Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
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Continuum Distorted Wave and Wannier Methods 52.2 Wannier Method 783
Expanding P in the same form as S, and restricting
the solution to finding P
0
,gives
P
0
=
C
ρ
5
ωu
1
u
2
2
, (52.63)
where C ∼C
12
E
1−m
12
. By solving these equations, Pe-
terkop extracted the Wannier cross section behavior by
matching the exact wave function with an approximate
one for which the energy dependence is known at some
arbitrarily finite value r
0
of ρ, giving the total cross
section as
σ
tot
∼ E
1.127
, (52.64)
as required. However, neither this method of Peterkop
nor the quantum mechanical approach of [52.47]were
able to deduce the constant of proportionality.
52.2.3 The Quantal Semiclassical
Approximation
As can be seen from the form of (52.63)forP
0
, u
2
vanishes in the double limit ρ →+∞, E → 0, and
so, the semiclassical theory breaks down at the very
configuration of importance. To avoid this problem,
Crothers [52.8] adopted a change of the dependent vari-
able to obtain a uniform JWKB approximation. Taking
α = π/4, (i. e., ∆α = 0) as the natural barrier, he set the
final-state wave function as
Ψ
−∗
=
x|sin(α −π/4)|
1/2
ρ
5/2
sin α cos α(sin θ
12
)
1/2
, (52.65)
so that
∂
2
∂ρ
2
+
1
ρ
2
sin |∆α|
∂
∂α
sin |∆α|
∂
∂α
+
1
ρ
2
sin
2
α cos
2
α
∂
2
∂θ
2
12
+2E +
2Z
ρ
+
(
1
4
+
1
4
csc
2
∆θ
12
)
ρ
2
sin
2
α cos
2
α
−
csc
2
|∆α|
4ρ
2
x = 0 ,
(52.66)
where |∆α| and θ
12
are, respectively, the polar and az-
imuthal angles. Near θ
12
= π and α = π/4, (i. e., at
∆θ
12
= ∆α = 0), the term (4ρ
2
)
−1
is negligible com-
pared with csc
2
θ
12
/(4ρ
2
).Alsotheθ
12
pseudopotential
is clearly attractive while the α potential is repulsive, and
both potentials are large just at the region of importance,
i. e., at ∆α = 0 = ∆θ
12
.
Again, following the method of Peterkop, the fi-
nal state wave function is written in the form (52.44),
(52.47)and(52.48), where now the action perturbation
expression S must be generalized to
S = s
0
ln |∆α|+s
1
ln(∆θ
12
) +S
0
(ρ)
+
1
2
S
1
(ρ)(∆α)
2
+
1
8
S
2
(ρ)(∆θ
12
)
2
+... ,
(52.67)
where the extra logarithmic phases indicate long-
range Coulomb potentials. By applying the Kohn
variational principle perturbatively, and invoking the Jef-
freys’ [52.48] connection formula on the Wannier ridge
with ρ = 0 as the classical turning point, the final state
wave function is [52.8]
Ψ
−∗
f
=
c
1/2
E
m
12
/2
ρ
m
12
/2+1/4
r(2Z
0
)
1/4
(−χ/2π)
1/2
(2Z
0
/ρ)
1/4
ρ
5/2
sin α cos α
× δ
ˆ
k
1
−
ˆ
r
1
δ
ˆ
k
2
−
ˆ
r
2
×exp
4i(8Z
0
ρ)
−1/2(∆θ
12
)
−2
×
exp
−i(8Z
0
ρ)
1/2
−
1
2
i(∆α)
2
× (2Z
0
ρ)
1/2
m
12
−
1
8
i(∆θ
12
)
2
×(2Z
0
ρ)
1/2
m
21
−
1
4
iπ
−c.c
,
(52.68)
where χ = 2πIm(m
21
).
