Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
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Electron–Atom, Electron–Ion, and Electron–Molecule Collisions 47.2 Electron–Molecule Collisions 721
Here, R represents the coordinates R
i
of all the nuclei,
r
m
represents the coordinates r
i
of the electrons in the
target molecule, and r represents the coordinates of the
scattered electron. The total energy E in (47.132) refers
to the frame of reference where the c.m. of the whole
system is at rest.
As in the case of electron–atom and electron–ion
collisions, introduce target eigenstates, and possibly
pseudostates Φ
i
, by the equation
Φ
i
|
H
m
|
Φ
j
=w
i
δ
ij
, (47.134)
and then expand the total wave function Ψ , in analogy
with (47.8), in the form
Ψ
j
= A
i
Φ
i
(
R, r
m
)
F
ij
(
r
)
+
i
χ
i
(
R, r
m
, r
)
b
ij
, (47.135)
where the spin variables have been suppressed for
notation simplicity, and we have not carried out
a partial-wave decomposition of the wave function
F
ij
representing the scattered electron. The subscripts
i and j now represent the rotational and vibrational states
of the molecule as well as its electronic states.
Coupled equations for the functions F
ij
can be ob-
tained by substituting expansion (47.135)into(47.132),
and projecting onto the target states Φ
i
and onto the
square integrable functions χ
j
. After eliminating the
coefficients b
ij
, the coupled integrodifferential equations
∇
2
+ k
2
i
F
ij
(
r
)
= 2
(
V
i
+ W
i
+ X
i
)
F
j
(
r
)
(47.136)
are obtained, where k
2
i
= 2
(
E − w
i
)
and where V
i
, W
i
,
and X
i
are the direct, nonlocal exchange, and nonlocal
correlation potentials. By expanding F
ij
in partial waves,
a set of coupled radial integrodifferential equations re-
sult, analagous to (47.10) for atoms and ions.
The scattering amplitude and cross section for
a transition from an initial state |i=|k
i
,Φ
i
,χ
1
2
m
i
to
a final state | j=|k
j
,Φ
j
,χ
1
2
m
j
is then given by (47.7),
where now the subscripts i and j refer collectively to
the ro-vibrational and electronic states of the molecule.
47.2.2 Molecular Frame Representation
The theory described in the previous section is com-
pletely general, and has been the basis of a number of
early calculations for simple diatomic molecules such
as H
2
. However, major computational difficulties arise
because of the very large number of rovibrational chan-
nels that need to be retained in expansion (47.135)for
all but the simplest low-energy calculations.
This difficulty can be overcome by making a Born–
Oppenheimer separation of the electronic and nuclear
motion. The electronic motion is first determined with
the nuclei held fixed. This is referred to as the fixed-
nuclei approximation. The molecular rotational and
vibrational motion is then included in a second step of
the calculation. This procedure owes its validity to the
large ratio of the nuclear mass to the electronic mass,
and can be adopted when the collision time is much
shorter than the periods of molecular rotation and vibra-
tion. Thus it is expected to be valid when the scattered
electron energy is not close to a threshold, or when the
energy does not coincide with that of a narrow reso-
nance. In these cases, further developments described
below are needed to obtain reliable cross sections.
In order to formulate the collision process in this
representation, adopt a frame of reference that is rigidly
attached to the molecule. The fixed-nuclei approxima-
tion then starts from the Schrödinger equation
(
H
el
+ T
e
+ V
)
ψ = Eψ, (47.137)
where H
el
is the electronic part of the target Hamiltonian
obtained by assuming that the target nuclei have fixed
coordinates denoted collectively by R. It follows that
H
el
is related to H
m
in (47.132)by
H
m
= H
el
+ T
R
, (47.138)
where T
R
is the kinetic energy operator for the rotational
and vibrational motion of the nuclei. The remaining
quantities T
e
and V are the same as in (47.132).
The solution of (47.137) proceeds in an analogous
way to the solution of (47.2) for electron collisions
with atoms and ions. We adopt an expansion similar
to (47.8), where we now expand the function repre-
senting the motion of the scattered electron in terms of
symmetry-adapted angular functions that transform as
an appropriate irreducable representation (IRR) of the
molecular point group (Burke et al. [47.97]). Substitut-
ing this expansion into (47.137), and projecting onto the
corresponding channel functions and onto the square
integrable functions, yields a set of coupled integrodif-
ferential equations with the form given by (47.10), where
now the channel indices i, j and represent the com-
ponent of the IRR, as well as the electronic state of the
target, and where Γ represents the conserved quantum
numbers that now include the IRR and the total spin.
