Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

Подождите немного. Документ загружается.

772 Part D Scattering Theory

51.8 New Developments

During the last eight years there have been some interest-

ing new developments [51.38–40] using hyperspherical

coordinates to describe the dynamics of ion–atom col-

lisions. Of particular interest for this chapter are the

calculations of Le et al. [51.40] for charge transfer in

the Si

/H(D) and Be

/H systems. Their calculated

cross sections are almost identical to those obtained by

Pieksma et al. [51.41] using the Thorson–Delos-type

[51.8] approximate Jacobi coordinates introduced in

Sect. 49.3. This conﬁrms the close connection between

the hyerspherical and the Thorson–Delos-type coordi-

nates which had already been observed by Gargaudet al.

[51.10] for two-state systems.

Another aspect of the theoretical formulation which

has been clariﬁed recently concerns the adiabatic–

diabatic transformation (51.20). The radial differential

equations (51.21) only take this simple form if it can be

assumed that

B =

dξ

A− A

. (51.35)

However, (51.35) is only strictly satisﬁed if the basis

set is complete. And indeed, it has been found [51.10]

from direct calculations of the matrix elements B

that

it is not well satisﬁed for any choice of Jacobi coor-

dinates. On the other hand, the calculations show that

(51.35) is well satisﬁed for a minimal basis set using the

Thorson–Delos reaction coordinates. This result con-

ﬁrms that convergence of an adiabatic basis set can

indeed be achieved using appropriate reaction coordi-

nates and also explains why the calculations of Le et al.

[51.40]andPieksma et al. [51.41, 42] agree so well.

References

51.1 A. Dalgarno, S. E. Butler: Comments At. Mol. Phys. 7,

129 (1978)

51.2 D. Pequignot, S. M. V. Aldrovandi, G. Stasinska: As-

tron. Astrophys. 63,313(1978)

51.3 M. Gargaud, J. Hanssen, R. McCarroll, P. Valiron:

J. Phys. B 14, 2259 (1981)

51.4 V. H. S. Kwong, Z. Fang: Phys. Rev. Lett. 71, 4127

(1993)

51.5 T. K. McLaughlin, S. M. Wilson, R. W. McCullough,

H. B. Gilbody: J. Phys. B 23, 737 (1990)

51.6 D. R. Bates, R. McCarroll: Proc. R. Soc. London A 245,

175 (1958)

51.7 S. B. Schneiderman, A. Russek: Phys. Rev. A 181,311

(1969)

51.8 W. R. Thorson, J. B. Delos: Phys. Rev. A 18, 117

(1978)

51.9 R. McCarroll, D. S. F. Crothers: Adv. At. Mol. Opt.

Phys. 32, 253 (1994)

51.10 M. Gargaud, R. McCarroll, P. Valiron: J. Phys. B 20,

1555 (1987)

51.11 L. F. Errea, L. Mendez, A. Riera: J. Phys. B 15, 2255

(1982)

51.12 L. F. Errea, C. Harel, H. Jouin, L. Mendez, B. Pons,

A. Riera: J. Phys. B 27, 3603 (1994)

51.13 M. Gargaud, R. McCarroll, L. Opradolce: J. Phys. B

21, 521 (1988)

51.14 M. Gargaud, F. Fraija, M. C. Bacchus-Montabonel,

R. McCarroll: J. Phys. B 29, 179 (1994)

51.15 P. Honvault, M. C. Bacchus-Montabonel, R. McCar-

roll: J. Phys. B 27, 3115 (1994)

51.16 P. Honvault, M. Gargaud, M. C. Bacchusmontabonel,

R. McCarroll: Astronom. Astrophys. 302, 931 (1995)

51.17 D. L. Cooper, M. J. Ford, J. Gerratt, M. Raimondi:

Phys. Rev. A 34, 1752 (1986)

51.18 S. Bienstock, T. G. Heil, A. Dalgarno: Phys. Rev. A

25, 2850 (1982)

51.19 C. Bottcher, A. Dalgarno: Proc. Soc. A 340,187(1974)

51.20 P. Valiron, R. Gayet, R. McCarroll, F. Masnou-

Seeuws, M. Philippe: J. Phys. B 12, 53 (1979)

51.21 R. Grice, D. R. Herschbach: Mol. Phys. 27,159(1974)

51.22 M. Gargaud, R. McCarroll: Phys. Scr. 51, 752 (1995)

51.23 M. Gargaud, R. McCarroll, L. Opradolce: Astron.

Astrophys. 208, 251 (1989)

