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

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732 Part D Scattering Theory

applicable here. We will discuss various theoretical ap-

proaches to this process in Sect. 48.2 and give references

to experimental results in Sect. 48.3.

Positronium formation in the ground state has

a threshold which is 1/4 of a Hartree (6.802 85 eV) below

the ionization threshold of the target. This means that

it is normally the lowest inelastic channel in positron

scattering from neutral atoms. For atoms with a small

ionization potential, such as the alkalis, this channel is

always open. The energy range between the positronium

threshold and the ﬁrst excited state of the atom is known

as the Ore gap. In this range, positronium formation is

the only possible inelastic process.

48.1.2 Annihilation

Annihilation is a process in which an electron–positron

pair is converted into two or more photons. It can oc-

cur either directly with a bound atomic electron or after

positronium formation has taken place. The direct anni-

hilation cross section for a positron of momentum k

colliding with an atomic or molecular target can be

written as [48.2]

eff



πa



(48.1)

where Z

eff

can be thought of as the effective number

of electrons in the target with which the positron can

annihilate. If Ψ(r

, r

,... ,r

, x) is the wave function

for the system of a positron, with coordinate x, colliding

with an N-electron target, then

eff



i=1



...dr



Ψ(r

, r

,... ,r

; r

)



(48.2)

While this formula can be naively derived by assuming

that the positron can only annihilate with an electron

if it is at the identical location, it actually follows

from a quantum electrodynamical treatment of the pro-

cess [48.3]. If the wave function Ψ is approximated by

the product of the undistorted target wave function times

a positron scattering function F(x),then

eff



drρ(r)



F(r)



, (48.3)

where ρ is the electron number density of the target.

Thus, in the Born approximation, where F is taken as

aplanewave,Z

eff

simply becomes the total number of

electrons Z in the target. However, a pronounced en-

hancement of the annihilation rate in the vicinity of the

Ps formation threshold due to virtual Ps formation was

predicted [48.4, 5]. Subsequently, the Born approxima-

tion was shown to be grossly inadequate by the San

Diego group, who found annihilation rates Z

eff

at room

temperature which are an order of magnitude larger for

some atoms and even up to ﬁve orders of magnitude

larger in large hydrocarbon molecules. Furthermore,

there is evidence that only the outer shell of electrons

takes part in the annihilation process. Two mechanisms

have been proposed in order to explain these large val-

ues for Z

eff

. One involves the enhancement of the direct

annihilation process below the Ps formation threshold

due to the attractive nature of the positron–electron in-

teraction, which increases the overlap of positron and

electron densities on the atom or molecule. The sec-

ond mechanism is referred to as resonant annihilation,

which occurs after the positron has been captured into

a Feshbach resonance, where the positron is bound to

a vibrationally excited molecule. A summary of the

above results can be found in the recent article by Barnes

et al. [48.6].

When a positron annihilates with an atomic elec-

tron, two 511 keV photons is the most likely result, if the

positron–electron pair are in a singlet spin state (para-

positronium). In the centre-of-mass frame of the pair, the

photons are emitted in opposite directions to conserve

momentum. However, in the laboratory frame the bound

electron has a momentum distribution which is reﬂected

in the photon directions not being exactly 180 degrees

apart. This slight angular deviation, called the angu-

lar correlation, can be measured, and gives information

about the momentum distribution of the bound electrons.

This quantity is given by [48.3]

S(q) =



i=1



...dr

i−1

i+1

...dr

(48.4)



dxe

iq·x



Ψ(r

, r

,... ,r

; x)δ(r

− x)



where q is the resultant momentum of the annihilating

pair. In evaluating this quantity, the positron is assumed

to be thermalized in the gas before undergoing anni-

hilation. Experimentally, only one component of q is

measured, so that S(q) is integrated over the other two

components of the momentum to obtain the measured

quantity. The spin triplet component of an electron–

positron pair (orthopositronium) can only decay with

the emission of three or more photons which do not

have well deﬁned energies. This is a much less proba-

ble process than the two photon decay from the singlet

component.