The final step is to solve these coupled integrodif-
ferential equations for each set of nuclear coordinates R
Part D 47.2
722 Part D Scattering Theory
of importance in the collision, using one of the methods
discussed in Sect. 47.1.5. This yields the K-matrices,
S-matrices and cross sections for fixed R. For scattering
calculations where only the ground electronic state has
been included in the expansion, a number of approaches
have been developed that replace the nonlocal exchange
and correlation potentials by local potentials [47.2].
These approaches have proved particularly important in
describing electronically elastic collisions of electrons
with polyatomic molecules.
47.2.3 Inclusion of the Nuclear Motion
This section discusses how observables involving the
nuclear motion, such as rotational and vibrational exci-
tation cross sections, and dissociative attachment cross
sections, can be obtained from the solutions of the
fixed-nuclei equations.
The most widely used approach is the adiabatic-
nuclei approximation [47.98–100]. In the case of
diatomic molecules in a
1
Σ state, the scattering am-
plitude for a transition between electronic, vibrational
and rotational states represented by iv jm
j
and i
v
j
m
j
is given by
f
i
v
j
m
j
,
iv jm
j
ˆ
k·
ˆ
r
=
χ
i
v
(R)Y
j
m
j
ˆ
R
f
i
i
ˆ
k·
ˆ
r; R
χ
iv
(R)Y
jm
j
ˆ
R
,
(47.139)
where f
i
i
ˆ
k·
ˆ
r; R
is the fixed-nuclei scattering am-
plitude, which depends parametrically on the nuclear
coordinates R,andχ
iv
and Y
jm
j
are the molecular vibra-
tional and rotational eigenfunctions, respectively. This
approximation is valid provided that the collision time is
short compared with the vibration and/or rotation times,
and is widely used in such situations.
The cross section is usually averaged over the de-
generate sublevels m
j
and summed over m
j
,giving
the cross section for the transition iv j to i
v
j
.This
leads to the relation
dσ
i
v
j
,iv j
dΩ
=
j+ j
j
t
=| j− j
|
j0 j
t
0 | jj
t
j
0
2
dσ
i
v
j
t
,iv0
dΩ
,
(47.140)
provided that the small differences in the wave numbers
for the different rotational channels can be neglected.
A similar relation holds for symmetric top molecules
such as NH
3
. For spherical top molecules such as CH
4
,
the equivalent relation is [47.101, 102]
dσ
i
v
j
,iv j
dΩ
=
2 j
+ 1
2 j + 1
j+ j
j
t
=| j− j
|
1
2 j
t
+ 1
dσ
i
v
j
t
,iv0
dΩ
.
(47.141)
The sum in (47.140)or(47.141) over the final rotational
state j
is independent of the initial state j. Also, if the
cross section is multiplied by the transition energy and
then sum over j
, the result, which is in the mean energy
loss by the incident electron, is still independent of j.
The adiabatic-nuclei approximation breaks down
close to threshold or in the neighborhood of narrow res-
onances [47.103]. A straightforward way of including
nonadiabatic effects that arise in vibrational excitation
is to retain the vibrational terms in the Hamiltonian, but
still to treat the rotational motion adiabatically. Hence,
instead of (47.137), the equation
(
H
el
+ T
vib
+ T
e
+ V
)
#
ψ = E
#
ψ (47.142)
is solved, where T
vib
is the kinetic energy operator for
the nuclear vibrational motion, and where the other
quantities have the same meaning as in (47.137). Adopt-
ing a frame of reference in which the molecule has
fixed spacial orientation, and separating out the angular
variables of the scattered electron, coupled integrodif-
ferential equations coupling the target vibrational states
as well the electronic states can be obtained. This ap-
proach has been adopted with success [47.104, 105],
but is computationally demanding since the number of
coupled channels can become very large.