Taking the total cross section for distinguishable
particles as
σ =
π
2
a
2
0
k
0
d
ˆ
k
1
d
ˆ
k
2
π Z
2
tanh χ
(2E)
1/2
f
ˆ
k
1
,
ˆ
k
2
2
×exp
−Z
2
4(2E)
1/2
(∆Θ
12
)
2
π tanh χ
, (52.69)
where f is the scattering amplitude, then the correspond-
ing triple-differential cross section is
d
3
σ
d
ˆ
k
1
d
ˆ
k
2
d
1
2
k
2
1
=
2π
2
a
2
0
k
0
d
dE
πZ
2
tanh χ
(2E)
1/2
×exp
−Z
2
4(2E)
1/2
(Θ
12
− Rπ)
2
π tanh χ
×
f
ˆ
k
1
,
ˆ
k
2
2
. (52.70)
Assuming the contribution from triplet states is negligi-
ble, | f|
2
can be written as
1
4
f
ˆ
k
1
,
ˆ
k
2
+ f
ˆ
k
2
,
ˆ
k
1
2
, (52.71)
Part D 52.2
784 Part D Scattering Theory
where f
ˆ
k
1
,
ˆ
k
2
(and by permutation f
ˆ
k
2
,
ˆ
k
1
, here-
after referred to as f and g respectively) is given
by
f
2i
π
dr
1
dr
2
dr
3
Ψ
−∗
f
φ
f
(2, r
3
)
× (H −E) exp(ik
0
· r
1
)ψ
i
(r
2
, r
3
). (52.72)
As a test of the above formulation for the process
e
−
+He → He
+
+2e
−
near the ionization threshold,
Crothers [52.8] used an independent-electron open-shell
wave function for the initial bound state helium target,
written as
ψ
i
(r
2
, r
3
) =
φ(r
2
, z
0
)φ(r
3
,β)+φ(r
3
, z
0
)φ(r
2
,β)
[2(1 +S)]
1/2
,
(52.73)
where
φ(r, z
0
) = z
3/2
0
π
−1/2
exp(−z
0
r), (52.74)
S =
φ(r, z
0
)φ(r,β)dr
2
=
4z
0
β
(z
0
+β)
2
3
, (52.75)
and z
0
and β take the physical values z
0
= 1.8072
1/2
and β = 2. The total singlet cross section was found to
be (in atomic units)
σ =2.37E
m
12
a
2
0
, (52.76)
in line with Wannier’s threshold law, and with ex-
periment [52.49], while the corresponding absolute
triple differential cross sections (TDCS) were expressed
as
d
3
σ
d
ˆ
k
1
d
ˆ
k
2
d
1
2
k
2
1
=
70cz
2
0
2
1/2
χ tanh χ
πR
∞
Z
1/2
0
m
12
| f +g|
2
×
d
dE
E
m
12
−1/2
exp
−Z
2
(Θ
12
−π)
2
π tanh χ
4(2E)
1/2
(52.77)
in units of 10
−19
cm
2
sr
−2
eV
−1
,where
c =
Γ(m
12
+3/2)Γ(m
12
+1)
2πZ
m
12
0
Γ(2m
12
+3/2)
,
(52.78)
and where f (and similarly g for Θ
2
)isgivenby
f =
∞
0
dρρ
3/2+m
21
/2+1/4
L
max
L=0
i
L
(2L +1) j
L
ρz
0
2
1/2
× P
L
(cos Θ
1
) exp
1
8
×Im
m
21
(Θ
12
−π)
2
(2Z
0
ρ)
1/2
×2cos
(8Z
0
ρ)
1/2
+
1
8
×Re
m
21
(Θ
12
−π)
2
(2Z
0
ρ)
1/2
× r(ρ, Θ
12
) (52.79)
with
(1 +S)
1/2
r(ρ, Θ
12
)
= exp
−
ρz
0
2
1/2
2
1/2
z
0
−
1
(1 −cos Θ
12
)
1/2
+exp(−2
1/2
ρ)
64(2)
1/2
(z
0
−1)
(z
0
+2)
3
+
32
(z
0
+2)
3
[2(2)
1/2
+ (z
0
+2)ρ]exp
−ρ(z
0
+2)
2
1/2
.
