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

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Mass Transfer at High Energies: Thomas Peak 57.6 Recent Developments 867

tion. If the width of the resonance is ∆E, then the

lifetime is τ =

/∆E. E and τ are conjugate vari-

ables. The Thomas peak is an overdamped resonance

in scattering angle, corresponding to a shift in the im-

pact parameter of the scattering event [57.13]. However,

unlike energy resonances, our Thomas resonance in

the scattering angle seems to have no classical ana-

log [57.14].

57.5 Destructive Interference of Amplitudes

It has already been noted that the location of the Thomas

peaks depends on the mass of the collision partners. For

the process

+ Atom → Atom + H , (57.10)

there are two separate Thomas peaks [57.2, 15]cor-

responding to cases A and B in the Lieber diagram

(Fig. 57.2). Experimental evidence exists for both peaks.

The standard Thomas peak occurs at small forward an-

gles [57.11], while the second peak [57.16] occurs at

about 60

◦

. If the mass of the projectile is reduced,

the positions of these Thomas peaks move toward one

another [57.17] as illustrated in Fig. 57.6.

When M

= M

, then both Thomas peaks occur

at 45

◦

. This occurs in positronium formation where

= M

= m

,i.e.,

+ He → Ps + He

+ e

−

. (57.11)

In cases A and B of Fig. 57.2,thetwoV

sec-

ond Born terms are of opposite sign because V

is of

opposite sign in diagrams A and B. This leads to de-

structive interference for 1s − 1s electron capture (which

is dominant at high velocities) as was ﬁrst discussed by

Shakeshaft and Wadehra [57.17]. Consequently, the ob-

served Thomas peak structure is expected [57.18, 19]

to be quite different for e

impact than for impact

of p

or other projectiles heavier than an electron.

The double ridge structure for transfer ionization of

helium by e

is expected to differ signiﬁcantly from

Thomas peak

60° Peak

60°

dσ

⍀

dσ

⍀

Destructive

interference

45°

Fig. 57.6 Change of position and nature of the Thomas

peaks with decreasing projectile mass

the structure shown in Fig. 57.3. Understanding such

destructive interference between resonant amplitudes

may give deeper insight into the physical nature of the

intermediate states in this special few-body collision

system.

57.6 Recent Developments

In the late 1990’s observations [57.20] of the Thomas

peak in the case of transfer ionization (where one

electron is ionized and another is transferred) differ-

ential in the momentum of the ejected electron provided

new speciﬁc detail on the kinematics of the two step

process [57.21]. In 2001 the Thomas peak was dis-

cussed [57.22] in the context of quantum time ordering.

In this case time ordering surprisingly is not signif-

icant at the center of the peak, in contradiction to

the classical picture that there is a deﬁnite order in

the two step process for transfer ionization. How-

ever, time ordering does contribute to the shape of the

Thomas peak. In 2002 the Stockholm group [57.23]

reported that at very high velocities, the ratio of trans-

Part D 57.6

868 Part D Scattering Theory

fer ionization to total transfer approaches the same

asymptotic limit as in double to single ionization in

(non-Compton) photoionization, namely 1.66%. This

was interpreted in terms of a common shake process oc-

curring when the wavefunction collapses after a sudden

collision.

References

57.1 L. H. Thomas: Proc. Soc. A 114, 561 (1927)

57.2 R. Shakeshaft, L. Spruch: Rev. Mod. Phys. 51,369

(1979)

57.3 K. Detmann, G. Liebfried: Z. Phys. 218,1(1968)

57.4 P. R. Simony: A second order calculation for charge

transfer. Ph.D. Thesis (Kansas State University,

Manhattan, KS, USA 1981)

57.5 M. Lieber: private communication (1987)

57.6 J. H. McGuire, J. C. Straton, T. Ishihara: The Appli-

cation of Many-Body Theory to Atomic Physics,ed.

by M. S. Pindzola, J. J. Boyle (Cambridge Univ. Press,

Cambridge 1994)

57.7 J. Palinkas, R. Schuch, H. Cederquist, O. Gustafsson:

Phys. Rev. Lett. 22, 2464 (1989)

57.8 J. H. McGuire, J. C. Straton, W. C. Axmann, T. Ishi-

hara, E. Horsdal: Phys. Rev. Lett. 62, 2933 (1989)

57.9 J. S. Briggs, K. Taulbjerg: J. Phys. B 12, 2565

(1979)

57.10 J. H. McGuire: Adv. At. Mol. Opt. Phys. 29, 217 (1991)

57.11 E. Horsdal-Pederson, C. L. Cocke, M. Stöckli: Phys.

Rev. Lett. 57, 2256 (1986)