Vibrational excitation and dissociative attachment
are particularly important in resonance regions when
the scattered electron spends an appreciable time in the
neighborhood of the molecule. As a result, a number
of approaches has been developed describing these pro-
cesses based on electron molecule resonance theories
(e.g., [47.106–110]). The basic idea is that a series of
fixed-nuclei resonance states ψ
(
r
)
n
are introduced for
a range of values of R, either by imposing Siegert
outgoing wave boundary conditions [47.111], or by in-
troducing Feshbach projection operators [47.15,16]. The
amplitude for a transition from an initial electronic–
vibrational state iv to a final state i
v
is then given
by
T
i
v
,iv
=
n
χ
i
v
R
ζ
ni
R
G
(
r
)
n
R
, R
×
ζ
ni
(
R
)
χ
iv
(
R
)
,
(47.143)
where χ
iv
are the vibrational eigenfunctions, ζ
ni
are the
“entry amplitudes” from the initial or final electronic
Part D 47.2
Electron–Atom, Electron–Ion, and Electron–Molecule Collisions 47.3 Electron–Atom Collisions in a Laser Field 723
states into the resonance states ψ
(
r
)
n
,andG
(
r
)
n
are the
Green’s functions that describe the propagation in the
intermediate resonance states ψ
(
r
)
n
. Dissociative attach-
ment can be described by a straightforward extension of
this theory.
The fixed-nuclei R-matrix method has also been ex-
tended to treat vibrational excitation and dissociative
attachment [47.112]. A generalized R-matrix is intro-
duced by an equation which, in analogy with(47.143),
can be written as
R
i
v
,iv
=
1
2a
k
χ
i
v
R
w
Γ
i
k
R
G
RM
k
R
, R
×
w
Γ
ik
(
R
)
χ
iv
(
R
)
,
(47.144)
where the surface amplitudes w
Γ
ik
(
R
)
are defined by
(47.76), and the Green’s function G
RM
k
now describes
the propagation in the intermediate R-matrix states de-
fined by (47.73). Once the generalized R-matrix has
been determined, the final step in the calculation is to
solve the collision problem in the external region which,
for diatomic molecules, is defined by the condition that
the scattered electron coordinate r is greater than some
given radius a, and the internuclear coordinate R is
greater than some given radius A.
Another approach that includes nonadiabatic effects
is the energy-modified adiabatic approximation intro-
duced by Nesbet [47.113]. In this approach, the S-matrix
elements connecting the vibrational states are defined by
S
i
v
,iv
=χ
i
v
| S
i
i
(
E − H
i
; R
)
| χ
iv
, (47.145)
where S
i
i
(
E − H
i
; R
)
is the S-matrix calculated in
the fixed nuclei approximation at the internuclear
separation R at an energy defined by the operator
H
i
= E
i
(
R
)
+ T
vib
. This has the effect of including the
internal energy of the target into the S-matrix elements,
giving the correct threshold energies.
Finally, we mention an off-shell T-matrix approach
for including nonadiabatic effects discussed first by
Shugard and Hazi [47.114]. This approach can also
extend the range of validity of the adiabatic-nuclei
approximation, while retaining much of its inherent
simplicity. This has recently been applied with suc-
cess to low-energy vibrational excitation of H
2
and
CH
4
[47.115].
47.2.4 Electron Collisions with Polyatomic
Molecules
In the last few years computer programs based on the ab
initio methods described in Sect. 47.1.5 have been de-
veloped and used to calculate cross sections for electron
collisions with polyatomic molecules of importance in
many applications.
Recent work includes electron collisions with ni-
trous oxide which is an important species in the
upper atmosphere where it plays a role in ozone
destruction. N
2
O lasers are also of importance.
Fixed-nuclei total cross sections, calculated using in-
dependent Bonn [47.116] and UK [47.117] polyatomic
R-matrix programs are in good agreement with experi-
ment [47.118], showing a
2
Π shape resonance near
2eV. The UK R-matrix program has also been used
to study electron collisions with the open-shell radical
OClO [47.119]. An important process in the strato-
spheric polar vortex involves the coupling of chlorine
and bromine chemistry, in which OClO plays a key
role. In this case OClO is formed by the reaction
BrO + ClO → Br + OClO. Fixed-nuclei total cross sec-
tions were calculated including eight electronic states in
the R-matrix expansion, giving good agreement with ex-
periment [47.120], reproducing a shoulder in the cross
section between 2 and 6 eV.