(52.80)
These results have been found to compare favorably
with both the relative experimental results of [52.50]
and the absolute experimental results of [52.51]. The
Crothers quantal semiclassical approximation has been
successfully applied to other threshold (e, 2e) and
(photon, 2e) collisions, namely two-electron photode-
tachment from H
−
[52.52], He
4
P
0
5/2
[52.53]and
K
−
[52.54]. Further investigations of the TDCS for
helium at threshold have since been carried out. The
3
P
0
triplet contribution to the TDCS was studied
in [52.55], where small but notable improvements
in the comparison with experimental results [52.50]
and [52.51] were achieved for most configurations
of the angles θ
1
and θ
2
. The inclusion of contri-
butions from
3
D
e,0
or
3
F
0
to the absolute TDCS
were found to be negligible in comparison with the
effect of the
3
P
0
[52.56], although the admittedly
non-Wannier effective-charge investigation of θ
12
= π
by Pan and Starace [52.57] suggests that
3
F
0
may
be important at θ
1
= π/6or5π/6, in line with
the experiment of Rösel et al. [52.51]. Another as-
pect which is thought to contribute to the TDCS is
explicit correlation in the initial bound state wave
Part D 52.2
Continuum Distorted Wave and Wannier Methods 52.2 Wannier Method 785
function for the helium target, in which the inter-
electronic distance r
23
is explicitly contained. Absolute
singlet triple-differential cross sections have been ob-
tained [52.58], using a helium ground state wave
function developed by Le Sech [52.59]. Again, excel-
lent agreement with the singlet results of [52.8]has
been achieved, and, in most configurations, notable
improvements with the corresponding experimental
data are obtained, as shown in Fig. 52.3 for scat-
tering angle θ
1
= 60
◦
(a) and 90
◦
(b), indicating
that electron correlation should also be considered
if a full picture of threshold ionization is to be
achieved.
The last eight years has seen significant new
developments and contributions to the Wannier the-
ory. One notable achievement has been the analytical
continuation of the uniform semi-classical wave func-
tion [52.8] to below the energy threshold to calculate
the complex eigenenergies for doubly excited states
of helium using a complex Bohr–Sommerfeld quan-
tization rule with at least one complex transition
point [52.4, 5]. The real parts of the eigenvalues
were found to be in good agreement with the ex-
perimental results of Buckman et al. [52.60, 61]for
the resonance positions while the imaginary parts
a) b)
(a) θ
1
= 60° (b) θ
1
= 90°
Fig. 52.3a,b Helium triply differential ionization cross section for coplanar geometry, E
1
= E
2
= 1eV and E = 2eV,
calculated (full curve) TDCS of Copeland and Crother s [52.58] in polar coordinates, with polar coordinates as θ
2
,for
scattering angles θ
1
= 60
◦
(a) and θ
1
= 90
◦
(b), in comparison with absolute experimental (circles)dataofRösel et al.
[52.44], and theoretical results (broken curve)ofCrothers [52.40]. The radius of each circle is 1.0×10
−19
cm
2
sr
−2
eV
−1
give the explicit widths of the resonances from
which the intensities have been estimated. The theory
in [52.4] was considered initially for the inaugu-
ral case of L = 0. Further investigation [52.5–7]has
extended the theory to include resonant states for
L = 1andL = 2. In the case of L = 1, an irra-
tional quantum number was obtained and attosecond
lifetimes were obtained. Excellent results were ob-
tained for the resonance positions, lifetimes, intensities,
and scaling rules in comparison with the experimen-
tal data [52.60]. This success persuaded Deb and
Crothers [52.62] to re-visit the problem of quantal
near-threshold ionization of He by electron impact.
In particular they re-examined the problem of above
threshold ionization of He by electron impact by re-
taining the term 2L(L +1)/ρ
2
in the hyperspherical
equation
1
ρ
5
∂
∂ρ
ρ
5
∂
∂ρ
+
1
ρ
2
sin
2
2α
∂
∂α
sin
2
2α
∂
∂α
+
4
ρ
2
sin θ
12
∂
∂θ
12
sin θ
12
∂
∂θ
12
+2E
+
2ζ(α, θ
12
)
ρ
2L(L +1)
ρ
2
Ψ = 0 .
(52.81)
Part D 52.2
786 Part D Scattering Theory
Following the procedure of Crothers [52.8], they ob-
tained
Ψ
−∗
f
=
c
1/2
E
m
12
/2
u
1/2
1
˜
ω
1/2
ρ
5/2
sin α cos α
δ
ˆ
k
1
−
ˆ
r
1
× δ
ˆ
k
2
−
ˆ
r2
exp
4i(8Z
0
ρ)
−1/2
(∆θ
12
)
−2
×exp
−i
S
0
+
1
2
S
1
(∆α)
2
+
1
8
S
2
(∆θ
12
)
2
+
π
4
−conjugate
,
(52.82)
where the classical action variables are given by
S
0
=
ρ
0
d
˜
ρ
˜
ω, (52.83)
S
i
= ρ
2
ω(ln u
i
)
, i = 1, 2 (52.84)
with
˜
ω =
ω
2
−ω
(
ln
u2
−ilnu
1
)
1/2
, (52.85)
ω
2
= 2E +2Z
0
/ρ −2L(L +1)/ρ
2
. (52.86)
The primes in (52.84)and(52.85) denote derivatives
with respect to ρ and
˜
ρ respectively. It is to be noted
here that the original work used the approximated form
of ω by dropping the L-dependent term in (52.86). The
inclusion of this angular momentum term moves the
classical turning point from the origin to ρ
+
,where
ρ
+
=
−Z
0
+
Z
2
0
+4EL(L +1)
2E
.