51.24 L. F. Errea, B. Herrero, L. Méndez, O. Mó, A. Riera:

J. Phys. B 24, 4049 (1991)

51.25 R. E. Olson, A. Salop: Phys. Rev. A 14,579(1976)

51.26 M. Kimura, T. Iwai, Y. Kaneko, N. Kobayashi, A. Ma-

tumoto, S. Ohtani, K. Okuno, S. Takagi, H. Tawara,

S. Tsurubuchi: J. Phys. Soc. (Japan) 53, 2224 (1984)

51.27 K. Taulbjerg: J. Phys. B 19, L367 (1986)

51.28 M. Gargaud: Transfert de charge entre ions multi-

chargés et hydrogène atomique (et moléculaire)

aux basses énergies. Ph.D. Thesis (Université de

Bordeaux, France 1987)

51.29 S. E. Butler, A. Dalgarno: Astrophys. J. 241,838

(1980)

51.30 F. T. Smith: Phys. Rev. 179, 111 (1969)

51.31 B. R. Johnson: J. Comput. Phys. 13, 445 (1973)

51.32 P. Valiron: Echange de charge des ions C

avec l’hydrogène atomique dans le milieu in-

terstellaire. Ph.D. Thesis (Université de Bordeaux,

France 1976)

51.33 F. Fraija, M. C. Bacchus-Montabonel, M. Gargaud:

Z. Phys. D 29, 179 (1994)

Part D 51

Ion–Atom Charge Transfer Reactions at Low Energies References 773

51.34 M. Gargaud, R. McCarroll, P. Valiron: Astron. Astro-

phys. 106, 197 (1982)

51.35 P. Roncin, C. Adjouri, N. Andersen, M. Barat,

A. Dubois, M. N. Gaboriaud, J. P. Hansen, S. E. Niel-

sen, S. Z. Szilagyi: J. Phys. B 27, 3079 (1994)

51.36 P. Roncin, C. Adjouri, M. N. Gaboriaud, L. Guille-

mot, M. Barat, N. Andersen: Phys. Rev. Lett. 65,

3261 (1990)

51.37 M. Gargaud, M. C. Bacchus-Montabonel, T. Groz-

danov, R. McCarroll: J. Phys. B 27, 4675 (1994)

51.38 A. Igarashi, C. D. Lin: Phys. Rev. Lett. 83,4041

(1999)

51.39 C.-N. Liu, A.-T. Le, T. Morishita, B. D. Esry, C. D. Lin:

Phys. Rev. A 67, 52705 (2003)

51.40 A.-T. Le, M. Hesse, T. G. Lee, C. D. Lin: Phys. Rev. A

67, 52705 (2003)

51.41 M. Pieksma, M. Gargaud, R. McCarroll, C. Havener:

Phys. Rev. A 54, 13 (1996)

51.42 D. Rabli: Extension de la méthode du potentiel

modèle pour traiter la dynamique des systèmes di-

atomiques. Application au transfert de charge dans

les collisions Si

+He et He

+He métastable.

Ph.D. Thesis (Université Pierre et Marie Curie, Paris

2001)

Part D 51

775

Continuum Dis

52. Continuum Distorted Wave and Wannier Methods

The continuum distorted wave model has been ex-

tensively applied to charge transfer and ionization

processes. We present both the perturbative and

variational capture theories as well as highlighting

the suitability of this model in describing the con-

tinuum ﬁnal states in both heavy and light particle

ionization. We then develop the Wannier theory

for threshold ionization, and further theoretical

work which led to the modern quantal semiclas-

sical approximation. This very successful theory

has provided the ﬁrst absolute cross sections

which are in good agreement with experiment.

52.1 Continuum Distorted Wave Method ....... 775

52.1.1 Perturbation Theory................... 775

52.1.2 Relativistic Continuum-Distorted

Waves...................................... 778

52.1.3 Variational CDW ........................ 778

52.1.4 Ionization ................................ 779

52.2 Wannier Method ................................. 781

52.2.1 The Wannier Threshold Law ........ 781

52.2.2 Peterkop’s Semiclassical

Theory ..................................... 782

52.2.3 The Quantal Semiclassical

Approximation.......................... 783

References .................................................. 786

The most recent developments include third-order

continuum distorted wave double-scattering 1s–1s tran-

sitions (Sect. 52.1.1), relativistic continuum distorted

waves (Sect. 52.1.2), and new theory on magnetic-

ally quantized continuum distorted waves [52.1, 2]

(Sect. 52.1.4). A novel ionization theory for low en-

ergies (below 80 keV) is also reported in which the

target is considered as a one electron atom and

the interactions between this active electron and the

remaining target electrons are treated by a model

potential including both short and long range ef-

fects. In the ﬁnal channel the usual product of

two continuum distorted wave functions, each asso-

ciated with a distinct electron–nucleus interaction, is

used [52.3].