Important advances have also been made in the theo-
retical treatment of vibrational excitation in polyatomic
molecules. Electron collisions with CO
2
molecules
were calculated in the fixed-nuclei approximation using
an electron–polyatomic molecular scattering program
based on the complex Kohn variational method [47.121].
At low energies the cross section is dominated by a vir-
tual state at threshold and a
2
Π
u
shape resonance at
3.8 eV. As the molecule bends from its ground state
linear configuration, the shape resonance splits into non-
degenerate
2
A
1
and
2
B
1
configurations. The fixed-nuclei
complex resonance energy surfaces are parametrized
and motion on these surfaces is computed using a gener-
alization of the Boomerang model [47.122]. The results
reproduce the oscillations resulting from the interfer-
ence between the nuclear and electronic motion seen
experimentally by Allen [47.123, 124].
47.3 Electron–Atom Collisions in a Laser Field
Electron–atom collisions in the presence of an intense
laser field have recently attracted considerable attention
because of the importance of these processes in applica-
tions such as laser plasma interactions, and also because
of their fundamental interest in atomic collision theory.
This section summarizes the basic theory, commenc-
Part D 47.3
724 Part D Scattering Theory
ing with the scattering of electrons by a potential in the
presence of a laser field. The discussion is then general-
ized to the scattering of electrons by complex atoms and
ions, where ‘dressing’ of the atomic eigenstates by the
laser field must be considered, and where simultaneous
electron photon excitation (SEPE) processes, defined
by
nhν + e
−
+ A
i
→ e
−
+ A
j
, (47.146)
can occur. Recent reviews of aspects of this subject
have been given by Mason [47.3], Newell [47.125]and
Mittleman [47.126].
47.3.1 Potential Scattering
We adopt a semiclassical description of the collision
process in which the electrons and the target atom are
described by the nonrelativistic Schrödinger equation,
and the laser field is described classically. This is valid
for most high intensity fields of current interest, where
a typical coherence volume of the field contains a very
large number of photons [47.126].
The time-dependent Schrödinger equation describ-
ing an electron scattered by a potential V
(
r
)
in the
presence of an external electromagnetic (laser) field is
then
i
∂
∂t
Ψ
(
r, t
)
=
−
1
2
∇
2
−
i
c
A
(
r, t
)
·∇
+
1
2c
2
A
2
(
r, t
)
+ V
(
r
)
!
Ψ
(
r, t
)
,
(47.147)
in the Coulomb gauge such that the vector potential
satisfies ∇ · A= 0. We also assume that the laser field
is monochromatic, monomode, linearly polarized, and
spacially homogeneous (i. e., its wavelength is large
compared with the range of the potential, or more
generally with the size of the atom). Hence we can
write
A
(
r, t
)
= A
(
t
)
=
ˆ
A
0
cos ωt , (47.148)
where
ˆ
is a unit vector along the field polarization di-
rection and ω is the angular frequency. The A
2
term in
(47.147) can be removed by the unitary transformation
Ψ
(
r, t
)
= exp
−
i
2c
2
t
A
2
t
dt
Ψ
V
(
r, t
)
,
(47.149)
where Ψ
V
satisfies the Schrödinger equation in the vel-
ocity gauge given by
i
∂
∂t
Ψ
V
(
r, t
)
=
−
1
2
∇
2
−
i
c
A
(
t
)
·∇+ V
(
r
)
!
× Ψ
V
(
r, t
)
.
(47.150)
The corresponding equation for the free electron with
V = 0 is readily solved to give the Volkov wave func-
tion [47.127]
χ
k
(
r, t
)
=
(
2π
)
−
3
2
exp
[
i
(
k· r − k· α
0
sin ωt − Et
)
]
,
(47.151)
where k is the wave vector and E =
1
2
k
2
the kinetic
energy. The quantity α(t) is defined by
α
(
t
)
=
1
c
t
A
t
dt
= α
0
sin ωt , (47.152)
where α
0
= E
0
/ω
2
, E
0
being the electric field strength.
To solve (47.150), introduce the causal Green’s func-
tion G
(+)
0
r, t; r
, t
satisfying the equation
i
∂
∂t
+
1
2
∇
2
+
i
c
A
(
t
)
·∇
G
(
+
)
0
r, t; r
, t
= δ
r −r
δ
t − t
.
