Appendix C. Quantum theory of scattering by a central potential.

Journal of the ICRU Vol 7 No 1 (2007) Report 77 Oxford University Press

doi:10.1093/jicru/ndm018

APPENDIX C. QUANTUM THEORY OF SCATTERING BY A CENTRAL POTENTIAL

Z0 Ze2 expðr=RÞ: VW ðrÞ ¼ r

¨ DINGER) C.1 NON-RELATIVISTIC (SCHRO THEORY Let us consider a scattering experiment (Figure 1.1) in which a beam of slow (non-relativistic) elec trons with momentum h k is scattered by the central field V(r). For the moment, we assume that the field has a finite range rc [i.e., V(r) ¼ 0 for r . rc]; this condition excludes the case of Coulomb and modified Coulomb fields, which will be considered later. For the sake of simplicity, we disregard in the non-relativistic formulation the spin of the electron; the electron spin will be properly accounted for in the relativistic Dirac theory. The deflected electrons are counted by means of a detector placed farther than the potential range and effectively screened from the incident beam. The situation is described by a DPW, i.e., a solution of the time-independent Schro¨dinger equation "

# 2 h 2 r þ VðrÞ ck ðrÞ ¼ Eck ðrÞ; 2me

ðh kÞ2 E¼ ; 2me

ðC:3Þ

with the asymptotic behavior

ck ðrÞ cinc ðrÞ þ csc ðrÞ; r!1

ðC:4Þ

where ðC:1Þ

cinc ðrÞ ¼

1 ð2pÞ3=2

expðik rÞ ; fk ðrÞ

ðC:5Þ

With the ‘screening radius’ R ’ a0 Z1=3 ;

ðC:2Þ

this potential provides a simple approximation to the scattering potential of neutral atoms of atomic number Z; in the limit R ! 1, the Wentzel potential reduces to a pure Coulomb potential.

is a plane wave, which represents the incident beam, and

csc ðrÞ ¼

1 ð2pÞ

3=2

expðikrÞ ^; kÞ f ðkr r

ðC:6Þ

is an outgoing spherical wave that represents the emerging flux of scattered particles. The adopted

# International Commission on Radiation Units and Measurements 2007

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Except for slow projectiles, with kinetic energy less than 100 eV, elastic scattering of electrons and positrons can be accurately described as scattering by a static central potential V(r). In this appendix, the fundamentals of the theory of scattering by central fields are reviewed. The appropriate theoretical framework is the one-electron Dirac theory, which provides a consistent description of relativistic (non-radiative) effects, including spin polarization. However, we shall start from the more familiar non-relativistic Schro¨dinger theory of the spinless electron. This approach provides an insight into the basic nature of the scattering process and also provides useful approximations that are easily generalized to the relativistic formulation. Occasional reference will also be made to the Klein-Gordon (or relativistic Schro¨dinger) theory, which describes the spinless relativistic electron. For the sake of concreteness, we shall limit our consideration to real potentials; partialwave calculations for optical-model potentials that include an absorptive imaginary part are performed by using essentially the same formulas as for real potentials (see, e.g., Salvat, 2003). To illustrate the use of various approximation methods, we shall frequently consider the example of the exponentially screened Coulomb potential, also known as the Wentzel (or Yukawa) potential, defined by Eq. (3.37),

ELASTIC SCATTERING OF ELECTRONS AND POSITRONS

operators L and Lz [which commute with the 2 /2me)r2 þ V(r)]. They have Hamiltonian H ¼ 2 (h the form

normalization for the DPW is such that ð

ck0 ðrÞck ðrÞdr ¼ dðk k0 Þ:

ðC:7Þ

cE‘m ðrÞ ¼ r1 PE‘ ðrÞY‘m ð^rÞ;

Notice that in the limit V ! 0 the DPW ck(r) reduces to the plane wave fk(r), Eq. (C.5). The function f(krˆ, k) is known as the scattering amplitude. For a given kinetic energy, E, the scattering amplitude is a function only of the angle u between the initial and final directions of the projectile. It can be shown (see, e.g., Joachain, 1983) that, with the adopted normalization for free states, f ðk ; kÞ ¼

4pme

h

2

ð

fk0 ðrÞVðrÞck ðrÞdr:

where Y‘m(rˆ) are spherical harmonics and the radial functions PE‘(r) are solutions of the radial Schro¨dinger equation "

¼

ðC:8Þ

jinc ;

r!1

k h ^: r me

ð

ðC:10Þ

cE0 ‘0 m0 ðrÞcE‘m ðrÞdr ¼

p dðk k0 Þd‘‘0 dmm0 : 2

ðC:15Þ

The radial equation (C.13) can be solved numerically to get the radial functions PE‘(r) and the phase shifts d‘. In the limit of zero field strength, PE‘(r) ¼ krj‘ (kr), where j‘ (x) is the spherical Bessel function, and d‘ ¼ 0. The phase shift d‘ quantifies the effect of the potential on the large-r behavior of the radial functions PE‘(r). For purely attractive (repulsive) fields, the phase shifts are positive (negative) reflecting that, within the range of the potential, PE‘(r) oscillates faster (slower) than the corresponding radial functions for a free particle. This is illustrated in Figure C.1 for the case of a Wentzel potential, Eq. (C.1), with different values of the ‘charge’ Z (Table C.1). The DPW ck(r) admits the following expansion in spherical waves,

ðC:11Þ

Thus, the scattering amplitude completely determines the DCS. C.1.1

ðC:14Þ

With this normalization, the spherical waves satisfy the orthonormality relation,

The number of scattered particles that enter the ˙ count ¼ ( jsc.rˆ)r 2 dV . From detector per unit time is N the definition (1.1), it follows that _ count N ds ¼ ¼ j f ðk0 ; kÞj2 : dV jjinc jdV

ðC:13Þ

p PE‘ ðrÞ sin kr ‘ þ d‘ : r!1 2

ðC:9Þ

At large distances r from the origin, the scattered wave has the form of a plane wave with wave vector k0 ¼ krˆ pointing along the radial direction rˆ. Hence, the current density of scattered particles that reach the detector is jsc ð2pÞ3 jf ðk0 ; kÞj2 r2

2 2 k h PE‘ ðrÞ 2me

satisfying the boundary condition PE‘(0) ¼ 0. At large radial distances (beyond the field range, r . rc), the radial function oscillates with a constant amplitude. We shall normalize the spherical waves in such a way that the radial function oscillates asymptotically with unit amplitude,

The current density of particles in the incident beam ( plane wave) is ih ½ðrfk Þfk fk ðrfk Þ 2me k h : ¼ ð2pÞ3 me

# 2 2 d2 ‘ð‘ þ 1Þ h h þ þ VðrÞ PE‘ ðrÞ 2me dr2 2me r2

Partial-wave series

Since the wave equation (C.3) is a partial differential equation depending on three variables, its direct numerical solution is impractical. An alternative is to expand the DPW ck(r) in the orthonormal basis of spherical wave functions, which can be easily (and accurately) calculated, and to derive the scattering amplitude from that expansion. The spherical waves are solutions of the Schro¨dinger equation (C.3) that are also eigenfunctions of the orbital angular momentum

1 ck ðrÞ ¼ k

rffiffiffiffi 2X ‘ ^ i expðid‘ ÞY‘m ðkÞcE‘m ðrÞ: p ‘;m

ðC:16Þ

To prove this equality, we first note that the righthand side satisfies Eq. (C.3). On the other hand, 132


0

ðC:12Þ

QUANTUM THEORY OF SCATTERING BY A CENTRAL POTENTIAL

not feel the scattering potential V(r). If the effective radial potential (i.e., the scattering potential plus the centrifugal potential) for a given partial wave ‘ has a sufficiently deep minimum at intermediate distances, the potential may support bound states with this angular momentum. The Levinson theorem (see, e.g., Schiff, 1968) relates the number n‘ of bound states and the corresponding phase shift d‘ (0) at zero energy,

with the aid of the asymptotic form (C.14), it can be shown that the function given by Eq. (C.16) has the required asymptotic behavior,

ck ðrÞ fk ðrÞ þ r!1

1 ð2pÞ

3=2

expðikrÞ f ðk^ r; kÞ; r

ðC:17Þ

and that the scattering amplitude can be expressed as a ‘partial-wave’ series, f ðk^ r; kÞ ¼

1 X ^ ð2‘þ1Þ½expð2id‘ Þ1P‘ ð^ r kÞ; 2ik ‘

lim d‘ ðEÞ ¼ pn‘ :

ðC:19Þ

E!0

For free states of certain energies E, the incident projectile can be temporarily trapped and stay about the minimum for a time much longer than the ordinary transit time. This situation is called a shape resonance; it manifests itself as a violent change of the phase shift and a corresponding fluctuation in the scattering amplitude and DCS at the energy E.