In addition, Sect. 52.2 on the Wannier Method re-

ports the major progress made over the last eight years.

These major advances include (a) the development of

below-threshold semiclassical theory for the study of

doubly excited states [52.4–7] (b) a more accurate vari-

ant of the semi-classical quantum-mechanical treatment

of Crothers [52.8].

52.1 Continuum Distorted Wave Method

52.1.1 Perturbation Theory

Continuum distorted wave theory (CDW) is one of

the most advanced and complete perturbative the-

ories of heavy particle collisions which has been

formulated to date. It was originally introduced by

Cheshire [52.9] to model the process of charge

transfer during the collision of an atom/ion with

an ion (speciﬁcally the resonant process of p +

H(1s) →H(1s) +p). These types of three-body colli-

sions are made amenable to the perturbative approach

when the ratio of the projectile impact velocity v

to the electron initial bound state mean velocity v

satisﬁes

 3 . (52.1)

The criterion for nonrelativistic collisions, in which

electron capture is a dominant process, is that both

v and v

are small compared with the speed of light.

The theoretical description of collisions which involve

the disturbance of a bound electron of mass m

the ﬁeld of a fast moving heavy particle of mass M

can be greatly simpliﬁed by exploiting the fact that

since the ratio m

/M is so small, the heavy par-

ticle follows a straight-line trajectory throughout the

collision.

Part D 52

776 Part D Scattering Theory

This allows the parametrization of the internuclear

vector R in terms of an impact parameter b, such that

R= b+vt .

(52.2)

This impact parameter picture (IPP) of the collision is

equivalent to the full quantal or wave treatment when

the eikonal criterion for small angle scattering is satis-

ﬁed [52.10].

It has become standard to work in a generalized

nonorthogonal coordinate system in which the vectors r

) from the target (projectile) to the electron are treated,

along with R, as independent variables [52.11]. Working

in the frame centered on the target nucleus and using

atomic units, the Lagrangian is given by

H −i

=−

∇

) −i

∂

∂t

−

∇

) +iv ·

∆

(R) −

∆

, (52.3)

where d/dt

refers to differentiation with respect to t,

keeping r

ﬁxed, and where

−Z

, V

−Z

, V

(52.4)

Since these potentials are pure Coulomb potentials, they

continue to affect the relevant wave functions even at

inﬁnity. These long range Coulomb boundary conditions

are deﬁned in (52.10)and(52.11). The +iv ·

∆

term

gives rise to the Bates–McCarroll electron translation

factors which are required to satisfy Galilean invariance.

The Lagrangian above has been written in such a way

as to highlight the three two-body decompositions ex-

ploited in CDW, with the −

∆

term, the so-called

nonorthogonal kinetic energy, coupling the systems. The

essence of CDW is to treat the bound electron as simul-

taneously being in the continuum of the other heavy

particle.

The initial wave function can be written as

= D

−v

)Φ

, t)C(R, t), (52.5)

where D

−v

is the distortion from the projectile, and

C is due to the internuclear potential, V

. The bound

state Φ

= φ

) exp

(

−i

)

,whereφ

) is the initial

eigenstate, and 

is the initial eigenenergy.

In this form, the action of the Lagrangian can be split

into three separate differential equations plus a residual

interaction. This gives the following solutions: for the

distortion D,

−v

= N(ζ

)

(iζ

;1;iv · r

+ivr

−



−v



∗

, (52.6)

with

N(ζ) = exp(πζ/2)Γ(1 −iζ) , ζ

T,P

= Z

T,P

/v ;

(52.7)

and for the internuclear function C,

C(R, t) = exp



ln(vR−v



. (52.8)

Similarly it can be shown that

= D

)Φ

, t)C

∗

(R, −t)

×exp



iv · r

−i



(52.9)

where iv · r

−i

t results from the Galilean transfor-

mation to the target frame. The superscripts plus and

minus refer to outgoing and incoming Coulomb bound-

ary conditions respectively. These are determined by the

asymptotic form of the wave functions

lim

t→−∞

∼ Φ

, t)