(47.153)
This Green’s function is given by
G
(+)
0
r, t; r
, t
=−iθ
t − t
χ
k
(
r, t
)
χ
∗
k
(r
, t
)dk , (47.154)
where θ(x) = 1forx > 0andθ(x) = 0forx < 0. The cor-
responding causal outgoing wave solution of (47.150)
is
Ψ
(+)
k
(r, t) = χ
k
(r, t) +
t
−∞
dt
dr
G
(+)
0
r, t; r
, t
× V
r
Ψ
(+)
k
r
, t
, (47.155)
and the S -matrix element for a transition k
i
→ k
f
in the
presence of the laser field is given by
S
k
f
,k
i
=−i
χ
k
f
V
Ψ
(+)
k
i
,
(47.156)
where an integration is carried out over all space and time
in this matrix element. The time integration in (47.156)
can be performed using the relation
exp
(
ix sin u
)
=
∞
n=−∞
J
n
(
x
)
exp
(
inu
)
,
(47.157)
Part D 47.3
Electron–Atom, Electron–Ion, and Electron–Molecule Collisions 47.3 Electron–Atom Collisions in a Laser Field 725
where J
n
(x) is an ordinary Bessel function of order n.
Then
S
k
f
,k
i
=−2πi
∞
n=−∞
δ
E
k
f
− E
k
i
− nω
T
n
k
f
,k
i
(47.158)
where the delta function ensures energy conservation,
and n is the number of photons absorbed or emitted.
The differential cross section for the scattering process
k
i
→ k
f
with exchange of n photons can then be defined
in terms of the T-matrix elements T
n
k
f
,k
i
by
dσ
n
dΩ
=
(
2π
)
4
k
f
k
i
T
n
k
f
,k
i
2
. (47.159)
There are two limiting cases in which consider-
able simplification in this expression occurs. First, at
high energies or for weak potentials, the first Born ap-
proximation can be used to describe scattering by the
potential V(r). In this case (47.159) reduces to
dσ
1,B
n
dΩ
=
(
2π
)
4
k
f
k
i
J
2
n
(
∆ · α
0
)
|V(∆)|
2
, (47.160)
where ∆ = k
i
− k
f
is the momentum transfer vector and
V(∆) = (2π)
−3
exp(i∆ · r)V(r) dr .
(47.161)
It follows immediately that
dσ
1,B
n
dΩ
= J
2
n
(∆ · α
0
)
dσ
1,B
dΩ
,
(47.162)
where dσ
1,B
/dΩ is the field-free first Born differential
cross section. Using the sum rule
∞
n=−∞
J
2
n
(x) = 1 , (47.163)
(47.162) immediately yields
∞
n=−∞
dσ
1,B
n
dΩ
=
dσ
1,B
dΩ
.
(47.164)
The second limiting case is the low frequency (soft
photon) limit, where the laser photon energy ω is small
compared with the electron energy E
k
i
. In this limit, the
T-matrix element is given by [47.128]
T
n
k
f
,k
i
= J
n
(∆ · α
0
)
χ
k
f
T
E
k
i
χ
k
i
+ O
w
2
,
(47.165)
where k
i
and k
f
are the shifted wave vectors
k
i
= k
i
+
nω
∆ · α
0
α
0
, k
f
= k
f
+
nω
∆ · α
0
α
0
(47.166)
and T (E
k
i
) is the T-operator in the absence of the
laser corresponding to the energy E
k
i
= k
2
i
/2. The dif-
ferential cross section for the transfer of n photons is
then
dσ
n
dΩ
=
k
f
k
i
J
2
n
(∆ · α
0
)
dσ
dΩ
k
f
, k
i
+ O
ω
2
,
(47.167)
where dσ(k
f
, k
i
)/dΩ refers to the transition k
i
→ k
f
in
the absence of the laser. If the frequency is small enough
to neglect the n dependence of k
i
and k
f
, then using the
sum rule (47.163), (47.167) becomes
∞
n=−∞
dσ
n
dΩ
=
dσ
dΩ
,
(47.168)
where dσ/ dΩ is the field-free differential cross sec-
tion. The Kroll–Watson result (47.167) has been found
to be surprisingly accurate, even for cases where
ω/E
k
i
≈ 0.5 [47.129, 130]. A nonrigorous extension
of the Kroll–Watson result to inelastic processes has
been considered by a number of authors, and has been
found to give qualitative agreement with experiments on
helium [47.131].