(52.87)
As a result, the lower limit of ρ integration will be
replaced by ρ
+
. The classical action variables S
1
and
S
2
are now evaluated without introducing the limit
Eρ → 0. Using the final state wave function in (52.82)
we have calculated first the direct ionization amplitude.
The exchange ionization amplitudes for the two indis-
tinguishable atomic electrons were then obtained by
interchanging the angles θ
1
and θ
2
in the direct ampli-
tude. Singlet and triplet contributions are then accounted
for in the usual ratio of 1:3.
Equation (52.82) is a more accurate variant of
(52.68) used in the original theory of Crothers [52.8].
Using this refinement of the wave function, all partial
wave contributions for singlets and triplets are accounted
for up to L = 6 for the case of He by electron impact
at an excess of 2 eV above threshold. It has been found
that within the co-planar geometry, both the symmetric
and asymmetric triple differential cross sections, peak-
ing at and near the Wannier ridge, are greatly improved
when compared with experiment [52.63]. However, far
away from the Wannier ridge the triple differential cross
sections tend to show qualitative differences from mea-
surement [52.63].
The improved theory [52.62] has also successfully
been applied to the calculation of total cross sections
of positron impact ionization of helium for energies
0.5–10 eV above threshold [52.64]. Excellent agreement
with available experimental data [52.65] was obtained
for the absolute theoretical calculation of positron im-
pact ionization [52.64].
Finally this recent work has answered some ques-
tions concerning near-threshold processes, a large and
interesting area of study and we firmly believe that fur-
ther development of this powerful theory will answer
many more.
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52.57 C. Pan, A. F. Starace: Phys. Rev. A 45, 4588 (1992)
52.58 F. B. M. Copeland, D. S. F. Crothers: J. Phys. B 27,
2039 (1994)
52.59 L. D. A. Siebbeles, D. P. Marshall, C. Le Sech: J. Phys.
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52.60 S. J. Buckman, P. Hammond, F. H. Read, G. C. King:
J. Phys B 16, 4039 (1983)
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52.63 T. Rösel, J. Röder, L. Frost, K. Jung, H. Ehrhardt,
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77, 1250 (1996)
Part D 52
789
Ionization in H
53. Ionization in High Energy Ion–Atom Collisions
Atomic species moving at high velocities form
an important component of ionizing radiation.
When these species interact with matter, they
effect chemical and biological changes which
originate with primary collision processes, usually
the ejection of electrons. This chapter gives an
overview of this primary process that has emerged
from studies of the energy and angular distribution
of electrons ejected by the impact of atomic or
ionic projectiles on atomic targets. It seeks to
highlight those features which are most
53.1 Born Approximation ............................ 789
53.2 Prominent Features ............................. 792
53.2.1 Target Electrons ........................ 792
53.2.2 Projectile Electrons .................... 796
53.3 Recent Developments........................... 796
References .................................................. 796
ubiquitous [53.1, 2]. Atomic units [53.3] are used,
although e, m,and
are sometimes exhibited
explicitly to show the connection with standard
treatments of the first Born approximation [53.1].
When high-velocity projectiles strike atomic targets,
electrons are ejected from the target atoms, and, if
the projectiles have some electrons, from the projec-
tile ions also. For partially ionized projectiles, there are
two groups of electrons, one from the target and one
from the projectile. In principle, these two groups of
electrons cannot be separately identified, but in prac-
tice each group often predominates in separate energy
and angular regions, and can be identified with the help
of computed distributions. These disstributions are ex-
pressed in terms of doubly differential cross sections
(DDCS). Two somewhat different cross sections are
used. The Galilean invariant cross sections are differ-
ential in the wave vectors k of the Schrödinger waves
for the ejected electrons and are denoted by d
3
σ/ dk
3
.
Thewavevectorsk refer to the laboratory or tar-
get frame. The Galilean invariant cross sections take
the same form in any reference frame, including the
projectile frame. In this frame the electron wave vec-
tors are denoted by primes, k
.TheDDCS is given
in terms of an alternative expression in Sect. 53.1,
(53.9).
