cannot, otherwise the energy spectrum would be unbounded from below. The full potential must necessarily deviate from the homogeneous form (85) in the vicinity of r = 0, e.g. in the form of a short-ranged repulsive term. For the moment we neglect the inAuence of such a short-ranged repulsive part in the potential and study the properties of the long-ranged attractive tail (85). For negative energies, E = −˝2 %2 =(2M), there is an outer classical turning point rout given by V&att (r) = −

rout = & (%& )−2=& ;

(86)

cf. Eq. (55). In analogy to Eqs. (66), (67) we now de>ne the reduced classical turning point as the ratio of the classical turning point rout and the quantum mechanical penetration depth %−1 , √ 1 1 1 |pas | rout ; (87) a = %rout = (%& )1−2=& = |E| 2 − & (C& )1=& 2M = ˝ ˝ where |pas | = ˝% is the absolute value of the asymptotic (r → ∞) local classical momentum (2), which is now purely imaginary. Again, a is a quantitative measure of the proximity to the semiclassical or the anticlassical limit, a → ∞ being the semiclassical and a → 0 the anticlassical limit.


385

|l2|=1

0

-1

conv. WKB

log10(error)

mod. quan. rule higher-order p.o. theory

-2

-3

-4 0

1

2

3

4 n

5

6

7

8

Fig. 4. Logarithmic plot of the error |En;approx − En;exact 1 | of various approximations of the energy eigenvalues En; |l2 | of the 1 circle billard for angular momentum number l2 = ±1. The >lled triangles are the errors obtained via conventional WKB quantization including the Langer modi>cation of the potential and the empty triangles are the errors of the eigenvalues obtained by Main [26] via higher-order periodic orbit theory. The >lled squares show the errors obtained via the modi>ed quantization rule (83).

We assume that the reAection phase r at the outer classical turning point is determined by the homogeneous tail (85) alone; its behaviour near the anticlassical limit can be derived [73] from the zero-energy solutions of the Schr4odinger equation in much the same way as for the repulsive potentials (53) in Section 3.3; for any power & ¿ 2 the result is,

√ 2(1 − 1 ) 2(1 − ') 1 a→0 ∼ a + 2'2' + ' − 3 &1 & tan sin(') a1+2' ; ' = : (88) 2 & 2(1 + ') &−2 2( 2 − & ) For E = 0, the classical turning point is at +∞ and the reAection phase is def (89) (0) =0 = + ' : 2 The zero-energy reAection phase 0 is one of three parameters characterizing the near-threshold properties of an attractive potential tail, as discussed in more detail in Section 4. Near the semiclassical limit, the leading behaviour of the reAection phase in the attractive homogeneous potential (85) is [73],

√ (& + 1)2( &1 ) a→∞ ∼ + a : (90) tan 2 & 12&2( 12 + &1 ) The dependence of on a in between the anticlassical (88) and semiclassical (90) limits can be derived from numerical solutions of the Schr4odinger equation and the results are shown in Fig. 5. The phases fall monotonically from the threshold value (=2) + ' towards the semiclassical expectation =2 at large a.

386


Fig. 5. Exact reAection phases for reAection by an attractive homogeneous potential (85) as function of the reduced classical turning point (87). From [73].

An algebraic approximation to the function (a) can now be obtained by generalizing the formula (79) to account for the fact that the value (0) in the anticlassical limit depends on the power & for the attractive potentials, 1 (91) (aR )2 + − aR : = + 2' 2 4 Here aR stands for a scaled reduced classical turning point, and the scaling factor is chosen such that the formula (91) reproduces the leading contributions (90) to in the semiclassical limit,

1 & + 1 2( &1 ) a ; R(&) = √ : (92) aR = (& − 2) tan R(&) & 3 & 2( 12 + &1 ) As a concrete example for the e?ectivity of the modi>ed quantization rule (83) we discuss the Lennard-Jones potential, which is a model for molecular interactions,

r 6 rmin 12 min : (93) −2 VLJ (r) = j r r The tail of the potential (93) has the homogeneous form (85) with & = 6; the strength parameter 6 is given by 1=4 4M(rmin )2 j = rmin (2BLJ )1=4 : (94) 6 = rmin ˝2 The potential (93) has its minimum value −j at r = rmin , and the energy eigenvalues, measured in units of j, depend only on the reduced strength parameter BLJ =2M(rmin )2 j=˝2 . The energy levels of the potential (93) and WKB approximations thereof were studied in considerable detail by Kirschner and Le Roy [71] and by Paulsson et al. [72]. Following Ref. [72] we choose a reduced strength


387

Fig. 6. The Lennard-Jones potential (93) with reduced strength parameter BLJ = 104 . Table 2 Selected energy eigenvalues, in units of j, for the Lennard-Jones potential (93) with a strength parameter BLJ = 104 . En represents the exact energy, WEnconv is the error of the conventional WKB approximation (84), and WEnR is the error in the modi>ed WKB quantization (83) with l = =2 and the algebraic approximation (91) for r , & = 6; ' = 14 and R(6) = 2:08287. For a table containing all eigenvalues see Ref. [73] En

109 × WEnconv

109 × WEnR

0 1

−0.941046032 −0.830002083

−85841 −82492

−17508 −16684

11 12

−0.147751411 −0.115225891

−46115 −42250

−7522 −6589

22 23

−0.000198340 −0.000002697

−4493 −1021

−100 +42

n

parameter BLJ = 104 , for which the potential supports 24 bound states corresponding to quantum numbers n = 0; 1; : : : ; 23. The potential is illustrated in Fig. 6. A selection of energy eigenvalues, in units of j, is listed in Table 2; complete lists are contained in Refs. [72,73]. Next to the exact eigenvalues we also show the errors of the conventional WKB eigenvalues, which are obtained via the conventional quantization rule (84), and the last column shows the results obtained with the modi>ed quantization rule (83) when the reAection phase l at the inner classical turning point is kept at =2 while the energy dependence of the outer reAection phase r is accounted for via the approximate algebraic formula (91), with ' = 14 and R(6) = 2:08287 for & = 6.

388

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 -2 conv. WKB mod. quan. rule

5

log10(relative error)

higher order WKB 4

-3

1

2

3

-4 0

5

10

15

20

n

Fig. 7. Relative errors (95) of the energy eigenvalues in the Lennard-Jones potential (93) with strength parameter BLJ =104 . The >lled triangles show the errors of conventional WKB quantization (84) and the >lled squares show the errors of the modi>ed quantization rule (83) when the reAection phase l at the inner classical turning point is kept at =2 while the energy dependence of the outer reAection phase r is accounted for via the approximate algebraic formula (91), with ' = 14 and R(6) = 2:08287 for & = 6. The open triangles numbered 1 to 5 show the relative error for the highest bound state, n = 23, as obtained in Ref. [72] with successive higher-order approximations involving terms up to S2N +1 in the expansion (20); the state becomes unbound for N ¿ 5.

Although the results of conventional WKB quantization seem quite satisfactory at >rst sight, allowing for the energy dependence of the reAection phase at the outer classical turning point improves the accuracy by a factor ranging from >ve for the low-lying states to 45 and 25 for the highest two states. The usefulness of the modi>ed quantization rule becomes clearer when looking at the errors relative to the level spacing, which decreases by a factor of 500 from the bottom to the top of the spectrum, WEn rel : WEn = (95) En − E n − 1 Fig. 7 shows the relative errors (95) obtained via conventional WKB quantization (>lled triangles) and via modi>ed quantization including the energy dependence of the outer reAection phase via Eq. (91) (>lled squares). The relative error of conventional WKB quantization grows by an order of magnitude as we approach the anticlassical limit at E = 0. In contrast, accounting for the energy dependence of the outer reAection phase keeps the relative error roughly constant at a comfortably low level. As shown in Ref. [72], higher-order WKB approximations involving terms up to S2N +1 in the expansion (20) substantially reduce the error for all but the last eigenstate, n = 23. The relative errors obtained in higher-order WKB approximations for the n = 23 state are shown as open triangles in Fig. 7. The relative error initially decreases, for N = 1; 2 and 3, but then increases with the order of the approximation. For N ¿ 5 the WKB series no longer yields a bound state with quantum


389

number n = 23. This is an illustrative demonstration of the asymptotic nature of the expansion (20) which breaks down towards the anticlassical limit. The observation that conventional semiclassical approximations deteriorate towards threshold in a typical molecular potential such as (93) sometimes causes surprise [74], because one might expect such approximations to improve with increasing quantum number. It is, however, not surprising and actually well understood [72,73,75–78], that semiclassical approximations break down near the anticlassical or extreme quantum limit, which is at energy zero for potentials such as (93). A more detailed discussion of quantization near the anticlassical limit is presented in Section 4.2. 3.5. Tunnelling When two classically allowed regions of (in our case one-dimensional) coordinate space are separated by a localized classically forbidden region, a barrier, then a classical particle approaching the barrier from one side is inevitably reAected and cannot be transmitted to the other side. In quantum mechanics there is generally a >nite probability PT for transmission, and the probability PR for reAection is correspondingly less than unity, PR + PT = 1 :

(96)

Although the following discussion focusses on tunnelling through a classically forbidden region, it can also be applied to the transmission through a classically allowed region, for which the probability can be less than unity when quantum e?ects lead to a >nite probability for classically forbidden reAection. Such quantum reAection requires the existence of a region of substantial quantality and is discussed in detail in Section 5. When a particle approaches the barrier (or the quantal region of coordinate space) from the left, the wave function to the left of this region can be expressed by the WKB wave functions r 1 i r i (r) ∼ (97) p(r ) dr + Rl exp − p(r ) dr exp ˝ rl ˝ rl p(r) with the reAection amplitude Rl , and the transmitted wave to the right is r 1 i p(r ) dr exp (r) ∼ Tl ˝ rr p(r)

(98)

with the transmission amplitude Tl . The points rl and rr are points of reference where the phases of the WKB wave functions vanish. When the particle is incident from the right, Eqs. (97) and (98) are replaced by r i 1 i r ; (99) p(r ) dr + Rr exp p(r ) dr (r) ∼ exp − ˝ rr ˝ rr p(r) 1 i r (100) (r) ∼ Tr p(r ) dr ; exp − ˝ rl p(r) where Eq. (99) applies for r values to the right and Eq. (100) for r values to the left of the barrier (or quantal region). The reAection and transmission amplitudes in Eqs. (97)–(100) are connected

390


through the reciprocity relations (see, e.g. Ref. [79]), T def Tr = Tl =T; Rr = −R∗l ∗ : (101) T If the potential becomes constant on one or both sides of the barrier, then the conventional ansatz with plane waves can be used to de>ne reAection and transmission amplitudes, e.g., 1 (r) ∼ √ {exp[ − ik(r − rr )] + Rr exp[ik(r − rr )]} ˝k

(102)

√ for a particle incident from the right. We have included the factor 1= ˝k to account for the velocity dependence of the particle density in accordance with the continuity equation. When the r dependence of the potential is negligible, the WKB waves in Eq. (99) are identical to the plane waves in (102) except for constant phase factors. The use of WKB waves to de>ne transmission and reAection amplitudes as in Eqs. (97)–(100) does not imply any semiclassical approximation of these amplitudes. The Schr4odinger equation should be solved exactly, and the exact wave functions matched to the incoming and reAected waves or the transmitted wave in the semiclassical regions on either side of the barrier (or quantal region). If there are no semiclassical regions, then the terms “incoming”, “reAected” and “transmitted” cannot be de>ned consistently. The ans4atze (97)–(100) involving WKB wave functions are more general than those using plane waves, because they can also be used when the potential depends strongly on r in the semiclassical region(s). The probabilities PT for transmission and PR for reAection are given by PT = |T |2 ;

PR = |R|2 ;

(103)

and they do not depend on whether the particle is incident from the left or the right, or on whether the amplitudes are de>ned via WKB or plane wave functions. The phase of the reAection amplitude depends on the direction of incidence according to Eq. (101), and also on whether WKB waves or plane waves are used as reference. For example, the reAection amplitude R(p) de>ned with reference to plane waves as in Eq. (102) is related to the r reAection amplitude R(WKB) de>ned via WKB waves according to Eq. (99) by r 1 r (p) (WKB) R r = Rr : (104) exp lim 2i k(r − rr ) − p(r ) dr r →∞ ˝ rr The exponential on the right-hand side simply accounts for the di?erent phases accumulated by the reference waves on their way to rr and back. The reAection amplitudes do not depend on whether plane waves or WKB waves are used to represent the transmitted wave. If 2M(E − V ) approaches the constant values ˝2 k 2 and ˝2 q2 for r → +∞ and r → −∞ respectively, then the transition amplitude T (p) de>ned via incoming and transmitted plane waves is related to the transition amplitude T (WKB) based on WKB waves by 1 r− (p) (WKB) T =T exp lim i q(r− − rl ) − p(r) dr r± →±∞ ˝ rl 1 r+ p(r) dr : (105) −i k(r+ − rr ) − ˝ rr


391

The phases of the reAection and transmission amplitudes also depend on the points of reference rl; r at which the reference waves have vanishing phase. In principle they could be chosen arbitrarily; in the presence of a potential barrier they are conveniently chosen as the left and right classical turning points, respectively. In the presence of a potential barrier, the wave function in the classically forbidden region is approximated by a superposition of exponentially increasing and decreasing WKB waves 1 r 1 r 1 : (106) A exp − |p(r )| dr + B exp + |p(r )| dr forb (r) = ˝ rl ˝ rl |p(r)| We assume that the classical turning points rl and rr are isolated, meaning that there is a region in the classically forbidden domain between rl and rr where the WKB representation (106) is valid. Instead of referring the WKB waves to the left classical turning point rl , we could equally have chosen rr as point of reference. For the derivation of an explicit expression for the transmission amplitude we consider the case that the particle is incident from the left and we make use of Eqs. (97), (98). In order to use the connection formulas (44) and (45) at the right turning point we rewrite Eq. (98) as r C 1 CU 1 r r U r (r) = √ cos + √ cos ; (107) p dr − p dr − p ˝ rr 2 p ˝ rr 2 where 2Tl e−ir =2 C= ; e−ir − e−iUr

U

2Tl e−ir =2 CU = − : e−ir − e−iUr

(108)

The cosines of Eq. (107) can now be matched to the growing and decaying exponentials in the WKB region under the barrier, Eq. (106), according to the connection formulas (44) and (45). The exponentials containing integrals with lower limit rl can be rewritten in terms of exponentials of integrals with upper limit rr by introducing the factor 1 rr ,B (E) = exp[I (E)]; I (E) = |p(r)| dr ; (109) ˝ rl this gives the coeScients A and B of Eq. (106), U A = NU r ,B C;

B=

Nr C : 2,B

(110)

The subscript r indicates application of the connection formulas at the right turning point. By using the connection formulas once again at the left turning point rl we get an expression of the wave function in the classical allowed region to the left of rl . Decomposing the cosines into exponentials and comparison with the WKB wave incident from the left, Eq. (97), gives the coeScient of the incoming WKB wave, which should be unity, as A il =2 B iUl =2 e + e : Nl 2NU l

(111)

392


Using Eqs. (110), (108), and (49) yields the general expression for the transmission amplitude Nl Nr 1 i(Ul +Ur )=2 −1 i(l +r )=2 − e : (112) T = iNl Nr ,B e NU l NU r 4,B This is a very general formula, but it still contains eight parameters, namely the unbarred and barred phases and amplitudes N at each of the two turning points, where the connection formulas have been applied. For a “dense” barrier, meaning that the exponentiated integral ,B as de>ned by Eq. (109) is large, we might choose to neglect the second term in the large brackets in Eq. (112) as subdominant. This leaves us with an approximate formula for the transmission amplitude T ≈ iNl Nr e−i(l +r )=2 =,B ;

