Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
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856 Part D Scattering Theory
56.8.4 Classical Charge Transfer Cross
Section
Applying the classical energy-change cross section
result (56.69)ofGerjuoy [56.16] to the case of charge-
transfer yields the four cases [56.17]
σ
CX
(v
1
,v
2
) ∼
∆E
u
∆E
σ
(eff)
∆E
(v
1
, v
2
)d∆E
=
πe
4
3v
2
1
v
2
$
−
2v
3
2
(∆E)
2
−
6v
2
/m
2
∆E
%
,
(56.243)
which is valid for 0 < ∆E < b,or
σ
CX
(v
1
,v
2
) =
πe
4
3v
2
1
v
2
3
v
1
/m
1
−v
2
/m
2
∆E
+
v
3
2
−v
3
2
−
v
3
1
+v
3
1
(∆E)
2
%
,
(56.244)
which is valid for b < ∆E < a,or
σ
CX
(v
1
,v
2
) =
πe
4
3v
2
1
v
2
$
−
2v
2
1
(∆E)
2
%
,
(56.245)
which is valid for ∆E > a, m
e
v
2
>(m
1
−m
e
)v
1
,or
otherwise σ
CX
=0when∆E > a, m
e
v
2
<(m
1
−m
e
)v
1
.
The above expressions for σ
CX
(v
1
,v
2
) must be averaged
over the speed distribution W(v
2
) before comparison
with experiment. See [56.17] for details. The constants
a and b above are as defined in Sect. 56.8.3,andthe
limits ∆E
,u
are given by
∆E
=
1
2
m
e
v
2
1
+U
A
−U
B
, (56.246a)
∆E
u
=
1
2
m
e
v
2
1
+U
A
+U
B
, (56.246b)
v
2
=
2U
A
/m
e
, (56.246c)
where U
A,B
are the binding energies of atoms A and B.
The expressions above for σ
CX
diverge for some v
1
> 0
if U
A
< U
B
.IfU
A
=U
B
then σ
CX
diverges at v
1
= 0.
To avoid the divergence, employ Gerjuoy’s modification,
∆E
=
1
2
m
e
v
2
1
+U
A
.
56.9 Born Approximation
See reviews [56.37,38], as well as any standard textbook
on scattering theory, for background details on the Born
approximation, and [56.39–42] for extensive tables of
Born cross sections.
56.9.1 Form Factors
The basic formulation of the first Born approximation
for high energy heavy particle scattering is discussed
in Sect. 53.1. For the general atom–atom or ion–atom
scattering process
A(i) + B(i
) → A( f) + B( f
), (56.247)
with nuclear charges Z
A
and Z
B
respectively, let K
i
and K
f
be the initial and final momenta of the
projectile A,and
q = K
f
− K
i
be the momentum
transferred to the target. Then (53.6) can be written in
the generalized form
σ
i
f
if
=
8πa
2
0
s
2
t
+
t
−
dt
t
3
Z
A
δ
if
−F
A
if
(t)
2
×
Z
B
δ
i
f
−F
B
i
f
(t)
2
, (56.248)
where the momentum transfer is t =qa
0
,ands =v/v
B
is the initial relative velocity in units of v
B
. The form
factors are
F
A
if
(t) =
Φ
A
f
N
A
k=1
exp(it · r
a
/a
0
)
Φ
A
i
,
(56.249)
where N
A
is the number of electrons associated with
atom A, and similarly for F
B
if
(t). The limits of integra-
tion are t
±
=|K
f
±K
i
|a
0
. For heavy particle collisions,
t
+
∼∞and
t
−
= (K
i
−K
f
)a
0
∆E
if
2s
1 +
m
e
∆E
if
4Ms
2
,
(56.250)
where M = M
A
M
B
/(M
A
+M
B
).
Limiting Cases. As discussed in Sect. 53.1, for the case
i = f , F
A
if
(t) → N
A
as t → 0, so that Z
A
−F
A
if
(t) →
Z
A
−N
A
. For the case i = f, F
A
if
(t) → 0ast → 0and
t →∞.