47.3.2 Scattering by Complex Atoms
and Ions
The time-dependent Schrödinger equation describing an
electron scattered by an N-electron atom or ion in the
presence of a laser field can be written in analogy with
(47.147)as
i
∂
∂t
Ψ
(
X
N+1
, t
)
=
$
H
N+1
−
i
c
N+1
i=1
A
(
r
i
, t
)
·∇
i
+
1
2c
2
N+1
i=1
A
2
(
r
i
, t
)
%
Ψ
(
X
N+1
, t
)
,
(47.169)
where H
N+1
is the (N + 1)-electron Hamiltonian de-
fined by (47.3)andX
N+1
represents the space and spin
coordinates of all N + 1 electrons defined as in (47.8).
With the same assumptions made in the reduction
of (47.147)to(47.150), (47.163) can be rewritten in the
velocity gauge form
i
∂
∂t
Ψ
V
(
X
N+1
, t
)
=
H
N+1
+
1
c
A
(
t
)
· P
N+1
Ψ
V
(
X
N+1
, t
)
,
(47.170)
Part D 47.3
726 Part D Scattering Theory
where
P
N+1
=
N+1
i=1
−i∇
i
(47.171)
is the total momentum operator.
The solution of (47.170)fore
−
–H scattering in
a strong laser field at high electron impact energies has
been considered by Francken and Joachain [47.132].
They discussed the use of the Born series and the eikonal
Born series (EBS) approximations, given by (47.104),
to describe the electron–atom collision, and included the
dressed wave function of the atomic hydrogen target to
first-order in the field strength E
0
. Important effects re-
lated to the dressing of the target are the appearance in
the cross sections of asymmetries between the absorp-
tion and the emission of a given number of laser photons,
as well as the appearance of new resonance structures in
the cross sections.
So far, detailed studies of low-energy electron–atom
and electron–ion collisions in laser fields have been very
limited. Pioneering work on e
−
–H
+
collisions by Dimou
and Faisal [47.133], and by Collinsand Csanak [47.134]
have shown important resonance effects caused by the
field coupling bound states to the continuum. In addition,
multichannel quantum defect theory has been applied
by Zoller and co-workers [47.135, 136] to study the be-
havior of Rydberg states in laser fields. We conclude
this section by briefly describing the R-matrix–Floquet
method [47.137] for treating electron collisions with
complex atoms and ions in a laser field, based on the
R-matrix method discussed in Sect. 47.1.5.
In this approach, configuration space is divided into
internal and external regions, as in the field-free case.
In the internal region, a further gauge transformation of
the field is made to the length gauge, so that the time-
dependent Schrödinger equation (47.170) now becomes
i
∂
∂t
Ψ
L
(X
N+1
, t)
=[H
N+1
+ E
(
t
)
· R
N+1
]Ψ
L
(
X
N+1
, t
)
, (47.172)
where R
N+1
=
N+1
i=1
r
i
, and we assume that the electric
field E
(
t
)
is given by
E (t) =−
1
c
d
dt
A(t) =
ˆ
E
0
cos ωt . (47.173)
In order to solve (47.172), we introduce the Floquet–
Fourier expansion [47.138, 139]
Ψ
L
(X
N+1
, t) = e
−iEt
∞
n=−∞
e
−inωt
Ψ
L
n
(X
N+1
).
(47.174)
Substituting this equation into (47.172), using (47.173)
and equating the coefficients of exp[−i(E + nω)t] to
zero gives
(H
N+1
− E − nω)Ψ
L
n
+ D
N+1
Ψ
L
n−1
+ Ψ
L
n+1
= 0 ,
(47.175)
where we have introduced the operator
D
N+1
=
1
2
E
0
ˆ
· R
N+1
. (47.176)
The functions Ψ
L
n
can be regarded as the components
of a vector Ψ
L
in photon space. Equation (47.175) can
then be written in this space as
(H
F
− EI)Ψ
L
= 0 , (47.177)
where the Floquet Hamiltonian H
F
is an infinite tridiag-
onal matrix.