57.12 J. D. Jackson: Classical Electrodynamics (Wiley, New

York 1975) p. 286

57.13 O. L. Weaver, J. H. McGuire: Phys. Rev. A 32,1435

(1985)

57.14 J. H. McGuire, O. L. Weaver: J. Phys. 17, L583 (1984)

57.15 J. H. McGuire: Indian J. Phys. 62B, 261 (1988)

57.16 E. Horsdal-Pederson, P. Loftager, J. L. Rasmussen:

J. Phys. B 15, 7461 (1982)

57.17 R. Shakeshaft, J. Wadehra: Phys. Rev. A 22,968

(1980)

57.18 A. Igarashi, N. Toshima: Phys. Rev. A 46, R1159

(1992)

57.19 J. H. McGuire, N. C. Sil, N. C. Deb: Phys. Rev. A 34,

685 (1986)

57.20 V. Mergel, R. Dörner, M. Achler, Kh. Khayyat,

S. Lencinas, J. Euler, O. Jagutski, S. Nüttgens,

M. Unverzagt, L. Spielberger, W. Ru, R. Ali, J. Ull-

rich, H. Cederquist, A. Salin, C. J. Wood, R. E. Olson,

Dz. Belkic, C. L. Cocke, H. Schmidt-Böcking: Phys.

Rev. Lett. 79, 387 (1977)

57.21 S. G. Tolmanov, J. H. McGuire: Phys. Rev. A 62,

032771 (2000)

57.22 A. L. Godunov, A. L. Godunov, J. H. McGuire,

P. B. Ivanov, V. A. Shipakov, H. Merabet, R. Bruch,

J. Hanni, Kh. Shakov: J. Phys. B 34,5055

(2001)

57.23 H. T. Schmidt, H. Cederquist, A. Fardi, R. Schuch,

H. Zettergren, L. Bagge, A. Kallberg, J. Jensen,

K. G. Resfelt, V. Mergel, L. Schmidt, H. Schmidt-

Boecking, C. L. Cocke: Photonic, Electronic and

Atomic Collisions, ed. by J. Burgdoerfer, J. S. Co-

hen, S. Datz, C. R. Vane (Rinton, Princeton, NJ 2002)

p. 720

Part D 57

869

Classical Trajec

58. Classical Trajectory and Monte Carlo Techniques

The classical trajectory Monte Carlo (CTMC)

method originated with Hirschfelder, who

studied the H + D

exchange reaction using

a mechanical calculator [58.1]. With the availability

of computers, the CTMC method was actively

applied to a large number of chemical systems

to determine reaction rates, and ﬁnal state

vibrational and rotational populations (see, e.g.,

Karplus et al. [58.2]). For atomic physics problems,

a major step was introduced by Abrines and

Percival [58.3] who employed Kepler’s equations

and the Bohr–Sommerﬁeld model for atomic

hydrogen to investigate electron capture and

ionization for intermediate velocity collisions

of H

+ H. An excellent description is given by

Percival and Richards [58.4]. The CTMC method has

a wide range of applicability to strongly-coupled

systems, such as collisions by multiply-charged

ions [58.5]. In such systems, perturbation methods

fail, and basis set limitations of coupled-channel

molecular- and atomic-orbital techniques have

difﬁculty in representing the multitude of active

58.1 Theoretical Background ....................... 869

58.1.1 Hydrogenic Targets .................... 869

58.1.2 Nonhydrogenic One-Electron

Models..................................... 870

58.1.3 Multiply-Charged Projectiles

and Many-Electron Targets......... 870

58.2 Region of Validity ................................ 871

58.3 Applications ........................................ 871

58.3.1 Hydrogenic Atom Targets............ 871

58.3.2 Pseudo One-Electron Targets ...... 872

58.3.3 State-Selective Electron Capture.. 872

58.3.4 Exotic Projectiles ....................... 873

58.3.5 Heavy Particle Dynamics ............ 873

58.4 Conclusions ......................................... 874

References .................................................. 874

excitation, electron capture, and ionization

channels. Vector- and parallel-processors now

allow increasingly detailed study of the dynamics

of the heavy projectile and target, along with the

active electrons.

58.1 Theoretical Background

58.1.1 Hydrogenic Targets

For a simple three-body collision system comprised of

a fully-stripped projectile (a), a bare target nucleus (b),

and an active electron (c), one begins with the classical

Hamiltonian for the system,

H = p

/2m

+ p

/2m

+ p

/2m

+ Z

, (58.1)

where p

are the momenta and Z

are the

Coulomb potentials between the individual particles.