56.9.2 Hydrogenic Form Factors
Bound–Bound Transitions. In terms of τ = t/Z,
|F
1s,1s
|=
16
4 +τ
2
2
, (56.251a)
Part D 56.9
Rydberg Collisions: Binary Encounter, Born and Impulse Approximations 56.9 Born Approximation 857
|F
1s,2s
|=2
17/2
τ
2
4τ
2
+9
3
, (56.251b)
|F
1s,2p
|=2
15/2
3τ
4τ
2
+9
3
, (56.251c)
|F
1s,3s
|=2
4
3
7/2
27τ
2
+16
τ
2
9τ
2
+16
4
, (56.251d)
|F
1s,3p
|=2
11/2
3
3
27τ
2
+16
τ
9τ
2
+16
4
, (56.251e)
|F
1s,3d
|=2
17/2
3
7/2
τ
2
9τ
2
+16
4
. (56.251f)
Bound–Continuum Transitions. In terms of the scaled
wave vector κ = ka
0
/Z for the ejected electron,
|F
1s,κ
|
2
=
2
8
κτ
2
1+3τ
2
+κ
2
exp
(
−2θ/κ
)
3
1+
(
τ −κ
)
2
3
1 +
(
τ +κ
)
2
3
1−e
−2π/κ
(56.252)
where θ = tan
−1
2κ/
1 +τ
2
−κ
2
. Expressions for
the bound–continuum Form Factors for the L-shell
(2 → κ)andM-shell(3 → κ) transitions can be
found in [56.43] and [56.44], respectively. See also
§4
of [56.45] for further details.
General Expressions and Trends
For final ns states
|F
1s,ns
|
2
=
2
4
n
(n−1)
2
+n
2
τ
2
n−1
τ
2
(n+1)
2
+n
2
τ
2
n+1
×sin
2
(n tan
−1
x +tan
−1
y), (56.253)
where
x =
2τ
n(τ
2
+1 −n
−2
)
, y =
2τ
τ
2
−1 +n
−2
.
(56.254)
For final n states [56.46]
F
1s,n
(τ)
= (iτ)
2
3(+1)
√
2 +1( +1)!
(n− −1)!
(n+)!
1/2
× n
+1
(n−1)
2
+n
2
τ
2
(n−−3)/2
(n+1)
2
+n
2
τ
2
(n++3)/2
a
n
C
(+2)
n−−1
(x)
− b
n
C
(+2)
n−−2
(x) +c
n
C
(+2)
n−−3
(x)
, (56.255)
with coefficients a
n
, b
n
and c
n
given by
a
n
= (n+1)
(n−1)
2
+n
2
τ
2
,
b
n
= 2n
(n−1)
2
+n
2
τ
2
(n+1)
2
+n
2
τ
2
,
c
n
= (n−1)
(n+1)
2
+n
2
τ
2
,
and argument
x =
n
2
−1 +n
2
τ
2
(n+1)
2
+n
2
τ
2
(n−1)
2
+n
2
τ
2
.
Summation over final states:
|F
1s,n
|
2
=
|F
1s,n
|
2
= 2
8
n
7
τ
2
1
3
n
2
−1
+n
2
τ
2
×
(n−1)
2
+n
2
τ
2
n−3
(n+1)
2
+n
2
τ
2
n+3
. (56.256)
which becomes for large n,
|F
1s,n
|
2
∼
2
8
τ
2
3τ
2
+1
3n
3
τ
2
+1
6
exp
−4
(τ
2
+1)
.
(56.257)
For initial 2s and 2 p states,
|F
2s,n
|
2
= 2
4
n
7
τ
2
−
1
3
+
1
2
n
2
−
3
16
n
4
+
1
48
n
6
+n
2
τ
2
1
3
−
2
3
n
2
+
19
48
n
4
+n
4
τ
4
5
3
−
7
6
n
2
+n
6
τ
6
×
1
2
n −1
2
+n
2
τ
2
n−4
1
2
n +1
2
+n
2
τ
2
n+4
, (56.258)
|F
2 p,n
|
2
=
2
4
n
9
τ
2
3
1
4
−
7
24
n
2
+
11
192
n
4
−n
2
τ
2
5
6
−
23
24
n
2
+
1
4
n
4
τ
4
×
1
2
n −1
2
+n
2
τ
2
n−4
1
2
n +1
2
+n
2
τ
2
n+4
. (56.259)
Part D 56.9
858 Part D Scattering Theory
Power Series Expansion. τ
2
1 [56.3]
|F
1s,ns
(τ)|
2
= A(n)τ
4
+ B(n)τ
6
+C(n)τ
8
+···,
(56.260)
where
A(n) =
2
8
n
9
(n−1)
2n−6
3
2
(n+1)
2n+6
,
B(n) =−
2
9
n
11
n
2
+11
(n−1)
2n−8
3
2
5(n+1)
2n+8
,
C(n) =−
2
8
n
13
313n
4
−1654n
2
−2067
3
2
5
2
7(n+1)
2n+10
× (n−1)
2n−10
.