ðC:18Þ where P‘ are Legendre polynomials. This analysis completes the proof of the equality (C.16). We see that the scattering amplitude is completely determined by the phase shifts d‘ and depends only on the variable rˆ . kˆ ¼ cos u. Owing to the spherical symmetry of the field, the scattering amplitude and the DCS are independent of the azimuthal scattering angle f (Figure 1.1). It is worth noting that each phase shift d‘, corresponding to a given angular momentum ‘, is determined independently of the other waves. For short-range fields and sufficiently small energies, only a modest number of low-‘ partial waves have appreciable phase shifts. This situation occurs because partial waves with higher angular momenta cannot penetrate the centrifugal poten 2 ‘(‘ þ 1)/(2mer 2) in Eq. (C.13) and do tial barrier h

Table C.1. Numerical values of the phase shifts d0 and d1 for the radial functions displayed in Fig. C.1.

133

Z

d0

d1

2 0 22 24 26

21.0241 0.0 1.9196 3.4572 4.7263

20.4308 0.0 0.6844 1.7586 2.9167


Figure C.1. Radial functions for electrons with E ¼ 1Eh (27.21 eV) and ‘ ¼ 0,1 in the Wentzel field VW (r) ¼ (Ze 2/r) exp(2r/a0). Apart from a constant shift, the vertical scale is the same for all radial functions. The arrows indicate the phase shifts, whose numerical values are given in Table C.1.


C.1.2

The Schro¨dinger equation (C.4) is a differential equation, and its DPW solutions are uniquely determined by the boundary condition (C.5). A differential equation with its boundary conditions can always be transformed into an equivalent integral equation. Thus, the DPWs satisfy the Lippmann – Schwinger equation (see, e.g., Joachain, 1983)

ck ðrÞ ¼ fk ðrÞ ð þ dr0 G0 ðE; r; r0 ÞVðr0 Þck ðr0 Þ

ðC:20Þ

ðB1Þ

ck

with the Green function

þ

me expðikjr r0 jÞ G0 ðE; r; r Þ ¼ 2 : jr r0 j 2ph 0

ðC:21Þ

2me 2 h

ð1

f j‘ ðkr0 ÞVðr0 ÞPE‘ ðr0 Þr0 dr0 :

ðB0Þ

0

ðrÞ ¼ fk ðrÞ:

ð2pÞ

3=2

expðikrÞ ðSBÞ f ðuÞ; r

ðC:25Þ

ð

me

expðiq rÞVðrÞ dr 2 2ph ð1 2me sinðqr0 Þ ¼ 2 Vðr0 Þr02 dr0 ; 0 qr h 0

ðu Þ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cos u q ¼ 2k sinðu=2Þ ¼ 2k : 2

ðC:26Þ

ðC:27Þ

From the asymptotic behavior of the Born wave function, it follows that the Born DCS is equal to jf (SB)(u)j2. The Born scattering amplitude can be expanded as a partial-wave series (Mott and Massey, 1965),

ðC:23Þ

ð

1

where h q¼h k2h k0 is the momentum transfer. In potential scattering, the magnitudes of the initial and final momenta are equal, and therefore

By inserting the function (C.23) into the right-hand side of Eq. (C.20), we obtain the wave function in the first Born approximation, ðB1Þ ck ðrÞ

ðSBÞ

ðC:22Þ

This expression involves the radial function PE‘(r) and therefore, it is not useful to calculate the phase shift. However, Eq. (C.22) offers a simple method for checking the consistency of computed phases and radial functions. The nth Born approximation results from considering the interaction field V(r) as a perturbation to order n. The Lippmann – Schwinger equation (C.20) can be used to obtain a formal expression for the scattering wave function in the nth Born approximation as follows. We start from the incident plane wave as the zeroth-order approximation,

ck

r!1

with the ‘Schro¨dinger-Born’ (SB) scattering amplitude given by

From the expansion of the Lippman – Schwinger equation in spherical waves, one can derive the following exact expression for the phase shifts, sin d‘ ¼

ðrÞ fk ðrÞ

f ðSBÞ ðuÞ ¼

1 X ðSBÞ ð2‘ þ 1Þð2id‘ Þ P‘ ðcos uÞ; 2ik ‘

ðC:28Þ

with the phase shifts,

ðB0Þ

¼ fk ðrÞ þ dr0 G0 ðE; r; r0 ÞVðr0 Þck ðr0 Þ ð ¼ fk ðrÞ þ dr0 G0 ðE; r; r0 ÞVðr0 Þfk ðr0 Þ:

ðSBÞ

d‘

¼

2me 2 h

k

ð1 0

j2‘ ðkr0 ÞVðr0 Þr02 dr0 :

ðC:29Þ

ðC:24Þ We expect the first Born approximation to be accurate only when the distorting effect of the scattering field on the incident plane wave is weak, i.e.,

Introducing this wave function into the right-hand side of Eq. (C.20) we obtain the second Born 134


approximation, c(B2)(r). Continuing in this way, we can generate approximations c(Bn)(r) with arbitrarily large orders n. In the limit n ! 1, we get the Born series, which is an exact representation of the scattering wave function. Since each iteration introduces a term with an additional three-dimensional integral, the difficulty of the calculation increases rapidly with the order. As a result, we shall normally limit consideration to the first Born approximation (and we shall frequently refer to it simply as ‘the Born approximation’). has the It can be shown that the function c(B1) k asymptotic behavior (for rrc)

Born approximation


approximation may not be valid under these circumstances. For fast projectiles (kR1), the condition (C.34) reduces to Z ln(kR)/(ka0)1, which indicates that, at least for projectiles with sufficiently high energy, the approximation is applicable. The Born DCS for the Wentzel potential,

when ð dr0 G0 ðE; r; r0 ÞVðr0 Þfk ðr0 Þ jfk ðr0 Þj ¼ ð2pÞ3=2

ðC:30Þ

or, equivalently, ð 2me 1 0 0 0 0 expðikr ÞVðr Þ sinðkr Þdr 1: 2 k 0 h

ðC:31Þ

When this approximation is valid, the radial function PE‘(r) approaches the spherical Bessel function, krj‘ (kr), and the phase-shift d‘ is small. Then, Eqs. (C.28) and (C.29) agree with the corresponding exact results, Eqs. (C.18) and (C.22). In practice, Eq. (C.29) provides fairly good estimates of those phase shifts that are small in magnitude, even when the first Born approximation for the scattering amplitude is not accurate. For a detailed analysis of the convergence of the Born series, see Byron and Joachain (1977). As an example, let us consider the case of the Wentzel potential, Eq. (C.1). The corresponding Born scattering amplitude is ðSBÞ f W ðu Þ

¼ ¼

2me

2

Ze 2 h 2Ze2 me 2 h

ð1 0

is frequently referred to as the Wentzel or ‘screened Rutherford’ DCS. This simple formula is useful for estimating the dependence of the elastic DCS on the atomic number Z, the energy E of the projectile, and the scattering angle u. In the limit R ! 1, the Wentzel potential reduces to the Coulomb potential, VC(r) ¼ Ze 2/r, and the corresponding Born DCS becomes ðSBÞ

ds C ¼ dV

0

sinðqr Þ expðr0 =RÞr0 dr0 qr0 1

2 h 2 2

Ze ¼ 2E

ðC:32Þ

ð1=RÞ2 þ q2

2 2Ze2 me

1 4k4 ð1

cos uÞ2

1 ð1 cos uÞ2

ðC:36Þ

;

which coincides with the classical Rutherford DCS. and the Born phase shifts are ðSBÞ dW;‘

¼

2Zme e2 2 h

ð1 k 0

j2‘ ðkr0 Þ expðr0 =RÞr0

! Z 1 ; ¼ Q‘ 1 þ ka0 2ðkRÞ2

dr

C.1.3 WKB approximation for the phase shifts

0

At large radial distances, the solution of the radial equation (C.13) can be approximated by means of the semiclassical WKB approximation (Merzbacher, 1970; Mott and Massey, 1965; Schiff, 1968), which gives

ðC:33Þ

2 /(mee 2) is the Bohr radius and Q‘ are where a0 ¼ h the Legendre functions of the second kind (Abramowitz and Stegun, 1974). For the Wentzel potential, the integral on the left-hand side of Eq. (C.31) can be evaluated analytically and the condition of validity of the Born approximation takes the form

ðWKBÞ ðrÞ PE‘

1=2

k‘ ðrÞ

sin

ð r

p ; k‘ ðr Þ dr þ 4 r0 0

0

ðC:37Þ

where C is a normalization constant, k‘(r) is the local wave number defined by

2Z 1 i 2 arctanð2kRÞ þ ln½1 þ 4ðkRÞ ka0 2 4 1:

¼

C

"

ð‘ þ 1=2Þ2 2me 2 VðrÞ k‘ ðrÞ ¼ k r2 h 2

#1=2 ;

ðC:38Þ

ðC:34Þ and r0 is the largest positive zero of k‘(r). Here, we have introduced the Langer correction, which consists in replacing ‘(‘ þ 1) by (‘ þ 1/2)2. This

For slow projectiles (kR1), this relation becomes 2ZR/a0 1, which means that the Born 135


ðSBÞ ds W ðSBÞ 2 ¼ f W ðu Þ dV 2 2Ze2 me 1 ¼ 2 2 h ½ð1=RÞ þ 2k2 ð1 cos uÞ2 2 2 Ze 1 ¼ ðC:35Þ 2E ½ð2kRÞ2 þ ð1 cos uÞ2


eikonal approximation, it is usually assumed that the interaction causes only small deflections of the projectile and that the potential energy of the projectile along classical trajectories with moderately large impact parameters is much smaller than its kinetic energy. With these assumptions, the eikonal DPWs for projectiles moving in directions parallel to the z-axis take the form

correction accounts for the fact that r varies from 0 to 1 in the radial equation, while in the derivation of the usual (one-dimensional) WKB approximation it is assumed that the independent variable ranges from 21 to 1 (Mott and Massey, 1965). Comparison of the approximation (C.37) with the exact asymptotic form of the radial function, Eq. (C.14), leads to the following WKB formula for the phase shift, ðWKBÞ

ime exp ikz 2 h k ð2pÞ3=2 ðz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 02 Vð x þ y þ z Þ dz : 1

ðeikÞ

ck^z ðrÞ ¼ ðC:39Þ

1

r0

ðC:43Þ

The WKB approximation is applicable when the potential V(r) varies so slowly with r that the momentum of the particle is practically constant over many ‘wave lengths’ [l(r) ¼ 2p/k‘(r)]. More explicitly, the phase-shift formula is expected to be accurate when (Schiff, 1968) 1 dk‘ 2k2 dr 1;

for r . r0 :

Inserting this expression into the right-hand side of Eq. (C.22) and recalling that we are considering only small deflections, the integral can be partially evaluated to give the following formula for the scattering amplitude f ðeikÞ ðuÞ ¼ ik

ðC:40Þ

ð1

J0 ðqbÞ

0

‘

fexp½ixðbÞ 1gb db; In general, the accuracy of the WKB phase shifts improves when the energy of the projectile increases. The WKB formula (C.39) is useful for understanding the dependence of the phase shifts on the potential V(r). It is easy to verify that if V ; 0, d(WKB) ¼ 0 (which is the exact value of the free‘ particle phase shift). If the potential is negative, the integrand is larger than for V ¼ 0 and the WKB phase shifts are positive. Conversely, if the are negative. In potential is positive, the d(WKB) ‘ general, when the potential increases, the WKB phase shifts decrease. C.1.4

where q ¼ 2k sin (u/2) is the magnitude of the momentum transfer and

xðbÞ ¼

ð1

2

h k

b

r dr VðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 b 2

u ðkaÞ1=2 ;

ðC:45Þ

ðC:46Þ

where a is the range of the potential. The review article of Byron and Joachain (1977) contains a very detailed study of the eikonal approximation and its relationship with the Born series. Wallace (1971) derived corrections to the eikonal phase in a systematic way. Introducing the leading correction, we obtain the Wallace-improved eikonal phase,

In the so-called eikonal approximation (see, e.g., Sakurai, 1997), the DPWs in a central field V(r) are approximated in the form ðC:41Þ

where SH(r) is the Hamilton principal function that satisfies the Hamilton – Jacobi equation of classical mechanics, 1 ½rSH ðrÞ2 þ VðrÞ ¼ E: 2me

2me

is the eikonal phase. The eikonal approximation is expected to be accurate for small scattering angles such that

The eikonal approximation

ck ðrÞ ’ exp½iSH ðrÞ=h ;

ðC:44Þ

x1 ðbÞ ¼

2me

ð1

VðrÞ 2 k b h

me dVðrÞ r dr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : VðrÞ þ r 1þ2 2 dr h k r2 b2 ðC:47Þ

ðC:42Þ

For the calculation of scattering DCSs within the 136


d‘

1 1 ‘þ kr0 ¼ 2 2 ð1 þ ½k‘ ðrÞ k dr:


The Coulomb DPW has the asymptotic behavior,

The Wallace correction consists of a simple modification of the interaction potential that, although depending on the energy of the projectile, is numerically inexpensive. Byron and Joachain (1977) have shown that, for the class of potentials found in studies of elastic scattering of charged particles by atoms, the Wallace-improved eikonal phase, Eq. (C.47), is substantially more accurate than the uncorrected phase, Eq. (C.45). For the Wentzel potential, Eq. (C.1), the Wallace-improved eikonal phase is 2

ð1

ðC:52Þ

with the scattering amplitude f ðCÞ ðuÞ ¼ h

Gð1 þ ihÞ exp½ih lnðsin2 ðu=2ÞÞ ; Gð1 ihÞ 2k sin2 ðu=2Þ ðC:53Þ

where u is the scattering angle (cos u ¼ rˆ . kˆ). The long range of the Coulomb field causes the logarithmic terms in the phases of the incident and scattered waves as well as the corrections in square brackets. These corrections vanish only when the observation point is far from the incident beam axis, i.e., when r(1 2 cos u) is large. The radial Schro¨dinger equation (C.13) for the Coulomb field, Eq. (C.49), can be recast in a dimensionless form by introducing the variable x ¼ kr,

Scattering by a Coulomb field

! d2 2h lðl þ 1Þ Fl ðh; xÞ ¼ 0; þ1 dx2 x x2

The case of the Coulomb field Ze r

exp½ik r

h2 þ i½kr k r exp½ikr ih lnð2krÞ þ f ðCÞ ðuÞ r #) " ð1 þ ihÞ2 þ 1þ i½kr k r

where K0(x) is the modified Bessel function of the second kind and zeroth-order. Inserting this expression in Eq. (C.44), the scattering amplitude f (eik)(u) can be evaluated numerically by a single quadrature.

VC ðrÞ ¼

ð2pÞ

rð1cos uÞ!1

þ ih lnðkr k rÞ 1 þ

expðr=RÞ Ze 2 k h b me Ze2 dr 12 expðr=RÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 h k R r b2 2 2 me Ze K0 ð2b=RÞ ; ¼ Ze2 K0 ðb=RÞ 2 hv h k2 R ðC:48Þ

C.1.5

3=2

2

ðC:54Þ

ðC:49Þ with l ¼ ‘. In non-relativistic theory, the angular momentum quantum number ‘ can only take positive integer values. For the sake of generality, however, we consider that l can take any real value larger than 21. As shown below, the Dirac-Coulomb radial functions can be expressed in terms of Coulomb functions with non-integer l values. Equation (C.54) admits a regular solution Fl(h, x) and an irregular solution Gl(h, x) that can be defined by their behavior near the origin

plays a fundamental role in scattering theory. The Schro¨dinger equation (C.3) with this potential can be solved analytically (see, e.g., Joachain, 1983). Distorted plane-wave solutions (i.e., free states satisfying the asymptotic boundary conditions of a scattering experiment) have the form ðCÞ

ck ðrÞ ¼ ð2pÞ3=2 expðph=2ÞGð1 þ ihÞ expðik rÞ ðC:50Þ 1 F1 ðih; 1; i½kr k rÞ;

Fl ðh; xÞ Cl ðhÞxlþ1 ; r!0

where G is the complex gamma function, 1F1 is the confluent hypergeometric function (Abramowitz and Stegun, 1974) and

Gl ðh; xÞ xl =½ð2l þ 1ÞCl ðhÞ; r!0

ðC:55Þ

where Ze2 me

Ze2 h¼ 2 ¼ hv h k

ðC:51Þ Cl ðhÞ ¼ 2l expðhp=2Þ

is the Sommerfeld parameter. As before, k and v denote the wave number and the velocity of the particle at large distances from the center of force.

jGðl þ 1 þ ihÞj Gð2l þ 2Þ

ðC:56Þ

With this normalization constant, the Coulomb functions have the following asymptotic behavior 137


x1;W ðbÞ ¼

2me

( ðCÞ ck ðrÞ


C.2

for large x, p Fl ðh; xÞ sin x l h ln 2x þ Dl ; 2 p Gl ðh; xÞ cos x l h ln 2x þ Dl ; 2