Part D 48.1

Positron Collisions 48.2 Theoretical Methods 733

48.2 Theoretical Methods

The basic theoretical approaches to the calculation of

positron scattering from atoms and molecules were orig-

inally developed for electron scattering and later applied

to the positron case. Thus, we emphasize here only the

differences that arise between the electron and positron

cases, both in the theoretical formulations, and in later

sections, in the nature of the results.

The lowest-order interaction between a free positron

and an atomic or molecular target is the repulsive static

potential of the target







(48.5)

where ψ

is the unperturbed target wave function and

V is the electrostatic interaction potential between the

positron and the target. Since this interaction has the

opposite sign from that for electron scattering, the static

potential also has the opposite sign in these two cases.

On the other hand, the next higher-order of interaction is

polarization, which arises from the distortion of the atom

by the incident particle. If we represent this distortion

of the target to ﬁrst-order by the wave function ψ

,asin

the polarized-orbital approximation, for example [48.9],

then the polarization interaction can be represented by

the potential







(48.6)

This potential is attractive for both positron and electron

scattering and has an asymptotic form with leading term

−α

/2r

,whereα

is the static dipole polarizability

(Sect. 23.2.3) of the target. Thus, the static and polariza-

tion potentials for positron scattering from ground state

systems are of opposite sign and tend to cancel one an-

other. This leads to very different behaviour from the

electron case where they are of the same sign. In par-

ticular, the elastic scattering cross sections for positron

scattering from an atom are much smaller than for elec-

tron scattering, and the phase shifts (Sect. 47.1.1)have

very different magnitudes and dependences on energy.

This is illustrated for the case of scattering from helium

in Figs. 48.1–48.3, where the results of the highly accu-

rate variational calculations for scattering by electrons

and positrons are shown. Note the difference in sign be-

tween the electron and positron s-wave phase shifts for

very small values of the incident momentum. The fact

that the positron phase shift goes through zero leads

to the Ramsauer minimum in the positron total cross

section, as shown in Fig. 48.3. The large difference in

magnitudes between the electron and positron s-and

s-wave phase shift

Momentum (a.u.)

0.0

–0.1

–0.5

–0.9

–1.3

0.4 0.8 1.2

Fig. 48.1 Variational s-wave phase shifts for electron [48.7]

(dashedline) and positron [48.8](solidline) scattering from

helium atoms

p-wave phase shifts leads to the large difference in the

total elastic cross sections as shown.

Higher-order terms in the interaction potential may

also give important contributions to scattering cross

sections. For a detailed discussion, see the article by

Drachman and Temkin [48.10].

A simple potential scattering calculation using the

sum of the static and polarization potentials, but with-

out the exchange terms that are present for the electron

case, can be applied to elastic scattering calculations for

closed shell systems (see Sect. 48.3.3).

The potentials deﬁned above can also be used in

a distorted-wave approximation (Chapt. 47) which can

be applied to excitation and ionization by positron im-

pact. Once again, the complicated exchange terms which

arise in electron scattering are absent here. Sienkiwicz

and Baylis [48.11] have included such potentials in

a Dirac–Fock formulation of positron scattering which

treats the positron as an electron with negative energy.

For positrons with high enough incident energies

(≈ 1 keV), the ﬁrst Born approximation will become

valid (Chapt. 47). Since the ﬁrst Born approximation

is independent of the sign of the charge of the incident

particles, this indicates that as the incident energy in-

creases, the corresponding cross sections for electron

Part D 48.2

734 Part D Scattering Theory

p-wave phase shift

Momentum (a.u.)