×exp



−1)

ln(vR−v



(52.10)

and

lim

t→+∞

−

∼ Φ

, t) exp



iv · r

−i



×exp



−i

−1)

ln(vR+v



(52.11)

Of course, ξ

and ξ

−

are not exact solutions of the

three-body Schrödinger equation; in fact,



H −i



= W

=−

∆

−v

∆

(52.12)

and



H −i



−

= W

−

=−e

iv·r

−i

∇

−

· ∇

(52.13)

The CDW transition amplitude is written as

=−i

+∞



−∞



−



CDW





(52.14)

Part D 52.1

Continuum Distorted Wave and Wannier Methods 52.1 Continuum Distorted Wave Method 777

A perturbative expansion via the distorted wave

Lippmann–Schwinger equation can be made for T

CDW

either in the post form

CDW

= W

†

(1 +G

) +T

CDW

(52.15)

or in the prior form

−

CDW



1 +W

†



†

−

CDW

(52.16)

where the Green functions are given by

i, f



− H +W

i, f

+i



−1

, (52.17)



− H +V +i



−1

, (52.18)

and V is any potential which ensures that the kernels of

the integral equations for T

CDW

are continuous [52.13].

By taking the ﬁrst term in the expansions (52.15)

and (52.16), we get the post and prior forms of the

CDW1 amplitude as used by Cheshire. When calculating

these amplitudes, the separable nature of the CDW wave

function is best exploited by using Fourier transforms to

move to the time-independent wave picture. A similar

transformation is not suitable in the coupled channel

approach discussed in Sect. 52.1.3.

Crothers [52.13], working in the wave treatment, has

calculated the second order CDW2 amplitude using var-

ious approximations for the Green functions, and has

shown that the CDW perturbation series has converged

very well to ﬁrst-order in most parts of the differen-

tial cross section. This is in contrast to the standard

Born or Brinkman–Kramers approximations which do

not start to converge until expanded to second-order.

CDW1 is the only ﬁrst-order perturbation theory, apart

from asymmetric hybrid models derived from it, which

produces a Thomas peak. Unfortunately, due to the ac-

cidental cancellation of the leading order terms, CDW1

has an extreme dip at the Thomas angle, a defect re-

movedinCDW2[52.14]. This is illustrated in Fig. 52.1

which also includes both folded and unfolded versions of

the asymmetric target CDW (TCDW) theory discussed

below, as well as experimental data [52.12].

Further work in this area has included the de-

velopment of the Thomas double-scattering electron

capture at asymptotically high velocity within the third-

order continuum distorted-wave perturbation theory for

1s–1s transitions in proton hydrogen collisions. It has

been shown [52.15] that at the critical proton scatter-

ing angles, namely the forward peak, Thomas double

encounter peak, small angles, and the interference min-

imum, the CDW series has converged at second order.

Moreover, it is proven that the third-order correction

makes no contribution to the velocity dependent v

−11

and v

−12

behavior of the Thomas double-scattering total

cross section at the leading angles. In contrast, it may

be seen in [52.15] that the Oppenheimer–Brinkman–

Kramers (OBK) travelling atomic orbital theory (which

in general suffers from a common phase factor which

embraces intermediate elastic divergences [52.16]in

the ﬁrst and higher-order terms) has not converged at

second order. It remains an open question as to whether

fourth-order terms or higher in the OBK approxima-

tion contribute to various differential cross-sections. It

is concluded that the CDW model gives a superior de-

scription of the Thomas double-scattering mechanism

when compared with the OBK model.

Anomalously large cross sections are obtained at low

energies if the CDW wave function is not normalized at

all times throughout the collision [52.11]. This is best

demonstrated by the presence of the N(ζ) terms in

lim

t→+∞

= N(ζ

)Φ

, t)

×exp



ln(vR−v



(52.19)

–18

–19

–20

–21

0.0 0.2 0.4 0.6 0.8

(dσ/d )

lab

(cm

/sr)

Ω

lab

(mrad)

Fig. 52.1 Differential cross sections for electron capture in

the collision H

+ H(1s) → H(1s) + H

as a function

of laboratory scattering angle (θ

lab

) for impact energy of

5MeV:solid line TCDW, folded over the experimental res-

olution of Vogt et al. [52.12]. Unfolded theoretical results:

dashed line TCDW; dotted line CDW1. Experimental data;

circles, Vogt et al. [52.12]