RELATIVISTIC (DIRAC) THEORY

Let us now consider the relativistic theory of electron scattering by a central field V(r) of finite range (V(r) ¼ 0 for r . rc). We assume an incident beam of electrons with momentum h k and kinetic energy

ðC:57Þ

where the Coulomb phase shift Dl is given by Dl ¼ arg Gðl þ 1 þ ihÞ:

E¼

ðC:58Þ

xþ1=2 ¼

ðCÞ

¼ Eckm ðrÞ;

r!1

ðC:66Þ

where

ðC:60Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E þ 2me c2 cinc ðrÞ ¼ fkm ðrÞ ¼ expðik rÞ 3=2 2E þ 2me c2 ð2pÞ 0 1 I2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi A @ ðC:67Þ E ^ xm sk 2 E þ 2me c 1

is a plane wave that represents the incident beam; here I2 is the 2 2 unit matrix and s is the vector of Pauli spin matrices, Eq. (B.4). The spherical wave

ðC:61Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðikrÞ E þ 2me c2 csc ðrÞ ¼ r 2E þ 2me c2 ð2pÞ3=2 0 1 I2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi AF ðk^ r; kÞxm @ E s r^ 2 E þ 2me c

ds C h2 ¼ jf ðCÞ ðuÞj2 ¼ dV 4k2 sin4 ðu=2Þ Z2 e4 1 ; 4E2 ð1 cos uÞ2

ðC:65Þ

ckm ðrÞ cinc ðrÞ þ csc ðrÞ;

The DCS for a Coulomb field takes the form,

¼

ðC:64Þ

with the asymptotic behavior

1 X ð2‘ þ 1Þ½expð2iD‘ Þ 1 2ik ‘ P‘ ðcos uÞ:

0 : 1

~ 1Þme c2 þ VðrÞckm ðrÞ ~ r þ ðb a ½ich

where F‘(h, r) are regular Coulomb functions and D‘ are Coulomb phase shifts. Moreover, the Coulomb scattering amplitude admits the partialwave expansion, f ðCÞ ðuÞ ¼

x1=2 ¼

The scattered particles are counted by means of a detector placed farther than the potential range rc and effectively screened from the incident beam. The situation is described by a Dirac DPW, i.e., a solution of the Dirac wave equation (see Appendix B),

1X ð2‘ þ 1Þi‘ kr ‘

^ expð+iD‘ ÞF‘ ðh; rÞP‘ ð^ r kÞ;

and

ðC:59Þ

A variety of procedures for computing Schro¨dinger– Coulomb functions can be found, e.g., in the papers of Fro¨berg (1955) and Barnett (1981). The subroutine FCOUL included in the RADIAL code package (Salvat et al., 1995) delivers values of the Coulomb functions, their derivatives, and Coulomb phase shifts with a high relative accuracy ( 10214). It can be shown that the spherical-wave expansion (C.16) is also valid for the Coulomb DPWs. That is,

ck ðrÞ ¼ ð2pÞ3=2

1 0

1

ðC:62Þ

ðC:68Þ

which is the classical Rutherford DCS. It is interesting to note that, for the Coulomb field, the first Born approximation yields the exact DCS (but the Born scattering amplitude differs from f (C)).

represents the emerging flux of scattered particles. The scattering amplitude F(krˆ, k), which now is a 138


1 nl ðxÞ ¼ Gl ð0; xÞ: x

ðC:63Þ

in a pure spin state represented by the spinor xm(m ¼ + 1/2),

In the limit h ! 0, the Coulomb functions reduce to spherical Bessel functions, jl(x), and spherical Neumann functions, nl(x), 1 jl ðxÞ ¼ Fl ð0; xÞ; x

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kÞ2 þ ðme c2 Þ2 me c2 ðch


2 2 matrix independent of r, completely characterizes the scattering of electrons by the field V(r). The normalization constants in Eqs. (C.66)–(C.68) are such that the DPWs satisfy the orthogonality relation ð

cyk0 m0 ðrÞckm ðrÞ dr ¼ dðk k0 Þdmm0 :

unit amplitude, p PEk ðrÞ sin kr ‘ þ dk : 2

With this normalization (see, e.g., Salvat et al., 1995),

ðC:69Þ

ð

If the electrons in the incident beam are in an arbitrary pure spin state xi, uþ1=2 u1=2

¼

X

dðk0 kÞdk0 ;k dm0 ;m :

ðC:70Þ

In the limit V ¼ 0, the radial functions become

2

juþ1=2 j þ ju1=2 j ¼ 1;

P0Ek ðrÞ ¼ krjk ðkrÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 0 krjk1 ðkrÞ; QEk ðrÞ ¼ E þ 2me c2

the associated DPW can be expressed as

cki ðrÞ ¼

X

um ckm ðrÞ

ðC:71Þ

if k . 0;

P0Ek ðrÞ ¼ krjk1 ðkrÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E Q0Ek ðrÞ ¼ krjk ðkrÞ; if k , 0; E þ 2me c2

m¼+1=2

and its asymptotic form is still given by Eqs. (C.67) and (C.68) with xm replaced by xi. It is important to note that, at large distances r, the scattered wave has the form of a plane wave with wave vector k0 ¼ krˆ pointing along the radial direction rˆ. Since the upper spinor component of a plane wave determines the lower one [see Eq. (B.29)], all observables in a scattering experiment can be calculated in terms of the initial spin state xi.

C.2.1

pðE þ me c2 Þ E þ 2me c2 ðC:75Þ

um xm ;

m¼+1=2

2

cyE0 k0 m0 ðrÞcEkm ðrÞ dr ¼

ðC:76Þ where j‘ (x) are spherical Bessel functions. The DPW cki (r) can be expressed as (see, e.g., Rose, 1961, p. 207) rffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 E þ 2me c2 X ‘ i expðidk Þ p 2E þ 2me c2 k;m

h iy ^ x c Vkm ðkÞ Ekm ðrÞ: i

1 cki ðrÞ ¼ k

ðC:77Þ

Partial-wave expansion It is worth noting that, when k is parallel to the z-axis, this expansion of the DPW reduces to the form used by Mott (1929), who obtained it by direct superposition of the free spherical waves introduced by Darwin (1928). With the aid of the asymptotic form (C.74), it can be shown after some tedious manipulations that the scattering-amplitude matrix F(k0 , k) is given by

As in the non-relativistic case, it is convenient to expand the DPW in the basis of spherical waves (see Section B.2), 1 cEkm ðrÞ ¼ r

PEk ðrÞVk;m ð^ rÞ ; ^Þ iQEk ðrÞVk;m ðr

ðC:72Þ

where Vk,m(rˆ) are the spherical spinors and the radial functions PEk(r) and QEk(r) satisfy the coupled system of differential equations, dPEk k E V þ 2me c2 ¼ P Ek þ QEk ; dr r ch dQEk EV k ¼ PEk þ QEk : dr r ch

^ Þ; F ðk0 ; kÞ ¼ f ðuÞ þ igðuÞðs n

ðC:78Þ

where u ; arccos(k0 . kˆ) is the polar scattering angle, ^¼ n

ðC:73Þ

^0 k ^ k ^ ^ 0 kj jk

ðC:79Þ

is a unit vector perpendicular to the ‘scattering plane’ (i.e., to the plane defined by the initial and

We shall normalize the spherical waves so that the radial function PEk(r) oscillates asymptotically with 139


xi ¼

ðC:74Þ


final directions, kˆ and k0 ) and f ðu Þ ¼

time, per unit solid angle, and per unit incident flux, is

1 X fð‘ þ 1Þ½expð2idk¼‘1 Þ 1 2ik ‘ þ ‘½expð2idk¼‘ Þ 1gP‘ ðcos uÞ;

^ 0 Þr2 ds ðj k ðxi Þ ; sc ¼ kxi jF y ðk0 ; kÞF ðk0 ; kÞjxi l dV jjin j

ðC:80Þ

¼ kxi jjf ðuÞj2 þ jgðuÞj2 i½f ðuÞg ðuÞ ^ Þjxi l f ðuÞgðuÞðs n

1 X gðuÞ ¼ ½expð2idk¼‘ Þ 2ik ‘

expð2idk¼‘1 ÞP1‘ ðcos

uÞ:

^ ; ðC:86Þ ¼ ðjf ðuÞj2 þ jgðuÞj2 Þ½1 Sp ðuÞP i n

ðC:81Þ

^ 0 ¼ ðsin u cos f; sin u sin f; cos uÞ k ^ ¼ ðsin f; cos f; 0Þ; n

Sp ðuÞ ¼ i

zÞ ¼ f ðuÞ þ gðuÞ F ðk ; k^ ¼

0

eif

eif

0

f ðuÞ

eif gðuÞ

eif gðuÞ

f ðu Þ

!