From (58.1), one obtains a set of 18 coupled, ﬁrst-order

differential equations arising from the necessity to de-

termine the time evolution of the Cartesian coordinates

of each particle,

/dt = ∂H/∂p

, (58.2)

and their corresponding momenta,

/dt =−∂H/∂q

. (58.3)

Five random numbers, constrained by Kepler’s equa-

tion, are then used to initialize the plane and eccentricity

of the electron’s orbit, and another is used to determine

the impact parameter within the range of interac-

tion [58.4, 5]. A fourth-order Runge–Kutta integration

method is suitable because of its ease of use and its abil-

ity to vary the time step-size. This latter requirement

is essential since it is not uncommon for the time step

to vary by three orders of magnitude during a single

trajectory.

In essence, the CTMC method is a computer experi-

ment. Total cross sections for a particular process are

determined by

= (N

/N)πb

max

, (58.4)

Part D 58

870 Part D Scattering Theory

where N is the total number of trajectories run within

a given maximum impact parameter b

max

,andN

is the

number of positive tests for a reaction, such as electron

capture or ionization. Angle and energy differential cross

sections are easily generalized from the above. As in

an experiment, the cross section given by (58.4)has

a standard deviation of

∆σ

= σ[(N − N

)/NN

]

1/2

, (58.5)

which for large N is proportional to 1/N

1/2

. Here lies

one of the major difﬁculties associated with the CTMC

method: it takes considerable computation time to deter-

mine minor or highly differential cross sections. Present

day desktop workstations can provide a partial remedy

of this statistics problem. To decrease the statistical er-

ror of a cross section by a factor of two, four times as

many trajectories must be evaluated.

58.1.2 Nonhydrogenic One-Electron Models

For many-electron target atoms, it is sometimes ade-

quate to treat the problem within a one-electron model

and employ the independent electron approximation to

approximate atomic shell structure [58.6]. For an ac-

curate calculation, it is necessary to use an interaction

potential that simulates the screening of the target nu-

cleus by the electrons. One can simply apply a Coulomb

potential with an effective charge Z

eff

obtained from, for

example, Slater’s rules. Then, the computational proce-

dure is the same as for the hydrogenic case. However,

the boundary conditions for the long- and short-range

interactions are poorly satisﬁed.

To improve the electronic representation of the

target, potentials derived from quantum mechanical

calculations are now routinely used. Here, the simple

solution of Kepler’s equation cannot be applied. How-

ever, Peach et al. [58.7]andReinhold and Falc

on [58.8]

have provided the appropriate methods that yield a target

representation that is correct under the microcanonical

distribution. The method of Reinhold and Falc

on is pop-

ular because of its ease of use and generalizability. For

the effective interaction potential, Garvey et al. [58.9]

have performed a large set of Hartree–Fock calculations,

and have parametrized their results in the form

V(R) =−[Z − NS(R)]/R ,

(58.6)

with the screening of the core given by

S(R) = 1 −{(η/ξ)[exp(ξR) − 1]+1}

−1/2

, (58.7)

where Z and N denote the nuclear charge and number

of nonactive electrons in the target core, and η and ξ are

screening parameters. Screening parameters are given

in [58.9] for all ions and atoms with Z ≤ 54. This

potential can also be used for the representation of

partially-stripped projectile ions.

58.1.3 Multiply-Charged Projectiles

and Many-Electron Targets

Multiple ionization and electron capture mechanisms

in energetic collisions between multiply-charged ions

and many-electron atoms is poorly understood because

major approximations must be made to solve a many-

electron problem associated with transitions between

two centers. For a representative collision system such

+ B → A

(q− j)+

+ B

+ (i − j)e

−

, (58.8)

it is essential that the theoretical method be able to

predict simultaneously the various charge states of the

projectile and recoil ions, and also the energy and angu-

lar spectra of the ejected electrons. Theoretical methods

based on the independent electron model fail because

the varying ionization energies of the electrons are not

well represented by a constant value, especially for outer

shells. To present, only the nCTMC method, which is

a direct extension of the hydrogenic CTMC method to

an n-electron system, has been able to make reasonable

predictions of the cross sections and scattering dynam-

ics of such strongly-coupled systems [58.10]. As such,

the number of coupled equations rises to 6n + 12, where

n is the number of electrons included in the calculation.

However, computing time does not increase linearly,

since modern vector-processors become very efﬁcient

with coupled equations for multiples of 64.

In the nCTMC technique, all interactions of the

projectile and target nuclei with each other and the elec-

trons are implicitly included in the calculations. The

inclusion of all the particles then allows a direct determi-

nation of their angular scattering, along with an estimate

of the energy deposition to electrons and heavy par-

ticles. Post collision interactions are included between

projectile and recoil ions with the electrons; however,

electron–electron interactions are introduced only in the

bound initial state via a screening factor in a central-

ﬁeld approximation. This theoretical model has been

very successful in predicting the single and double dif-

ferential cross sections for the ionized electron spectra.