53.1 Born Approximation
The first Born approximation, often referred to as FBA or
B1, provides an excellent framework to understand qual-
itatively most of the prominent features observed in fast
ion–atom collisions. For a bare ion projectile of charge
Z
P
impinging upon a neutral atom target of nuclear
charge Z
T
, the Hamiltonian of the system is written as
H = H
0
+ H
1
, (53.1)
where
H
0
=
P
2
P
2µ
+
Z
T
j=1
p
2
j
2m
−
Z
T
e
2
r
j
+
Z
T
i> j
e
2
|r
i
−r
j
|
,
H
1
=
Z
P
Z
T
e
2
R
−
Z
T
j=1
Z
2
P
|R−r
j
|
,
(53.2)
and where µ is the projectile reduced mass, m the
electron mass, R the position vector of the pro-
jectile with respect to the target nucleus, P
P
the
corresponding momentum operator, r
j
the position
vector of the ith target electron and p
j
the cor-
responding momentum operator. The first Born ap-
proximation consists of breaking H as in (53.1):
the wave functions employed are antisymmetric
products of the target eigenfunctions of H
0
and
a plane wave of relative motion for the target
and projectile. Transitions are induced by the in-
teraction term H
1
. For an inelastic collision, the
target wave function is orthogonal to the initial state
wave function and the matrix element of the first
term in H
1
, Z
P
Z
T
e
2
/R , vanishes. Using Bethe’s
Part D 53
790 Part D Scattering Theory
integral
e
iq· R
|R−r|
d
3
r =
4π
q
2
e
iq· R
, (53.3)
the inelastic cross section becomes [53.4]
σ
if
=
4µ
2
e
4
4
K
f
K
i
dΩ
K
f
Z
2
P
q
4
× |Φ
f
|
j
exp(iq· r
j
)|Φ
i
|
2
, (53.4)
where K
i
and K
f
are the initial and final momenta of
the projectile,
q = (K
f
−K
i
) is the momentum trans-
ferred from the projectile to the target, Φ
( f,i)
represents
the wave functions for the final ( f ) and initial (i) states
of the target electrons, Ω
K
f
is the solid angle of the
scattered projectile, j sums over all target electron coor-
dinates r
j
, and an average over initial and sum over final
states is assumed [53.4]. The integral over dΩ
K
f
can be
performed by making the change of variables
dΩ
K
f
≡ sin θ
K
f
dθ
K
f
dφ
K
f
=
1
K
i
K
f
q dqdφ
K
f
(53.5)
and substituting K
i
= µv
i
to obtain
σ
if
=
8πa
2
0
Z
2
P
(v
i
/v
B
)
2
×
K
+
K
−
dq
q
|Φ
f
|
Z
T
j=1
exp(iq· r
j
)|Φ
i
|
2
(qa
0
)
2
,
(53.6)
where v
i
is the initial relative velocity and v
B
= αc is
the atomic unit of velocity. (v
i
/v
B
)
2
= T/R in the no-
tation of [53.4]. From the definition of q, the limits of
integration are K
±
=|K
i
±K
f
|. The cross section is
independent of the mass of the incident projectile.
The dimensionless generalized oscillator strength
(GOS) is defined by [53.5]
f
if
(q) ≡
∆E
R
∞
|Φ
f
|
j
exp(iq· r
j
)|Φ
i
|
2
(qa
0
)
2
, (53.7)
where ∆E is the energy lost by the projectile in the
collsion; again, an average over initial and sum over final
magnetic substates is assumed. In terms of the GOS,the
cross section becomes
σ
if
=
8πa
2
0
Z
2
P
(v
i
/v
B
)
2
K
+
K
−
f
if
(q)
∆E/R
∞
dq
q
.
(53.8)
The DDCS, as defined, is related to the Galilean invariant
cross section d
3
σ/ dk
3
by [53.6]
d
3
σ
if
dE
k
dΩ
k
=
m
2
k
d
3
σ
if
d
3
k
,
(53.9)
but it is more useful for describing the low energy
ejected electrons since it connects smoothly with cross
sections for discrete excitations of the target. The first
Born approximation for both excitation and ionization
by a bare projectile scale as Z
2
P
. The advantage of the
GOS is that, in the limit q → 0, it approaches the op-
tical oscillator strength – a relationship that connects
ionization in fast ion–atom collisions with photoioniza-
tion [53.7].