For analytical expressions for A(n), B(n) and C(n) for
final npand nd states see [56.47, 48].
General Trends in Hydrogenic Form Factors [56.49].
The inelastic form factor |F
n→n
| oscillates with
on
an increasing background until the value
max
= min
$
(n
−1), n
2(n+3)
(n+1)
1/2
−
1
2
%
(56.261)
is reached, after which a rapid decline for >
max
occurs. See [56.49] for illustrative graphs.
56.9.3 Excitation Cross Sections
Atom-Atom Collisions [56.50, 51]
Single Excitation. For the process
A(i) + B → A( f ) + B ,
(56.262)
(56.248) reduces to
σ
if
=
8πa
2
0
s
2
∞
t
−
dt
t
3
F
A
if
2
Z
B
−F
B
i
i
2
. (56.263)
Double Excitation. For the process
H(1s) +H(1s) → H(n) +H(n
), (56.264)
σ
1s,n
l
1s,n
=
8πa
2
0
s
2
∞
t
−
dt
t
3
F
1s,n
2
F
1s,n
2
. (56.265)
Special cases are [56.52]
σ
1s,2s
1s,2s
=
2
30
πa
2
0
880t
4
−
+396t
2
−
+81
495s
2
4t
2
−
+9
11
,
(56.266a)
σ
1s,2p
1s,2p
=
2
30
3
4
πa
2
0
11s
2
4t
2
−
+9
11
, (56.266b)
σ
1s,2p
1s,2s
=
2
29
3
2
44t
2
−
+9
55s
2
4t
2
−
+9
11
, (56.266c)
with t
2
−
=
9/
16s
2
1 +3m
e
/
4Ms
2
+···
.
Ion–Atom Collisions. For the proton impact process
H
+
+H(1s) → H
+
+H(n) , (56.267)
(56.248) reduces to
σ
1s,n
=
8πa
2
0
s
2
∞
t
−
dt
t
3
F
1s,n
(t)
2
, (56.268)
with t
−
=
1 −n
−2
/(2s).
Asymptotic Expansions
σ
1s,ns
=
4πa
2
0
n
2
−1
|X
1s→ns
|
2
24s
2
n
2
C
s
(n) −
1
s
2
+
n
2
+11
20n
2
s
4
+
313n
4
−1654n
2
−2067
8400n
4
s
6
,
(56.269)
σ
1s,np
=
2
10
πa
2
0
n
7
(n−1)
2n−5
3s
2
(n+1)
2n+5
C
p
(n) +ln s
2
+
n
2
+11
10n
2
s
2
+
313n
4
−1654n
2
−2067
5600n
4
s
4
,
(56.270)
σ
1s,nd
=
2
11
πa
2
0
n
2
−4
n
5
n
2
−1
2
(n−1)
2n−7
3
2
5s
2
(n+1)
2n+7
×
C
d
(n) −
1
s
2
+
11n
2
+13
28n
2
s
4
,
(56.271)
where C
s
(2) = 16/5, C
s
(3) = 117/32, C
s
(4) ≈ 3.386,
and
γ
n
C
p
(n) =
1.3026
n
3
+
1.7433
n
5
+
16.918
n
7
, (56.272)
γ
n
C
d
(n) =
2.0502
n
3
+
7.6125
n
5
, (56.273)
with
γ
n
≡
2
8
n
7
(n−1)
(2n−5)
3(n+1)
(2n+5)
. (56.274)
Part D 56.9
Rydberg Collisions: Binary Encounter, Born and Impulse Approximations 56.9 Born Approximation 859
Further asymptotic expansion results can be found
in [56.2–5].