! :

ðC:83Þ

uþ1=2 u1=2 eif u1=2 uþ1=2 eif h ¼ ðjf ðuÞj2 þ jgðuÞj2 Þ 1 iSp ðuÞ uþ1=2 u1=2 i ðC:88Þ eif u1=2 uþ1=2 eif :

Recalling that (s . kˆ)(s . kˆ) ¼ I2, the current density associated with the incident wave can be expressed as v ð2pÞ

3

xyi xi ¼

v ð2pÞ3

ðC:87Þ

ds ðx Þ ¼ jf ðuÞj2 þ jgðuÞj2 þ ½f ðuÞg ðuÞ f ðuÞgðuÞ dV i h i

Differential cross-section

jin ¼ cyinc cinc v ¼

:

It is interesting to observe that the DCS exhibits a ˆ] right-left asymmetry [see the definition (C.79) of n that is caused by the fact that the polarization sets a preferred direction in space. The asymmetry disappears when the beam polarization is parallel or antiparallel to the direction of incidence kˆ (since ˆ ¼ 0 in this case). Pi . n The angular and spin dependences of the scattering can be made more explicit by taking the direction of incidence parallel to the spin quantification (z-) axis. In this case, the scattering amplitude takes the form given by Eq. (C.83) and the DCS, Eq. (C.86), becomes

ðC:82Þ

The functions f(u) and g(u) are called the ‘direct’ and ‘spin-flip’ scattering amplitudes. C.2.2

jf ðuÞj2 þ jgðuÞj2

and

where u and f are the polar and azimuthal angles of the ‘final’ direction k0 , respectively. From Eq. (C.78), we obtain 0

f ðuÞg ðuÞ f ðuÞgðuÞ

;

ðC:84Þ

We see that when the two components u þ1/2 and u 21/2 of the initial spin state xi are different from zero (i.e., when the polarization Pi has a nonvanishing projection on the xy-plane), the DCS depends on the azimuthal scattering angle. If one of these components vanishes, the beam polarization is parallel or antiparallel to the quantification axis, and the DCS is axially symmetric. The DCS can also be expressed as

k/(E þ mec 2) is the velocity of the where v ¼ c 2h electrons in the beam. Similarly, the current density of scattered electrons at the detector entrance is

^ 0 Þ ¼ cy ðrk ^ 0 Þc ðrk ^ 0 Þvk ^0 jsc ðrk sc sc v 1 y y 0 ^ 0: ¼ xi F ðk ; kÞF ðk0 ; kÞxi k ð2pÞ3 r2

X ds ðx i Þ ¼ kxi jF y ðk0 ; kÞjxm l kxm jF ðk0 ; kÞjxi l dV m¼+1=2 X jkxm jF ðk0 ; kÞjxi lj2 : ðC:89Þ ¼

ðC:85Þ The DCS, which is defined as the number of scattered particles that enter the detector per unit

m

140


where Pi ¼ kxijsjxil is the polarization of the incident beam and we have introduced the asymmetry function, or the Sherman function,

Here P‘(cos u) and P1‘ (cos u) are Legendre polynomials and associated Legendre functions, respectively. F(k0 , k) can be expressed in a more explicit form by adopting a reference frame with the z-axis along the initial direction, zˆ ¼ kˆ. Then,


amplitudes. In the special case of an unpolarized beam (Pi ¼ 0), we have

The two terms, dsðupÞ ðxi Þ ¼ kxþ1=2 jF ðk0 ; kÞxi lj2 dV 2 ¼ uþ1=2 f ðuÞ u1=2 eif gðuÞ

P f ¼ Sp ðuÞ^ n:

¼ juþ1=2 f ðuÞj2 þ ju1=2 gðuÞj2 2 Re uþ1=2 f ðuÞu1=2 eif gðuÞ ðC:90Þ

2 dsðdownÞ ðxi Þ ¼ kx1=2 F ðk0 ; kÞxi l dV 2 ¼ uþ1=2 eif gðuÞ þ u1=2 f ðuÞ ¼ juþ1=2 gðuÞj2 þ ju1=2 f ðuÞj2 þ 2Re uþ1=2 eif g ðuÞu1=2 f ðuÞ ; ðC:91Þ represent the DCSs for scattering to spin-up and spin-down states, respectively. Notice that the interference terms vanish when u þ1/2 ¼ 0 or u 21/2 ¼ 0, i.e., when xi is an eigenstate of s3 or, equivalently, when the polarization of the incident beam is along the z-axis. In this case, the DCSs for direct and spin-flip scattering are equal to jf(u)j2 and jg(u)j2, respectively. The DCS for an incoherent beam consisting of various pure states xi with weights wi is given by

C.2.3

As in the non-relativistic formulation, the scattering wave function satisfies the LippmannSchwinger equation,

cki ðrÞ ¼ fkiþ ðrÞ ð þ G0 ðW; r; r0 ÞVðr0 Þcki ðr0 Þ dr0 ;

where fkiþ (r) is a positive energy plane wave [see Eq. (B.23)]. The Green function G0 (W,r, r0 ) is a 4 4 object that depends on the total energy W ¼ E þ mec 2. This function can be expressed as (see, e.g., Strange, 1998) G0 ðW; r; r0 Þ ¼

ðC:93Þ

~ me c2 expðikjr r0 jÞ W þ c~ apþb : 2 jr r0 j 4pc2 h ðC:97Þ

From the foregoing discussion, it is clear that Fxi represents the spin state of the scattered wave. Therefore, the polarization of the scattered electrons is given by kx jF y sF jxi l Pf ¼ i kxi jF y F jxi l

ðC:96Þ

ðC:92Þ

The DCS for an unpolarized beam, represented as a compensated mixture of states x þ1/2 and x 21/2, is dsunpol 1 ds 1 ds ¼ ðxþ1=2 Þ þ ðx Þ dV 2 dV 2 dV 1=2 ¼ jf ðuÞj2 þ jgðuÞj2 :

Born approximation

When jr 2 r0 j is large, ~ me c2 expðikrÞ W þ ch k~ a r^ þ b 2 r!1 r 4pc2 h 0 ^ r Þ: ðC:98Þ expðikr

G0 ðW; r; r0 Þ

ðC:94Þ

and depends on the polarization Pi of the incident beam and on the angular deflection u, f. With some spin algebra, we can obtain a general, rather involved expression for Pf in terms of the scattering

The wave function in the first Born approximation can be obtained by replacing cki (r0 ) on the right-hand side of Eq. (C.96) by the undistorted 141


That is, the Sherman function gives the degree of polarization of scattered electrons from unpolarized beams. For certain combinations of target, incident energy and scattering angle, the absolute value of Sp(u) is close to unity; under these circumstances, elastic scattering produces highly polarized electron beams (although their intensity is relatively small). The Sherman function can be experimentally determined by means of double-scattering experiments. A detailed analysis of electron polarization phenomena and double-scattering experiments can be found in the book of Kessler (1976); see also Kessler (1969). The textbook of Landau (1990) contains a simple derivation of the most relevant formulas.

and

ds X ds ¼ ðx Þ: wi dV dV i i

ðC:95Þ


plane wave fkiþ (r0 ),

PEk(r) and QEk(r) by those of the free electron, P0Ek(r) and Q0Ek(r), and substitute dk(DB) for sin dk, to obtain

ðB1Þ

cki ðrÞ ¼ fkiþ ðrÞ ð þ G0 ðW; r; r0 ÞVðr0 Þfkiþ ðr0 Þ dr0 :

dðDBÞ ¼ k ðC:99Þ

E þ 2me c2 2 c2 h k

ð1 0

f½P0Ek ðrÞ2

þ ½Q0Ek ðrÞ2 gVðrÞr dr:

ðC:105Þ

(B1) cki (r),

we To analyze the asymptotic behavior of can use the form (C.98) of the Green function. After some tedious manipulations, we find that

ðDBÞ

g þ 1 ðSBÞ g 1 ðSBÞ d‘ þ d‘1 ; 2 2 g þ 1 ðSBÞ g 1 ðSBÞ ¼ d‘ þ d‘þ1 : 2 2

dk¼‘ ¼ ðC:100Þ

ðDBÞ

dk¼‘1

These formulas were obtained by Parzen (1950) by a different method, which makes use of the fact that the Born scattering amplitudes, Eqs. (C.102) and (C.103), can be expressed as