(113)

which contains only the well-de>ned (unbarred) connection parameters. With the standard semiclassical choice (46) for the unbarred connection parameters, Eq. (113) leads to the standard WKB expression for the tunnelling probability, PTWKB (E) = |T |2 = (,B )−2 :

(114)

This formula fails near the top of a barrier, where the two turning points rl and rr coalesce. The expression (114) gives unity at the top of a barrier, whereas the exact result is generally smaller than unity. An improved formula, 1 PTKemble (E) = ; (115) 1 + (,B )2 is due to Kemble [80,81], and it is actually exact at all energies for an inverted quadratic potential, V (r) ˙ −r 2 ; in particular, PTKemble is exactly one half at the top of the barrier. This result is accurate for all barriers which can be approximated by an inverted parabola in a range of r values reaching from the top of the barrier into the semiclassical regions on either side. If we include the subdominant term in Eq. (112) and >x the connection parameters according to conventional WKB theory, Nl; r = NU l; r = 1, l; r = −U l; r = =2, we obtain the result 1 −1 ; (116) T = ,B + 4,B which can be found, e.g., in Ref. [45]. We now look at the behaviour of the tunnelling amplitude and probability near the base of a barrier, where the potential becomes asymptotically constant on at least one side. The limit of small excess energy, E =˝2 k 2 =(2M) → 0, corresponds to the anticlassical limit of the Schr4odinger equation if the potential approaches its asymptotic limit faster than 1=r 2 . For this case it can be shown [82], that both the amplitude N in the connection formula (44) and the amplitude NU in the connection √ formula (45) become proportional to k for k → 0. For potential tails falling o? faster than 1=r 2 the exponentiated √ integral (109) remains >nite at E = 0, so the transmission amplitude (112) is proportional to k near the threshold E → 0, if E = 0 represents the base of the barrier on just one side. The transmission probability through an asymmetric barrier with two di?erent asymptotic levels is thus proportional to the square root of the excess energy above the higher level, in the limit that this excess energy becomes small. If the potential approaches the same asymptotic limit


393

on both sides and does so faster than 1=r 2 , then the transmission amplitude (112) is proportional to k and the transmission probability is proportional to E near the base. 2 For a potential barrier √ with tails vanishing faster than 1=r , the quantum mechanical tunnelling probability vanishes as E or as E at the base, E = 0. This behaviour is not reproduced by any of the semiclassical formulas (114), (115) or (116), which all predict a >nite tunnelling probability at the base. For a symmetric barrier, the left and right connection parameters are the same and we can drop the subscripts. Eq. (112) then simpli>es to −1 N 2 1 iU 2 i T = iN ,B e − 2 e : (117) NU 4,B For the calculation of the transmission probability through a symmetric barrier the phases in Eq. (117) are eliminated via Eq. (49) giving − 1 2 ,B 1 2 − +1 : (118) PT (E) = |T | = N2 4,B NU 2 If we keep only the dominant term, PT (E) ≈

N4 : (,B )2

As an example consider a symmetric rectangular barrier of length L, 2 ˝ (K0 )2 =(2M) for |r| ¡ L=2 ; V (r) = 0 for |r| ¿ L=2 :

(119)

(120)

This corresponds to a sharp-step potential at both classical turning points; the phase and ampli2 2 tude N are given in Eq. (52) as functions of the inverse √ penetration depth % = (K0 ) − k on the classically forbidden side and the wave number k = 2ME=˝ on the classically allowed side of the step. For the sharp-step potential, the WKB waves are exact except at the classical turning point, so the exponentially increasing wave function in the classically forbidden region can be uniquely de>ned; this allows the determination of the barred parameters in the connection formula (45), N NU = ; 2

U = − :

(121)

With Eqs. (52), and (121) for the connection parameters at both turning points and the explicit expression for the exponentiated WKB integral (109), ,B = e%L , the transmission amplitude (117) is T=

e%L (k

+

i%)2

4ik% : − e−%L (k − i%)2

The resulting transmission probability is −1 2 %2 + k 2 sinh %L + 1 PT = ; 2%k

(122)

(123)

394


a standard textbook result. The low-energy (k → 0) behaviour of the expression (123) is, PT ≈

1 16k 2 + O(k 4 ) ; 2 (K0 ) (,B − 1=,B )2

(124)

as could also be calculated via Eq. (118). Now consider symmetric barriers decaying asymptotically as an inverse power of the coordinate, V (r)

|r |→∞

∼

V&rep (|r|) =

˝2 (& )&−2 ; 2M |r|&

(125)

with & ¿ 2 and & positive. The turning points of the homogeneous potential, V& (|r|), are given by rl; r = ∓& (k& )−2=& . In order to calculate the phases and amplitudes in the low energy limit it suSces to consider a potential step which shows the same behaviour as the potential (125) for r → +∞ and stays classically forbidden for energies near zero on the left side, r → −∞. It follows that we can use the results for the reAection by singular repulsive potentials, Section 3.3. After taking into account the proportionality factors in Eqs. (68) and (70) we get the leading contribution to the amplitude factor N , k →0

N ∼

2'&'=2 k& ; 2(1 + ')

'=

1 ; &−2

(126)

which can be inserted into Eq. (119) to give for the tunnelling probability at the base of the barrier k →0

PT ∼

16'2&' (k& )2 : 2(1 + ')4 (,B )2

(127)

The formula (127) is accurate as long as the homogeneous behaviour (125) of the tails of the barrier continues far enough into the barrier, i.e., until the WKB wave functions (106) are accurate. Under these conditions, more accurate values for N going beyond the leading contribution (126) can be obtained by numerically solving the Schr4odinger equation for the homogeneous tail and matching the exact solution to the WKB waves on either side of the turning point. As an example we have calculated N for a homogeneous tail proportional to 1=r 8 and applied the formula (119) to the potential V (r) =

V0 : 1 + (r=)8

(128)

Fig. 8 shows the resulting tunnelling probabilities in a doubly logarithmic plot. The solid line is the exact numerically calculated tunnelling probability and the thick dashed line shows the prediction of the formula (119), with N calculated for the homogeneous tail proportional to 1=r 8 . The straight-line behaviour for small energies demonstrates the proportionality of the tunnelling probability to energy near the base of the barrier, see Eq. (127). The thin dashed line shows the conventional semiclassical result (114), which fails near the base of the barrier, because it remains >nite. For further examples see Refs. [60,82–84]. An accurate description of tunnelling is important for the understanding of energy levels in potentials with two or more wells, e.g. the energy splitting between two almost degenerate levels in a


395

√ Fig. 8. Numerically calculated exact tunnelling probabilities (solid line) for the potential (128) with MV0 =˝=5 together with the prediction (119) based on values of N (k) calculated numerically for a homogeneous potential tail proportional to 1=r 8 (thick dashed line). The thin dashed line shows the prediction of the conventional WKB formula (114). From [60].

symmetric double well is essentially determined by the tunnelling probability through the potential barrier separating the wells [43,85]. Modi>ed WKB quantization techniques utilizing the generalized connection formulas (44), (45) have been applied to the quantization in multiple-well potentials by several authors in the last few years, and substantial improvements over the predictions of the standard WKB procedure have been achieved. For details see Refs. [86–91].

4. Near the threshold of the potential In this section we consider potentials which vanish asymptotically, r → ∞, and we focus on the regime of small positive or negative energies near the threshold E = 0. For potentials falling o? faster than 1=r 2 ; E = 0 corresponds to the anticlassical or extreme quantum limit of the Schr4odinger equation; conventional semiclassical methods are not applicable in this case, but modi>ed methods involving exact wave functions in the quantal regions of coordinate space and WKB wave functions elsewhere can give reliable and accurate results. The Schr4odinger equation (1) contains the energy in order O(E), but the leading near-threshold behaviour of the wave functions is often determined by terms of lower order in E, see e.g. Eqs. (76), (88) for the near-threshold reAection phases in repulsive or attractive homogeneous potentials or Eq. (122) for the near-threshold transmission amplitude through a rectangular potential barrier. If exact solutions of the Schr4odinger equation are known at threshold, E = 0, then they also determine the leading behaviour of the solutions near threshold to order less than E, because the energy can be treated as a perturbation of (higher) order E in the Schr4odinger equation. Examples involving continuum states above threshold and discrete bound states below threshold are presented in this section.

396


4.1. Scattering lengths Consider a potential V (r), which vanishes faster than 1=r 2 for r → ∞. For positive energies, E = ˝2 k 2 =(2M) ¿ 0, the regular solution reg (r) of the Schr4odinger equation, which is de>ned by the boundary condition reg (0) = 0, behaves asymptotically as reg (r)

r →∞

˙ sin(kr + 1) ;

(129)

and 1(k) is the phase shift of the exact wave function reg relative to the free wave sin(kr), as already discussed in Section 3.3. The phase shift approaches an integral multiple of at threshold, as long as the potential falls o? faster than 1=r 2 . For a potential falling o? faster than 1=r 4 , the leading terms of the low-energy behaviour of the phase shift are given by the e?ective range expansion [61,92] 1 1 k →0 k →0 2 + k 2 re? ; 1 ∼ n − ka0 + O(k 3 ) : (130) k cot 1 ∼ − a0 2 The parameter a0 in Eq. (130) is the scattering length and re? the e?ective range of the potential V (r). For a potential behaving asymptotically as C4 ˝2 (4 )2 = − ; (131) r4 2M r 4 the leading near-threshold behaviour of the phase shift is given [93] by k →0 1(k) ∼ n − ka0 − (k4 )2 ; (132) 3 so the e?ective range expansion (130) begins with the same constant term −1=a0 on the right-hand side, but there is a term linear in k preceding the quadratic one. For all potentials falling o? faster than 1=r 3 , the scattering length a0 dominates the low-energy properties of the scattering system. It is, e.g., a crucial parameter for determining the properties of Bose-Einstein condensates of atomic gases, see [36,94]. For a potential behaving asymptotically as r →∞

V (r) ∼

−

C3 ˝ 2 3 = − ; r3 2M r 3 the leading near-threshold behaviour of the phase shift is given [92] by r →∞

(133)

k →0

(134)

V (r) ∼ −

1(k) ∼ n − k3 ln(k3 ) ;

and it is not possible to de>ne a >nite scattering length. Scattering lengths depend sensitively on the positions of near-threshold bound states, so they are determined by the potential in the whole range of r values, not only by the potential tail. The mean value of the scattering length—averaged, e.g. over a range of well depths—is, however, essentially a property of the potential tail. If there is a WKB region of moderate r-values, where the exact solutions of the Schr4odinger equation (at near-threshold energies) are accurately approximated by WKB wave functions, explicit expressions for the scattering length can be derived [95–97] as brieAy reviewed below. The Schr4odinger equation at threshold (E = 0) has two linearly independent solutions, 0 and 1 , whose asymptotic (r → ∞) behaviour is given by 0 (r)

r →∞

∼ 1 + o(r −1 );

1 (r)

r →∞

∼ r + o(r 0 ) :

(135)


397

In the WKB region, the exact threshold solutions 0 ; 1 can be written in WKB form, ∞ 1 0; 1 1 ; p0 (r ) dr − cos 0; 1 (r) = D0; 1 ˝ r 2 p0 (r)

(136)

with well de>ned amplitudes D0 and D1 and the phases 0 and 1 . Here p0 (r) is the local classical momentum at threshold, (137) p0 (r) = −2MV (r) : Since 0 (r) is the solution which remains bounded for r → ∞, the phase in the WKB form of this wave function in the WKB region is just the threshold value of the reAection phase at the outer classical turning point, which lies at in>nity for E = 0, i.e. the zero-energy reAection phase 0 , cf. Eq. (89) in Section 3.4. The asymptotic behaviour (129) of the regular solution of the Schr4odinger equation becomes reg (r)

k →0

˙ sin[k(r − a0 )] ∼ k(r − a0 )

√

(138)

when we insert the near-threshold behaviour (130) of the phase shift 1(k). To order k ˙ E, the regular solution of the Schr4odinger equation for small k thus corresponds to the following linear superposition of the zero-energy solutions 0 and 1 : reg (r)

˙ k[

1 (r)

− a0

0 (r)]

:

(139)

For values of r in the WKB region, 0 and 1 are WKB wave functions (136), so ∞ ∞ k 1 1 1 0 − a0 D0 cos p0 (r ) dr − p0 (r ) dr − D1 cos reg (r) ˙ ˝ r 2 ˝ r 2 p0 (r) ∞ 1 + 1 ˙ p0 (r ) dr − cos −8 ; (140) ˝ r 4 p0 (r) where 8 is an angle de>ned by − a0 + D1 =D0 ; tan tan 8 = a0 − D1 =D0 4

(141)

and + ; − stand for the sum and di?erence of the phases in Eq. (136), + = 0 + 1 ;

− = 0 − 1 :

Coming from the inner turning point rin , the WKB wave function r 1 1 in (E) cos p(r ) dr − WKB (r) = ˝ rin (E) 2 p(r)

(142)

(143)

is expected to be an accurate approximation of reg (r) for r values in the WKB region. The reAection phase in at the inner turning point will be near =2 if the conditions for conventional matching are ful>lled in the neighbourhood, see Section 3.2, but, even if this is not the case, in can be expected to be a smooth analytical function of the energy E near threshold, in (E) = in (0) + O(E) :

(144)

398


The inner turning point rin also depends weakly and smoothly on E, and for r values between rin and a value r in the WKB region, the local classical momentum p(r ) in Eq. (143) di?ers from its threshold value (137) only in order E near threshold. So to order less than E we can assume E = 0 in Eq. (143), r 1 in (0) 1 : (145) (r) ≈ (r) ˙ p (r ) dr − cos reg WKB 0 ˝ rin (0) 2 p0 (r) Eqs. (140) and (145) are compatible if and only if the cosines agree at least to within a sign. This leads to an explicit expression for the angle 8 in terms of the threshold value ∞ S(0) = p0 (r) dr (146) rin (0)

of the action integral, namely − S(0) in (0) + − − − n = (nth − n) + : (147) ˝ 2 4 4 Here we have introduced the threshold quantum number nth which ful>lls the modi>ed WKB quantization rule (83) at E = 0, 8=

S(0) in (0) 0 − = nth : − (148) 2 2 ˝ nth is an upper bound to the quantum numbers n = 0; 1; 2 : : : of the negative energy bound states and is usually not an integer. An integer value of nth indicates a zero-energy bound state. The number of negative energy bound states is [nth ] + 1 where [nth ] is the largest integer below nth . Resolving Eq. (141) for a0 and using Eq. (147) gives D1 tan(nth + − =4) + tan(− =4) D0 tan(nth + − =4) − tan(− =4) 1 D1 − 1 : = sin + D0 2 tan(− =2) tan(nth )

a0 =

(149)

The exact zero-energy solutions 0 , 1 of the Schr4odinger equation may be known for a tail-region of the potential. The amplitudes D0; 1 and phases 0; 1 of the WKB form of the exact solutions can then be derived from these wave functions, if the tail-region, where the Schr4odinger equation is accurately solved by the known forms of 0 and 1 , overlaps with the WKB region, where these wave functions can be matched to the WKB form (136). The amplitudes and phases are thus tail parameters, which depend only on the potential in the tail region and not on its shape inside the WKB region, or at even smaller r values. The factor D1 0 − 1 (150) b= sin D0 2 in front of the square bracket in Eq. (149) has the physical dimension of a length and is a characteristic property of the potential tail beyond the WKB region. The threshold quantum number nth , on the other hand, is related to the total number of bound states supported by the well and depends