Moreover, since a ﬁxed target nucleus approximation is

not used, this method has been the only one available to

help understand and predict the results for the new ﬁeld

of recoil-ion momentum spectroscopy [58.11].

Part D 58.1

Classical Trajectory and Monte Carlo Techniques 58.3 Applications 871

Only a modest amount of work has been com-

pleted on molecular targets. At present, only an H

target has been formulated for application to the CTMC

method [58.12]. For H

, a ﬁxed internuclear axis is

assumed which is then randomly orientated for each tra-

jectory. The electrons are initialized in terms of two

one-electron microcanonical distributions constructed

from the quantum mechanical wave function for the

ground state of H

. Total cross sections for a variety

of projectiles are in reasonable accord with experiment.

The effect of the orientation of the molecular axis on the

cross sections was also investigated, along with tests as

to the validity of assuming that the cross sections for H

are simply the product of twice the H values.

58.2 Region of Validity

The CTMC method has a demonstrated region of ap-

plicability for ion–atom collisions in the intermediate

velocity regime, particularly in the elucidation of both

heavy-particle and electron collision dynamics. The

method can be termed a semiclassical method in that

the initial conditions for the electron orbits are deter-

mined by quantum mechanically determined interaction

potentials with the parent nucleus. Since the method

is most applicable to strongly-coupled systems, it has

been applied successfully to a variety of intermediate

energy multiply-charged ion collisions. Figure 58.1 de-

scribes pictorially the regions of validity of theoretical

models. Both the atomic orbital (AO)andmolecularor-

bital (MO) basis set expansion methods (Chapt. 50)work

well until ionization strongly mediates the collision,

since the theoretical description of the ionization con-

tinuum is not well-founded and relies on pseudostates

to span all ejected electron energies and angles. We

have arbitrarily limited these methods to a projectile

charge to target charge ratio of, Z

,  8, since

above this value the number of terms in the basis set

becomes prohibitively large. The CTMC method does

not include molecular effects, and thus it is restricted

from low velocities, except in the case of high-charge-

state projectiles that capture electrons into high-lying

Rydberg states which are well-described classically.

Likewise, at high velocities the method is inapplica-

ble in the perturbation regime where quantum tunneling

is important, and thus is restricted to strongly cou-

100

0.1

0.1 1 10 100

v/v

CTMC

CDW

BORN

Fig. 58.1 Approximate regions of validity of various theo-

retical methods. Z

is the ratio of the projectile charge

to the target charge, and v/v

is the ratio of the collision

velocity to the velocity of the active target electron. The-

oretical methods: molecular orbital (MO), atomic orbital

(AO), classical trajectory Monte Carlo (CTMC), contin-

uum distorted wave (CDW), and ﬁrst-order perturbation

theory (BORN)

pled systems. The continuum distorted wave (CDW)

method (Chapt. 52) greatly extends the region of ap-

plicability of ﬁrst-order perturbation methods and has

demonstrated validity in high-charge state ionization

collisions.

58.3 Applications

58.3.1 Hydrogenic Atom Targets

The original application of the CTMC method to atomic

physics collisions were done on the H

+ Hsys-

tem [58.3]. Here, the electron capture and ionization

total cross sections were found to be in very good accord

with experiment. The Abrines and Percival procedure

casts the coupled equations into the c.m. coordinate sys-

tem to reduce the three-body problem to 12 coupled

equations. However, this reduction complicates exten-

Part D 58.3

872 Part D Scattering Theory

sions of the code to laser processes and collisions in

electric ﬁelds or with many electrons.

An ideal application for the CTMC method is

for collisions involving excited targets. Such pro-

cesses are well-described classically, and basis set

expansion methods show limited applicability due to

computer memory constraints. Considerable early work

has been done on Rydberg atom collisions which in-

cludes state-selective electron capture, ionization, and

electric ﬁelds [58.13–15]. Presently, there is a resur-

gence of work on Rydberg atom collisions because new

crossed-ﬁeld experimental techniques allow the produc-

tion of these atoms with speciﬁc spatial orientations and

eccentricities [58.16].

For hydrogenic ion–ion collision processes, one

must be careful to apply the CTMC method only

for projectile charges Z

≥ Z

because after the ini-

tialization of the active electron’s orbit and energy,

there is no classical constraint on the orbital energy of

a captured electron. For a low-charge-state ion collid-

ing with a ground state highly charged ion, one will

obtain unphysical results because a captured electron

will tend to preserve its original binding energy. Thus,

excess probability will be calculated for electron or-

bits that lead to unrealistic deeply bound states of the

projectile.