The DB scattering-amplitude matrix is given by ^ Þ; F ðDBÞ ðk0 ; kÞ ¼ f ðDBÞ ðuÞI 2 igðDBÞ ðuÞðs n ðC:101Þ

f ðDBÞ ðuÞ ¼

with f ðDBÞ ðuÞ ¼

gðDBÞ ðuÞ ¼

gþ1 g1 þ cos u f ðSBÞ ðuÞ; 2 2

g1 sin u f ðSBÞ ðuÞ; 2

E þ 2me c2

ðC:107Þ gðDBÞ ðuÞ ¼

ðC:103Þ

1 h i 1X ðDBÞ ðDBÞ dk¼‘ dk¼‘1 P1‘ ðcos uÞ; ðC:108Þ k ‘¼0

given by Eqs. (C.106). The DB DCS for with d(DB) k unpolarized electrons is dsðDBÞ ¼ jf ðDBÞ ðuÞj2 þ jgðDBÞ ðuÞj2 dV 1 b2 sin2 ðu=2Þ ðSBÞ 2 ¼ f ðuÞ ; 1 b2

ðC:109Þ

where b ¼ v/c. For the Wentzel field, Eq. (C.1), the DCS given by Eq. (C.109) takes the form

ð1

½P0Ek ðrÞPEk ðrÞ 2 c2 h k 0 þ Q0Ek ðrÞQEk ðrÞVðrÞr dr;

1 h i 1X ðDBÞ ðDBÞ ð‘ þ 1Þdk¼‘1 þ ‘dk¼‘ P‘ ðcos uÞ; k ‘¼0

ðC:102Þ

where f (SB)(u) is the non-relativistic Born scattering amplitude and g ¼ 1 þ E/(mec 2) is the total energy of the projectile in units of its rest mass. Since the Born scattering amplitudes f (SB)(u) and g (SB)(u) are real, the Sherman function Sp(u), Eq. (C.87), vanishes. As a consequence, the first Born approximation does not describe spin-polarization effects. Transforming the differential equations (C.73), with the boundary condition (C.74) into an integral equation (see, e.g., Rose, 1961, pp. 244 – 245), one finds that sin dk ¼

ðC:106Þ

ðDBÞ

ds W dV

ðC:104Þ

where P0Ek(r) and Q0Ek(r) are the radial functions for V ¼ 0, Eq. (C.76). This relation is exact and reduces to the non-relativistic result Eq. (C.22) when c ! 1. Eq. (C.104) can be used to derive a formula for the relativistic phase shifts in the Born approximation. Proceeding by strict analogy with the nonrelativistic case, we replace the radial functions

2 1 b2 sin2 ðu=2Þ 2Ze2 me ¼ 2 1 b2 h 1 ; 2 ½ð1=RÞ þ 2k2 ð1 cos uÞ2

ðC:110Þ

which is known as the ‘screened Rutherford’ DCS. This simple formula is useful for understanding the qualitative dependence of the DCS on the atomic number and the projectile energy. In the limit R ! 1, expression (C.110) reduces to the DB 142


expðikrÞ ðB1Þ cki ðrÞ fkiþ ðrÞ þ ð2pÞ3=2 r!1 r 0 1 I2 ^ 0 AF ðDBÞ ðk0 ; kÞxi : @ ch ks k W þ m e c2

Inserting the expressions (C.76) and recalling the definition (C.29) of the non-relativistic Born phase shifts, we have


with h(r) ; [E 2 V(r) þ 2mec 2]/(2mec 2), and eliminating the small-component radial function QEk(r). The resulting equation is

DCS for the Coulomb field,

ðDBÞ

ds C dV

¼

Ze2 m e v2

2

1 b2 ð1 cos uÞ2

! d2 ‘ð‘ þ 1Þ 2m e e k ðrÞ PðrÞ þ k2 2 V dr2 r2 h

ðC:111Þ

½1 b2 sin2 ðu=2Þ:

¼ 0;

The DB phase shifts for the Wentzel field are given by [see Eqs. (C.33) and (C.106)] "

where the effective Dirac potential,

!

Z gþ1 1 Q‘ 1 þ ka0 2 2ðkRÞ2 !# g1 1 Q‘1 1 þ ; þ 2 2ðkRÞ2 " ! Z gþ1 1 Q‘ 1 þ ¼ ka0 2 2ðkRÞ2 !# g1 1 Q‘þ1 1 þ : þ 2 2ðkRÞ2

V 2 ðrÞ ~ k ðrÞ ; VðrÞ 1 þ E V 2 2me c m e c2 " # 2 k h0 3 h0 2 1 h00 h þ þ 2me r h 4 h 2h

dW;k¼‘ ¼

ðDBÞ

dW;k¼‘1

depends on the energy and the relativistic quantum number k. For large r values, h becomes a constant, i.e., P becomes proportional to PEk, and therefore, the phase shifts may be computed by solving the Schro¨dinger Equation (C.115) as in the non-relativistic case. In particular, the WKB approximation with the Langer correction yields

ðC:112Þ

From the property X ‘ 1 2 1 Q‘ ð1 þ xÞ ¼ ln 1 þ þ Oð‘2 xÞ; 2 x m m¼1

dðWKBÞ k

it follows that, in the high energy limit ( g 1, kR1), ðDBÞ

ðC:117Þ

where "

ð‘ þ 1=2Þ2 2me ~ 2 V k ðrÞ kk ðrÞ ; k r2 h

ðC:113Þ

2

Dalitz (1951) derived an analytical formula for the DCS of the Wentzel potential, Eq. (C.1), in the second-order DB approximation. Gorshkov (1961, 1962) applied the same approximation to a more general analytical potential, expressed as a linear superposition of Wentzel fields; in the case of a single Wentzel potential, the Gorshkov formula reduces to Dalitz’s result (see also Motz et al., 1964).

#1=2 ðC:118Þ

is the local wave-number and r0 is the largest positive zero of kk(r). Tables C.2 and C.3 display Dirac phase shifts for scattering of 500 keV electrons by attractive and repulsive Wentzel potentials, Eq. (C.1), respectively, that were calculated from the DB and WKB approximations and using the DPWA subroutine package (‘numerical’) described in Section 4.2. It is seen that, at this energy, the WKB approximation gives excellent results for all phase shifts except the ones with low jkj values. It may be noted that the WKB formula approximates the ‘true’ value of dk, whereas the numerical ‘exact’ values are indeterminate in a multiple of p. For large jkj, the phase shifts have small absolute values and are well reproduced by the Born approximation, even when the Born approximation for the DCS is not very accurate.

C.2.4 WKB approximation for the phase shifts The radial Dirac Eq. (C.73) may be reduced to Schro¨dinger form by introducing the substitution (Mott and Massey, 1965; Walker, 1971) PEk ðrÞ ¼ h1=2 ðrÞPðrÞ;

1 1 ‘þ ¼ p kr0 2 2 ð1 þ ½kk ðrÞ k dr; r0

jX kj1

Zg 1 1 lnð2kRÞ ka0 2jkj m m¼1 ! 2 k : þO 2ðkRÞ2

dW;k ¼

ðC:116Þ

ðC:114Þ 143


ðDBÞ

ðC:115Þ

ELASTIC SCATTERING OF ELECTRONS AND POSITRONS Table C.2. Dirac phase shifts for scattering of 500 keV electrons by an attractive Wentzel potential, Eq. (C.1), with Z 5 280 and R 5 0.2a0 (which approximately corresponds to the field of Hg atoms).

d 2 jkj

d þ jkj

Numerical

WKB

Born

Numerical

WKB

Born

1 2 3 4 5 10 15 20 30 40 50 75 100 150 200 250 300 350 400 450 500 600 700

23.016Eþ0 2.464Eþ0 2.099Eþ0 1.862Eþ0 1.688Eþ0 1.183Eþ0 9.117E21 7.313E21 5.000E21 3.572E21 2.617E21 1.279E21 6.569E22 1.867E22 5.594E23 1.727E23 5.432E24 1.731E24 5.573E25 1.807E25 5.896E26 6.361E27 6.954E28






C.2.5

the projectile. The relativistic eikonal scattering amplitude can then be obtained directly from the corresponding non-relativistic formulas (C.44) and (C.45). We have

The eikonal approximation

Molie`re (1948) was the first to use the relativistic eikonal approximation to compute DCSs for scattering of fast projectiles by atoms. To account for relativistic kinematic effects, he considered the Klein–Gordon (or relativistic Schro¨dinger) equation for the potential V(r) (see, e.g., Schiff, 1968), which describes particles without spin. In the case of free states, the Klein– Gordon equation can be stated in the form (

2 1 h r2 þ VðrÞ 1 VðrÞ 2gme 2gme c2 ) ðh kÞ2 c ðrÞ ¼ 0; 2gme k

ð ibgme c 1 J0 ðqbÞ h 0 fexp½ixðbÞ 1gb db:

f ðeikÞ ðuÞ ¼

q ¼ 2bgmec sin(u/2) and with h

ð 2 1 xðbÞ ¼ VðrÞ h bc b

1 dVðrÞ r dr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1þ 2 VðrÞ þ r 2 dr b gme c r2 b2 ðC:122Þ