For an ionization process where an electron of
momentum
kis ejected from the target, the DDCS,dif-
ferential in ejected electron energy E
k
and angle Ω
k
,
denoted by d
3
σ
if
/dE
k
dΩ
k
,isalsogivenby(53.4)
with the proviso that the Φ
f
becomes Φ
−
f
, representing
a target ion plus an ejected (continuum) electron with
incoming wave boundary conditions [53.5], normalized
on the energy scale [53.3], i. e.,
Φ
−
f
(E)
Φ
−
f
E
= δ
E − E
.
(53.10)
If the energy is in Rydbergs, the asymptotic form of the
radial part, r →∞, of the continuum wave function of
angular momentum is given by [53.7]
1
r
1
πk
1
2
sin
kr −π/2 +k
−1
ln 2kr +σ
+δ
,
(53.11)
where σ
≡ arg Γ( +1 −i/k) is the Coulomb phase
shift, and δ
is the non-Coulomb phase shift due to
the short range part of the potential. For normalization
on the energy scale, in atomic units (1 a.u. = 2R
∞
),
this asymptotic form is multiplied by
√
2, and for
normalization on the k-scale, (53.11) is multiplied by
√
2k [53.3].
One can also obtain the single differential cross sec-
tion (SDS), dσ
if
/dE
k
, by integrating over Ω
k
.This
cross section can be expressed in terms of the differential
GOS density in the continuum [53.5],
d f
if
(q)
dE
k
=
∆E
R
∞
1
(qa
0
)
2
×
|Φ
−
f
|
Z
T
j=1
exp(iq· r
j
)|Φ
i
|
2
dΩ
k
,
(53.12)
Part D 53.1
Ionization in High Energy Ion–Atom Collisions 53.1 Born Approximation 791
as
dσ
if
dE
k
=
8πa
2
0
(v
i
/v
B
)
2
d f
if
(q)/dE
k
∆E/R
dq
q
,
(53.13)
with ∆E = E
k
+I
T
,andI
T
is the ionization energy of
the target. Finally, the total ionization cross section is
obtained by integration of dσ
if
/dE
k
over E
k
,
σ
if
=
8πa
2
0
Z
2
P
(v
i
/v
B
)
2
Z
2
P
d f
if
(q)/dE
k
∆E/R
dq
q
dE
k
.
(53.14)
Let us now consider a collision in which the projectile
brings in its own N
P
electrons. The Hamiltonian for the
system becomes
H = H
0
+ H
1
H
0
=
p
2
P
2µ
+
Z
T
j=1
p
2
j
2m
−
Z
T
e
2
r
j
+
Z
T
i> j
e
2
|r
i
−r
j
|
+
N
P
k=1
p
2
k
2m
−
Z
P
e
2
r
k
+
N
P
i>k
e
2
|r
i
−r
k
|
H
1
=
Z
P
Z
T
e
2
R
−
Z
T
j=1
Z
P
e
2
|R−r
j
|
−
N
P
k=1
Z
T
e
2
|R+r
k
|
+
Z
T
j=1
N
P
k=1
e
2
|R+r
k
−r
j
|
(53.15)
where p
k
and r
k
refer to projectile electrons. Under
these conditions, the solutions of H
0
include the wave
function of the projectile electrons in the antisymmetric
product. For an inelastic collision, the matrix element
of the first term of H
1
vanishes as before. However,
there are two extra terms in H
1
: the interaction of the
projectile electrons with the target nucleus, and the inter-
action of the target electrons with projectile electrons.
These projectile electrons open physically distinct, al-
ternative possibilities for the ionization process [53.2]:
(a) target ionization, projectile remains in initial state;
(b) target ionization, projectile excited (including ioniza-
tion); (c) projectile ionization, target remains in initial
state; (d) projectile ionization, target excited (including
ionization).
For process (a), evaluation of the first Born cross sec-
tion gives (53.4) with the projectile charge Z
2
P
replaced
by |Z
P
−F
ii
(q)|
2
, where the elastic form factor F
ii
(q) is
given by
F
ii
(q) =
Φ
P
i
N
P
k=1
exp(iq· r
k
)
Φ
P
i
,
(53.16)
and Φ
P
( f,i)
represents the wave function for the final and
initial electron state of the projectile. Thus, the effect
of the projectile electrons on process (a) is to screen
the projectile nucleus, and the dynamical screen-
ing depends upon the momentum transfer q. Clearly
from (53.16), F
ii
(0) = N
P
and F
ii
(∞) = 0. Thus,
for small energy transfer, which implies small q
(large impact parameter), Z
P
→ Z
P
− N
P
,i.e.,full
screening of the projectile by its electrons. For
large energy transfer, which implies q (small im-
pact parameter), Z
P
remains unmodified, i. e., no
screening.