Part D 52.1

778 Part D Scattering Theory

and

lim

t→−∞

−

= N

∗

(ζ

)Φ

, t) exp



iv · r

−i



×exp



−i

ln(vR+v



(52.20)

Using

lim

v→0

| N(Z/v)|

∼

2πZ

(52.21)

it is clear that the problem gets worse as v decreases. It

can be corrected by deﬁning

i, f

= ξ

i, f



i, f



i, f



−1/2

. (52.22)

Simpler distorted wave models can be generated through

further approximations. Two of the most popular

are Target CDW [D

−v

) → 1] and Projectile CDW

[

) → 1

]

. These approximations are justiﬁed when

> Z

and Z

> Z

, respectively, and are particularly

simple to calculate when the simple Born-like residual

interaction is used rather than the full CDW form.

The asymptotic forms of the CDW wave functions

can be used throughout the collision, ensuring nor-

malization, and this leads to the eikonal or symmetric

eikonal models.

52.1.2 Relativistic Continuum-Distorted

Waves

The CDW model can be naturally extended to the two-

center time-dependent Dirac equation, so that a Lorentz

invariant theory is obtained. When the electron orbital

velocity αZc, or the collision velocity v, approaches the

speed of light c, the kinematics are modiﬁed by time

dilation. In addition, the particle interactions change be-

cause of retardation and the fact that spin-orbit effects are

now important. Moreover, vacuum interactions such as

radiative emissions and electron–positron pair produc-

tion begin to play a role. A comprehensive account of

atomic processes in relativistic heavy-particle collisions

can be found in two recent books [52.17, 18]. At high

collision energies, γ ≡



1 −v



−1/2

 1, and high-

charge states of the ions, the Dirac sea of negative energy

states becomes energetically accessible and strongly

coupled. The process of electron capture, for exam-

ple, may be mediated by spin-ﬂip transitions [52.19], or

spontaneous X-ray emission (radiative electron capture)

[52.20] and even electron capture via pair produc-

tion [52.21, 22]. Although the importance of vacuum

processes diminishes with energy, these mechanisms

dominate in the extreme relativistic regime. Indeed the

last of these processes was used to produce antihydrogen

in the laboratory [52.23]atGeVu

−1

energies.

At relativistic energies, the principal inelastic

process is collisional ionization [52.17, 18]. The ex-

tensions of the distorted-wave theory to accommodate

Lorentz invariance has been developed by Rivarola and

Deco [52.24, 25]andCrothers and coworkers [52.19]

following work on the Born series [52.26]andim-

pulse approximation [52.27]. In practical applications

to electron capture cross sections, the symmetric semi-

relativistic CDW theory of Glass et al. [52.19]was

found to be in very good agreement with experiments

in the GeV u

−1

energy range with charges Z

P, T

∼ 6–80.

For non-radiative electron capture, it was found that

second-order retardation dominates at extreme relativis-

tic velocities so that σ ∼ γ

−1

(ln γ)

[52.28]. However,

the momentum transfer kinematics for this process are

unfavorable, and a more efﬁcient mechanism based on

electron–positron pair production with capture of the

created electron is more strongly coupled. Theoretical

estimates of this process using relativistic CDW [52.29]

compared with experiments [52.30] are in very good

agreement.

The same model has been applied to estimate yields

of antihydrogen following antiproton impact with neu-

tral high-Z atoms [52.23] following experiments at

CERN and Fermilab. The virtual photon model of

Baur [52.30] gives cross sections that agree well with

the limited data [52.23]. However, these estimates are

roughly ten times larger than the relativistic CDW results

[52.17] and one hundred times the ﬁrst-Born estimate

[52.31]. It appears that additional studies, both experi-

mental and theoretical, would be worthwhile in order to

understand this process more fully.