0.00

0.3

0.2

0.1

0.0

0.40 0.80 1.20

Fig. 48.2 Variational p-wave phase shifts for elec-

tron [48.7](dashed line) and positron [48.15](solid line)

from helium atoms

and positron scattering will eventually merge. From ﬂux

conservation arguments, this means that the positron-

ium formation cross section will rapidly decrease as the

incident energy increases. In fact, from experimental

measurements, the total cross sections (summed over all

possible channels) for electron and positron scattering

appear to merge at a much lower energy than the cross

sections for individual channels [48.12]. More elaborate

calculations for high energy scattering have been car-

ried out in the eikonal-Born series [48.13](Chapt.47).

These approximations allow for both polarization and

absorption (i. e., inelastic processes) and yield good

agreement with elastic experimental measurements of

differential cross sections at energies above 100 eV.

A detailed analysis [48.13, 14] of the various contri-

butions to the scattering indicates that absorption effects

due to the various open inelastic channels plays a much

more important role here than for electron scattering.

A more elaborate treatment of positron scattering is

based on the close-coupling approximation (Chapt. 47),

where the wave function for the total system of positron

plus target is expanded using a basis set comprised of

the wave functions of the target. Once again, there are no

exchange terms involving the positron and, in principle,

a complete expansion including the continuum states of

the target would include the possibility of positronium

formation. However, such an expansion is not practi-

Momentum (a.u.)

0.00

0.40 0.80

0.1

Total cross section (πa

)

Fig. 48.3 Total elastic cross sections for electron (dashed

line) and positron (solid line) from helium atoms calculated

from the phase shifts shown in Figs. 48.1 and 48.2.The

higher-order phase shifts were calculated from effective

range theory (Sect. 47.1.1)

cable if one wants to calculate explicit cross sections

for positronium formation. Even in cases where such

cross sections are not required, the considerable effect

that the positronium formation channels can have on

the other scattering cross sections is best included by

a close-coupling expansion that includes terms repre-

senting positronium states plus the residual target ion.

There is a problem of double counting of states in

such an expansion but, in practice, this does not ap-

pear to be a problem if the number of states in the

expansion is not large. Also, in many cases, additional

pseudostates have to be included in the expansion in

order to correctly represent the long-range polarization

interaction.

A close-coupling expansion including positronium

states is a two-centre problem, i. e., it includes the cen-

tres of mass of both the target and the positronium states.

Since positronium is a light system, the semi-classical

approach often used to treat rearrangement collisions

between heavy systems (Chapt. 50) is not applicable

here. This means that one is faced with a problem of

considerable computational complexity [48.16–21].

Another way to take into account the effects of open

inelastic channels without the complications of a full

close-coupling approach is to use optical potentials.

These are often based on a close-coupling formal-

Part D 48.2

Positron Collisions 48.3 Particular Applications 735

ism [48.14, 22] and lead to a complex potential, the

real part of which represents distortions of the target

(such as polarization) while the complex part allows for

absorption (i. e., ﬂux into open channels not explicitly

represented).

Brayand Stelbovics[48.23] have applied the conver-

gent close-coupling method to the scattering of positrons

from atoms. This method includes contributions from

the continuum states of the target and sufﬁcient terms in

the expansion are included to ensure numerical conver-

gence. At the present time, however, positronium states

have not been explicitly included.

Finally, there is the variational method (Chapt. 47),

which uses an analytic form of trial wave function to

represent the total system. The parameters of this an-

alytic function are determined as part of the method.

Given a trial wave function with sufﬁcient ﬂexibility and

a large enough number of parameters, essentially exact

results can be obtained in the elastic energy range and

the Ore gap. Because the complexity of the trial func-

tion increases as the square of the number of electrons in

the target, only positron scattering from hydrogen, he-

lium and lithium and the hydrogen molecule have been

treated by this method, to date [48.8, 24].

In the case of ionization there appears to be quite

distinct threshold behaviour of the cross sections for

electron and positron collisions. For electrons, the Wan-

nier threshold law (Sect. 52.2.1) has exponent 1.127,

while a similar analysis for positrons [48.25] yields an

exponent of 2.651. However, the existence of the positro-

nium formation channel leaves in question whether

this analysis will give the dominant term at thresh-

old. For a fuller discussion see [48.26], and references

therein.