399

on the whole potential via the action integral S(0). It is likely that there is an inner WKB region in the well when nth is a large (but >nite) number; this is not a necessary condition however, as shown by the example of a shallow step potential, see below. For a uniform distribution of values of nth , the second term in the square bracket in Eq. (149) will be distributed evenly between positive and negative values, so the >rst term de>nes a mean scattering length aU0 , b b ; a0 = aU0 + : (151) aU0 = tan[(0 − 1 )=2] tan(nth ) 4.1.1. Example: Sharp-step potential The radial sharp-step potential is de>ned as, 2 K0 ; ˝2 (K0 )2 ˝2 Vst (r) = − ,(L − r) = − 2M 2M 0;

for 0 ¡ r 6 L ;

(152)

for r ¿ L :

The quantal region where the WKB approximation is poor is restricted to the single point r = L where the potential is discontinuous. The WKB approximation is exact for 0 ¡ r ¡ L regardless of whether the potential be deep or shallow, and the tail region of the potential can be any interval of r values that includes the discontinuity at r = L. The zero-energy solutions of the Schr4odinger equation with the asymptotic (here: r ¿ L) behaviour (135) are given, in the WKB region 0 ¡ r ¡ L, by st 0 (r)

= cos[K0 (L − r)] ; L st1 st cos K0 (L − r) − 1 (r) = cos(st1 =2) 2

st1 with tan 2

=−

1 : K0 L

(153)

The zero-energy reAection phase 0 at the outer turning point (here at r = L) is zero. The tail parameter b and the mean scattering length are thus given by 1 bst = ; aUst0 = L : (154) K0 The reAection phase at the inner turning point r = 0 is corresponding to reAection at a hard wall, so the threshold quantum number de>ned by Eq. (148) is given by nstth = K0 L − =2. Indeed, the number of bound states in the sharp-step potential (152) is the largest integer bounded by nstth + 1. With Eq. (151) and using tan(nstth ) = −1=tan(K0 L) we obtain the well known result [98] for the scattering length of the sharp-step potential, tan(K0 L) ast0 = L − : (155) K0 4.1.2. Example: Attractive homogeneous potentials More important and realistic examples are the homogeneous potential tails, C& ˝2 (& )&−2 = − ; (156) r& 2M r& which are characterized by a power & and a strength parameter & with the physical dimension of a length. The zero-energy wave functions which solve the Schr4odinger equation with the potential V&att (r) = −

400


(156) and have the asymptotic behaviour (135) are essentially Bessel functions of order ±1=(& − 2) [55,73,76], 2(1 + ') r (&) (&) ' (r) = J (z); (r) = 2(1 − ')' & rJ−' (z) ; ' 0 1 '' & 1=(2') & 1 ; z = 2' : (157) '= &−2 r For a suSciently rapid fall-o? of V&att (r), namely & ¿ 2, the WKB approximation becomes increasingly accurate for r → 0, see Section 2.4. The WKB region corresponds to small values of r=& and hence to large arguments of the Bessel functions in Eq. (157), so we can use their asymptotic expansion [59] to write 0 and 1 in the WKB form (136). This yields the phases and amplitudes, 0(&) = + '; 1(&) = − ' ; 2 2 ˝ 2(1 + ') ˝& (&) (&) 2(1 − ')'' : ; D1 = (158) D0 = ' '& ' ' The length parameter (150) is now given by b(&) = & '2'

2(1 − ') '1+2' sin(') = & ; 2(1 + ') 2(1 + ')2

and the mean scattering length is 2(1 − ') cos(') : aU0(&) = & '2' 2(1 + ')

(159)

(160)

It is interesting to observe, that the formula (160) for the mean scattering length of the attractive homogeneous potential (156) is very similar to the formula (74) for the true scattering length of the repulsive homogeneous potential (53) discussed in Section 3.3. It simply contains an additional factor cos('). The true scattering length for a potential with an attractive tail such as (156) depends on the whole potential via the threshold quantum number nth according to Eq. (151). If we take the threshold value in (0) of the inner reAection phase in Eq. (148) to be =2, then nth = S(0)=˝ − (1 + ')=2 and the true scattering length is S(0) ' − : (161) a0(&) = aU0(&) 1 − tan(')tan ˝ 2 The formulas (160) and (161) for homogeneous potential tails were >rst derived by Gribakin and Flambaum [95]. Together with Harabati these authors also derived an expression for the e?ective range re? in the next term of the expansion (130) for the phase shift [96]. For the discussion in this section we assumed that the potential falls o? faster than 1=r 3 asymptotically. Indeed, for a potential behaving asymptotically as (133), the low-energy behaviour of the phase shift is given by Eq. (134), so a >nite scattering length cannot be de>ned. Note, however, that the expression (159) for the tail parameter b(&) remains >nite for ' → 1 corresponding to & → 3, and also for higher integer values of ' corresponding to powers & = 2 + 1=' between 2 and 3.


401

Table 3 Ratios of the tail parameter (159) and the mean scattering length (160) to the strength parameter & for attractive homogeneous potential tails (156) &

3

4

5

6

7

8

&→∞

b(&) =& aU(&) 0 =&

—

1 0

0.6313422 0.3645056

0.4779888 0.4779888

0.3915136 0.5388722

0.3347971 0.5798855

=& 1

The mean scattering length (160) diverges with 2(1−') for positive integers ', but b(&) remains well de>ned and >nite. The ratio b(&) =& expressing the tail parameter b(&) of a homogeneous potential in units of the strength parameter & is tabulated in Table 3 for integer powers & from 3 to 8. For & = 4; : : : ; 8 Table 3 also shows the mean scattering length aU0(&) in units of & ; for homogeneous potentials (156), aU0(&) and b(&) are related by aU0(&) = b(&) =tan('). 4.2. Near-threshold quantization and level densities The generalized quantization rule as introduced in Section 3.4 reads S(E) def 1 rout (E) in out = p(r) dr = n + + : ˝ ˝ rin (E) 2 2

(162)

This assumes that there is a WKB region between the inner classical turning point rin and the outer one rout , where WKB wave functions are accurate solutions of the Schr4odinger equation. The reAection phases in and out account for the phase loss of the WKB wave at the respective turning points. They are equal to =2 when the conditions for the conventional connection formulae are well ful>lled near the turning points, but they can deviate from =2 away from the semiclassical limit. For a potential V (r) with a deep well and an attractive tail, the reAection phase in at the inner turning point may—or may not—be close to =2; in any case it can be expected to be a smooth analytic function of the energy E, and there is nothing special about the threshold E = 0, cf. Eq. (144). The situation is di?erent near the outer turning point rout which, for a smoothly vanishing potential tail, moves to in>nity at threshold. When the potential is attractive at large distances and vanishes more slowly than 1=r 2 , then the action integral S(E) grows beyond all bounds as E → 0; the potential well supports an in>nite number of bound states and conventional WKB quantization, with out = =2 at the outer turning point, becomes increasingly accurate towards threshold. For a potential behaving asymptotically as V&att of Eq. (156) with 0 ¡ & ¡ 2, and energies E = −˝2 %2 =(2M) close enough to threshold, the action integral can be written as rout (E) S(E) (& )&−2 = C+ − %2 dr & r ˝ r0 √ F(&) 2(1=& − 12 ) %→0 ; (163) ; F(&) = ∼ C + (%& )(2=&)−1 2& 2(1=& + 1)

402


which leads to the near-threshold quantization rule, F(&) n→∞ n ∼ C + : (%& )(2=&)−1

(164)

The point r0 in Eq. (163) is to be chosen large enough for the potential to be accurately described by the leading asymptotic term proportional to 1=r & . The constants C, C and C in Eqs. (163) and (164) depend on the potential at shorter distances r ¡ r0 , but the energy dependent terms depend only on the potential tail beyond r0 , i.e. only on the power & and the strength parameter & determining the leading asymptotic behaviour of the potential tail. For a Coulombic potential tail, & = 1, F(1) = =2 we recover the Rydberg formula, En = −

R ˝2 %(n)2 =− ; 2M (n − C )2

R=

˝2 ; 2M(21 )2

(165)

with Bohr radius 21 , Rydberg constant R and quantum defect C − 1. The level density is de>ned as the (expected) number of energy levels per unit energy. If the quantum number n is known as a function of energy, then the level density is simply the energy derivative of the quantum number, dn=dE. Simple derivation of Eq. (164) with respect to E = −˝2 %2 =(2M) gives the near-threshold behaviour of the level density, 1 − 1 1 + 1 & 2 ˝2 dn E →0 F(&) 1 1 1 & 2 = : (166) − dE & 2 2M(& )2 |E| For Coulombic tails, & = 1, this reduces to the well known form, √ dn E →0 1 R = : dE 2 |E|3=2

(167)

When the potential vanishes faster than 1=r 2 at large distances, then the action integral S(E) remains bounded at threshold. The number of bound states is >nite, and conventional WKB quantization deteriorates towards threshold, see Section 3.4. Based on the exact zero-energy solutions (135) introduced in Section 4.1, it is however possible to derive a modi>ed quantization rule which becomes exact in the limit E = −˝2 %2 =(2M) → 0. To order below O(E) = O(%2 ), the wave function b (r)

=

0 (r)

−%

1 (r)

r →∞

∼ 1 − %r

(168)

solves the Schr4odinger equation and, to order below O(%2 ), it has the correct asymptotic behaviour r →∞ required for a bound state at energy E, namely b ˙ exp(−%r). If there is a WKB region of moderate r values, where we can write the WKB expressions (136) for 0 (r) and 1 (r), then in this region the bound state wave function (168) has the form ∞ 1 1 + −; ; (169) cos p0 (r ) dr − b (r) ˙ ˝ r 4 p0 (r) in analogy to Eq. (140). In Eq. (169), ; is the angle de>ned by − D1 1 + %D1 =D0 − 2 = tan 1 + 2% tan ; = tan + O(% ) ; 1 − %D1 =D0 4 4 D0

(170)


so ; =

+ % sin( 2− )D1 =D0 + O(%2 ), and ∞ 1 0 1 2 − %b + O(% ) ; p0 (r ) dr − cos b (r) ˙ ˝ r 2 p0 (r)

403

− 4

(171)

where b is the length parameter already introduced in Section 4.1, Eq. (150). Comparing this with the WKB wave function for the bound state at energy E, rout (E) 1 1 out (E) ; (172) cos p(r ) dr − WKB (r) ˙ ˝ r 2 p(r) yields an explicit expression for the reAection phase out at the outer turning point, namely out (E) = 0 + 2

S(E) − S(0) + 2%b + O(%2 ) : ˝

(173)

In deriving Eq. (173) we have exploited the fact, that the di?erence between the action integrals (162) at >nite energy E and (146) at energy zero is given, to order less than E, entirely by the tail parts of the integrals beyond the point r in the WKB region. Contributions from smaller distances than r to the action integral can be expected to depend smoothly and analytically on E near threshold, so their e?ect on the di?erence S(E) − S(0) will only be of order E ˙ %2 . Inserting the expression (173) for the outer reAection phase and (144) for the inner reAection phase into the quantization rule (162) yields b % + O(%2 ) ; (174) where nth is the threshold quantum number already introduced in Section 4.1, Eq. (148). Note that the leading energy dependence of the outer reAection phase near threshold exactly cancels with the energy dependence of the action integral, so the near-threshold quantization rule (174) has a universal form with a leading energy dependent term proportional to |E|. The parameter b determining the magnitude of the leading energy dependent term in the near-threshold quantization rule is just the tail parameter already de>ned in Eq. (150). The near-threshold quantization rule (174) applies for potential tails falling o? faster than 1=r 2 asymptotically. For potentials falling o? faster than 1=r 2 but not faster than 1=r 3 , the exact zero energy solutions (135) have next-to-leading asymptotic terms whose fall-o? is not a whole power of r faster than the leading term 1 or r. The wave function 0 remains well de>ned and its deviation from unity vanishes asymptotically. The wave function 1 may have asymptotic contributions corresponding to a nonnegative power of r (less than 1); a possible admixture of 0 would be of equal or lower order, so it seems that the de>nition of 1 has some ambiguity regarding possible admixtures of 0 . This does not a?ect the derivation of the expression (171), however, because 1 enters with a small coeScient % in the wave function (168) and only its leading term is relevant. The length parameter (150) is actually invariant with respect to possible ambiguities in the choice of 1 . From the universal form (174) of the near-threshold quantization rule, we immediately derive the leading behaviour of the near-threshold level density, dn E →0 1 2Mb2 + O(E 0 ) : = (175) dE 2 ˝2 |E| %→0

n ∼ nth −

404


Eq. (175) is also quite universal, in that it holds for all potential tails falling o? faster than 1=r 2 . The leading contribution to the near-threshold level density is quite generally proportional to 1= |E|, and its magnitude is determined by the tail parameter b. Even though the number of bound states in a potential well with a short-ranged tail is >nite and there usually is a >nite interval below threshold with no energy level at all, the level density at threshold becomes in>nite as 1= |E| towards E → 0. The probability density for >nding a bound state near E = 0 diverges as 1= |E|, but the expected number of states in a small energy interval below threshold, which is obtained by integrating this probability density, has a leading term proportional to |E|. It is worth mentioning that the considerations above, and in particular the universal formulas (174) for the near-threshold quantization rule and (175) for the near-threshold level density, apply for all potential wells with tails vanishing faster than 1=r 2 , irrespective of whether the leading asymptotic part of the tail is attractive or repulsive. The only condition is, that on the near side of the tail there be a WKB region in the well where the WKB approximation is good for near-threshold energies. The threshold properties are derived via the zero-energy solutions (135) which can be written as WKB wave functions (136) in this region and involve four independent parameters, two amplitudes D0; 1 and two phases 0; 1 . Due to the freedom to choose the overall normalization of the wave functions (143), (171), there remain three independent tail parameters, which determine the near-threshold properties of the potential and depend only on the potential tail beyond the semiclassical region. Three physically relevant tail parameters derived from D0; 1 and 0; 1 are: (i) the characteristic parameter b given by Eq. (150), which enters the universal near-threshold quantization rule (174) and determines the leading, singular contribution to the near-threshold level density (175); (ii) the mean scattering length aU0 given by Eq. (151); (iii) the zero-energy reAection phase 0 , which enters into the de>nition of the threshold quantum number nth , see Eq. (148), and is an important ingredient of the universal near-threshold quantization rule (174). The characteristic parameter b and the mean scattering length aU0 are related (151) via the di?erence − = 0 − 1 of the phases, aU0 = b=tan(− =2), so any two of the parameters b, aU0 and − contain essentially the same information when tan(− =2) is >nite. 4.2.1. Example: The Lennard-Jones potential To demonstrate the validity of the near-threshold quantization rule (174) we take a closer look at the threshold of the Lennard-Jones potential (93) which was studied in Ref. [73] and discussed in Section 3.4,

r 6 rmin 12 min ; (176) −2 VLJ (r) = j r r it has its minimum value, −j, at r =rmin . The attractive tail of the potential is of the form (156) with & = 6, and the strength parameter 6 is given by 6 = rmin [4M(rmin )2 j=˝2 ]1=4 . The energy eigenvalues En , measured in units of j, depend only on the reduced strength parameter BLJ = 2M(rmin )2 j=˝2 , and for BLJ = 104 the potential well supports 24 bound states, n = 0; 1; : : : ; 23. In this speci>c case, the energy of the highest bound state, n=23, is near −2:7×10−6 , and its distance to the threshold is less than one seventieth of its separation to the second most weakly bound state at E22 =−0:198 : : :×10−3 , see Table 2 in Section 3.4. It is, however, still not so close to the anticlassical limit, E = 0, as can be seen by noting that the reduced classical turning point a = %rout = (%6 )2=3 , which vanishes in the anticlassical limit, is still as large as 1.56 at the energy E23 .