52.1.3 Variational CDW

As the ratio of v/v

decreases, perturbation theory starts

to fail. This is due to the effective interaction time be-

tween the projectile and target atoms being long enough

for strong three-body coupling. In this environment

variational methods have proved successful. This pro-

cedure ensures both gauge invariance and unitarity –

two fundamental attributes perturbation theory usually

cannot guarantee. Continuing in the IPP, we use the Sil

variational principle, which gives

+∞



−∞

dtΨ |H −i

|Ψ =0 . (52.23)

Part D 52.1

Continuum Distorted Wave and Wannier Methods 52.1 Continuum Distorted Wave Method 779

In the two-state CDW approximation we may assume

CDW

= c

(t)ξ

−

(52.24)

subject to the boundary conditions c

(−∞) = 1and

(−∞) = 0. Variation of c

∗

and c

∗

gives the standard

coupled equations

+iN

= H

+ H

, (52.25)

+iN

= H

+ H

, (52.26)

where









−



−







−



= N

∗

and





H −i









H −i



−





−



H −i







−



H −i



−



Equations (52.25)and(52.26) can clearly be written as

a matrix equation,

c = Hc,

(52.27)

which is then easily generalized for larger expansions

of Ψ . By using an orthogonalized basis set of normalized

functions in the manner of Löwdin [52.32,33], the N ma-

trix reduces to the unit matrix. This will be understood

when considering expansions for Ψ from now on.

Another interesting, but potentially ruinous, result

of the asymptotic forms (52.19)and(52.20) is their fail-

ure to obey the correct long-range Coulomb boundary

conditions; compare this with their expressions at the op-

posite time extreme in (52.10)and(52.11). This has no

consequence until second-order VCDW is calculated in

which divergent integrals arise as a direct result of this

feature of the wave functions. These terms are analo-

gous to the well-known intermediate elastic divergences

which occur in Born-type expansions which do not have

the correct Coulomb phases.

A novel way to avoid this problem [52.34, 35]isto

split the time plane into two parts, allowing the well-

behaved set {ξ

} to be used exclusively for t ≤0, while

the set {ξ

−

}forms the basis for t ≥0. This phase integral

halfway house VCDW is based on the factorization of

the scattering matrix S into a product of two Møller

matrices,

S= Ω

†

−

Ω

, (52.28)

where Ω

†

−

represents the propagation of the initial state

from t =−∞to t = 0

−

, while Ω

represents the prop-

agation of the ﬁnal state from t =+∞to t = 0

The total wave function is similarly split into two

expansions over an orthogonal basis ψ, with

−

= c

−

, t < 0 , (52.29)

= c

−

, t > 0 , (52.30)

where the superscripts on the Ψ correspond to the re-

spective heavy-particle motion. This in turn divides the

coupled equations into two sets

−

= H

−

, t < 0 , (52.31)

= H

−−

, t > 0 , (52.32)

where

±±

=Ψ

| H −i

| Ψ

 . (52.33)

The coefﬁcients c

then have to be matched over a lo-

cal discontinuity in the total wave function at t = 0,

such that c

(0) =c

−

(0). Halfway house VCDW has all

the appealing attributes of a variational theory but, by

explicitly satisfying the long-range Coulomb boundary

conditions, it is divergence free.

52.1.4 Ionization

CDW, by treating the Coulomb interactions to such

a high degree, has obvious attractions for modelling

the ionization process. Single ionization of an electron

from an atom by a high-energy projectile is a pertur-

bative process and the Born approximation will match

experimental total cross sections rather well. How-

ever CDW-like representations of the initial and ﬁnal

states generate better results at lower energies, as well

as producing features in the differential cross sec-

tions which are beyond the reach of the ﬁrst Born

approximation.

In full CDW ionization theory, the initial state is

given by the usual charge transfer wave function ξ

Part D 52.1

780 Part D Scattering Theory

(52.5), while the ﬁnal state takes the form

−

= (2π)

−2/3

exp



ik· r

−i



×exp



−i

ln(vR+v · R)



× N

∗

/k)

× (−iZ

/k; 1;−ik· r

−ikr

)

× N

∗

/ p)

× (−iZ

/ p; 1;−ip· r

−ipr

), (52.34)

where k ( p =k−v) is the momentum of the electron

relative to the target (projectile) nucleus.

CDW ionization theories presented prior to 1982

produced spuriously large results due to the unnormal-

ized initial state. Since the matrix element ξ

| ξ

 is

computationally expensive to calculate as a function of

b and t, a very successful alternative is to take the initial

state as an eikonal distorted state [52.36], thus ensuring

normalization. The initial state in this CDW-EIS theory

is taken to be

EIS

−v

)Φ

, t)C(R, t), (52.35)

where now

−v

) = exp



−i

ln(vr

+v · r

)



. (52.36)

The ﬁnal state remains as in (52.34).