58.3.2 Pseudo One-Electron Targets

Collisions involving alkali atoms are of interest be-

cause of their relevance to applied programs, such as

plasma diagnostics in tokamak nuclear fusion reac-

tors. They are also a testing ground for theoretical

methods since experimental benchmarks are difﬁcult

to realize with hydrogenic targets, but are amenable

for such cases as alkali atoms. In such collisions, it

is essential that a theoretical formalism be used that

correctly simulates the screening of the nucleus by the

core electrons (Sect. 58.1.2), since a simple −Z

eff

Coulomb potential is inadequate for both large and

small R.

One can also apply the methods of Sect. 58.1.2 to

partially or completely ﬁlled atomic shells. This works

reasonably well for collisions with a low charge state

ion such as a proton, but fails for strong collisions in-

volving multiply-charged ions. The reason is that the

independent electron approximation [58.6] must be ap-

plied to the calculated transition probabilities in order to

simulate the shell structure. This latter method can only

maintain its validity if the transition probability is low.

Otherwise, it will greatly overestimate the multiple elec-

tron removal processes since the ﬁrst ionization potential

is inherently assumed for each subsequent electron that

is removed from the shell, leading to an underestimate

of the energy deposition.

58.3.3 State-Selective Electron Capture

One of the powers of the CTMC method is that it

can be applied to electron capture and excitation of

high-lying states that are not accessible with basis

set expansion techniques [58.17]. For the C

+ Li

system, where AO calculations and experimental data

exist, the CTMC method agrees quite favorably with

both [58.18].

The procedure is ﬁrst to deﬁne a classical number n

related to the calculated binding energy E of the ac-

tive electron to either the projectile (electron capture) or

target nucleus (excitation) as

E =−Z





(58.9)

Then, n

is related to the principal quantum number n

of the ﬁnal state by the condition [58.19]





n −



(n− 1)



1/3

< n

≤





n +



(n+ 1)



1/3

(58.10)

From the electron’s normalized classical angular mo-

mentum l

= (n/n

)(r× k), l

is related to the orbital

quantum number l of the ﬁnal state by

l < l

≤ l + 1 . (58.11)

The magnetic quantum number m

is then obtained from

− 1

2l+ 1

≤

+ 1

2l+ 1

(58.12)

where l

is the z-projection of the angular momentum

obtained from calculations [58.20]. In principle, it is also

possible to analyze the ﬁnal-state distributions from the

effective quantum number

∗

= n− δ

, (58.13)

where δ

is the quantum defect. In this latter case,

it is necessary to sort the angular momentum quan-

tum numbers ﬁrst, and then sort the principal quantum

numbers.

The CTMC method has been widely applied to col-

lisions of multiply-charged ions and hydrogen targets

in the nuclear fusion program. Here, the calculated nl

Part D 58.3

Classical Trajectory and Monte Carlo Techniques 58.3 Applications 873

charge exchange cross sections are used to predict the

resulting visible and UV line emissions arising after

electron capture to high principle quantum numbers.

These line emission cross sections are routinely used as

a diagnostic for tokamak fusion plasmas [58.21]. Like-

wise, for low energy collisions, CTMC results have been

used to provide an explanation for the X-ray emission

discovered from comets as they orbit through our solar

system [58.22, 23].

58.3.4 Exotic Projectiles

The study of collisions involving antimatter projectiles,

such as positrons and antiprotons, is a rapidly growing

ﬁeld which is being spurred on by recent experimental

advances. Such scattering processes are of basic inter-

est, and they also contribute to a better understanding of

normal matter-atom collisions. Antimatter-atom studies

highlight the underlying differences in the dynamics of

the collision, as well as on the partitioning of the overall

scattering. In the Born approximation, ionization cross

sections depend on the square of the projectile’s charge

and are independent of its mass. Thus, the comparison

of the cross sections for electron, positron, proton and

antiproton scattering from a speciﬁc target gives a di-

rect indication of higher-order corrections to scattering

theories.

Early work using the CTMC method concentrated

on the spectra of ionized electrons for antimatter pro-

jectiles [58.24]. Later work focused on the angular

scattering of the projectiles during electron removal col-

lisions, such as positronium formation, on ratios of the

electron removal cross sections, and on ejected elec-

tron ‘cusp’ and ‘anticusp’ formation. A recent review

that compares various theoretical results and available

experiments is given in [58.25].