405

-5

log10|E23|

-6

-7

exact

-8

conv. WKB near-thr. qu. rule

-9

-10 0.0

0.5

a

1.0

1.5

Fig. 9. Binding energy |E23 | of the highest bound state in the Lennard-Jones potential (176) as function of the reduced classical turning point a = %rout = (%6 )2=3 , as the reduced strength parameter BLJ is varied in the range between 9800 (corresponding to a ≈ 0) and 104 (a = 1:56). The exact energies are shown as >lled circles, the >lled triangles show the prediction of conventional WKB quantization (84) and the open squares show the results obtained with the near-threshold quantization rule (174).

The immediate vicinity of the anticlassical limit can be studied by gradually reducing the depth of the potential well in order to push the highest bound state closer to threshold. When applying the near-threshold quantization rule (174), we assume the reAection phase at the inner classical turning point, which enters in the de>nition (148) of nth , to be =2, which is not exactly true. For the characteristic parameter b we ignore deviations of the potential tail from the homogeneous −1=r 6 form, i.e., we assume b = b(6) = 0:4779888 × 6 (see Table 3 in Section 4.1). The results are shown in Fig. 9 where the binding energy |E23 | of the highest bound state (in units of j) is plotted as a function of the reduced outer classical turning point a at the energy of this state. The largest a value in the >gure, a = 1:56, corresponds to a reduced strength parameter BLJ = 104 and a binding energy (in units of j) of 2:6969 × 10−6 . By gradually reducing BLJ , the exact binding energy of the n = 23 state, shown as >lled circles in Fig. 9, gets smaller and smaller, and it vanishes for BLJ ≈ 9800. At the same time, the reduced outer classical turning point a at the energy E23 decreases from its initial value 1:56 for BLJ = 10 000 to zero for BLJ ≈ 9800. The binding energy obtained via conventional WKB quantization (>lled triangles), which is based on reAection phases =2 at both inner and outer classical turning points, is almost 40% too large for a ≈ 1:5; it decreases much more slowly and reaches a >nite value near 2 × 10−7 when the exact binding energy vanishes. In contrast, the prediction of the universal near-threshold quantization rule (open squares) is not so accurate when a is larger than unity, but it improves rapidly as a decreases. For the three smallest values of a in Fig. 9 which lie between 0.05 and 0.2, the absolute error of the prediction of the near-threshold quantization rule (174) is less than 10−11 times the potential depth. This is smaller than the level spacing to the second highest state, n = 22, by a factor 10−7 and represents an improvement of

406


a factor of 10−4 over the performance of conventional WKB quantization. Note that applying the universal near-threshold quantization rule (174) is no more involved than applying conventional WKB quantization (84); it is in fact less so, because the action integral only has to be calculated at threshold, E = 0. The direct numerical integration of the one-dimensional Schr4odinger equation is, of course, always possible, but close to threshold it can be a nontrivial and subtle exercise, and it is de>nitely more time consuming than the direct application of a quantization rule. The universal near-threshold quantization rule (174) can thus be of considerable practical use, e.g., when a problem involves many repetitions of an eigenvalue calculation near threshold. 4.3. Nonhomogeneous potential tails The near-threshold properties of a potential tail falling o? faster than 1=r 2 are determined by three independent tail parameters, the length parameter b, the mean scattering length aU0 (which is, however, not de>ned for a potential falling o? as 1=r 3 ) and the zero-energy reAection phase 0 . This statement implies, that there is a region of moderate r values, where the WKB approximation is accurate for near-threshold energies. The tail parameters are then properties only of the potential tail beyond the WKB region; they do not depend on the potential in the WKB region or at even smaller r values. The fact that the leading asymptotic (r → ∞) behaviour of a potential is proportional to 1=r & does not necessarily mean, that the tail parameters are as given by Eqs. (158) [see also Eq. (89)], (159) and (160) for homogeneous tails (156). For these results to be valid, the homogeneous form (156) of the potential must be accurate not only in the limit of large r values, but all the way down to the WKB region. If the potential tail beyond the WKB region deviates signi>cantly from the homogeneous form (156), then the tail parameters di?er from the tail parameters of the homogeneous tails. The extent to which such nonhomogeneous contributions quantitatively a?ect the tail parameters was >rst studied by Eltschka et al. [82,97]. The tail parameters can be derived from the zero-energy solutions of the Schr4odinger equation in the tail region of the potential, which are then expressed in WKB form in the WKB region as described in Sections 4.1 and 4.2. For a given (nonhomogeneous) potential tail, the tail parameters can always be derived, at least numerically, from the known zero-energy wave functions. If the zero-energy solutions of the Schr4odinger equation are known analytically for the tail region of the potential, then the tail parameters can be derived analytically. Several examples are given in this section. Consider a potential tail consisting of two homogeneous terms, (& )&−2 (&1 )&1 −2 C & C &1 ˝2 V&; &1 (r) = − & − &1 = − ; &1 ¿ & ¿ 2 : + (177) r r 2M r& r &1 In contrast to homogeneous potentials, potential tails consisting of two homogeneous terms contain an intrinsic, energy independent, length scale which can, e.g., be chosen as the position L, where the two contributions have equal magnitude, L=

C &1 C&

1 &1 − &

(&1 )&1 −2 = (& )&−2

1 &1 − &

;

(178)


407

they also have an intrinsic scale of depth, which can be chosen as the magnitude of one of the terms at r = L and expressed in terms of a depth parameter K0 , ˝2 (K0 )2 C& C&1 (& )(&−2)=2 (&1 )(&1 −2)=2 = & = &1 ; K0 = = : 2M L L L&=2 L&1 =2 The dimensionless parameter (&−2)(&1 −2) 2(&1 −&) & < = K0 L = & 1

(179)

(180)

is a useful measure of the relative importance of the two contributions proportional to 1=r & and to 1=r &1 , respectively [82]. When the powers &, &1 in the potential (177) ful>ll the condition &1 − 2 = 2(& − 2) ;

(181)

zero-energy solutions of the Schr4odinger equation are available analytically [99], and analytical expressions for the tail parameters b, 0 and 1 are given in Refs. [82,97]. Zero-energy solutions are also available for the special case, & = 4;

&1 = 5 ;

(182)

which does not ful>ll the condition (181), and analytical expressions for the tail parameters as functions of < are given in Refs. [100,101]. A further example of a nonhomogeneous potential tail for which analytic zero-energy solutions of the Schr4odinger equation are known is, −1 −1 3 r r4 ˝2 r 3 r4 + =− + ; (183) V1 (r) = − C3 C4 2M 3 (4 )2 which resembles a homogeneous tail for r-values either much larger or much smaller than the characteristic length L=

C4 (4 )2 = : C3 3

(184)

For rL the potential (183) resembles a −1=r 3 potential, as in the van der Waals interaction between a polarizable neutral atom and a conducting or dielectric surface; for rL it resembles a −1=r 4 potential. The potential (183) was used by Shimizu [41] as a model for describing the e?ects of retardation in atom–surface interactions [57] and has been further studied in Refs. [56,101], see also Section 5.3. A natural de>nition of the intrinsic strength of the potential (183) is via the wave number K0 given by (3 )2 ˝2 (K0 )2 C3 C4 ; = 3 = 4 ⇒ K0 = 2M L L (4 )3

(185)

and the dimensionless parameter measuring the relative importance of the large-r and the smaller-r parts of the potential is, √ 3 2M C3 √ < = K0 L = = : (186) 4 ˝ C4

408


We thus have analytical results for three di?erent potential tails whose leading asymptotic (r → ∞) behaviour is proportional to −1=r & with & = 4: the two-term sum (177) with condition (181) ful>lled implying &1 = 6, the case (182) where it is not ful>lled, and Shimizu’s potential (183). A −1=r 4 potential occurs in several physically important situations, such as in the retarded atom– surface interaction mentioned above, and also as the leading contribution to the interaction between a charged projectile (electron or ion) and a neutral polarizable target (atom or molecule), when retardation e?ects are not taken into account. The strength of the leading −1=r 4 term in an ion-atom potential depends on the polarizability of the neutral atom [102]. For any two-term tail ful>lling (181), the tail parameters b, 0 and − are given as functions of the parameter (180) in Refs. [82,97] (where < is called (). For the special case (&; &1 ) = (4; 6)—for which < = (4 =6 )2 —the parameters b, aU0 = b=tan(− =2) and 0 are 2( 3 − 1 ined to within a factor consisting of an integer power of the right-hand side of Eq. (194); multiplying E0 by an integer

power of exp 2= g − 14 does not a?ect the energies in the dipole series (193) except for an appropriate shift in the quantum number n labelling the bound states. One interesting feature of dipole series is that, for a given strength ( ¡ − 14 of the attractive potential tail, the limit of large quantum numbers, n → ∞, does not coincide with the semiclassical limit. As discussed in Section 2.1, the semiclassical limit for an inverse-square potential can be approached for large absolute values of the potential strength, but not by varying the energy. As a consequence, the energy eigenvalues of a dipole series obtained via conventional WKB quantization (84) do not become more accurate in the limit n → ∞, not even when the Langer modi>cation (64) is used in the WKB calculation. The WKB energies obtained with the Langer modi>cation acquire a constant relative error in the limit n → ∞, whereas the error grows exponentially with n if they are calculated without the Langer modi>cation [109]. A potential with an attractive inverse-square tail (191) no longer supports an in>nite series of bound states, when the strength parameter ( is equal to (or larger than) − 14 . This can be expected from the breakdown of formulas such as (193), (196) and (200) when −( = g = 14 . It is also physically reasonable, considering that the inverse-square potential V(=−1=4 is the s-wave (l2 = 0) centrifugal potential for a particle moving in two spatial dimensions, see Eq. (59). The fact that the radial Schr4odinger equation for a particle moving in a plane with zero angular momentum actually contains an attractive centrifugal force has lead Cirone et al. [110] to investigate possibilities of a state being bound by the associated force, which the authors call “anticentrifugal”. Kowalski et al. [111] have contributed to a clari>cation of the issue by drawing attention to the topological e?ects occurring when a singular point is extracted from the plane as an origin for the de>nition of the polar coordinates. It is diScult to imagine a physical mechanism that would bind a free particle in a Aat plane, so the discontinuation of dipole series of bound states at the value − 14 of the strength parameter ( seems more than reasonable. A potential well with a weakly attractive inverse-square tail, i.e. with a strength parameter in the range −

1 6(¡0 ; 4

(201)

can support a (>nite) number of bound states if supplemented by an additional attractive potential. If the additional potential is regular at the origin, then the action integral from the origin to the outer classical turning point diverges because of the −1=r 2 singularity of the potential at r = 0, so a naive application of the generalized quantization rule (162) does not work. This can be overcome by shifting the inner classical turning point to a small positive value and adjusting the reAection phase in accordingly [70].


415

The near-threshold quantization rule for a weakly attractive inverse-square tail has been studied in some detail by Moritz et al. [112], and analytical results have been derived for tails of the form 1 ˝2 (m )m−2 g r →∞ (weak) V (r) ∼ Vg; m (r) = − + 2 ; m ¿ 2; g 6 : (202) m 2M r r 4 For g ¡ 14 (i.e., excluding the limiting case g = 14 ), the near-threshold quantization rule is [112], n = nth − with

(%m =2)20 + O((%m )40 ) + O(%2 ) ; 2 2' sin(0)(m − 2) 0'[2(0)2(')]

(203)

20 1 1 −g= (+ and ' = : (204) 0= 4 4 m−2 The >nite but not necessarily integer threshold quantum number nth in Eq. (203) is given by m−2 m 2 1 r 2 in (0) ' − − : (205) p0 (r ) dr + − nth = ˝ rin (0) m−2 r 2 4 2

As in the discussion of Eqs. (196) and (198), the point r de>ning the upper limit of the action integral must lie in a region of the potential well where the WKB approximation is suSciently accurate and the potential is dominated by the −1=r m term, so the inverse-square contribution can be neglected; the sum of the integral and the term proportional to 1=r (m−2)=2 in (205) is then independent of the choice of r. When we express % in terms of the energy E = −˝2 %2 =(2M), the near-threshold quantization rule (203) becomes n = nth − B(−E)0 ;

(206)

with B=

(M(m )2 =(2˝2 ))0 : sin(0) (m − 2)2' 0'[2(0)2(')]2

(207)

The limiting case (=− 14 corresponding to 0=0 and '=0 requires special treatment; the near-threshold quantization rule in this case is [112], ˝2 2=(m − 2) 1 n = nth + ; B = : (208) +O ln(−E=B) 2M(m )2 [ln(−E=B)]2 Again, nth is given by the expression (205); note that ' vanishes in this case. We now have a very comprehensive overview of near-threshold quantization in potential wells with attractive tails. Potentials falling o? as −1=r & with a power 0 ¡ & ¡ 2 support an in>nite number of bound states, and the limit of in>nite quantum numbers is the semiclassical limit. The near-threshold quantization rule (164) contains a leading term proportional to 1=(−E)1=&−1=2 in the expression for the quantum number n. For & = 2, the threshold E = 0 no longer represents the semiclassical limit of the Schr4odinger equation, but the potential still supports an in>nite number of bound states, if the attractive inverse-square tail is strong enough, Eq. (192); the near-threshold quantization rule now contains

g − 14 ln(−E) in the expression for the quantum number n, see Eq. (193). The attractive

416


inverse-square tail ceases to support an in>nite series of bound states at the value g = −( = 14 of the strength parameter, which corresponds to the strength of the (attractive) s-wave centrifugal potential for a particle in a plane. In the near-threshold quantization rule, the leading term in the expression for the quantum number now is a >nite number nth related to the total number of bound states,and 1

the next-to-leading term contains the energy as 1=ln(−E) for ( = − 14 [Eq. (208)], or as (−E) (+ 4 for ( ¿ 14 , see Eq. (206). It is interesting to note, that the properties of potential wells with short-ranged tails falling o? faster than 1=r 2 >t smoothly into the picture elaborated for inverse-square tails when we take the strength of the inverse-square term to be zero. The near-threshold quantization rule (203) acquires the form (174) when ( = 0, 0 = 12 , and the coeScient of % becomes b(m) = with b(m) given by Eq. (159) when we also insert ' = 1=(m − 2). The discussion of weakly attractive inverse-square tails, de>ned by the condition (201), can be continued without modi>cation into the range of weakly repulsive inverse-square tails, de>ned by strength parameters in the range 3 0¡(¡ : (209) 4 The parameter 0 = ( + 14 determining the leading energy dependence on the right-hand sides of Eqs. (203) and (206) then lies in the range 1 ¡0¡1 ; (210) 2 and the leading energy dependence (−E)0 expressed in these equations is still dominant compared to the contributions of order O(E), which come from the analytical dependence of all short-ranged features on the energy E and were neglected in the derivation of the leading near-threshold terms. We can thus complete the comprehensive overview of near-threshold quantization by extending it to repulsive potential tails. For weakly repulsive inverse-square tails (209), the formulas (203) and (206) remain valid. The upper boundary of this range is given by 3 1 (= ; 0= (+ =1 ; (211) 4 4 which corresponds to the p-wave centrifugal potential in two spatial dimensions, l2 = ±1, see Eq. (59). At this limit, the near-threshold quantization rule has the form, n = nth − O(E) ;

(212)

and this structure prevails for more strongly repulsive inverse-square tails, ( ¿ 34 , and for repulsive potential tails falling o? more slowly than 1=r 2 . Repulsive tails falling o? more rapidly than 1=r 2 comply with the case ( = 0, i.e. of vanishing strength of the inverse-square term in the potential, and, provided there is a suSciently attractive well at moderate r values, the quantization rule has the form (174) with a tail parameter b and a threshold quantum number nth which also depends on the shorter-ranged part of the potential. Note that the condition (211) also de>nes the boundary between systems with a singular and a regular level density at threshold. For attractive potential tails and for repulsive potential tails falling o? more rapidly than 1=r 2 or as an inverse-square potential with ( ¡ 34 , the level density dn=dE