10
–18
10
–19
10
–20
10
–21
10
–22
0 5 10 15 20 25 30 35
ε (Ry)
υ = 60°
He
++
He
+
H
+
eV sr
cm
2
δ
2
σ
δεδ⍀
Fig. 53.1 Theoretical double differential cross sections
(DDCS) for the ionization of He by equal velocity H
+
,
He
++
and He
+
(target ionization with no projectile exci-
tation only) as a function of ejected electron energy in Ry
for an ejection angle of 60
◦
. The incident velocity cor-
responds to 0.5MeV H
+
and 2.0MeV He
+
and He
++
which all have the same velocity as a 20 Ry (272 eV)
electron
Part D 53.1
792 Part D Scattering Theory
To illustrate this effect, a calculation of process (a) is
presented in Fig. 53.1 for the DDCS for 2 MeV He
+
+He
collisions, along with a DDCS calculation for equal
velocity H
+
and He
++
projectiles [53.8]. From these
results it is seen that for small energy transfer, the He
+
projectile behaves almost like a heavy proton. With in-
creasing ejected electron energy, however, it is seen to
approach He
++
-like behavior. Note that this result is
process (a) alone.
Looking at process (b), the situation is rather dif-
ferent. Now two electrons change their state so that
transition matrix elements of all terms except the
electron–electron term in H
1
vanish. In this case, Z
2
P
in
(53.4) is replaced by the square of the inelastic projectile
form factor |F
fi
|
2
,where
F
fi
(q) =
Φ
P
f
N
P
k=1
exp(iq· r
k
)
Φ
P
i
.
(53.17)
Owing to the orthogonality of initial and final projec-
tile states, this form factor vanishes in both the limits,
q → 0andq →∞. Thus, the total target DDCS is
process (a) plus the sum over all of the possible pro-
jectile excitations (including projectile ionization) of
process (b). Since there is an infinite number of pro-
jectile excitations, it is useful to have a method for
summing over all of them. Using the closure relation,
the sum over all projectile excitations at fixed q is given
by [53.2]
f=i
|F
fi
(q)|
2
= N
P
−|F
ii
(q)|
2
+
Φ
P
i
×
N
P
j, j
, j=j
exp[iq· (r
j
−r
j
)]
Φ
P
i
,
(53.18)
where only the initial state wave function of the pro-
jectile electrons appears. While this sum rule is exact,
it cannot be substituted exactly into (53.4) because of
the transformation between cos θ
K
f
and q, and hence
the limits of integration over q vary with the excitation
energy of the final state of the projectile. Various ways
of choosing approximate integration limits have been
suggested [53.9].
Projectile ionization alone, and with target excita-
tion, process (c) and (d) above, are handled exactly as
(a) and (b), but in the projectile reference frame. Thus,
ionization of the projectile by the target is calculated
using the methodology detailed above, after which the
results are transformed into the laboratory frame using
the invariance of d
3
σ
if
/d
3
k and k
L
= k
P
+K
i
/µ.The
subscripts L and P refer to the laboratory and projectile
frame, respectively.
53.2 Prominent Features
Aplotofd
3
σ/ dk
3
superimposes electrons from the tar-
get and electrons from the projectile. To sort out the
main features of the two groups of electrons, first con-
sider impact of bare ions on neutral targets, where the
DDCS exhibits only electrons ejected from the target.
Figure 53.2 shows a computed cross section for pro-
tons on H. The wave vectors are resolved into two
components: a component k
, which is parallel to the
wave vector K
i
of the projectile in the laboratory frame,
and a component k
⊥
perpendicular to K
i
.Inmost
cases, integration is over the direction of K
f
so that
K
i
is an axis of symmetry. Then the DDCS is in-
dependent of the direction of k
⊥
. Because Fig. 53.2
was computed using an approximate theory described
in Sect. 53.2.1, the details are not necessarily accu-
rate, but these computations are qualitatively reliable
over an electron energy and angular range which ex-
cludes very low electron energies, i. e., the region around
k =0.