58.3.5 Heavy Particle Dynamics

A major attribute of the CTMC method is that it inher-

ently includes the motion of the heavy particles after the

collision. A straight-line trajectory for the projectile is

not assumed, nor is the target nucleus constrained to be

ﬁxed. This allows one to compute easily the differential

cross sections for projectile scattering or the recoil mo-

menta of the target nucleus. As a computational note, the

angular scattering of the projectile should be computed

from the momentum components, not the position coor-

dinates after the collision, since faster convergence of the

cross sections using the projectile momenta is obtained.

For recoil ion momentum transfer studies, one must ini-

tialize the target atom such that the c.m. of the nucleus

plus its electrons has zero momentum so that there is no

initial momentum associated with the target. A common

error is to initialize only the target nucleus momenta

to zero. Then the target atom after a collision has an

artiﬁcial residual momentum that is associated with the

Compton proﬁle of the electrons because target–electron

interactions are included in the calculations. Examples

of recoil and projectile scattering cross sections are given

in [58.10,11].

The ﬁeld of recoil ion momentum spectroscopy is

rapidly expanding and the CTMC theoretical method has

impacted the interpretation and understanding of experi-

mental results because the method inherently provides

a kinematically complete description of the collision

products. For the studied systems, primarily He targets

because of experimental constraints, it is necessary that

a theoretical method be able to follow all ejected elec-

trons and the heavy particles after a collision. As an

example, it has been possible to observe the backward

recoil of the target nucleus in electron capture reac-

tions, which is due to conservation of momentum when

the active electron is transferred from the target’s to

the projectile’s frame of [58.26]. Theoretical methods

are being tested further with the recent development of

magneto-optical-taps (MOT) that provide frozen alkali

metal atomic targets (T  1 mK) from which to perform

recoil ion studies [58.27].

For three- and four-body systems, it is now possible

to measure the momenta of all collision products. These

observations provide a severe test of theory, since all

projectile and target interactions must be included in

calculations. The CTMC method includes all projectile

interactions with the target nucleus and electrons. Thus,

it is possible to calculate fully differential cross sections.

It is of interest that recent triply differential cross sec-

tions calculated using the CTMC method compare very

favorably with sophisticated continuum distorted wave

methods [58.28].

The CTMC technique allows one to incorporate

electrons on both the projectile and target nuclear cen-

ters. All interactions between centers are included. The

only interaction that needs to be approximated is the

electron–electron interactions on a given center. Here,

simple screening parameters derived from Hartree–Fock

calculations are employed to eliminate nonphysical

autoionization. Within this many-electron model, the

signatures of the electron–electron and electron–nuclear

interactions on the dynamics of the collisions have

been observed [58.29, 30]. Further, projectile ionization

studies can be undertaken [58.31].

Part D 58.3

874 Part D Scattering Theory

58.4 Conclusions

In many ways it is surprising that a classical model

can be successful in a quantum mechanical world, es-

pecially since the classical radial distribution for the

hydrogen atom is described so poorly. However, hy-

drogen’s classical momentum distribution is exactly

equivalent to the quantum one, and since collision

processes are primarily determined by velocity match-

ing between projectile and electron, reasonable results

can be expected. Moreover, the CTMC method pre-

serves conservation of ﬂux, energy, and momentum;

and Coulomb scattering is the same in both quantal and

classical frameworks.

Of signiﬁcant importance is that the CTMC method

is not restricted to one-electron systems and can eas-

ily be extended to more complicated systems involving

electrons on both projectile and target. For these latter

cases, multiple electron capture and ionization reactions

can be investigated.

References

58.1 J. Hirschfelder, H. Eyring, B. Topley: J. Chem. Phys.

4,170(1936)

58.2 M. Karplus, R. N. Porter, R. D. Sharma: J. Chem.

Phys. 43, 3259 (1965)

58.3 R. Abrines, I. C. Percival: Proc. Phys. Soc. 88,861

(1966)

58.4 I. C. Percival, D. Richards: Adv. At. Mol. Phys. 11,1

(1975)

58.5 R. E. Olson, A. Salop: Phys. Rev. A 16, 531 (1977)

58.6 J. H. McGuire, L. Weaver: Phys. Rev. A 16, 41 (1977)

58.7 G. Peach, S. L. Willis, M. R. C. McDowell: J. Phys. B

18, 3921 (1985)

58.8 C. O. Reinhold: Phys. Rev. A 33, 3859 (1986)

58.9 R. H. Garvey, C. H. Jackman, A. E. S. Green: Phys.

Rev. A 12, 1144 (1975)

58.10 R. E. Olson, J. Ullrich, H. Schmidt-Böcking: Phys.

Rev. A 39, 5572 (1989)

58.11 C. L. Cocke, R. E. Olson: Phys. Rep. 205, 153 (1991)

58.12 L. Meng, C. O. Reinhold, R. E. Olson: Phys. Rev. A

40, 3637 (1989)