A general expression for Born excitation and ioniza-
tion cross sections for hydrogenic systems in terms of
a parabolic coordinate representation (Chapt. 9)isgiven
in [56.53].
Number of Independent Transitions N
i
Between
Levels n and n
. [56.53]
N
i
= n
2
n
−
n
3
+
n
3
.
(56.275)
Validity Criterion. The Born approximation is valid
provided that [56.54]
nE ln
4E
J
n
(56.276)
for transitions n → n
when n, n
1and|n −n
|∼1.
The constant J
n
is undetermined (see [56.54] for details)
but is generally taken to be the ionization potential of
level n.
56.9.4 Ionization Cross Sections
e
−
(k) +H → e
−
(k
) +H
+
+ e
−
(κ) . (56.277)
The general expression for the Born differential ioniza-
tion cross section can be evaluated in closed form using
screened hydrogenic wave functions. The differential
cross section per incident electron scattered into solid
angle dΩ
k
, integrated over directions κ for the ejected
electron (treated as distinguishable) is [56.55–57]
I(θ, φ) dΩ
k
dκ
=
4k
kq
4
a
0
˜
Z
4
B
F
n,κ
2
(q)dΩ
k
dκ
,
(56.278)
where the form factor is given by (56.253)) for the case
n = 1s, with the ejected electron wa venumber κ and
momentum transferred q in the collision, κ
= κa
0
/
˜
Z
B
,
q = (k
−k)a
0
/
˜
Z
B
, being scaled by the screened nu-
clear char ge
˜
Z
B
appropriate to the n-shell from which
the electron is ejected. The total Born ionization cross
section per electron is
σ
B
ion
=
κ
max
0
dκ
k+k
k−k
I(q,κ
) dq , (56.279)
which is generally evaluated numerically.
56.9.5 Capture Cross Sections
Electron Capture.
A
+
+ B(n) → A(n
) +B
+
. (56.280)
In the Oppenheimer–Brinkman–Kramers (OBK)ap-
proximation [56.58],
σ
n,n
=
M
2
2π
3
v
f
v
i
1
−1
d(cos θ)
F
n→n
2
,
(56.281)
where v
i
= v
i
ˆ
n
i
, v
f
= v
f
ˆ
n
f
, θ is the angle between
ˆ
n
i
and
ˆ
n
f
, M = M
A
M
B
/(M
A
+M
B
),and
F
n,n
=
dr ds ϕ
i
(r)ϕ
∗
f
(s)
Z
A
e
2
r
×e
i(α·r+β·s)
, (56.282)
with
α = k
f
ˆ
n
f
+k
i
ˆ
n
i
M
A
M
A
+m
e
,
β =−k
i
ˆ
n
i
−k
f
ˆ
n
f
M
B
M
B
+m
e
,
k
i
=
v
f
M
B
(M
A
+m
e
)
(M
A
+M
B
+m
e
)
,
k
f
=
v
f
(M
B
+m
e
)M
A
(M
A
+M
B
+m
e
)
.
The Jackson–Schiff correction factor [56.59]is
γ
JS
=
1
192
127 +
56
p
2
+
32
p
4
−
tan
−1
1
2
p
48 p
83 +
60
p
2
+
32
p
4
+
tan
−1
1
2
p
2
24 p
2
31 +
32
p
2
+
16
p
4
,
(56.283)
and the capture cross section is
σ(n
i
i
, n
f
f
) =
γ
JS
πa
2
0
p
2
C(n
i
i
, n
f
f
)
×
∞
x
F(n
i
i
, n
f
f
;x) dx , (56.284)
with
p =
mv
i
, a =
Z
A
n
i
, b =
Z
B
n
f
,
x =
p
2
+(a+b)
2
p
2
+(a−b)
2
0
4p
2
.