In order to solve (47.177) in the internal region,
the components Ψ
L
n
are expanded in a basis which, in
analogy with (47.69), has the form
Ψ
L
kn
(X
N+1
)
= A
Γ ij
Φ
Γ
i
x
1
,... ,x
N
;
ˆ
r
N+1
σ
N+1
r
−1
N+1
× u
j
(
r
N+1
)
a
Γ
ijkn
+
Γ i
χ
Γ
i
(
x
1
,... ,x
N+1
)
b
Γ
ikn
, (47.178)
where the summation over Γ is required since the to-
tal orbital angular momentum L and the total parity
π in (47.9) are no longer conserved. The coeffi-
cients a
Γ
ijkn
and b
Γ
ikn
are then determined by diagonaliz-
ing H
F
+ L
b
,whereL
b
is an appropriate Bloch operator.
In the external region, the wave function de-
scribing the scattered electron is transformed to the
velocity gauge, and the corresponding coupled equa-
tions integrated outwards from the internal region
boundary for each energy of interest [47.140]. Af-
ter a further transformation to the acceleration frame
(or Kramers–Henneberger frame [47.141]) the wave
function can be fitted to an asymptotic form to yield
the K-matrix, S-matrix and collision cross sections.
This approach has been used to calculate laser-assisted
electron scattering by H and He atoms [47.142, 143]
and a detailed discussion of the theory has been
given [47.144].
Part D 47.3
Electron–Atom, Electron–Ion, and Electron–Molecule Collisions References 727
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Part D 47
731
Positron Collis
48. Positron Collisions
The positron is the antiparticle of the electron,
having the same mass but opposite charge.
Positrons undergo collisions with atomic and
molecular systems in much the same way as
electrons do. Thus, the standard scattering theory
for electrons (see Chapt. 47) can also be applied to
positron scattering. However, there are a number
of important differences from electron scattering
which we outline below.
Since the positron is a distinct particle from
the atomic electrons, it cannot undergo an
exchange process with the bound electrons during
a collision, as is possible with electrons. Thus,
the nonlocal exchange terms which arise in the
description of electron scattering are not present
for positrons. This leads to a simplification of the
scattering equations from those for electrons.
However, there are scattering channels available
with positron scattering which do not exist with
electrons. These are dealt with in Sect. 48.1.
Historically beams of low-energy positrons
were difficult to obtain and consequently there is
considerably less experimental data available for
positrons than for electrons. This was particularly
true for quantities which required large incident
48.1 Scattering Channels ............................. 731
48.1.1 Postronium Formation ............... 731
48.1.2 Annihilation ............................. 732
48.2 Theoretical Methods ............................ 733
48.3 Particular Applications ......................... 735
48.3.1 Atomic Hydrogen....................... 735
48.3.2 Noble Gases.............................. 735
48.3.3 Other Atoms ............................. 736
48.3.4 Molecular Hydrogen .................. 737
48.3.5 Other Molecules ........................ 737
48.4 Binding of Positrons to Atoms .............. 737
48.5 Reviews .............................................. 738
References .................................................. 738
positron fluxes, such as differential scattering cross
sections and coincidence parameters. However,
the recent development of cold trap-based
positron beams with high resolution and high
brightness by the San Diego group [48.1]hasthe
potential to revolutionize this field and put it on
a par with electron scattering.
Throughout this chapter we will employ atomic
units unless otherwise noted.
48.1 Scattering Channels
Positrons colliding with atomic and molecular sys-
tems have the same scattering channels available as
for electrons, viz., elastic, inelastic, ionization, and for
molecules, dissociation. However, two channels exist
for positrons which do not exist for electrons, viz.,
positronium formation and annihilation.
48.1.1 Postronium Formation
Positronium, a bound state of an electron–positron
pair (Chapt. 27), can be formed during the collision
of a positron with an atomic or molecular target. The
positronium ‘atom’ can escape to infinity leaving the
target in a ionized state with a positive charge of one.
Thus, this process can be difficult to distinguish exper-
imentally from true ionization where both the incident
positron and the ionized electron are asymptotically free
particles. The positronium atom can exist in its ground
state or in any one of an infinite number of excited states
after the collision. The level structure of positronium is,
to order α
2
,whereα is the fine-structure constant, iden-
tical to that of hydrogen but with each level having half
the energy of the corresponding hydrogenic state.
Positronium formation is a rearrangement channel,
and thus, is a two-centre problem. Because positronium
is a light particle, having a reduced mass one-half of
that of an electron, the semi-classical type of approxi-
mations used in ion–atom collisions (Chapt. 50) are not
Part D 48