ðC:119Þ

k ¼ bgmec is the momentum of the projectile. where h This equation has the same form as the nonrelativistic Schro¨dinger equation (C.3) with the relativistic mass gme instead of the electron mass, and with the effective potential

VKG ðrÞ ¼ VðrÞ 1

1 VðrÞ : 2gme c2

ðC:121Þ

This approximation, without the Wallace correction (the factor in braces), was used by Molie`re (1948) and Zeitler and Olsen (1964) to study screening effects in elastic scattering of fast electrons by atoms. Figure C.2 shows DCSs for scattering of electrons with the indicated kinetic energies by a Wentzel potential, Eq. (C.1), with Z ¼ 2 80 and R ¼ 0.2a0 (which approximates the field of Hg atoms). The displayed curves were obtained by using the Dirac partial-wave expansion, which provides nominally exact results, the DB approximation, and the

ðC:120Þ

Because we are considering only small scattering angles (i.e., large impact parameters), we can assume that V(r) gmec2 along the classical trajectories of 144


jkj

QUANTUM THEORY OF SCATTERING BY A CENTRAL POTENTIAL Table C.3. Dirac phase shifts for scattering of 500 keV positrons by a repulsive Wentzel potential, Eq. (C.1), with Z 5 1 80 and R 5 0.2a0 (which approximately corresponds to the potential of Hg atoms).

d 2 jkj

d þ jkj

Numerical

WKB

Born

Numerical

WKB

Born

1 2 3 4 5 10 15 20 30 40 50 75 100 150 200 250 300 350 400 450 500 600 700







with Coulomb spherical waves

eikonal approximation. The close agreement between the eikonal and partial-wave results at small angles (up to 308) is noteworthy. It is seen that the eikonal approximation yields results that are much more accurate than those from the DB approximation for all energies. The same is true for more realistic atomic fields. As expected, the accuracy of the Born and eikonal approximations improves when the kinetic energy increases and/or the strength of the potential decreases. C.2.6

ðCÞ cEkm ðrÞ

ðuÞ

fEk ðrÞ Vk;m ð^ rÞ ðlÞ

rÞ ifEk ðrÞ Vk;m ð^

! :

ðC:124Þ

The radial Dirac-Coulomb functions f(u,l) Ek (r) satisfy the equations

Scattering by a Coulomb field

The scattering of Dirac particles by a pure Coulomb field VC(r), Eq. (C.49), was first studied by Mott (1929) and is occasionally referred to as Mott scattering (not to be confused with the scattering of identical particles, which is known with the same name). The Dirac wave equation (C.65) for VC(r) cannot be solved analytically, but the Coulomb DPW can still be represented in the form (C.77), ðCÞ cki ðrÞ

1 ¼ r

ðuÞ

dfEk k ðuÞ E VC þ 2me c2 ðlÞ ¼ fE k þ fEk ; dr r ch ðlÞ

dfEk E VC ðuÞ k ðlÞ ¼ fE k þ fE k : dr ch r

rffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 E þ 2me c2 p 2E þ 2me c2

h iy X ‘ ðCÞ ^ i expðidk Þ Vkm ðkÞ xi cEkm ðrÞ

1 ¼ k

ðC:125Þ

The spherical waves in the expansion (C.123) correspond to the regular solution at r ¼ 0, for which f(u,l) Ek (0) = 0. For the numerical calculation of phase shifts, it is useful to consider also an independent solution of the radial equations that is irregular at r=0, and which will be denoted by g(u,l) Ek (r).

k;m

ðC:123Þ 145


jkj


Coulomb functions, Eq. (C.55),

The regular and irregular radial Dirac-Coulomb functions are given by Salvat et al. (1995)

Ze2 x ¼ kr; h ¼ ; hv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ k2 z2 ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 þ h2 kh c Fl ðh; xÞ þzðlme c2 kWÞ Fl1 ðh; xÞ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðlÞ c Fl ðh; xÞ fEk ðrÞ ¼ N z l2 þ h2 kh

ðuÞ fEk ðrÞ

¼ N ðk þ lÞ

z¼

Ze2 Z

; c 137 h ðC:127Þ

and

þðk þ lÞðlme c2 kWÞ Fl1 ðh; xÞ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðuÞ c Gl ðh; xÞ gEk ðrÞ ¼ N ðk þ lÞ l2 þ h2 kh þzðlme c2 kWÞ Gl1 ðh; xÞ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðlÞ c Gl ðh; xÞ gEk ðrÞ ¼ N z l2 þ h2 kh þðk þ lÞðlme c2 kWÞ Gl1 ðh; xÞ ;

N¼

i1=2 ð1ÞSðz;kÞ h 2 z ðE þ 2me c2 Þ2 þ ðk þ lÞ2 ðkh cÞ2 ; l ðC:128Þ

with S(z,k) ¼ 1 if z , 0 and k , 0, and S(z,k) = 0 otherwise. Asymptotically, the upper-component radial functions behave as p ðuÞ fEk ðrÞ sin kr ‘ h ln 2x þ Dk ; 2 p ðuÞ gEk ðrÞ cos kr ‘ h ln 2x þ Dk ; 2

ðC:126Þ

where Fl(h,x) and Gl(h,x) are the non-relativistic 146

ðC:129Þ


Figure C.2. Differential Cross-sections for scattering of electrons with the indicated kinetic energies by a Wentzel potential, Eq. (C.1), with Z ¼ 280 and R ¼ 0.2a0. The curves were obtained using the numerical partial-wave expansion method (solid line), the eikonal approximation (dashed line), and the Born approximation (dot-dashed line).


functions,

where

ðuÞ

gEk ðrÞ ¼ kr nk ðkrÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ðlÞ kr nk1 ðkrÞ; gEk ðrÞ ¼ E þ 2me c2

k Dk ; arg½zðE þ 2me c2 Þ iðk þ lÞch p ðl ‘ 1Þ þ arg Gðl þ ihÞ Sðz; kÞp 2 ðC:130Þ

ðuÞ

gEk ðrÞ ¼ kr nk1 ðkrÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ðlÞ kr nk ðkrÞ; gEk ðrÞ ¼ E þ 2me c2

ðC:134Þ The asymptotic form of Coulomb DPWs differs from the expressions (C.66)–(C.68) by a logarithmic phase factor in the incident and scattered wave, but the scattering-amplitude matrix keeps the form (C.78) with the following direct and spinflip amplitudes,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E p ðlÞ h ln 2x þ D ; cos kr ‘ fEk ðrÞ k r!1 2 E þ 2me c2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E p ðlÞ h ln 2x þ D sin kr ‘ gEk ðrÞ k : r!1 2 E þ 2me c2 ðC:131Þ

n:r:

ðuÞ fEk ðxÞ ! F‘ ðh; xÞ;

f ðCÞ ðuÞ ¼

convention, in the nonE mec 2, l ! jkj), the and g(l) Ek vanish and the to the non-relativistic

n:r:

1 1 X fð‘ þ 1Þ½expð2iD‘1 Þ 1 2ik ‘¼0

þ ‘½expð2iD‘ Þ 1gP‘ ðcos uÞ;

gðCÞ ðuÞ ¼

ðC:135Þ

1 1 X fexpð2iD‘ Þ 2ik ‘¼0

expð2iD‘1 ÞgP1‘ ðcos uÞ:

ðuÞ gEk ðxÞ ! G‘ ðh; xÞ;

if k , 0:

ðC:136Þ

ðC:132Þ The DCS for Mott scattering is

where ‘ stands for the orbital angular momentum quantum number (‘ ¼ k if k . 0 and ‘ ¼ 2 k 2 1 if k , 0). In the limit of zero field strength (z ! 0, l ¼ jkj), the Dirac-Coulomb phase shifts vanish. With the aid of the identities (C.59), we obtain the familiar result that the regular radial Dirac functions of a free particle are the spherical Bessel function (see, e.g., Rose, 1961, p. 161).

ds M ¼ jf ðCÞ ðuÞj2 þ jgðCÞ ðuÞj2 : dV

Although Dirac-Coulomb phase shifts are known analytically, it is not possible to give closed analytical formulas for the relativistic scattering amplitudes, and we must rely on their partial-wave expansions (C.135) and (C.136). Algorithms to sum these series have been described by a number of authors (e.g., Sherman, 1956; Walker, 1971). For angles larger than 18, the Coulomb partial-wave series can be summed directly with the aid of the reduced-series method of Yennie et al. (1954) (Section 4.2.2); for smaller angles, these series converge very slowly (they do diverge for u ¼ 0) and the numerical summation of the partial-wave series must be abandoned. It is convenient to express the Mott DCS in the form