Part D 56.9
860 Part D Scattering Theory
Table 56.2 Coefficients C(n
i
i
→ n
f
f
) in the Born cap-
ture cross section formula (56.284)
n
f
f
C(1s → n
f
f
)
1s 2
8
Z
5
A
Z
5
B
2s 2
5
Z
5
A
Z
5
B
2 p 2
5
Z
5
A
Z
7
B
3s 2
8
Z
5
A
Z
5
B
/3
3
3 p 2
13
Z
5
A
Z
7
B
/3
6
3d 2
15
Z
5
A
Z
9
B
/3
9
4s 2
2
Z
5
A
Z
5
B
4 p 5Z
5
A
Z
7
B
4d Z
5
A
Z
9
B
4 f Z
5
A
Z
11
B
/20
C(2s → n
f
f
) = C(1s → n
f
f
)/8
C(2 p → n
f
f
) = C(1s → n
f
f
)/24
The coefficients C in (56.284)aregiveninTable56.2,
while the functions F are given in Table 56.3 [56.58]. In
Table 56.3 Functions F(n
i
i
→ n
f
f
;x) in the Born cap-
ture cross section formula (56.284)
n
f
f
F(n
i
i
,n
f
f
; x)
1s x
−6
2s (x −2b
2
)
2
x
−8
2 p (x −b
2
)x
−8
3s (x
2
−
16
3
b
2
x +
16
3
b
4
)
2
x
−10
3 p (x −b
2
)(x −2b
2
)
2
x
−10
3d (x −b
2
)
2
x
−10
4s (x −2b
2
)
2
(x
2
−8b
2
x +8b
4
)
2
x
−12
4 p (x −b
2
)(x
2
−
24
5
b
2
x +
24
5
b
4
)
2
x
−12
4d (x −b
2
)
2
(x −2b
2
)
2
x
−12
4 f (x −b
2
)
3
x
−12
F(2s, n
f
f
;x) = (x −2a
2
)
2
x
−2
F(1s, n
f
f
;x)
F(2 p, n
f
f
;x) = (x −a
2
)x
−2
F(1s, n
f
f
;x)
Table 56.3, the appropriate value of a and b is indicated
by the quantum numbers n
i
,
i
and n
f
,
f
.
References
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56.12 A. Burgess, I. C. Percival: Adv. At. Mol. Phys. 4,109
(1968)
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56.15 M. Matsuzawa: J. Phys. B 8, 2114 (1975)
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56.21 A. Omont: J. de Phys. 38, 1343 (1977)
56.22 B. Kaulakys: J. Phys. B 17, 4485 (1984)
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56.24 V. S. Lebedev: Soviet Phys. (JETP) 76, 27 (1993)
56.25 J. P. Coleman: Case Studies in Atomic Collision
Physics I, ed. by E. W. McDaniel, M. R. C. McDowell
(North Holland, Amsterdam 1969) Chap. 3
56.26 J. P. Coleman: J. Phys. B 1, 567 (1968)
56.27 M. R. Flannery: Phys. Rev. A 22, 2408 (1980)
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56.29 M. R. C. McDowell: Proc. Phys. Soc. 89, 23 (1966)
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56.33 M. R. Flannery: Ann. Phys. 79, 480 (1973)
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Rydberg Collisions: Binary Encounter, Born and Impulse Approximations References 861
56.36 B. N. Roy, D. K. Rai: J. Phys. B 5, 816 (1973)
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Part D 56
863
Mass Transfer
57. Mass Transfer at High Energies: Thomas Peak
Thomas peaks correspond to singular second-
order quantum effects whose location may be
determined by classical two step kinematics. The
widths of these peaks (or ridges) may be estimated
using the uncertainty principle. A second-order
quantum calculation is required to obtain the
magnitude of these peaks. Thomas peaks and
ridges have been observed in various reactions
in atomic and molecular collisions involving mass
transfer and also ionization.