417

Table 5 Summary of near-threshold quantization rules for attractive and repulsive potential tails. The second column gives the leading term(s) to the quantization rule in the limit of vanishing energy, E = −˝2 %2 =(2M) → 0. The third column lists equations where explicit expressions for the constants appearing in the second column can be found; these can apply quite generally, as in the >rst row, or to special models of potential tails with the asymptotic behaviour given in the >rst column V (r) for r → ∞ ˝2 − 2M (& )&−2 =r & , ˝2 2M

(=r 2 , ( ¡ −

Quantization rule for E → 0 0¡&¡2

1 4

(¿

3 4

3 4

˙ ±1=r , & ¿ 2

|(| −

n ∼ nth + A=ln(−E=B)

˙ +1=r & , 0 ¡ & ¡ 2 &

F(&)=(%& )

√

n ∼ nth − B(−E) n ∼ nth − O(E) n ∼ nth − O(E) 1 n ∼ nth − b%

Refs. for constants

(2=&)−1

1 n ∼ − 2 ln(−E=E0 )=

( = − 14 − 14 ¡ ( ¡

n∼

1

(+1=4

F(&): Eq. (163) 1 4

E0 : (196), (200) nth : (205) A, B: (208) nth : (205) B: (207) nth : (205)

nth : (148) b: (150), (154), (159), (187), (188), (189), Tables 3, 4

is singular at threshold, and the leading singular term is determined by the tail of the potential. For a repulsive inverse-square tail with ( ¿ 34 , and for a repulsive tail falling o? more slowly than 1=r 2 , the level density is regular at threshold, and the leading (constant) term depends also on the short-ranged part of the potential. A summary of the near-threshold quantization rules reviewed in the last three subsections is given in Table 5. 4.5. Tunnelling through a centrifugal barrier A potential with a repulsive tail at large distances and a deep attractive part at small distances forms a barrier through which a quantum mechanical particle can tunnel. If the repulsive tail falls o? more slowly than 1=r 2 , then the threshold represents the semiclassical limit (in the tail region), and the conventional WKB formula for the tunnelling probability (114), 1 rout (E) −2I (E) WKB ; I (E) = |p(r)| dr ; (213) PT (E) = e ˝ rin (E) involving the action integral between the inner classical turning point rin (E) and the outer classical turning point rout (E) is expected to work well near threshold, provided the conditions of the semiclassical limit are also well ful>lled around rin (0). As we saw in Section 3.5, the WKB formula (213) fails near the base of a barrier with a tail falling o? faster than 1=r 2 , because this corresponds to the anticlassical limit of the Schr4odinger equation; the exact tunnelling probability vanishes at the base whereas Eq. (213) produces a >nite result. Potential barriers involving the centrifugal term in

418


the radial Schr4odinger equation fall o? as 1=r 2 for large r, and hence lie on the boundary between these two regimes. When the potential on the near side (smaller r) of the barrier supports a WKB region where WKB wave functions are good approximate solutions of the Schr4odinger equation for near-threshold energies, then the amplitudes for transmission through and reAection by the barrier can be obtained in a way quite analogous to the methods for deriving the near-threshold quantization rules as described in Section 4.4, see Refs. [84,108,112]. Analytical results have been derived for a potential tail of the form ( ˝2 (m )m−2 r →∞ V (r) ∼ V(; m (r) = ; m¿2 : (214) − 2M r 2 rm For suSciently large r values, the −1=r m term in the potential tail (214) can be neglected, and the wave function on the far side of the barrier at energy E = ˝2 k 2 =(2M) ¿ 0 can be approximated by the analytically known solution of the Schr4odinger equation with the 1=r 2 potential alone. In the barrier region, the analytically known zero-energy solutions for the potential (214) solve the Schr4odinger equation to order less than O(E), and a unique solution is obtained by matching to the asymptotic wave function just mentioned. In the WKB region on the near side of the barrier, the unique solution constructed as above can be written as a superposition of inward and outward travelling WKB waves. Comparing the amplitudes of inward and outward travelling waves on both sides of the barrier yields the transmission amplitude to order less than O(E), and the leading contribution to the transmission probability PT is, k →0

PT ∼ P(m; ()(km )20 ;

(215)

with the coeScient P(m; () =

42 =220 ; (m − 2)2' 0'[2(0)2(')]2

(216)

0 = ( + 14 and ' = 20=(m − 2) are already de>ned in Eq. (204), but now, in Eqs. (215) and (216), ( can also assume nonnegative values. When the inverse-square tail originates from a centrifugal potential in three dimensions, its strength parameter ( is related to the angular momentum quantum number l3 by, 1 1 (217) ( = l3 (l3 + 1); 0 = ( + = l3 + ; 4 2 and the energy dependence of the transmission probability (215) is simply an expression of Wigner’s threshold law [113], according to which probabilities Pl3 which are suppressed by a centrifugal barrier of angular momentum quantum number l3 generally behave as E →0

Pl3 ˙ E 0 = E l3 +1=2 :

(218)

Note that the formulae (215) and (216) can be continued to negative strength parameters in the range of weakly attractive (201) or vanishing inverse-square tails; in fact, they hold for any ( ¿ − 14 . For − 14 ¡ ( 6 0, the potential tail (214) no longer contains a barrier, so transmission between the asymptotic (large r) region and the inner WKB region is classically allowed at all (positive) energies. The probability for this classically allowed transmission is, however, less than unity, because incoming


419

waves can be reAected in the region of high quantality, where the WKB approximation is not good, see Section 2.3. Such “quantum reAection” will be discussed in more detail in Section 5. The probability for transmission from the outer asymptotic region to the inner WKB region actually goes to zero at threshold, even for weakly attractive inverse-square tails, − 14 ¡ ( 6 0, for which there is no barrier. Wigner’s threshold law (218) can be formally extended down to negative angular momentum quantum numbers in the range − 12 ¡ l3 6 0. We now discuss the accuracy of the WKB formula (213) for tunnelling probabilities through a centrifugal barrier at near-threshold energies. For an inverse-square tail (191) with ( ¿ 0, the WKB √ integral I (E) is dominated by the tail near outer classical turning point rout = (=k at near-threshold √ energies, and it diverges as ln( (=k) for k → 0. This means that the leading contribution to the √ √ WKB tunnelling probability (213) is proportional to k 2 ( ˙ E ( , in contradiction to the exact result (218) which obeys Wigner’s threshold law. This contradiction can be resolved by invoking the Langer modi>cation (64) when applying the WKB formula, but a residual error remains, because the WKB expression does not necessarily give the right coeScient of the E 0 term. For potential tails of the form (214), Moritz [84,108] found an upper bound for the WKB integral entering the expression (213), namely 20 m0 2−4=m ln Iapprox = − 1 + O((km ) ) ; (219) m−2 (km )1−2=m so the leading term for the corresponding tunnelling probability, 2m0=(m−2) e (km )20 ; PTWKB; approx = e−2Iapprox = 20

(220)

is a lower bound for the leading term of the WKB tunnelling probability (213). It turned out [114] that the expression (220) is not only a lower bound but becomes equal to the WKB tunnelling probability (213) in the limit E → 0. In the near-threshold limit, the WKB tunnelling probability (213) thus overestimates the exact tunnelling probability, which is given by Eqs. (215) and (216), by a factor G, 0+' e−2Iapprox e 2(0)2(') 2 def G = lim = : (221) k →0 PT 200−1=2 ''−1=2 For large values of the strength ( of the 1=r 2 term in the potential, 0 and ' are also large and we can express the gamma functions in Eq. (221) via Stirling’s formula [59]. This gives 1 m (→∞ +O ; (222) G ∼ 1+ 120 02 showing that the WKB treatment gives the correct leading behaviour of the near-threshold tunnelling probabilities in the limit of large angular momenta. The dependence of the factor G on 0 ≡ l3 + 12 is illustrated in Fig. 13 for powers m of the attractive term in the potential tail (214) ranging from m = 3 to m = 7. The WKB results become worse as 0 decreases and as m increases. For the realistic example 0 =3=2; m=6, which corresponds to angular momentum quantum number l3 =1 and an inter-atomic van der Waals attraction, the WKB formula (213) overestimates the exact tunnelling probabilities by 38% near threshold. Transmission probabilities can also be calculated for attractive inverse-square tails, if there is a WKB region of moderate r values where the WKB approximation is good. The equations (215)

420


2.5 m=3 m=4 m=5 m=6 m=7

2.0 G

1.5

1.0 0

2

4

µ

6

8

10

Fig. 13. Behaviour of G [Eq. (221)] as function of 0 for m = 3; : : : ; 7. For a potential barrier (214) consisting of an attractive −1=r m potential and a centrifugal term corresponding to angular momentum quantum number l3 = 0 − 12 ; G gives the factor by which the conventional calculation of transmission probabilities via the WKB formula (213), including the Langer modi>cation (64), overestimates the exact result (215), (216) for near-threshold energies. From [108].

and (216) for the exact transmission probabilities through the potential tail (214) are also valid for weakly attractive or vanishing inverse-square terms, − 14 ¡ ( 6 0. In this range of values of (, the Langer modi>cation (64) actually produces an asymptotically repulsive potential with a barrier to tunnel through, so the conventional WKB formula (213) can be applied for near-threshold energies. Note, however, that the factor (221) by which the conventional WKB result overestimates the exact result is quite large in the range − 14 ¡ ( 6 0 corresponding to 0 ¡ 0 ¡ 12 and 0 ¡ ' ¡ 1=(m − 2), and diverges to +∞ for ( → − 14 corresponding to 0; ' → 0. For vanishing strength of the inverse-square term, ( = 0; 0 = 12 ; ' = 1=(m − 2), the near-threshold probability (215), (216) for transmission through the potential tail (214) reduces to, 4'1+2' PT = km P(m; ( = 0) = km = 4kb(m) ; (223) 2(1 + ')2 where b(m) is the length parameter (159), which determines the near-threshold quantization rule (174) and the level density just below threshold for a homogeneous −1=r m potential tail. Eq. (223) thus formulates a connection between the near-threshold properties of bound states at negative energies and those of continuum states at positive energies. This connection applies not only for the special case of vanishing strength of the inverse-square term, but to all strengths within the range of weak inverse-square tails [112], 1 0¡0 = ( + ¡1 : (224) 4 If we write the near-threshold quantization rule (203) as %→0

n ∼ nth − C¡ (%)20

(225)


421

for small negative energies E = −˝2 %2 =(2M), and the expression for near-threshold transmission probabilities (215), (216) at small positive energies E = +˝2 k 2 =(2M) as k →0

PT ∼ C¿ (k)20

(226)

then the constants C¡ and C¿ are related by C¿ = 4 sin(0)C¡ :

(227)

The constant in Eqs. (225), (226) can be any (common) length; it is included so that the coef>cients C¡ and C¿ are dimensionless. For vanishing strength of the inverse-square term, i.e. for potentials falling o? faster than 1=r 2 , we have 20 = 1, and the product C¡ is simply the length parameter b of the potential tail as de>ned in Section 4.1, Eq. (150), see also the bottom block of Table 5. In this case, Eqs. (225) and (226) reduce to %→0

n ∼ nth − %b;

k →0

PT ∼ 4kb ;

(228)

with the same length parameter b appearing in both equations. The relation (227) is independent of the power of the shorter-ranged attractive contribution to the potential tail (214). It seems reasonable to assume that it is a universal relation connecting the near-threshold states at positive and negative energies for potentials with weak inverse-square tails, and that this is also true for the special case (228) of potentials falling o? faster than 1=r 2 . The behaviour for this latter case is con>rmed in Section 5.1, see Eq. (241). 5. Quantum re ection Just as quantum mechanics can allow a particle to tunnel through a classically forbidden region, it can also lead to the reAection of a particle in a classically allowed region where there is no classical turning point. The term “quantum reAection” refers to such classically forbidden reAection. Quantum reAection can only occur in a region of appreciable quantality, i.e. where the condition (36) is violated. In regions where Eq. (36) is well ful>lled, motion is essentially (semi-) classical, and, in the absence of a classical turning point, the particle does not reverse its direction. Quantum reAection can occur above a potential step or barrier, or in the attractive long-range tails of potentials describing the interaction of atoms and molecules with each other or with surfaces. For potentials falling o? faster than 1=r 2 , the probability for quantum reAection approaches unity at threshold, so it is always an important e?ect at suSciently low energies. Quantum reAection had been observed to reduce the sticking probabilities in near-threshold atom-surface collisions more than twenty years ago [115–120], and the recent intense activity involving ultracold atoms and molecules has drawn particular attention to this phenomenon [121–125]. As described in Section 3.5, transition and reAection amplitudes, T and R, can be de>ned using WKB wave functions for the incoming, transmitted and reAected waves in the semiclassical regions, or plane waves if the potential tends to a constant asymptotically. The transmission and reAection probabilities (103), PT = |T |2 ;

PR = |R|2 ;

(229)

are independent of the choice of WKB waves or plane waves, but the phases of the transmission and reAection amplitudes depend on this choice [see Eqs. (104), (105)] and also on the choice of

422


reference points, at which the reference waves have vanishing phase. Throughout this section, the reAection amplitude is always de>ned with respect to incoming plane waves incident from the right as in Eq. (102). Whenever the phase of R is important, we retain the subscript “r” to remind us of this choice. The point of reference is taken to be at r = 0 unless explicitly stated otherwise. For brevity, we shall refer to the absolute value of the reAection amplitude, i.e. to the square root of the reAection probability, as the re?ectivity. 5.1. Analytical results A fundamentally important example [46] is the Woods-Saxon step potential, V (r) = −

˝2 (K0 )2 =(2M) ; 1 + exp(r=)

(230)

which is treated in detail in the textbook of Landau and Lifshitz [43]. The reAection amplitude, de>ned with reference to plane waves (102) is, 2(2ik) 2(−ik − iq) 2 q + k Rr = (231) ; q = (K0 )2 + k 2 : 2(−2ik) 2(ik − iq) q−k From the properties of the gamma functions of imaginary argument [59] the absolute value of the reAection amplitude (231) is sinh[(q − k)] |R| = ; (232) sinh[(q + k)] as already used in Section 2.1, Eq. (14). For small values of the di?useness, → 0, the Woods-Saxon potential (230) approaches the sharp step potential already discussed in Sections 2.4 and 4.1, and the reAection amplitude becomes [cf. Eq. (39)] q−k Rr = − : (233) q+k The large-r tail of the Woods-Saxon potential (230) is an exponential function, ˝2 (K0 )2 exp(−r=) : (234) 2M The exponential potential (234) is an interesting example in itself, because the semiclassical approximation becomes increasingly accurate for r → −∞, although the r-dependence of the potential gets stronger and stronger. The Schr4odinger equation for the potential (234) is solved analytically [59] by any Bessel function of order ' = 2ik and argument z = 2K0 exp[ − r=(2)]. The solution which merges into a leftward travelling WKB wave for r → −∞ is the Hankel function (r) ˙ H'(1) (z), and matching this solution to a superposition of incoming and reAected plane waves yields the reAection amplitude, 2(2ik) (K0 )−4ik exp(−2k) : Rr = (235) 2(−2ik) V (r) = −

For any given value of k, the reAection amplitude (231) of the Woods-Saxon step actually becomes equal to the result (235) in the limit that the relative diBuseness [63] K0 is large, K0 → ∞.