58.13 R. E. Olson: J. Phys. B 13,483(1980)

58.14 R. E. Olson: Phys. Rev. Lett. 43, 126 (1979)

58.15 R. E. Olson, A. D. MacKellar: Phys. Rev. Lett. 46,1451

(1981)

58.16 D. Delande, J. C. Gay: Europhys. Lett. 5, 303 (1988)

58.17 R. E. Olson: Phys. Rev. A 24, 1726 (1981)

58.18 R. Hoekstra, R. E. Olson, H. O. Folkerts, W. Wolfrum,

J. Pascale, F. J. de Heer, R. Morgenstern, H. Winter:

J. Phys. B 26, 2029 (1993)

58.19 R. C. Becker, A. D. MacKellar: J. Phys. B 17,3923

(1984)

58.20 S. Schippers, P. Boduch, J. van Buchem, F. W. Bliek,

R. Hoekstra, R. Morgenstern, R. E. Olson: J. Phys. B

28, 3271 (1995)

58.21 H. Anderson, M. G. von Hellermann, R. Hoek-

stra, L. D. Horton, A. C. Howman, R. W. T. Konig,

R. Martin, R. E. Olson, H. P. Summers: Plasma Phys.

Control. Fusion 42, 781 (2000)

58.22 P.Beiersdorfer,R.E.Olson,G.V.Brown,H.Chen,

C. L. Harris, P. A. Neill, L. Schweikhard, S. B. Ut-

ter, K. Widmann: Phys. Rev. Lett. 85,5090

(2000)

58.23 P. Beiersdorfer, C. M. Lisse, R. E. Olson, G. V. Brown,

H. Chen: Astrophys. J. 549, 147 (2001)

58.24 R. E. Olson, T. J. Gay: Phys. Rev. Lett. 61, 302

(1988)

58.25 D. R. Schultz, R. E. Olson, C. O. Reinhold: J. Phys. B

24, 521 (1991)

58.26 V. Frohne, S. Cheng, R. Ali, M. Raphaelian,

C. L. Cocke, R. E. Olson: Phys. Rev. Lett. 71,696

(1993)

58.27 J. W. Turkstra, R. Hoekstra, S. Knoop, D. Meyer,

R. Morgenstern, R. E. Olson: Phys. Rev. Lett. 87,

123202 (2001)

58.28 J. Fiol, R. E. Olson: J. Phys. B 35, 1759 (2002)

58.29 H. Kollmus, R. Moshammer, R. E. Olson, S. Hag-

mann, M. Schulz, J. Ullrich: Phys. Rev. Lett. 88,

103202 (2002)

58.30 J.Fiol,R.E.Olson,A.C.F.Santos,G.M.Sigaud,

E. C. Montenegro: J. Phys. B 34, 503 (2001)

58.31 R. E. Olson, R. L. Watson, V. Horat, K. E. Zaharakis:

J. Phys. B 35, 1893 (2002)

Part D 58

875

Collisional Bro

59. Collisional Broadening of Spectral Lines

One-photon processes only are discussed and

aspects of line broadening directly related to

collisions between the emitting (or absorbing)

atom and one perturber are considered. Molecular

lines and bands are not considered here. Pointers

to other aspects are included and a comprehensive

bibliography of work on atomic line shapes,

widths, and shifts already exists [59.1–7]. The

perturber may be an electron, a neutral atom or

an atomic ion and can interact weakly or strongly

with the emitter. The emitter is either a hydrogenic

or nonhydrogenic atom that is either neutral or

ionized. In general, transitions in nonhydrogenic

atoms can be treated as isolated, that is the

separation between neighboring lines is much

greater than the width of an individual line.

When the emitter is hydrogen or a hydrogenic

ion, the additional degeneracy of the energy

levels with respect to orbital angular momentum

quantum number means that lines overlap and

are coupled.

Pressure broadening is a general term that

describes any broadening and shift of a spectral

line produced by ﬁelds generated by a background

gas or plasma. The term Stark broadening implies

that the perturbers are atomic ions and/or

electrons, and collisional broadening implies

that the ‘collision’ model is appropriate; this

term is often used to describe an isolated line

perturbed by electrons. Neutral atom broadening

indicates neutral atomic perturbers; this implies

short-range emitter-perturber interactions which

in turn inﬂuence the approximations made.