There has been an investigation [48.27] of the be-

haviour of the elastic cross sections at the positronium

formation threshold which predicts the occurrence of

a Wigner cusp for the lighter noble gases.

48.3 Particular Applications

48.3.1 Atomic Hydrogen

Because of the difﬁculty of making measurements in

atomic hydrogen, the available experimental data is re-

stricted so far to total cross sections, as well as to

total ionization and positronium formation cross sec-

tions. Essentially exact variational calculations have

been carried out in the elastic energy regime and the

Ore gap [48.8].

Ionization cross sections have been measured by

both the Bielefeld and London groups [48.28, 29](and

references therein) and have been calculated in a number

of approximations [48.23, 30–32]. However, disagree-

ments between the experimental measurements mean

that there is at present no reliable way of assessing the

various approximations used. More elaborate calcula-

tions with asymptotically correct wave functions have

been used to determine triple differential cross sections

for ionization [48.33,34]. However, the task of integrat-

ing these to produce total cross sections is a formidable

one.

The total positron–hydrogen cross section has also

been measured by the Detroit group [48.35], and is in

quite good agreement with calculations based upon the

coupled-pseudostate method [48.32] except at very low

energies where the experimental uncertainties are the

greatest.

In order to determine reliable positronium formation

cross sections, the explicit positronium states have to

be included. Several such calculations have been carried

out [48.17–20]. These indicate the necessity of explicitly

including positronium formation channels in the expan-

sion of the total wave function in order to obtain accurate

results, even for elastic scattering. The most recent cal-

culations [48.32, 36] are in quite good agreement with

various experiments [48.28,37] over the majority of the

energy range.

As is the case for electron scattering, positron cross

sections exhibit resonances (Sect. 47.1.3). These have

been extensively studied by Ho [48.38] using variational

and complex rotation methods.

48.3.2 Noble Gases

Because the noble gases are convenient experimental

targets, a good deal of effort has gone into calculations

for these targets, particularly for elastic scattering, ion-

ization, and Ps formation. In the purely elastic energy

range, i. e., for energies below the positronium forma-

tion threshold, the simple potential scattering approach

using the static and polarization potentials deﬁned above

yields quite good results. Since the long-range behaviour

of the sum of the potentials is attractive, the scattering

phase shifts for positrons must be positive for sufﬁ-

Part D 48.3

736 Part D Scattering Theory

ciently low energies. However, as its incident energy

increases, the positron probes the repulsive inner part

of the potential and the phase shifts become negative.

This behaviour leads to the well-known Ramsauer min-

imum in the integral elastic cross sections (Chapt. 47)

for the lighter noble gases (helium, neon and possibly

argon), but not for krypton and xenon [48.15, 39](and

references therein). This differs from electron scatter-

ing, where some low-energy phase shifts can be negative

(modulo π) because of the existence of bound orbitals

of the same symmetry.

Another difference between positron and electron

scattering is exhibited by the differential cross sections

(Chapt. 47). As a result of the low intensity and width

of positron beams, most measurements of differential

cross sections are relative, with the normalization of-

ten being made to theoretical calculations at speciﬁc

angles. For electrons, the shape of the cross section is

determined by a few dominant phase shifts, whereas

for positrons, many phase shifts contribute to the ﬁnal

shape [48.40]. Because of this behaviour, the differ-

ential cross sections for positron scattering have much

less overall structure than for electron scattering. How-

ever, the differential cross sections for positrons for

many of the noble gases have a single minimum at

relatively small angles, both below and above the ﬁrst

inelastic channel. These have been reviewed by Kaup-

pila et al. [48.41]. At intermediate energies, the simple

potential scattering approximation is no longer sufﬁ-

cient and the inelastic channels have to be taken into

account via, for example, the use of an optical po-

tential [48.13, 14, 42]. Furthermore, in the inelastic

scattering regime, the existence of open channels has

a much more marked effect on the shape of the dif-

ferential cross sections for positron scattering than for

electrons [48.43].