423

In the near-threshold region, the leading behaviour of the reAection amplitude (231) is k →0

Rr ∼ − [1 − 2k coth(K0 ) − 4ik((E + R{ where (E = 0:57721 : : : is Euler’s constant and contributions to the reAectivity are, |R| = 1 − 2kb + O((k)2 );

2

2 (−iK0 )})]

;

(236)

= 2 =2 is the digamma function [59]. The leading

b = coth(K0 ) :

(237)

This means, that the reAectivity is unity at threshold, and its initial decrease from unity is linear in k, i.e. in the asymptotic velocity of the particle on the side where this velocity goes to zero. This threshold behaviour of the quantum reAectivity is very general and holds for all potential tails falling o? faster than 1=r 2 [55,119] as can be shown using the methods applied in Sections 4.1 and 4.2. Consider an attractive potential going to zero faster than 1=r 2 for large r with a semiclassical WKB region at moderate or small r values. Using the zero-energy solutions (135) of the Schr4odinger equation, which go unity resp. r for large r and behave as (136) for small r, we can construct a wave function (r) =

D0

1 (r)exp(i0 =2)

− D1 0 (r)exp(i1 =2) ; D0 D1 sin[(0 − 1 )=2]

(238)

which is proportional to a leftward travelling WKB wave of the form (100) in the semiclassical region r → 0. Matching the asymptotic (r → ∞) form of the wave function (238) to the superposition 1 kr →0 √ [exp(−ikr) + Rr exp(ikr)] ˙ 1 + Rr − ikr(1 − Rr ) ˝k

(239)

gives the leading near-threshold contribution to the reAection amplitude, k →0

Rr ∼ −

1 − ik exp[ − i(0 − 1 )=2]D1 =D0 : 1 + ik exp[ − i(0 − 1 )=2]D1 =D0

For the reAectivity |R|, Eq. (240) implies k →0

2

|R| ∼ 1 − 2kb = exp(−2kb) + O(k );

D1 0 − 1 : b= sin D0 2

(240)

(241)

Eq. (241) implies that the probability for quantum reAection behaves as 1 − 4kb near threshold, and k →0 this is consistent with the result PT ∼ 4kb given for the transmission probability in Eq. (228) at the end of Section 4. The characteristic length parameter of the quantal region of the potential tail, namely b as de>ned in Eq. (150), determines not only the near-threshold quantization rule (174) and the near-threshold level density (175), but also the reAection and transmission properties of the potential tail near threshold. The near-threshold behaviour of the phase of the reAection amplitude also follows from (240) and the result is, k →0

arg(Rr ) ∼ − 2k aU0 ;

aU0 =

b : tan [(0 − 1 )=2]

(242)

The parameter aU0 determining the near-threshold behaviour of the phase of the reAection amplitude is just the mean scattering length de>ned in Eq. (151). Scattering lengths and mean scattering lengths

424


are only de>ned for potentials falling o? faster than −1=r 3 . For a potential proportional to −1=r 3 , V (r) = −

˝ 2 3 C3 = − ; r3 2M r 3

(243)

the wave function which is proportional to a leftward travelling WKB wave of the form (100) in the semiclassical region r → 0 is (r) ˙ H1(1) (z)=z with z = 2 3 =r, and matching to the asymptotic waves (239) gives k →0

arg(Rr ) ∼ − 2k3 ln(k3 ) :

(244)

Note that the formula (241) for the near-threshold reAectivity holds for all potentials falling o? faster than −1=r 2 , even for those such as (243), where the phase of the reAection amplitude becomes divergent at threshold. The energy dependence of the phase of the reAection amplitude can be related to the time gain or delay of a wave packet during reAection [126]. If the momentum distribution of the incoming wave packet is sharply peaked around a mean momentum ˝k0 , then the shape of the reAected wave packet is essentially the same and the time shift can be calculated in the same way as in partial-wave scattering [127,128] where the reAection amplitude R is replaced by the partial-wave S-matrix. The derivative of arg[R(k)] with respect to k, taken at k0 , describes an apparent shift Wr in the point of reAection, Wr = −

1 d [arg(Rr )]k=k0 : 2 dk

(245)

The time evolution of the reAected wave packet corresponds to reAection of a free wave at the point r = Wr rather than at r = 0. For a free particle moving with the constant velocity v0 = ˝k0 =M this implies a time gain Wt =

M d d 2Wr [arg(Rr )]k=k0 = −˝ [arg(R)]E=˝2 k02 =(2M) : =− v0 ˝k0 d k dE

(246)

For a positive (negative) value of Wr the reAected wave packet thus experiences a time gain (delay) relative to a free particle (with the same asymptotic velocity v0 ) travelling to r = 0 and back. Note however, that the classical particle moving under the accelerating inAuence of the attractive potential is faster than the free particle; the quantum reAected wave packet may experience a time gain with respect to a free particle but nevertheless be delayed relative to the classical particle moving in the same potential (see Section 5.2). Eq. (242) implies that the near-threshold behaviour of the space shift (245) and of the time shift (246) is k →0

Wr 0∼ aU0 ;

k →0

Wt 0∼

2M aU0 : ˝k0

(247)

The near-threshold behaviour of the time shift due to reAection for a wave packet with a narrow momentum distribution is determined by the mean scattering length aU0 . Near threshold, the quantum reAected wave packet evolves as for a free particle reAected at r = aU0 .


425

0 α=3 α=4

-2

α=5 α=6 log Rα

-4

-6

-8

5

10

15

20

25

kα Fig. 14. Natural logarithm of the reAectivity of the homogeneous potential (248) as function of k& for various values of &. From [56].

5.2. Homogeneous potentials The threshold behaviour of the reAection amplitude as summarized by Eqs. (241) and (242) is determined by the two parameters b and aU0 which were obtained analytically for a variety of attractive potential tails in Section 4. They are given for homogeneous potentials, V&att (r) = −

C& ˝2 (& )&−2 = − r& 2Mr &

(248)

in Eqs. (159) and (160) and are tabulated in Table 3. For homogeneous potentials (248) the properties of the Schr4odinger equation do not depend on the energy E = ˝2 k 2 =(2M) and potential strength parameter & independently, but only on the product k& . For energies above the near-threshold region, analytical solutions of the Schr4odinger equation are not available (except for & = 4), and the reAection amplitudes have to be obtained numerically. Figs. 14 and 15 show the real and imaginary parts of ln Rr , namely ln |R| and arg Rr , as functions of k& for various values &. For attractive potential tails such as (248) or the Casimir-Polder-type potentials studied in the next section, the quantality function (36) tends to have its maximum absolute value near the position rE where the absolute value of the potential is equal to the total energy [56], |V (rE )| = E =

˝2 k 2 : 2M

(249)

426

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 4 3

φ(k)

2 1 0 -1 -2 0

1

2

3

4

5

kβα

Fig. 15. Phase = arg Rr of the quantum reAection amplitude for the homogeneous potential (248). From top to bottom the curves show the results for & = 3, 4, 5, 6 and 7. From [126].

In the corresponding repulsive potential −V (r), the point rE is the classical turning point. For the homogeneous potential (248) we have rE = & (k& )−2=& :

(250)

In the limit of large energies, we may use a semiclassical expression for the reAection amplitudes which was derived by Pokrovskii et al. [129,130]. We use the reciprocity relation (101) to adapt the formula of Refs. [129,130] to the reAection amplitude Rr de>ned via the boundary conditions (102),   rt 2i k →∞ p(r) dr  : (251) Rr (k)∗ ∼ i exp ˝ Here rt is the complex turning point with the smallest (positive) imaginary part. For a homogeneous potential (248) it can be written as rt = (−1)1=& rE = ei=& rE ;

(252)

where rE is de>ned by Eq. (250) and lies close to the maximum of |Q(r)|. Real values of the momentum p(r) only contribute to the phase of the right-hand side of Eq. (251), so the reAectivity |R| is una?ected by a shift of the lower integration point anywhere along the real axis. Integrating along the path r=rE = cos(=&) + i; sin(=&) with ; = 0 → 1 gives the result [56] k →∞

|R| ∼ exp(−B& krE ) = exp[ − B& (k& )1−2=& ] ;

−&

1 R + i; sin B& = 2 sin 1 + cos d; : & & & 0

(253)


427

Table 6 The coeScients B& , which are given by Eq. (253) and appear before (k& )1−2=& in the exponents describing the high-energy behaviour of the reAectivities of attractive homogeneous potential tails &

3

4

5

6

7

8

&→∞

B&

2.24050

1.69443

1.35149

1.12025

0.95450

0.83146

2=&

0 α=3 α=4

-2

α=5 α=6 log Rα

-4

-6

-8

2

4

6

8

(kβα) 1− 2/α

Fig. 16. Natural logarithm of the reAectivity of the homogeneous potential (248) as function of (k& )1−2=& for various values of &. The straight lines correspond to the behaviour (253) with the values of B& as listed in Table 6. From [56].

In terms of the energy E, the particle mass M and the strength parameter C& of the potential (248), the energy-dependent factor in the exponent is √ 1 1 1 pas rE ; (254) (k& )1−2=& = E 2 − & (C& )1=& 2M = ˝ ˝ where pas = ˝k is the asymptotic (r → ∞) classical momentum. The high-energy behaviour (253) of the reAectivity as function of ˝ is an exponential decrease typically expected for an analytical potential which is continuously di?erentiable to all orders, see the discussion in Section 2.4. Numerical values of the coeScients B& are listed in Table 6. Fig. 16 shows a plot of the logarithm of |R| as function of (k& )1−2=& for various values &, and the obvious convergence of the curves to the straight lines is strong evidence in favour of the high-energy behaviour (253). The phase of the right-hand side of Eq. (251) depends more sensitively on the choice of lower integration limit, which is not speci>ed in Refs. [129,130]. The k-dependence of the integral in the exponent is determined by the complex classical turning point (252), rt = rE [cos(=&) + i sin(=&)].

428


4 2

φ(k)

0 -2 -4 -6 0

1

2 (kβα)

3

4

5

1- 2/α

Fig. 17. Phase = arg Rr of the reAection amplitude for the homogeneous potential (248) as function of (k& )1−2=& for various values of &. From top to bottom the curves show the results for & = 3, 4, 5, 6 and 7.

If we assume that the real part of the integral becomes proportional to ˝k × R(rt ) = ˝krE cos(=&) for large k, then the high-energy behaviour of the phase of the reAection amplitude is k →∞

arg Rr ∼ c − c0 krE = c − c0 (k& )1−2=&

(255)

with real constants c; c0 . This conjecture is supported by numerical calculations as demonstrated in Fig. 17. Eq. (255) implies that the space shift (245) is given for large energies by 2 k0 →∞ c0 1− rE : (256) Wr ∼ 2 & The space shifts (245) obtained from the numerical solutions of the Schr4odinger equation are plotted in Fig. 18 as functions of k0 & for & = 3, 4, 5, 6 and 7. Except for & = 3 and values of k0 3 less than about 0.15, the space shifts are always positive: according to Eq. (246) this corresponds to time gains relative to the free particle reAected at r =0. For & =3 and energies close to threshold there are signi>cant time delays. Note, however, that the classical particle accelerated under the inAuence of the attractive potential is faster than the free particle [with the same asymptotic velocity v0 =˝k0 =M], and its time gain is ∞ 2M 1 1 dr = − B(&)rE ; (257) (Wt)cl = 2M ˝k0 p(r) ˝k0 0 where B(&) depends only on &, 1 1 1 1 + 2 1− : B(&) = √ 2 2 & & Numerical values of B(&) are given in Table 7.

(258)


429

0.6 0.4

∆r/βα

0.2 0 -0.2 -0.4 -0.6 -0.8

0

1

2

3

4

5

k0βα Fig. 18. Space shift (245) for quantum reAection by the homogeneous potential (248) as function of k0 & for various values of &. From bottom to top the curves show the results for & = 3, 4, 5, 6 and 7. From [126]. Table 7 Numerical values of B(&) as de>ned in Eq. (258) &

3

4

5

6

7

8

&→∞

B(&)

0.862370

0.847213

0.852623

0.862370

0.872491

0.881900

1

The time gain (257) corresponds to the space shift v0 (Wt)cl = B(&)rE ; (259) (Wr)cl = 2 the classical particle which is accelerated in the potential and reAected at r = 0 eventually returns at the same time as a free particle reAected at (Wr)cl . The classical space shifts (259) are generally larger than the space shifts of the quantum reAected wave, as illustrated in Fig. 19 for the example & = 4. At high energies both the classical space shifts (259) and the quantum space shift (256) show the same dependence on k0 & , i.e., proportionality to rE , but the coeScient B(&) in the classical case is larger than the coeScient in the quantum case. At small energies, the classical space shift diverges as rE [Eq. (250)], whereas the quantum space shift remains bounded by a positive distance of the order of the potential strength parameter & , see Figs. 18, 19. Although the quantum reAected wave may experience a time gain relative to the free particle reAected at r = 0, it is always delayed relative to the classical particle which is accelerated in the attractive potential [126]. 5.3. Quantum re?ection of atoms by surfaces The probability for quantum reAection of atoms by a surface is directly accessible to measurement, because the (elastically) reAected atoms return with their initial kinetic energy, whereas those atoms

430


1.25 1

∆r/βα

0.75 0.5 0.25 0 0

1

2

k0βα

3

4

5

Fig. 19. Space shift (245) for quantum reAection by the homogeneous potential (248) with & = 4. The solid line shows the space shift of the quantum reAected wave while the dot-dashed line shows the classical space shift (259). Similar results are obtained for other powers & ¿ 3. From [126].

which are transmitted through the quantal region of the potential tail and hence approach the surface to within a few atomic units are usually inelastically scattered or adsorbed (sticking). Quantum reAection has been observed in the scattering of thermal hydrogen atoms from liquid helium surfaces [118,120], and in the scattering of laser-cooled metastable neon atoms from smooth [41] or structured [131] silicon surfaces. Beyond the region of very small distances of a few atomic units, the interaction between a neutral atom (or molecule) and a conducting or dielectric surface is well described by a local van der Waals potential, corrected for relativistic retardation e?ects as described in the famous paper by Casimir and Polder in 1948 [57]. A compact expression has been given in Refs. [132,133]; in atomic units, the atom-surface potential is ∞ (&fs )3 ∞ V@ (r) = − &d (i!)!3 exp(−2!rp&fs )h(p; @) dp d! ; (260) 2 0 1 where h(p; @) =

s − @p s−p + (1 − 2p2 ) ; s+p s + @p

with s =

@ − 1 + p2 ;

(261)

&fs ≡ 1=c=0:007297353 : : : is the >ne-structure constant and @ is the dielectric constant of the surface; &d is the frequency-dependent dipole polarizability of the projectile atom in its eigenstate labelled n0 with energy En0 [49], | n0 | Zj=1 zj | n |2 2(En − En0 ) : (262) &d (i!) = (En − En0 )2 + !2 n


431

For a perfectly conducting surface, a simpler formula is obtained by taking @ → ∞ in Eq. (261) and integrating over p in Eq. (260), ∞ 1 V∞ (r) = − &d (i!)[1 + 2&fs !r + 2(&fs !r)2 ]exp(−2&fs !r) d! 4r 3 0 ∞ x 1 [1 + 2x + 2x2 ]exp(−2x) d x : =− &d i (263) 4&fs r 4 0 &fs r For small r values, we can put r = 0 in the upper line of Eq. (263) and obtain the van der Waals potential between the atom and a conducting surface, ∞ C3 (∞) 1 vdW V∞ (r) = − ; C (∞) = &d (i!) d! : (264) 3 r3 4 0 For >nite values of the dielectric constant @, the derivation of the small-r behaviour of the potential is a bit more subtle, but the result is quite simple [132,134], V@vdW (r) = −

C3 (@) ; r3

C3 (@) =

@−1 C3 (∞) : @+1

(265)

For large r values, we can assume the argument of &d in the lower line of Eq. (263) to be zero and perform the integral over x. This gives the highly retarded limit of the Casimir-Polder potential between the atom and a conducting surface, ret V∞ (r) = −

C4 (∞) ; r4

C4 (∞) =

3 &d (0) : 8 &fs

(266)

For >nite values of the dielectric constant @, we have [133,134] V@ret (r) = −

C4 (@) ; r4

C4 (@) =

@−1 (@)C4 (∞) ; @+1

(267)

∞ where (@) = 12 ((@ + 1)=(@ − 1)) 0 h(p + 1; @)(p + 1)−4 dp is a well de>ned smooth function which for @ = 1 to unity for @ → ∞. Explicit expressions for increases monotonically from the value 23 30 (@) and a table of values are given in Ref. [133]. The atom-surface potential behaves as −C3 =r 3 for “small” distances [Eqs. (264), (265)] and as −C4 =r 4 for large distances [Eqs. (266), (267)]. The ratio L=

C4 (4 )2 = C3 3

(268)

de>nes a length scale separating the regime of “small” r values, rL, from the regime of large r values, rL. In Eq. (268) we have introduced the parameters 3 and 4 which express the potential strength in the respective limit in terms of a length, as for the homogeneous potentials (248). The expressions (260) and (263) have been evaluated for the interaction of a hydrogen atom with a conducting surface by Marinescu et al. [135] and for the interaction of metastable helium 21 S and 23 S atoms with a conducting surface (@ = ∞) and with BK-7 glass (@ = 2:295; (@) = 0:761425) and fused silica (@ = 2:123; (@) = 0:760757) surfaces by Yan and Babb [134]. A list of the potential parameters determining the “short”-range and the long-range parts of the respective potentials is given in Table 8.