57.1 The Classical Thomas Process ................ 863
57.2 Quantum Description ........................... 864
57.2.1 Uncertainty Effects .................... 864
57.2.2 Conservation
of Overall Energy and Momentum 864
57.2.3 Conservation
of Intermediate Energy .............. 865
57.2.4 Example:
Proton–Helium Scattering .......... 865
57.3 Off-Energy-Shell Effects ....................... 866
57.4 Dispersion Relations ............................ 866
57.5 Destructive Interference
of Amplitudes ..................................... 867
57.6 Recent Developments........................... 867
References .................................................. 868
Transfer of mass is a quasiforbidden process. Simple
transfer of a stationary mass M
2
to a moving mass M
1
is forbidden by conservation of energy and momen-
tum. If M
1
< M
2
then M
1
rebounds, if M
1
= M
2
then
M
1
stops and M
2
continues on, and if M
1
> M
2
then
M
2
leaves faster than M
1
. In none of these cases
do M
1
and M
2
leave together. Thomas [57.1] under-
stood this in 1927 and further realized that transfer of
mass occurs only when a third mass is present and
all three masses interact. The simplest allowable pro-
cess is a two-step process now called a Thomas process
[57.2, 3].
Quantum mechanically [57.4], the second Born term
at high energies is the largest Born term and corresponds
to the simplest allowed classical process, namely the
Thomas process. While the classically forbidden first
Born term is not zero (saved by the uncertainty prin-
ciple), the first Born cross section varies at high v as
v
−12
, in contrast to the second Born cross section which
varies as v
−11
, thus dominating. Higher Born terms cor-
respond to multistep processes that are unlikely in fast
collisions where there is not enough time for compli-
cated processes. The higher Born terms (n > 2) are also
smaller than the second Born term.
57.1 The Classical Thomas Process
The basic Thomas process is shown in Fig. 57.1. Here,
the entire collision is coplanar since particles 1 and 2
go off together (that is what is meant by mass trans-
fer). We assume that all the masses and the incident
velocities v are known. Thus, there are six unknowns,
v
, v
f
and v
3
, each with two components. Conservation
of momentum gives two equations of constraint for each
collision. Conservation of overall energy gives a fifth
constraint, and conservation of energy in the interme-
diate state (which only holds classically) gives a sixth
constraint. With six equations of constraint, all six un-
knowns may be completely determined. The allowed
values of v
, v
f
and v
3
depend on the masses M
1
,
M
2
and M
3
. For example, in the case of the transfer
of an electron from atomic hydrogen to a proton, i. e.,
p
+
= H → H + p
+
, it is easily verified that the angles
are α = (M
2
/M
1
) sin 60
◦
, β = 60
◦
and γ = 120
◦
,where
m
= M
2
= m
e
and M
1
= M
3
= M
p
= 1836 m
e
.Forthe
case e
+
+ H → Ps + p
+
, it may be verified that α = 45
◦
,
β = 45
◦
and γ = 90
◦
.
Part D 57
864 Part D Scattering Theory
1 + (2, 3) (1, 2) + 3
Z
1
M
1
v
m
f
v
Z
f
Z
3
M
3
v
3
Z
f
Z
f
M
f
v
f
m
f
v
f
α
γ
β
α
Fig. 57.1 Diagram for mass transfer via a Thomas two-step
process
In general, the intermediate mass m
may be equal
to M
1
, M
2
or M
3
. We shall regard these as different
Thomas processes, and label them B, A, and C respec-
tively. The standard Thomas process (the one actually
considered by Thomas in 1927) is case A, and corre-
sponds to the first example given above. Lieber diagrams
for the Thomas processes A, B, and C [57.5] are illus-
trated in Fig. 57.2. In the Lieber diagram, mass regions
in which solutions exist for processes A, B, and C are
shown. (An equivalent diagram was given earlier by
3
2
1
∞
123
∞
M
3
/M
2
M
1
/M
2
A and C
allowed
A and B
allowed
C only
Two-step forbidden
Two-step forbidden
A
B
C
M
1
M
2
M
3
M
1
M
2
M
3
M
1
M
3
M
2
B only
Fig. 57.2 Lieber diagram for mass transfer
Detmann and Liebfried [57.3].) There are some regions
in which two-step processes are forbidden. In these re-
gions the theory of mass transfer is not fully understood
at present.
57.2 Quantum Description
57.2.1 Uncertainty Effects
In quantum mechanics, energy conservation in the in-
termediate states may be violated within the limits of
the uncertainty principle, ∆ E ≥
/∆t,where∆t is the
uncertainty in time of mass transfer. It is not possible
to determine if mass transfer actually occurs at the be-
ginning, in the middle, or at the end of the collision.