The CDW ﬁnal state is most effective when differ-

ential cross sections are studied. The most interesting

features are in the forward θ = 0 ejection angle; i. e., the

soft collision peak (k  0), the electron capture to the

continuum peak ECC (k  v) and the binary encounter

peak (k  2v). CDW theories are especially suited to

the description of the ECC peak which results theoret-

ically from the presence of the N(Z

/ p) factor in the

wave function. This peak can be analyzed in detail via

a multipole expansion and much theoretical work has

centered on the dipole parameter β.Anegativeβ is

strongly suggested by both experiment and physical in-

tuition. CDW, CDW-EIS and halfway house VCDW all

predict different values for β with the last theory be-

ing the only one which gives a high energy limit which

remains negative [52.37].

Magnetically quantized continuum distorted wave

theory also has been considered [52.1] in the description

of ionization in ion–atom collisions. This generalizes

the CDW-EIS theory of Crothers and McCann [52.2]

to incorporate the azimuthal angle dependence of each

CDW in the ﬁnal state wave function. This is accom-

plished by the analytic continuation of hydrogenic-like

wave functions from below to above threshold, using

parabolic coordinates and quantum numbers, includ-

ing magnetic quantum numbers, thus providing a more

complete set of states. The continuation applies to ex-

citation, charge transfer, ionization, and double and

hybrid events for both light- and heavy-particle colli-

sions. It has successfully been applied to the calculation

of double differential cross sections for the single ioniza-

tion of the hydrogen atom and for a hydrogen molecule

by a proton for electrons ejected in the forward di-

rection at a collision energy of 50 keV and 100 keV,

respectively.

It is well known that the CDW-EIS models are

the best suited to the intermediate and high energy re-

gions. Recent results for proton-argon total ionization

cross sections [52.38] highlight large discrepancies be-

tween CDW-EIS theory and experiment for energies

below 80 keV. This problem has recently been ad-

dressed [52.3]. Here, following the theory of [52.39]the

authors in [52.3] use a Born initial state wave function.

In the ﬁnal channel, the usual product of two continuum

distorted wave functions each associated with a distinct

electron–nucleus interaction is used. In their treatment

the target is considered as a one-electron atom and the

interactions between this active electron and remain-

ing target electrons are treated by a model potential

including both short- and long-range effects. The suc-

cess of this new theory for low energies is shown in

Fig. 52.2. Here it is clear that the calculation in [52.3]

gives good agreement for the total cross sections in

the energy range 10–300 keV with the measurements

of Rudd et al. [52.40].

Double ionization in general remains an extremely

difﬁcult area for theoretical models based on perturba-

tive expansions – even those with explicit distortions

built in. In this process, the explicit correlation between

the electrons in the target atom is vitally important.

However, one example where CDW theory can over-

come these problems is in the bound state wave function

of Pluvinage [52.41]. This very successful treatment,

which also includes a variationally determined param-

eter, is just the CDW analogy in bound state theory,

although this appeared well before Cheshire’s paper on

scattering. The Pluvinage wave function for the ground

state of a two-electron atom is given by

φ = c(κ)





exp

[

−Z

) +iκr

]



1 +

2iκ

;2;2iκr



(52.37)

Part D 52.1

Continuum Distorted Wave and Wannier Methods 52.2 Wannier Method 781

0.1

10 100 1000

Total cross sections (10

–16

)

Incident proton energy (keV)

where r

1,2

are the distances of the two electrons from the

target nucleus, and r

is the inter-electron separation.

Fig. 52.2 Total ionization cross section for the proton-

impact single ionization of Ar: solid line theoretical results

of [52.3], dashed line theoretical results of OPM [52.38]

and dotted line HFS [52.38]. Experimental data; circles,

Rudd et al. [52.40]

The constant κ is variationally determined to be 0.41,

giving the normalization constant c =(0.364 05)

1/2

An analogous wave function for two electrons

in the target continuum was derived [52.11]and

implemented later in a successful CDW treatment

of ionization by electrons and positrons [52.42].

This BBK theory, so named in deference to the

authors, demonstrates the high resolution CDW ﬁ-

nal states obtained, right down to triply differential

cross sections. However, this model suffers from the

low-energy normalization problems associated with

CDW, and is also unable to describe the thresh-

old effects which are in the domain of Wannier

theory.

52.2 Wannier Method

52.2.1 The Wannier Threshold Law

In 1953, Wannier [52.43] deduced the relationship be-

tween the cross section at the threshold of a reaction,

and the excess-of-threshold energy of the incident par-

ticle for a three-body ionization problem, where the ﬁnal

state consists of a residual unit positive charge and two

electrons, with each body moving in the continuum of

the other two. This extended to three bodies the earlier

two-body threshold law derived by Wigner [52.44](see

Sect. 60.2.1).