432


Table 8 Parameters determining the “short”-range behaviour (264), (265) and the long-range behaviour (266), (267) of the atom-surface potentials calculated by Marinescu et al. [135] for hydrogen and by Yan and Babb [134] for metastable helium. The length L is the distance (268) separating the regime of “small” distances from the regime large distances; < is the parameter (273) determining the relative importance of the “small”-distance regime and the large-distance regime for quantum reAection. All quantities are in atomic units Atom

H

He(21 S)

He(23 S)

@

∞

∞

2.295

2.123

∞

2.295

2.123

C3 C4 3 4 L
ed, but it still lies beyond the regime of really small distances of a few atomic units, where more intricate details of the atom–surface interaction involving the microscopic structure of the atom and of the surface become important. The energies at which quantum reAection becomes important are given by wave numbers of the order of 1=3 and 1=4 , i.e. typically below 10−4 atomic units for metastable helium atoms. This corresponds to velocities of the order of centimetres per second, which are very small indeed, but not beyond the range of modern experiments involving ultra-cold atoms [41,42,131,136,137]. The “high”-energy behaviour of the reAection amplitude discussed in Section 5.2 in connection with Eqs. (251), (253), (255), (256) and Figs. 16, 17 refers to high energies relative to this near-threshold regime; these can still be well within the range of ultra-cold atoms. For the potential (263) between the atom and a conducting surface, we can also make some general statements about the next-to-leading terms at large and small separations. For large separations we can exploit the fact that the dipole polarizability (262) is an even function of the imaginary part of its argument, so V∞ (r) as given in the second line of Eq. (263) is an even function of 1=r and the next term in the large-distance expression (266) must fall o? at least as 1=r 6 , C4 1 r →∞ V∞ (r) ∼ − 4 + O 6 : (269) r r For small distances r we can calculate a correction to the expression (264) via a Taylor expansion of the integral in the >rst line of Eq. (263), and prudent use of the Thomas-Reiche-Kuhn sum rule


433

[49] yields [138], Z&fs r ; (270) 4 where Z is the total number of the electrons in the atom. The second term on the right-hand side of Eq. (270) is the leading retardation correction to the van der Waals potential at “small” distances, but “small” means small compared to the lengths L listed in Table 8, and this can still be quite large in atomic units. It was >rst derived by Barton for one-electron atoms in 1974 [139]. An intriguing feature of the correction (270) to the van der Waals potential between an atom and a conducting surface is, that it is universal: it depends only on the number Z of electrons in the atom and not on its eigenstate n0 . The shape of the atom-surface potential in between the “short”-range behaviour (264), (265) and the long-range behaviour (266), (267) can be expressed in terms of a shape function vshape (r=L),

r

r C4 C3 = − 4 vshape ; (271) V (r) = − 3 vshape L L L L r →0

r 3 V∞ (r) ∼ − C3 +

x→0

x→∞

whose “short”- and long-range behaviour is given by vshape (x) ∼ 1=x3 and vshape (x) ∼ 1=x4 . Shimizu [41] analyzed his experimental data with the simple shape function v1 (x) = 1=(x3 + x4 ). This gives the potential V1 (r) [see Eq. (183)], whose near-threshold properties were already discussed in Section 4.3. The potential between a hydrogen atom and a conducting surface as calculated by Marinescu et al. [135] is well represented by a rational approximation, vH (x) =

1 + Jx ; x3 + 8x4 + Jx5

8 = 1; J = 0:31608 :

(272)

The coeScients 8 and J are actually determined by the expressions (269) and (270) respectively, so the formula (272) contains no adjusted parameters; it reproduces the tabulated values [135] of the exact hydrogen-surface potential to within 0.6% in the whole range of r values. 1 In numerical calculations it is generally advisable to work with smooth potentials which are continuous in all derivatives. Otherwise, e.g. when using a spline interpolation of tabulated potential values, discontinuities in higher derivatives of the potential can lead to remarkably irregular spurious contributions to the quantum reAection amplitude, see e.g. the discussion in Section 2.4. A detailed study of quantum reAection probabilities for potentials behaving as (264), (265) for “small” distances and as (266), (267) for large distances has been given in Ref. [56]. Which part of the potential dominantly determines the reAection probability depends on a crucial parameter √ 2MC3 3 gure shows the “high”-energy behaviour expected for a homogeneous −1=r 4 potential according to Eq. (253) with 4 = 11400 a:u. The straight line in the bottom-left part of the >gure shows the near-threshold behaviour (241) for b = 4 = 11400 a:u. The curves were obtained by numerically solving the Schr4odinger equation with potential shapes given by the shape function v1 de>ning Shimizu’s potential (183) and with the shape function vH for the potential of Marinescu et al. [135] for the interaction of a hydrogen atom with a conducting surface, Eq. (272); the value of 3 de>ning the “short”-range van der Waals part of the potential was either 11400 a:u: (< = 1) or 114000 a:u: (< = 10). From [56].

a hydrogen atom and a conducting surface. The values of b lie within 5% of the large-< limit 4 when < ¿ 2 and approach the small-< limit 3 = 4 =< for < → 0. The most pronounced shape dependence is observed around < = 1. Recent measurements of quantum reAection were carried out by Shimizu for metastable neon atoms reAected, e.g., by a silicon surface [41]. The transition from the linear dependence of ln |R| on k near threshold to the proportionality of −ln |R| to k 1−2=& at “high” energies is nicely exposed by plotting ln(−ln |R|) as a function of ln k, see Fig. 22. At the “high”-energy end of the >gure, the data clearly approximate a straight line with gradient near 12 corresponding to & = 4. Fitting a straight line of gradient 12 through the last six to ten data points yields ln(−ln |R|) = 5:2 + 12 ln k (straight line in top-right corner of Fig. 22), and comparing this with ln(−ln |R|) = ln B4 + 12 ln k + 12 ln 4 according to Eq. (253) yields 4 ≈ 11400 a.u. The corresponding near-threshold behaviour (241), ln(−ln |R|) = ln(24 ) + ln k is shown as a straight line in the bottom-left corner of Fig. 22 and >ts the data well within their rather large scatter. Also shown in Fig. 22 are the results obtained by numerically solving the Schr4odinger equation with a potential given by two of the three shapes already introduced in connection with Fig. 21, and with the above value of 4 and two di?erent choices of the crucial parameter (273), namely rming that the quantum reAectivity is essentially that of the highly retarded −1=r 4 potential (267) in this case too. In fact, large values of


1E-4

2

En e

0

rg y

1 Lo w

ln(-ln|R|2)

Reflection Coefficient, |R|2

1E-3

E igh

H

1

0.01

gy

ner

3

0.1

437

-1 -7

-6

-5

-4 -3 ln(kia)

-2

-1

0

1E-5 1E-6 1E-7 1E-8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 kia

Fig. 23. Quantum reAection probabilities |R|2 observed by Druzhinina and DeKieviet in the scattering of ground state 3 He atoms by a rough quartz surface. The solid line shows the reAection probabilities calculated with Shimizu’s potential (183) with 3 = 650 a:u: and 4 = 350 a:u. The straight line in the bottom left-hand corner of the inset shows the low-energy behaviour (241), and the straight line with gradient 1=3 in the top right-hand corner shows the “high”-energy behaviour (253) expected for the nonretarded −1=r 3 part of the potential. From [42], courtesy of M. DeKieviet.

the crucial parameter (273) are ubiquitous in realistic systems, so quantum reAection data provide a conspicuous and model-independent signature of retardation e?ects in atom-surface potentials [56]. In a more recent experiment, Druzhinina and DeKieviet [42] measured the probability for quantum reAection of (ground state) 3 He atoms by a rough quartz surface. The helium atoms which are transmitted all the way to the surface are reAected di?usely because of the surface roughness and contribute only negligibly to the yield of specularly reAected atoms; the specularly reAected atoms thus represent the quantum reAection yield. The results of Ref. [42] are reproduced in Fig. 23. In the label of the abscissa, ki is the incident wave number perpendicular to the surface and a=5 a.u. is the location of the minimum of a realistic atom-surface potential [141]. The authors analyzed their data using Shimizu’s potential (183); the strength parameter 4 was >xed via the known polarizability of the helium atoms [cf. Eq. (266)] and the dielectric constant of the quartz surface [cf. Eq. (267)] to be 4 = 350 a.u., and the strength parameter 3 was determined by >tting the calculated probabilities to the experimental data. This gave 3 = 650 a.u. corresponding to L = 190 a.u. and a crucial parameter < = 1:9. The straight line in the bottom left-hand corner of the inset in Fig. 23 is close to the near-threshold reAectivity (241) with b ≈ 4 = 350 a.u. as expected for the −1=r 4 part of the potential, and the straight line with slope 1=3 in the top right-hand corner shows the “high”-energy behaviour (253) expected for the −1=r 3 part of the potential with 3 = 650 a.u.

438


0 -1

log10PR

-2 -3 -4 -5 -6 -7 -8 0.0

0.1

0.2 ka

0.3

Fig. 24. Quantum reAection probabilities for Shimizu’s atom surface potential (183) with 3 = 650 a:u: and 4 = 350 a:u: (solid line) in comparison with the predictions of the associated homogeneous potentials (248) proportional to −1=r 3 (dashed line) and to −1=r 4 (dotted line).

An important aim of the work in Ref. [42] was to measure quantum reAection probabilities so far above the threshold, that they are signi>cantly inAuenced by the nonretarded van der Waals part of the atom-surface potential. The high-energy data can actually be seen to approach the straight line in the upper right-hand corner of the inset in Fig. 23, but the quantum reAection probabilities are still substantially larger than the predictions for a pure −1=r 3 potential. This is illustrated in Fig. 24 comparing the quantum reAection probabilities predicted for the realistic (Shimizu’s) potential (solid line), which >t the data, with those obtained for the −1=r 3 potential alone (dashed line) and for the −1=r 4 potential alone (dotted line). The transition point k = (B3 =B4 )6 rst and second derivatives of U vanishes. This is, e.g. the case if the coupling potentials Vij ; j = i vanish for small r, or if the diagonal potentials become dominant with respect to the coupling potentials, r →0

e.g. if Vii ˙ − 1=r & whereas the coupling potentials remain bounded. In these cases, U becomes the unit matrix at small distances. If both the diagonal potentials and the coupling terms behave as an inverse power of r for small r, then the asymptotic (r → 0) decoupling of the channels depends on the relation of the powers involved. If, e.g. in a two-channel example, V11 (r) and V22 (r) are proportional to −1=r & for r → 0, and V12 (r) and V21 (r) proportional to −1=r & , then we obtain decoupled channels for small r, if and only if |& − &| = 1 [143]. If the diagonal potentials and the coupling potentials have the same spatial dependence for small r, dfij =0 ; (279) Vij (r) = f(r) × fij ; dr then there is a decoupling of channels, but U = 1 so the decoupled channels will be superpositions of the diabatic channels i which are uncoupled for r → ∞. Decoupling of channels occurs in the limit r → 0 for many-body systems described in hyperspherical coordinates. In these coordinates, the hyperradius r stands for the root-mean-square average of the radial coordinates of all particles involved, and the remaining coordinates, the hyperangles, encompass not only all angular degrees of freedom, but also “mock angles” de>ned by the ratios of the individual radial coordinates [144]. Note however, that in applications to many-electron atoms, the diagonal potentials are proportional to 1=r at small values of the hyperradius r, and there need not be a semiclassical WKB region in the regime where the channels decouple (see the discussion in the last paragraph of Section 5.3). For the waves transmitted to small r values, r → 0 or r → −∞, the appropriate generalization of the one-channel boundary conditions (100) is, r 1 i j (r) ∼ T1j (280) qj (r ) dr ; exp − ˝ rl qj (r) where ˝qj (r) = 2M[E − Wjj (r)] is the local classical momentum in the adiabatic channel j. The form (280) implies, that the adiabatic channel j is open for transmission, i.e., that E − Wjj (r) ¿ 0 for small r. Some general statements can be made about the threshold behaviour of the quantum reAection amplitudes when potentials and coupling terms approach their asymptotic limits faster than 1=r 2 . If the incoming channel is energetically lowest and nondegenerate (E1 ¡ Ei for all i = 1) then the near-threshold behaviour of the elastic reAectivity |R11 | follows the pattern (241), k →0

|R11 | 1∼ 1 − 2bk1 = exp(−2bk1 ) + O((k1 )2 ) ;

(281)

with a characteristic length parameter b, depending only on the tails of the potentials and coupling terms Vij in the quantal region of coordinate space. SuSciently close to threshold only the elastic


441

reAection channel and a certain number of transmission channels are open, and particle number conservation requires the sum of the transmission probabilities to grow proportional to 1 − |R11 |2 ∼ 4bk1 + O((k1 )2 ); this actually holds for each transmission probability individually, k1 →0 k1 →0 (PT )ij ˙ k1 ; T1j ˙ k1 : (282) This is a straightforward generalization of the second equation (228) to the coupled-channel situation. Analytical solutions for two coupled step potentials with a step-like coupling term are given in Ref. [142], and smoother Woods-Saxon steps as well as inverse-power potentials are discussed in Ref. [143]. Exponential potentials and coupling terms have been studied in considerable detail for isolated special cases by Osherov and Nakamura [145,146], while more general cases were treated in Refs. [147,148] using semiclassical methods. When two diagonal diabatic potentials, V11 (r) and V22 (r) cross at a point r0 , then coupling of the channels 1 and 2 leads to an avoided crossing of the corresponding adiabatic potentials W11 and W22 . In an adiabatic process, the incoming and reAected or transmitted waves remain on the potential energy curve Wii associated with the respective adiabatic channel, but the avoided crossing can be overcome by a nonadiabatic transition. The probabilities for such nonadiabatic transitions have been a topic of great interest for more than seventy years [149–152]. With the assumptions that the diabatic potential curves are linear at the crossing and that the coupling potential is constant one obtains the semiclassical Landau-Zener formula, V12 (r0 )2 ; (283) (P1→1 )LZ = 1 − exp −2 ˝v0 WF where v0 is the velocity at the crossing point, Mv02 =2 = E − V11 (r0 ) and WF is the di?erence of the slopes of the two crossing curves, WF = |V11 (r0 ) − V22 (r0 )|. Eq. (283) actually describes the probability that the incoming wave in the channel labelled 1 (wave function 1 (r) for large r) remains on the adiabatic potential curve and is transmitted to the transmission channel 1 , which is the channel labelled 2 in the diabatic basis. Many improvements of the simple Landau-Zener formula (283) have been proposed over the years [152], but we shall focus on one aspect, namely the quenching of the curve crossing probability due to quantum reAection [153]. Consider a system of two Woods-Saxon step potentials, Ui + (i − 1)E0 ; i = 1; 2 ; Vii (r) = − (284) 1 + exp(ai r) with a Gaussian coupling potential, V12 (r) = V21 (r) = U12 exp[ − (a12 )2 (r − r0 )2 ] ;

(285)

as illustrated in Fig. 25. The performance of the simple Landau-Zener formula is illustrated in Fig. 26 for an example of very small coupling. The transition probability P1→1 = |T12 |2 , obtained by numerically solving the two-channel Schr4odinger equation (275) with the boundary conditions (276), (280) is plotted as function of k1 ≡ k. The solid line shows the exact result, which goes to zero at threshold according to Eq. (282). This vanishing transmission probability is not accounted for in the conventional Landau-Zener formula (283)—illustrated as dashed line in Fig. 26—nor in the numerous improvements introduced over the years [152]. It is due to the fact, that the incoming wave only reaches

442

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 5

Channel 2 Coupling Potential

Potential

0

Channel 1

Channel 2’

-5 -10 -15 -20

Channel 1’ -20

-10

0 r

10

20

Fig. 25. Woods-Saxon potentials (284) with the coupling potential (285) (dashed line). The parameters are U1 = 5:5, U2 = 26, U12 = 0:5, E0 = 3 and a1 = a2 = a12 = 1. The dotted line shows the quantality function (36) for the elastic channel just above threshold (k = 0:2). From [153].