53.2.1 Target Electrons
The Bethe Ridge
Figure 53.2 provides a useful starting point to exam-
ine electron energy and angular distributions. The most
prominent feature is the semicircular ridge, called the
Bethe ridge, surrounding a valley. The center of the semi-
circle is at k = v,wherev = K
i
/M
P
is the velocity of
the projectile in the lab frame. In the projectile frame,
where k
= k−v, the center of the circle is at k
= 0and
the ridge is at k
= v. This region corresponds to ion-
ization events where the momementum J =q−k of the
recoiling target ion vanishes. Momentum conservation
implies that
K
i
= K
f
+k+ J , (53.19)
while energy conservation is expressed as
K
2
i
2M
P
+ε
i
=
K
2
f
2M
P
+
k
2
2
+
J
2
2M
T
, (53.20)
Part D 53.2
Ionization in High Energy Ion–Atom Collisions 53.2 Prominent Features 793
where M
T
is the mass of the ionized target. At the Bethe
ridge, J ≈ 0sothat(53.19)and(53.20) combine to
determine the value of k at this point, called k
B
;
[(1 +m/M
P
)k
B
−v]
2
= v
2
+(1 +m/M
P
)ε
i
(53.21)
where ε
i
is the initial electron eigenenergy. In the ap-
proximation that m/M
P
is set equal to zero and the
inital binding energy is small compared with 1/2v
2
,
(53.21) shows that the prominent ridge seen in Fig. 53.2
at k
= v corresponds to collisions where all of the mo-
mentum lost by the projectile is transferred to the ejected
electron. This ridge extends from E
k
= 0uptoelectron
energies of the order of 2v
2
. Generally, ejected elec-
trons move in the combined field of both the target
and the projectile. Electrons with momenta k
in the
region 0 < k
<vZ
P
/(Z
P
+ Z
P
) will be referred to as
low energy electrons since they move in regions where
the target potential is stronger than that of the projec-
tile. For very fast projectiles, many more electrons are
in the low energy region than at higher energies, thus
this region is of special interest. Since the target poten-
tial influences the motion significantly, the ionization
process in this region represents a continuation of ex-
citation across the ionization threshold. To analyze this
region, cross sections differential in E
k
and angle Ω
k
in the target frame are preferable since they connect
smoothly with cross sections for exciting target states
(53.4). Because the DDCS is independent of the az-
imuthal angle ϕ
k
,theDDCS integrated over ϕ
k
is often
employed.
As the variables E
k
, θ
k
suggest, the main features
of the low energy electrons emerge in plots of energy
and angular distributions. For fixed energy, the Bethe
ridge appears at an angle θ
B
givenby(53.21) and can be
written as a relation between cos θ
B
and E
k
as
cos θ
B
=
E
k
+ε
i
√
2E
k
1
v
.
(53.22)
For fixed E
k
= 0, the DDCS maximizes at an angle
that approaches 90
◦
as v →∞; in particular this an-
gle is obtained when E
k
=−ε
i
. Figure 53.3 shows
the angular distribution of 13.6 eV electrons ejected
from He by 5 MeV proton impact [53.10]. The cross
section maximizes at, and is nearly symmetric about,
90
◦
. This feature is understandable from the Born
approximation.
For small deflection angles, (53.20)gives
q
≈ K
i
−K
f
≈
E
k
+ε
i
v
.
(53.23)
15
10
5
0
–5
–10
–15
0
2
4
6
8
10
12
14
2
0
–2
–4
–6
–8
–10
–12
–14
k
P
parallel
k
P
perpen-
dicular
log (DDCS
(arb. units))
Fig. 53.2 Plot of the Galilean invariant DDCS for electrons ejected
from atomic hydrogen by 1.5 keV proton impact computed in the
distorted wave strong potential Born approximation (DSPB)
For electron energies E
k
of the order of the initial
binding energy |ε
i
| and large v, the parallel compon-
ent of q is much smaller than the magnitude given
by (53.19), so that q is predominantly perpendicular
to the beam direction. From (53.4), electrons ejected
2
1.5
1
0.5
0
0 40 80 120 160
Angle (deg)
DDCS(10
–24
m
2
/sr)
Fig. 53.3 Plot of the angular distribution of 13.6 eV elec-
trons ejected from He by 5 keV proton impact [53.10]. The
solid curve is the Born approximation and the points are
the measured values
Part D 53.2