ðuÞ

fEk ðrÞ ¼ kr jk ðkrÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ðlÞ kr jk1 ðkrÞ; fEk ðrÞ ¼ E þ 2me c2

if k . 0;

ðuÞ

fEk ðrÞ ¼ kr jk1 ðkrÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ðlÞ kr jk ðkrÞ; fEk ðrÞ ¼ E þ 2me c2

ðC:137Þ

if k , 0; ðC:133Þ

ds M ds R ¼ RMR ðuÞ; dV dV

and the irregular functions are spherical Neumann 147

ðC:138Þ


is the Dirac-Coulomb phase-shift. The asymptotic behavior of the lower-component functions can be obtained from the first of the Dirac equations (C.125), which gives

With the adopted sign relativistic limit (c ! 1, small radial functions f(l) Ek large functions reduce Coulomb functions

if k . 0;


To determine the phase shifts dk, the radial functions PEk(r) and QEk(r) must be obtained by solving the radial equations (C.73) numerically. The numerical solution is started at r ¼ 0 and extended outwards up to a certain distance rm beyond the range rc of Vsr(r) (rm . rc) by some suitable integration method. This procedure yields unnormalized radial functions P(r) and Q(r) that differ from the true (normalized) radial functions by a constant A, i.e.,

where ds R ¼ dV

Ze2 m e v2

2

1 b2

ðC:139Þ

ð1 cos uÞ2

RMR ðuÞ ¼ 1

Gð1=2 izÞ Gð1 þ izÞ þ pb2 zRe Gð1=2 þ izÞ Gð1 izÞ

PEk ðrÞ ¼ APðrÞ;

ðC:142Þ

For r . rm, the field is purely Coulombian, and the normalized radial Dirac functions can be expressed as ðuÞ ðuÞ PEk ðrÞ ¼ cos d^k fEk ðrÞ þ sin d^k gEk ðrÞ; ðlÞ ðlÞ QEk ðrÞ ¼ cos d^k fEk ðrÞ þ sin d^k gEk ðrÞ;

ðC:143Þ

(u,l) where f(u,l) Ek (r) and gEk (r) stand for the regular and irregular Dirac-Coulomb functions for Z ¼ Z1. From the asymptotic behavior of the Dirac-Coulomb functions, Eq. (C.129), it follows that

sinðu=2Þ: ðC:140Þ

p PEk ðrÞ sin kr ‘ h ln 2kr þ dk ; r!1 2 ^ dk ¼ Dk þ dk : ðC:144Þ

This is all we need to calculate the DCS at small angles. We see that the DCS diverges at small angles in the same way as the relativistic Rutherford formula. For polarization studies, we may also need to know the asymmetry function S(u); at small angles, and for sufficiently high energies, this function takes very small values and can be set to zero.

C.2.7

QEk ðrÞ ¼ AQðrÞ:

That is, PEk(r) asymptotically oscillates with unit amplitude and, therefore, the ‘outer’ solution (C.143) is adequately normalized. To determine the ‘inner’ normalization constant A and the phase-shift dˆk, the inner and outer solutions are matched by requiring continuity of PEk(r) and QEk(r) at rm,

Modified Coulomb fields ðuÞ ðuÞ APðrm Þ ¼ cos d^k fEk ðrm Þ þ sin d^k gEk ðrm Þ;

Let us finally consider the relativistic theory for electron scattering by a modified Coulomb field of the form Z1 e2 þ Vsr ðrÞ; VðrÞ ¼ r

ðlÞ ðlÞ AQðrm Þ ¼ cos d^k fEk ðrm Þ þ sin d^k gEk ðrm Þ:

ðC:145Þ

These two equations give A and tandˆk. The value of dˆk is then indeterminate in a multiple of p, which has no effect on the scattering amplitudes. We can always reduce the calculated values of dˆk to the interval (2 p/2, þ p/2) so as to get rid of this indeterminacy when jdˆkj , p/2.

ðC:141Þ

where Vsr(r) is a short-range potential that vanishes for r . rc; beyond rc the potential is Coulombian [see Eq. (C.49)] with Z ¼ Z1. As for the case of a pure Coulomb field, the asymptotic form of the DPW ckm of this potential differs from the expression (C.66) by a logarithmic phase factor in the incident and scattered waves. These DPWs can be expanded in the form (C.77), and the scattering amplitude matrix has the usual form (C.78) with the ‘direct’ and ‘spin-flip’ scattering amplitudes given by Eqs. (C.80) and (C.81).

Notice that the ‘inner’ phase shift dˆk is only due to the short-range distortion Vsr(r); the effect of the Coulomb field is accounted for by the logarithmic phase, 2 h ln 2kr, and the Coulomb phase shift Dk. For a pure Coulomb field (Vsr ; 0), dˆk ¼ 0. As in the non-relativistic case, attractive (repulsive) shortrange fields give positive (negative) inner phase shifts. 148


is the the so-called relativistic Rutherford DCS. This DCS results from applying the first Born approximation to the Klein-Gordon equation, which describes a spinless electron [notice that it differs from the DB DCS for the Coulomb field, Eq. (C.111)]. The factor RMR(u), which varies smoothly with u and approaches unity when u ! 0, accounts for the effect of spin. For small scattering angles, where the convergence of the partial-wave series (C.135) and (C.136) is too slow, we can use the following asymptotic formula attributed to Bartlett and Watson (Motz et al., 1964),


f ðuÞ ¼ fsr ðuÞ þ f ðCÞ ðuÞ; gðuÞ ¼ gsr ðuÞ þ gðCÞ ðuÞ;

ðC:146Þ

where f (C)(u) and g (C)(u) are the scattering amplitudes for the Coulomb field with Z ¼ Z1, and fsr(u) and gsr(u) are the contributions of the short-range component of the field, 1 n 1 X ð‘þ1Þexpð2iD‘1 Þ½expð2i^d‘1 Þ1 2ik ‘¼0 o þ‘expð2iD‘ Þ½expð2i^d‘ Þ1 P‘ ðcos uÞ;

fsr ðuÞ ¼

ðC:147Þ 1 n X

1 expð2iD‘ Þ½expð2i^d‘ Þ1 2ik ‘¼0 o expð2iD‘1 Þ½expð2i^d‘1 Þ1 P1‘ ðcos uÞ:

gsr ðuÞ ¼

ðC:148Þ

149


These last series converge as rapidly as the partialwave series for the short-range field alone. As a consequence, the number of inner phase shifts to be calculated is normally much less than the number of Coulomb phase shifts required for the convergence of the partial-wave series for fC(u) and gC(u), Eqs. (C.135) and (C.136). In practice, when Z1 = 0, the small-angle scattering (which corresponds to large impact parameters in a semiclassical picture) is determined by the Coulomb tail and, as a consequence, the DCS for u less than 18 practically coincides with the Mott DCS (and diverges at u ¼ 0).

The convergence of the partial-wave series (C.80) and (C.81) is similar to that of a pure Coulomb field, i.e., slow for small angles. To simplify the computations, it is convenient to write

A finite-element visualization of quantum reactive scattering. II. Nonadiabaticity on coupled potential energy surfaces.

Theory of Graphene Raman Scattering.

A central theory of biology.

Toward a physical theory of quantum cognition.

Theory of Interacting Quantum Gases.

Generalizing Prototype Theory: A Formal Quantum Framework.

Theory of Thomson scattering in inhomogeneous media.

classical theory and comparison with quantum results.

Comments on quantum probability theory.

classical theory of rotationally and vibrationally inelastic scattering in space-fixed and body-fixed reference frames.

Cavity-enhanced coherent light scattering from a quantum dot.

Theory of light emission from quantum noise in plasmonic contacts: above-threshold emission from higher-order electron-plasmon scattering.

The hierarchy of evidence and quantum theory.

A bond-order theory on the phonon scattering by vacancies in two-dimensional materials.

Infinite-time average of local fields in an integrable quantum field theory after a quantum quench.

Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective.

Appendix a.

Analysis of temporal evolution of quantum dot surface chemistry by surface-enhanced Raman scattering.

Emergent "Quantum" Theory in Complex Adaptive Systems.

Medcine: Diagnostic Palpation of the Vermiform Appendix, &c.

Phonon lifetime investigation of anharmonicity and thermal conductivity of UO2 by neutron scattering and theory.

Quantum scattering calculations for ro-vibrational de-excitation of CO by hydrogen atoms.

Erratum: Theory of Thomson scattering in inhomogeneous media.

Appendix C: reports produced by the german national research center for environment and health (gsf).