59.1 Impact Approximation ......................... 875

59.2 Isolated Lines...................................... 876

59.2.1 Semiclassical Theory .................. 876

59.2.2 Simple Formulae ....................... 877

59.2.3 Perturbation Theory................... 878

59.2.4 Broadening

by Charged Particles .................. 879

59.2.5 Empirical Formulae ................... 879

59.3 Overlapping Lines ................................ 880

59.3.1 Transitions in Hydrogen and

Hydrogenic Ions ........................ 880

59.3.2 Infrared and Radio Lines............ 882

59.4 Quantum-Mechanical Theory ................ 882

59.4.1 Impact Approximation ............... 882

59.4.2 Broadening by Electrons ............ 883

59.4.3 Broadening by Atoms ................ 884

59.5 One-Perturber Approximation .............. 885

59.5.1 General Approach and Utility...... 885

59.5.2 Broadening by Electrons ............ 885

59.5.3 Broadening by Atoms ................ 886

59.6 Uniﬁed Theories and Conclusions .......... 888

References .................................................. 888

General reviews of the theory of pressure

broadening have been given [59.8–10], and

Chapt. 2,Chapt.10,Chapt.14,Chapt.19,Chapt.45,

Chapt. 47,andChapt.86 discusstopicsrelevantto

the theory of collisional broadening of spectral

lines. The International Conference on Spectral

Line Shapes (ICSLS) is devoted exclusively to this

subject.

59.1 Impact Approximation

If the perturbers are rapidly moving, the broad-

ening and shift of the line arise from a series

of binary collisions between the atom and one

of the perturbers. The theory assumes that al-

though weak collisions may occur simultaneously,

strong collisions are relatively rare and only occur

one at a time. The impact approximation is valid

wτ 1 ,

V τ/

 1 , (59.1)

where w is the half width at half maximum, τ is the

average time of collision and

V is the average emitter-

Part D 59

876 Part D Scattering Theory

perturber interaction. It is not only widely applicable

to electron and neutral atom broadening, but also, for

certain plasma conditions, to broadening by atomic ions.

The power radiated per unit time and per unit interval in

circular frequency ω, in terms of the line proﬁle I

(

)

,is

(

)

(

)

. (59.2)

For an isolated line produced by a transition from an up-

per energy level i to a lower level f , the line proﬁle is

Lorentzian with a shift d:

(

)



∗

∆

∗





ω −ω

−d



(59.3)

and if the proﬁle is for a transition between an upper

set of levels i, i



and a lower set f, f



with quantum

numbers J

, J



, J

, and J



,then

(

)









∗

∆



∗



(59.4)





∗





w +id−i



ω−ω





−1



∗





where I is the unit operator and w and d are width and

shift operators. In (59.4) ∆ is an operator corresponding

to the dipole line strength deﬁned by



∗

∆



∗



≡ q











(59.5)

where r represents the internal emitter coordinates, q

(

4π

)

is the square of the electronic charge, and 

is the permittivity of vacuum in SI units of J m. On taking

the average over all degenerate magnetic sublevels,

(

)









∗



∆





∗



(59.6)





∗





w +id−i



ω −ω





−1



∗





in terms of reduced matrix elements that are independent

of magnetic quantum numbers. They are deﬁned by



∗

∆



∗



= D

ifi





∗



∆





∗



(59.7)

where

ifi





(

−1

)



−M





1 J

−M

µ M





1 J



−M



µ M





(59.8)





∗





w +id−i(ω −ω





∗







ifi







∗





w +id−i(ω −ω





∗



(59.9)

with s =−1, 1. For the line proﬁle (59.3), the width and

each have a single matrix element:

γ = 2w =



∗



, d =



∗



(59.10)

where γ is the full width at half maximum.

Throughout the rest of this article it will be as-

sumed that collisions only connect the set of upper

levels i, i



or the set of lower levels f, f



which is valid

when w  ω; pressure broadening of spectral lines is

also assumed to be independent of Doppler broaden-

ing. However, for microwave spectra of molecules, w

can be of the order of ω and collisions connecting the

upper to the lower levels become important; for further

details see Ben-Reuven [59.11, 12]. Also for microwave

spectra, pressure broadening and Doppler broadening

cannot be considered to be independent effects and

a generalized theory has been developed by Ciuryło and

Pine [59.13].

59.2 Isolated Lines

59.2.1 Semiclassical Theory

The motion of the perturber relative to the emitter is

treated classically and is assumed to be independent

of the internal states of the emitter and perturber. This

common trajectory is speciﬁed by an emitter–perturber

separation

R≡ R

(

b, v, t

)

, b· v = 0 ,

(59.11)

where v is the relative velocity and b is the impact

parameter. The time-dependent wave equation for the

emitter-perturber system is

dΨ

= HΨ

(59.12)

and the eigenfunctions ψ

for the unperturbed emitter

obey

= E

, i = 0, 1, 2,... . (59.13)

Part D 59.2