Thus, we choose ∆t =
¯
r/
¯
v,where
¯
r is the size of the
collision region and
¯
v is the mean collision velocity.
Taking
¯
r ≈ a
0
/Z
target
and
¯
v ≈ v, the projectile velocity,
we have
∆E ≈
∆t
=
¯
v
¯
r
=
v
a
0
/Z
target
=
Z
target
a
0
. (57.1)
Here, a
0
is the Bohr radius and Z
target
is the nuclear
charge of the target in units of the electron charge.
Within this range of energy ∆ E, the constraint of en-
ergy conservation in the intermediate state does not
apply.
57.2.2 Conservation of Overall Energy
and Momentum
Conservation of overall energy and momentum then
gives three equations of constraint on the four un-
knowns v
f
and v
3
[57.6], namely,
M
1
v =
M
f
+
˜
M
f
v
f
+ M
3
v
3
, (57.2)
M
1
v
2
=
M
f
+
˜
M
f
v
2
f
+ M
3
v
2
3
, (57.3)
where M
f
˜
M
f
is the mass of the upper (lower) particle
in the final state of the bound system shown in Fig. 57.1,
in which m
is the mass of the intermediate particle
M
1
, M
2
or M
3
.From(57.2)and(57.3)itmaybeshown
that the velocity of the recoil particle is constrained by
the condition
2v
3
·
ˆ
v = 2cosγ =
M
1
+ M
2
+ M
3
M
1
v
3
v
−
M
2
M
3
v
v
3
.
(57.4)
Thus, the magnitude and the direction of v
3
are not
independent. Specifying either v
3
or
ˆ
v
3
is sufficient, to-
Part D 57.2
Mass Transfer at High Energies: Thomas Peak 57.2 Quantum Description 865
gether with the equations of constraint, to determine the
energies and directions of all particles in the final state.
Similarly, one may express the equations of constraint
in terms of v
f
and
ˆ
v
f
.
57.2.3 Conservation of Intermediate Energy
In a classical two-step process, the projectile hits
a particle in the target, and the intermediate mass m
then propagates and subsequently undergoes a second
collision. Quantum mechanically, this corresponds to
a second-Born term represented by V
1
G
0
V
2
,whereV
represents an interaction and G
0
is the propagator of the
intermediate state, namely,
G
0
= (E − H
0
+ i)
−1
=−iπδ(E − H
0
) + ℘
1
E − H
0
, (57.5)
where ℘ is the Cauchy principal value of G
0
which
excludes the singularity at E = H
0
. This singularity cor-
responds to conservation of energy in the intermediate
state. It is this singularity which gives rise to the weaker
secondary ridge in Fig. 57.3 at v
3
= v. The width of the
secondary ridge is given approximately by ∆E =
/∆t,
as discussed above. The intersection of the ridges is
the Thomas peak. At very high collision velocities, the
Thomas peak dominates the total cross section for mass
transfer.
10
–23
10
–28
10
–33
v–10v
0
v+
10v
0
Recoil speed,
v
3
Recoil angle, γ
π
4
π
2
3π
2
v
p + He H + He
++
+ e
(arb. unit)
d
2
σ
dv
3
Fig. 57.3 Counting rate (or cross section) on the vertical
axis versus recoil speed v
3
and recoil angle γ for a Thomas
process in which a proton (projectile) picks up an electron
from helium (target). The captured electron bounces off the
second target electron
The constraint imposed by conservation of inter-
mediate energy may be expressed by replacing the
speed of the recoil particle v
3
by the scaled variable
K = M
3
v
3
/m
v. Then it may be shown that the conser-
vation of intermediate energy may be expressed in the
form [57.6]:
M
3
v
3
m
v
2
≡ K
2
= 1 . (57.6)
The constraints of conservation of overall energy and
momentum, i. e., (57.4), may be easily written in terms
of K as
2cosγ = r
m
M
2
K −
M
2
m
1
K
,
(57.7)
where r = (M
1
+ M
2
+ M
3
)M
2
/(M
1
M
3
).