The ﬁrst absolute differential cross sections were

measured for argon and krypton at very low energies

using a magnetized beam of cold positrons [48.44, 45].

These results are in excellent agreement with a variety

of different theoretical predictions [48.46–48].

There is relatively very little experimental data for

the excitation of the noble gases. Some experimen-

tal work has been carried out for the lighter noble

gases, helium, neon and argon [48.49] (and references

therein), and there is satisfactory agreement between

these measurements and close-coupling [48.50], as well

as distorted-wave [48.51] calculations.

The ﬁrst state-resolved absolute excitation cross sec-

tions for the 4s [1/2]

and 4s [3/2]

states of argon and

the 6s [3/2]

state of xenon have been measured by the

San Diego group [48.6, 52]. Relativistic distorted-wave

calculations are in satisfactory agreement with the ex-

periment for argon [48.53], but less so for xenon [48.54].

The total ionization and positronium cross sections

have been measured extensively for all of the noble

gases. In general, there is good agreement amongst the

various experiments for the ionization cross section,

but much less so for the positronium formation cross

section. A summary of the experimental work on the

ionization and positronium cross sections for neon, ar-

gon, krypton and xenon can be found in the article by

Laricchia et al. [48.55] (and references therein), while

a more detailed analysis of these cross sections in ar-

gon can be found in recent articles by the San Diego

group [48.6, 56].

There has also been an extensive investigation of

the energy dependence of the elastic and positronium

formation cross section near the Ps formation threshold

for all the inert gases [48.57] (and references therein).

There exist some measurements of Z

eff

and angular

correlation parameters for these gases [48.58], mainly at

room temperature, and calculations for them have been

made in the polarized-orbital approximation [48.59].

48.3.3 Other Atoms

In the case of positron scattering from the alkali atoms,

the positronium formation channel is always open

and the simple potential scattering approach does not

yield reliable results. The Detroit group has meas-

ured the total cross section, as well as upper and

lower bounds to the positronium formation cross sec-

tion for sodium, potassium, and rubidium. Although

early close-coupling calculations of the elastic and ex-

citation cross sections [48.60–62] were in surprisingly

good agreement with the experimental total cross sec-

tion, these calculations did not include the positronium

formation channel. Subsequently, much more sophis-

ticated calculations were carried out by the Belfast

group using the coupled-pseudostate method, which

included both eigenstates of the target as well as positro-

nium [48.32, 63–65]. These calculations also showed

the increasing importance of positronium formation

in excited states for the alkalis potassium, rubidium,

and caesium. The overall agreement between experi-

mental results and those from the coupled-pseudostate

method are quite good for both potassium and rubid-

ium, however, for sodium, the experimental positronium

formation cross section is signiﬁcantly above these the-

oretical calculations. A summary of the experimental

work on the alkalis can be found in [48.35], while

Part D 48.3

Positron Collisions 48.4 Binding of Positrons to Atoms 737

a corresponding summary of theoretical work is given

in [48.32].

Substantial resonance features have been found in

these positron–alkali atom cross sections [48.66].

The positronium formation threshold for magnesium

is very low, only 0.844 eV, and hence, the elastic and

positronium formation cross sections will dominate in

the low energy region. Upper and lower bounds to the

Ps formation cross section in magnesium have been de-

termined [48.35, 67], and are in agreement with both

close-coupling calculations [48.68] and the results of

many-body theory [48.69].