QM LZ

0.00012

P1→1′

0.0001 8e-05 6e-05 4e-05 2e-05 0 0

0.2

0.4

0.6

0.8

1

k Fig. 26. Transition probability P1→1 = |T12 |2 for the coupled Woods-Saxon potentials (284), (285). The parameters are U1 = 5:5, U2 = 26, U12 = 0:01, E0 = 3, a1 = a2 = 1, a12 = 0:05. The solid line shows the exact result and the dashed line is the prediction of the simple Landau-Zener formula (283). From [153].

the curve-crossing region with a small probability, because quantum reAection in the quantal region of the incoming-channel potential becomes dominant towards threshold. This region is indicated by the dotted line in Fig. 25, which shows the quantality function (36) for the potential V11 at a near-threshold momentum, k = 0:2. For k larger than about 0.4, the e?ect of quantum reAection is negligible, and the conventional Landau-Zener formula reproduces the exact result to within a rather constant error of a few per cent. A straightforward improvement of the Landau-Zener formula (283) is to account for the e?ect of quantum reAection by multiplying the probability (283) by the probability for transmission through


443

0.00012 0.0001

P1→1′

8e-05 6e-05

QM LZM

4e-05 2e-05 0

0

0.2

0.4

0.6

0.8

1

k Fig. 27. Transition probability P1→1 = |T12 |2 for the coupled Woods-Saxon potentials (284), (285). The parameters are U1 = 5:5, U2 = 26, U12 = 0:01, E0 = 3, a1 = a2 = 1, a12 = 0:05. The solid line shows the exact result and the dashed line is the prediction of the modi>ed Landau-Zener formula (286). From [153].

the quantal region of the potential tail, (P1→1 )LZM = (P1→1 )LZ (1 − |R|2 ) :

(286)

In the very-weak-coupling example in Fig. 26, the quantum reAectivity in the elastic channel is insensitive to the coupling, so we can take |R| to be given by the reAectivity (232) of an uncoupled Woods-Saxon potential. As shown in Fig. 27, the modi>ed Landau-Zener formula (286) does indeed account correctly for the e?ects of quantum reAection in this case and leads to much better agreement with the exact result. 6. Conclusion Although the semiclassical WKB approximation is generally expected to be most useful near the semiclassical limit, where quantum mechanical e?ects are small, semiclassical WKB wave functions can be used to advantage far from the semiclassical limit, even near the anticlassical, the extreme quantum limit of the Schr4odinger equation. This is because the conditions for the accuracy of the WKB wave functions are inherently local. Under conditions which are far from the semiclassical limit for the Schr4odinger equation as a whole, there may still be large regions of coordinate space where WKB wave functions are highly accurate approximations of the exact quantum mechanical wave functions; they may even be asymptotically exact. For example, for homogeneous potentials proportional to 1=r & with & ¿ 2, the threshold E = 0 represents the anticlassical limit of the Schr4odinger equation, but WKB wave functions become exact for r → 0 at all energies, in particular at threshold and in the near-threshold region. The (local) condition for the accuracy of WKB wave functions such as (25) is conveniently expressed via the quantality function (36): |Q(r)|1. The regions of coordinate space, where |Q(r)| is not negligibly small are quantal regions where quantum e?ects such as classically forbidden tunnelling or reAection can be generated. In many physically important situations, exact or highly

444


accurate quantum mechanical wave functions are available in the quantal regions and globally accurate wave functions can be constructed by matching these to WKB wave functions in the semiclassical regions. In this way, several highly accurate, sometimes asymptotically exact results have been derived via WKB waves far from the semiclassical limit of the Schr4odinger equation. Numerous examples have been given in the preceding sections. WKB wave functions are singular at a classical turning point and the connection formulas relating the oscillating WKB waves on the classically allowed side to the exponential waves on the forbidden side are usually formulated under restrictive assumptions for the potential (linearity), which are not related to whether or not WKB wave functions may be accurate away from the turning point. Allowing more general connection formulas (44), (45) greatly widens the range of applicability of WKB wave functions. Correct choice of the phase of the WKB wave (56) on the allowed side of a classical turning point via an appropriate de>nition of the reAection phase is a vital ingredient for the construction of accurate WKB wave functions in the classically allowed region. This leads, e.g. to a formula for the scattering phase shifts for repulsive inverse-power potentials which is highly accurate at all energies, and to an exact expression (74) for the scattering lengths, which determine the behaviour of the phase shifts in the anticlassical limit, see Section 3.3. For bound state problems, a generalization (83) of the conventional WKB quantization rule to allow an appropriate de>nition of the reAection phases at the classical turning points leads to greatly improved accuracy without complicating the procedure, see e.g. Figs. 3, 4 and 7, in Section 3.4. The generalized connection formulas (44), (45) also lead to more precise expressions for tunnelling probabilities, in particular near the base of a barrier, where conventional WKB expressions fail when the potential tail falls o? faster than 1=r 2 , see Section 3.5. For deep potential wells, where WKB wave functions are accurate in some region of small or moderate r values, the properties of bound states for E ¡ 0 and continuum states for E ¿ 0 are largely determined by the tail of the potential beyond this WKB region. For potentials falling o? faster than 1=r 2 , the threshold E = 0 represents the anticlassical limit of the Schr4odinger equation and the near-threshold properties of the wave functions are determined by three independent tail parameters, which can be derived by matching zero-energy solutions of the Schr4odinger equation to WKB waves in the WKB region. These three parameters are the characteristic length b, Eq. (150), the mean scattering length aU0 , Eq. (151), and the zero-energy reAection phase 0 , which is the phase loss of the WKB wave due to reAection at the outer classical turning point at threshold, see Eq. (136) in Section 4.1. Immediately below threshold, the quantization rule acquires a universal form (174) which becomes exact for E → 0 and contains the tail parameter b as well as the threshold quantum number nth , which depends on 0 and also on the threshold value of the action integral over the whole of the classically allowed region, see Eq. (148). The characteristic length b determines the leading singular contribution to the near-threshold level density according to Eq. (175), and also the near-threshold behaviour of the quantum reAectivity of the potential tail according to Eq. (241) in Section 5.1. The mean scattering length aU0 determines the near-threshold behaviour of the phase of the amplitude for quantum reAection according to Eq. (242), and hence also the time and space shifts involved in the quantum reAection process. With the correct tail parameters, Eqs. (174), (175), (241) and (242) are asymptotically exact relations for the near-threshold behaviour of the bound and continuum states. When the zero-energy solutions of the Schr4odinger equation for the potential tail beyond the WKB region are known analytically, analytical expressions for the tail parameters b, aU0 and 0 can


445

be derived. A summary for various potential tails is given in Table 4 in Section 4.3. If analytical solutions of the Schr4odinger equation are not available, the tail parameters can be obtained from numerical zero-energy solutions of the Schr4odinger equation. Knowing the asymptotic (E → 0) behaviour of the bound and continuum states near threshold is of considerable practical value, because the direct numerical integration of the Schr4odinger equation for small >nite energies is an increasingly nontrivial exercise as the energy approaches zero. For potential tails falling o? more slowly than 1=r 2 , the threshold E = 0 represents the semiclassical limit of the Schr4odinger equation, and semiclassical methods are expected to work well in the near-threshold region, e.g. in the derivation of the near-threshold quantization rule (164) for the energies of the bound states in the limit n → ∞. Potential tails vanishing as 1=r 2 represent the boundary between long-ranged potentials, which support in>nitely many bound states, and shorter-ranged potentials which can support at most a >nite number of bound states. The behaviour of potentials with inverse-square tails (191) depends crucially on the strength parameter (, as discussed in detail in Section 4.4. For ( ¡ − 14 , i.e. for potential tails more attractive than the s-wave centrifugal potential in two dimensions (see Eq. (59) in Section 3.3), the potential supports an in>nite dipole series of bound states (193), in which the energy depends exponentially on the quantum number near threshold. Potentials with weak inverse-square tails, − 41 6 ( ¡ 34 , support at most a >nite number of bound states, but the leading energy dependence in the near-threshold quantization rule is still of order less than O(E), so the near-threshold level density is still singular at E =0. The properties of short-ranged potential tails falling o? faster than 1=r 2 appear as a special case of weak inverse-square tails with ( = 0. A summary of quantization rules is given in Table 5 in Section 4.4. Probabilities for transmission through the quantal region of an inverse-square tail with ( ¿ − 14 1

are proportional to E (+ 4 , which is a generalization of Wigner’s threshold law (218). For weak inverse-square tails, − 14 6 ( ¡ 34 , the transmission probability above threshold is related to the leading energy dependence in the near-threshold quantization rule below threshold, Eq. (227). For all potentials which are asymptotically (r → ∞) less repulsive than the p-wave centrifugal potential in two dimensions ((= 34 ), the leading near-threshold energy dependence in the quantization rule is of order less than O(E) and can be derived from the tail of the potential. For potentials with more repulsive tails, i.e. for inverse-square tails with ( ¿ 34 or for repulsive tails falling o? more slowly than 1=r 2 , the leading energy dependent terms in the near-threshold quantization rule are of order O(E) and include e?ects of the potential for smaller r values; they cannot be derived from the properties of the potential tail alone. These results have been derived using WKB waves to approximate the quantum mechanical wave functions only in regions where such an approximation is highly accurate or asymptotically exact. The results do not depend on the conditions of the semiclassical limit being ful>lled for the Schr4odinger equation as a whole. Indeed, the comprehensive results for the near-threshold region refer to the immediate vicinity of the anticlassical or extreme quantum limit in the case of potential tails falling o? faster than 1=r 2 . Many of the results summarized in this article are of direct practical importance in various >elds, e.g. in atomic and molecular physics, where the intense current interest in ultra-cold atoms and molecules has drawn attention to quantum e?ects speci>c to small velocities and low energies. The near-threshold phenomena studied in Sections 4 and 5 are explicit examples of such quantum e?ects. The quantum reAection of atoms moving as slowly as a few centimetres per second towards

446


a surface occurs several hundreds or thousands of atomic units from the surface and can be observed in present-day experiments. Understanding this and similar phenomena is important for technical developments such as the construction of atom-optical devices. The theoretical considerations and practical applications discussed in this review refer mostly to the Schr4odinger equation for one degree of freedom. Matching the exact or highly accurate solutions of the Schr4odinger equation in the “quantal region” of coordinate space to WKB waves which are accurate in the “WKB region” can occur at any point where these regions overlap. An extension to systems with more than one degree of freedom does not seem straightforward, because matching between quantal and WKB regions would have to occur on a subspace of dimension one or more, and it is not clear whether this is easy to do in general. A more promising >eld for generalising the techniques reviewed in this article is that based on coupled ordinary Schr4odinger equations, such as the coupled channel equations used in the description of scattering and reactions in nuclear, atomic and molecular physics. Multicomponent WKB waves have been used in the treatment of coupled wave equations with the individual equations referring to spin components of Pauli or Dirac particles or di?erent Born-Oppenheimer energy surfaces in a molecular system [154–158]. Generalizing such theories to allow for signi>cant deviations from the semiclassical limit may greatly enhance their range of applicability. A simple example is the inAuence of quantum reAection on Landau-Zener curve crossing probabilities as described in Section 5.4. Acknowledgements The authors express their gratitude to the current and former collaboraters who have contributed to the results presented in this review, namely Kenneth G.H. Baldwin, Robin CôtPe, Christopher Eltschka, Stephen T. Gibson, Xavier W. Halliwell, Georg Jacoby, Alexander Jurisch, Carlo G. Meister and Michael J. Moritz. Harald Freidrich also wishes to thank the members of the Department of Theoretical Physics and of the Atomic and Molecular Physics Laboratories, in particular Brian Robson, Steve Buckman, Bob McEachran and Erich Weigold, for hospitality and enlightening discussions during his stay at the Australian National University during spring and summer 2002/2003. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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451

CONTENTS VOLUME 397 P.S. Landa, P.V.E. McClintock. Development of turbulence in subsonic submerged jets J.S.M. Ginges, V.V. Flambaum. Violations of fundamental symmetries in atoms and tests of unification theories of elementary particles

1

63

C. Hanhart. Meson production in nucleon–nucleon collisions

155

D.V. Bugg. Four sorts of meson

257

H. Friedrich, J. Trost. Working with WKB waves far from the semiclassical limit

359

Contents of volume

451

doi:10.1016/S0370-1573(04)00246-7

Physics Reports vol.397

Physics Reports vol.342

Physics Reports vol.302

Physics Reports vol.347

Physics Reports vol.107

Physics Reports vol.351

Physics Reports vol.326

Physics Reports vol.369

Physics Reports vol.381

Physics Reports vol.361

Physics Reports vol.321

Physics Reports vol.318

Physics Reports vol.359

Physics Reports vol.394

Physics Reports vol.316

Physics Reports vol.307

Physics Reports vol.434

Physics Reports vol.356

Physics Reports vol.308

Physics Reports vol.363

Physics Reports vol.319

Physics Reports vol.346

Physics Reports vol.312

Physics Reports vol.435

Physics Reports vol.380

Physics Reports vol.343

Physics Reports vol.355

Physics Reports vol.331

Physics Reports vol.396

Physics Reports vol.314

Physics Reports vol.375

Physics Reports vol.397

Physics Reports vol.342

Physics Reports vol.302

Physics Reports vol.347

Physics Reports vol.107

Physics Reports vol.351

Physics Reports vol.326

Physics Reports vol.369

Physics Reports vol.381

Physics Reports vol.361

Physics Reports vol.321

Physics Reports vol.318

Physics Reports vol.359

Physics Reports vol.394

Physics Reports vol.316

Physics Reports vol.307

Physics Reports vol.434

Physics Reports vol.356

Physics Reports vol.308

Physics Reports vol.363

Physics Reports vol.319

Physics Reports vol.346

Physics Reports vol.312

Physics Reports vol.435

Physics Reports vol.380

Physics Reports vol.343

Physics Reports vol.355

Physics Reports vol.331

Physics Reports vol.396

Physics Reports vol.314

Physics Reports vol.375

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