For ﬁnal states with L = 0, Wannier employed hy-

perspherical coordinates (ρ,α,θ

),where

,α= tan

−1





= cos

−1

(

). (52.38)

Here we assume that the residual ion is inﬁnitely mas-

sive and at rest with respect to the two electrons. By

converting the two-electron problem to the case of mo-

tion of a single point in six-dimensional space, we

can take the hyperradius ρ as the ‘size’ of the hy-

persphere, α as the radial correlation of the electrons

and θ

as their angular correlation. This allows the

Schrödinger equation for the ﬁnal state to be written

(in a.u.) as



−





cos

−





sin

+2E

2Z(α, θ

)



Ψ(r

, r

) = 0 , (52.39)

where

∂

∂ρ



∂

∂ρ



sin

α cos

∂

∂α



sin

α cos

∂

∂α



(52.40)

and

Z(α, θ

) =

cos α

sin α

−

(1 −cos θ

sin 2α)

(52.41)

is the potential surface on which the particle is mov-

ing [52.45].

The most likely conﬁguration of the electrons lead-

ing to double escape at threshold corresponds to the

region r

=−r

, i. e., the two electrons escape in oppo-

site directions from the reaction zone, corresponding to

the saddle point of the potential surface Z(α, θ

),and

deﬁned as the Wannier ridge. Also, dynamic screening

Part D 52.2

782 Part D Scattering Theory

between the two electrons means there would be equal

partitioning of the available energy for the two particles,

and so they would have equal but opposite velocities on

escape. In hyperspherical coordinates, the most impor-

tant region for double escape is, therefore, α =π/4, and

= π.

The main conclusion of Wannier’s theory is that the

total cross section for electron impact single ionization

scales as

σ =kE

, (52.42)

where E is the excess-of-threshold energy,

=−



100Z −9

4Z −1



, (52.43)

and Z the residual charge; m

is 1.127 for unit residual

charge, with m

→ 1asZ →∞.

52.2.2 Peterkop’s Semiclassical Theory

The Wannier threshold law has been veriﬁed both semi-

classically [52.46] and quantum mechanically [52.47].

Peterkop [52.46] adopted a semiclassical JWKB ap-

proach to the problem, by using the three-dimensional

WKB ansatz

= P

exp





(52.44)

for the ﬁnal-state wave function, where S and P are,

respectively, the solutions of the Hamilton–Jacobi equa-

tion,

(∇

+(∇

= 2(E −V), (52.45)

and the continuity equation,

∇

(P∇

S) +∇

(P∇

S) = 0 . (52.46)

In hyperspherical coordinates, these equations become



∂S

∂ρ





∂S

∂α



sin

2α



∂S

∂θ



= 2E +

(52.47)

and



∂S

∂ρ







∂S

∂α



+ D



∂S

∂θ



= 0 ,

(52.48)

where

f =

∂

∂ρ

(ρ

f ), (52.49)

f =

sin

2α

∂

∂α



f sin

2α



, (52.50)

f =

sin

2α

sin θ

∂

∂θ

( f sin θ

). (52.51)

Following Wannier’s hypothesis, solutions of these

equations are found in the region α = π/4,θ

= π.

Taking the Taylor expansion for Z(α, θ

) as

Z(α, θ

) = Z

(∆α)

(∆θ

)

+··· ,

(52.52)

where ∆α = α −π/4and∆θ

= θ

−π, it follows

from (52.41)that

√

, Z

√

, Z

=−

√

(52.53)

Similarly, taking the solution of (52.47) in the form

S = S

(ρ) +

(ρ)(∆α)

(ρ)(∆θ

)

+···

(52.54)

gives

dρ

= ω,

(52.55)

dρ

, i = 1, 2 ,

(52.56)

where ω =

(

2E +2Z

/ρ

)

. The solutions are

= ρω +

ρ(χ +ω)

, (52.57)

= ρ

dρ

, i = 1, 2 ,

(52.58)

where χ = (2E)

and

= C

, (52.59)

= ρ



, m

+1;2m

;

−Eρ



(52.60)

=−

−

, m

=−

, (52.61)



1 +



. (52.62)

where, for i = 2, the principal branch is understood.

Part D 52.2