57.2.4 Example: Proton–Helium Scattering
The effect of the constraints of conservation of overall
energy and momentum may be seen in Fig. 57.3,where
a sharp ridge is evident in the reaction
p
+
+ He → H + He
++
+ e
−
, (57.8)
and M
1
= M
p
, M
2
= M
3
= m
e
.Herev
3
is the speed
of the recoiling ionized target electron, and the target
nucleus is not directly involved in the reaction. The width
10
–25
10
–26
10
–27
10
–28
60 90 120
Emission angle (deg)
(cm
2
/sr eV)d
2
σ/dEd
⍀
Fig. 57.4 Observation of a slice of the Thomas ridge struc-
ture in p
+
+ He → H + He
++
+ e
−
at 1 MeV by Palinkas
et al. [57.7]
Part D 57.2
866 Part D Scattering Theory
of the sharp ridge is due to the momentum spread of the
electrons in helium and may be regarded as being caused
by the uncertainty principle since this momentum (or
velocity) spread corresponds to ∆ p =
/∆r where ∆r
is taken as the radius of the helium atom.
The locus of the sharp ridge in Fig. 57.3, correspond-
ing to conservation of overall energy and momentum,
is given by (57.8). The locus of the weaker ridge,
corresponding to the conservation of energy in the in-
termediate state, is given by (57.7). The intersection of
these two loci gives the unique classical result suggested
by Thomas. The width of these ridges may be estimated
from the uncertainty principle as described above.
Experimental evidence for the double ridge structure
has been reported by Palinkas et al. [57.7] corresponding
to the calculations given in Fig. 57.3, but at a collision
energy of 1 MeV, as shown below.
ThedatainFig.57.4 corresponds to a slice across
the sharp ridge of Fig. 57.3 at v = v
3
. The solid line is
a second Born calculation [57.8,9]at1MeV.Thebump
of data above a smooth backgroud is the indication of
the ridge structure.
57.3 Off-Energy-Shell Effects
In (57.6), the Green’s function G
0
contains an energy-
conserving term iπδ(E − H
0
) which is imaginary, and
a real energy-nonconserving term ℘[1/(E − H
0
)].The
latter does not occur classically; it is permitted by the
uncertainty principle and represents the contribution of
virtual (off-the-energy-shell or energy-nonconserving)
states within ±∆E =
/∆t about the classical value
E = H
0
. This quantum term also represents the effect
of time-ordering in the second Born amplitude [57.10].
In plane wave second-Born calculations, the off-
energy-shell term gives the real part of the scattering
amplitude f
2
, while the on-shell (energy conserving)
term gives the imaginary part of f
2
. These two contri-
butions are shown in Fig. 57.5.
Half of the Thomas peak comes from energy-non-
conserving contributions which are not included in
a classical description. Also, the energy-nonconserving
contribution plays a significant role in determining the
shape of the standard Thomas peak, which has been
observed [57.11].
2.0
1.0
0.0
–1.0
12345
λ = (4M sin )
2
θ
–
2
Scattering amplitude (10
–6
a. u.)
Thomas peak
ReT
2
(off-shell)
50 MeV
p+H H+p
ImT
2
(on-shell)
Fig. 57.5 Energy-conserving (on-shell) and energy-non-
conserving contributions to the second Born scattering
amplitude
57.4 Dispersion Relations
Because of the form of the Green’s function of (57.6), the
second Born contribution f
2
to the scattering amplitude
has a single pole in the lower half of the complex plane.
Consequently it obeys the dispersion relation
Re[ f
2
(λ)]=−
1
π
℘
+∞
−∞
Im[ f
2
(λ
)]
λ − λ
dλ
,
Im[ f
2
(λ)]=
1
π
℘
+∞
−∞
Re[ f
2
(λ
)]
λ − λ
dλ
. (57.9)
where Re f
2
and Im f
2
denote the real and imaginary
parts of f
2
. Thus the energy-nonconserving part of f
2
is related to an integral over the energy-conserving part
and vice versa. In the case of the dielectric constant
it is well known that the real and imaginary parts of
are also related by a dispersion relation, namely the
Kramers–Kronig relatio [57.12].
Resonances are usually a function of energy E.The
width of a resonance gives the lifetime τ of the reso-
nance. Classically, τ is how long the projectile orbits
the target before it leaves, corresponding to a delay
or shift in time of the projectile during the interac-
Part D 57.4