48.3.4 Molecular Hydrogen

By its fundamental nature, molecular hydrogen has at-

tracted considerable attention both experimentally and

theoretically. The total elastic cross section has been

measured by both the Detroit group [48.37] and the Lon-

don group [48.70], with good agreement between both

sets of data. There have been several theoretical calcu-

lations of this cross section by a variety of methods:

Kohn variational [48.71], R-matrix [48.72], distributed

positron model [48.73], and recently, a Schwinger multi-

channel method [48.74], with [48.72, 73]beingin

satisfactory agreement with experiment. Once again,

the elastic cross section is strongly inﬂuenced by the

positronium formation channel near threshold.

Quite recently the vibrational (0 → 1) excitation

cross section of molecular hydrogen has been measured

between 0.55 and 4 eV by Sullivan et al. [48.75]. Their

data are in quite good agreement with theoretical cal-

culations. The San Diego group has also measured the

electronic excitation of the B

Σ state from threshold

to 30 eV [48.52]. Their data are in reasonable agree-

ment with the Schwinger multichannel calculation of

Lino et al. [48.76]. Interestingly, the measured positron

excitation cross section appears to be larger than that

determined for electron excitation.

The ionization cross section has been determined

over a wide range of energies by a number of different

groups [48.77–81] (and references therein). Since all of

the above measurements are relative, they must be nor-

malized to one another at particular energies. Although

there are some differences between these experiments,

near threshold the positron cross section increases less

rapidly, in general, than the corresponding electron cross

section, in accordance with the Wannier law. Theoret-

ical calculations are in satisfactory agreement with the

measurements [48.82] (and references therein).

The positronium formation cross section has also

been measured by a number of different groups with

coupled-channels calculations being carried out for this

process [48.83].

48.3.5 Other Molecules

For diatomic and triatomic molecules, most of the ex-

perimental and theoretical work has been carried out for

CO, CO

,andN

. Total cross sections for O

and CO

have been measured from threshold to several

hundred eV [48.70] (and references therein). Relative

differential cross sections have been measured for CO,

,aswellasN

O, on both sides of the

positronium formation threshold [48.84]. Absolute dif-

ferential cross sections have been measured for CO at

6.75 eV [48.45]. At low energies the gases N

,and

CO exhibit a minimum in the DCS at small angles, as

per the heavier noble gases. This minimum gradually

disappears as the energy increases.

Vibrational excitation cross sections for CO and CO

have been measured [48.75] and are in excellent agree-

ment with the theoretical calculations of [48.85]for

CO, and in satisfactory agreement with theory [48.86]

for CO

. Electron excitation of the a

Π and a

1

states of N

have been measured from threshold to

20 eV [48.52]. Interestingly, the positron cross sec-

tion near threshold is approximately double that for

electrons.

For polyatomic molecules, the majority of exper-

imental and theoretical work has been carried out

for CH

. This includes the total cross section and

quasi-elastic (summed over vibration–rotational levels)

differential cross sections. At low energies there is a min-

imum in these DCS at small angles, as per the heavier

noble gases which, in turn, also disappears at higher en-

ergies. The positronium formation cross section has also

been measured.

48.4 Binding of Positrons to Atoms

There have been many recent investigations of the

possible binding of positrons to a variety atoms. As

was mentioned in Sect. 46.2.2, such binding could

greatly enhance the annihilation cross section and help

to explain the large measured values of Z

eff

for both

atoms and molecules. It has been shown theoretically

Part D 48.4

738 Part D Scattering Theory

that a positron will bind to a large number of one-

and two-electron atomic systems [48.87] (and refer-

ences therein). For one-electron systems, where the

ionization potential is less than 6.802 85 eV, the dom-

inant conﬁguration is a polarized positronium (Ps)

cluster moving in the ﬁeld of the residual positive

ion, while for two-electron systems, with an ioniza-

tion potential greater than 6.802 85 eV, the dominant

conﬁguration involves a positron orbiting a polarized

neutral atom [48.88]. So far, there is no experimen-

tal evidence for these positronic atoms. However, there

is considerable evidence that positrons will bind to

large hydrocarbon molecules [48.6] (and references

therein).

48.5 Reviews

For a number of years a Positron Workshop has been

held as a satellite of the International Conference on the

Physics of Electronic and Atomic Collisions. Their pro-

ceedings [48.89–100] give an excellent summary of the

state of positron scattering research, both experimen-

tal and theoretical, including such additional topics as

positronium scattering from atoms, the formation of an-

tihydrogen, inner shell ionization, and applications to

astrophysics.

There are several review articles on positron

scattering in gases, including the early historical

development [48.101], more comprehensive arti-

cles [48.2, 102–104], as well as a more recent

review [48.105], which also discusses the future for

positron physics.

A recent book [48.106] discusses various as-

pects of both experimental and theoretical positron

physics.

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Part D 48

741

Adiabatic and

49. Adiabatic and Diabatic Collision Processes

at Low Energies

Adiabatic and diabatic electronic states of a sys-

tem of atoms are deﬁned and their properties

are described. Nonadiabatic interaction for slow

quasiclassical motion of atoms is discussed

within two-state common-trajectory approx-

imation. Analytical formulae for nonadiabatic

transition probabilities are presented for par-

ticular modles with reference to single and

double passage of coupling regions (Landau–

Zener–Stückelberg, Rosen–Zener–Demkov, Nikitin

models). Generalization for multiple passage is

described.

49.1 Basic Deﬁnitions.................................. 741

49.1.1 Slow Quasiclassical Collisions ...... 741

49.1.2 Adiabatic

and Diabatic Electronic States ..... 742

49.1.3 Nonadiabatic Transitions:

The Massey Parameter ............... 742

49.2 Two-State Approximation .................... 743

49.2.1 Relation Between Adiabatic

and Diabatic Basis Functions ...... 743

49.2.2 Coupled Equations

and Transition Probabilities

in the Common Trajectory

Approximation.......................... 744

49.2.3 Selection Rules

for Nonadiabatic Coupling.......... 745

49.3 Single-Passage Transition Probabilities:

Analytical Models ................................ 746

49.3.1 Crossing

and Narrow Avoided Crossing

of Potential Energy Curves:

The Landau–Zener Model

in the Common Trajectory

Approximation.......................... 746

49.3.2 Arbitrary Avoided Crossing

and Diverging Potential

Energy Curves: The Nikitin Model

in the Common Trajectory

Approximation.......................... 747

49.3.3 Beyond the Common Trajectory

Approximation.......................... 748

49.4 Double-Passage Transition Probabilities

and Cross Sections ............................... 749

49.4.1 Mean Transition Probability

and the Stückelberg Phase ......... 749

49.4.2 Approximate Formulae

for the Transition Probabilities.... 750

49.4.3 Integral Cross Sections

for a Double-Passage

Transition Probability ................ 751

49.5 Multiple-Passage

Transition Probabilities ........................ 751

49.5.1 Multiple Passage

in Atomic Collisions ................... 751

49.5.2 Multiple Passage

in Molecular Collisions ............... 751

References .................................................. 752

49.1 Basic Deﬁnitions

49.1.1 Slow Quasiclassical Collisions

Slow collisions of atoms or molecules (neutral or

charged) are deﬁned as collisions for which the velocity

of the relative motion of colliding particles v is substan-

tially lower than the velocity of valence electrons v

v/v

 1 . (49.1)

If v

is estimated as v

≈ 1a.u.≈ 10

cm/s, then (49.1)

is fulﬁlled for medium mass nuclei (∼ 10 amu) up to

several keV.

Quasiclassical collisions are those for which the

de Broglie wavelength λ

for the relative motion is

substantially smaller than the range parameter a of the

interaction potential

 a . (49.2)

The two conditions (49.1)and(49.2) deﬁne the energy

range within which collisions are slow and quasiclas-

sical. For medium mass nuclei, this energy range covers

collision energies above room temperature and below

hundreds of eV. The paramater a should not be confused

with another important parameter L

which character-

Part D 49