Collisions of charged particles with atoms
Author: Marc Barroso Mancha
Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗
Advisor: Dr. Francesc Salvat
Abstract: Elastic collisions of charged particles with atoms are studied by using the classical
and quantum theories for the scattering of particles in central potentials. A simple calculation
scheme has been adopted and implemented in a Fortran program. Numerical results illustrate some
characteristic features of the classical and quantum differential cross sections, and allow verifying
the Bohr condition for the validity of the classical theory.
I. INTRODUCTION
Elastic collisions of fast charged particles with atoms
cause large angular deflections of the trajectories of these
particles. An accurate description of elastic collisions is
required for Monte Carlo simulation of radiation trans-
port, which finds applications in electron microscopy,
medical physics, dosimetry, and in the design and quan-
tification of radiation detection devices. Except for pro-
jectiles with low energies, elastic collisions can be de-
scribed by means of the static-field approximation, that
is, as the scattering of the projectile by the electrostatic
field of the target atom. In this report we briefly formu-
late the classical and quantum theories of scattering by
central potentials, and their implementation in a Fortran
program for the case of electron scattering by an atomic
potential model. Numerical results are used to reveal
some characteristic features of the classical and quantum
DCSs, and to analyse the applicability of Bohr’s condi-
tion for the validity of the classical scattering theory.
A. The interaction potential
We consider elastic collisions of charged particles of
mass M and charge Z1e with neutral atoms of atomic
number Z. For simplicity, the target atom is assumed to
have a point nucleus much heavier than the projectile and
fixed at the origin of coordinates, and a spherical electron
cloud. The interaction with the projectile is considered
to be purely electrostatic (static-field approximation),
V (r) =
Z1Ze
2
r
Φ(r). (1)
The screening function Φ(r) represents the shielding of
the nuclear charge by the atomic electrons, it equals unity
at r = 0 and decreases monotonously with r tending to
zero at large radii.
To ease the calculations we use analytical screening
∗Electronic address: mbarroma8@alumnes.ub.edu
functions of the form
Φ(r) =
3∑
i=1
Ai exp(−air) (2)
with parameters Ai and ai determined by Salvat et al.
[1], which approximate closely the atomic potentials ob-
tained from self-consistent Dirac-Hartree-Fock-Slater cal-
culations, and lead to an analytical expressions for the
scattering amplitude and the phase shifts in the first Born
approximation.
II. CLASSICAL COLLISION THEORY
We briefly review here the classical theory of scatter-
ing by central potentials. We assume that the projectile
starts its motion very far from the target, with linear mo-
mentum ~pi parallel to the polar axis and kinetic energy
E = p2i /2M . The angular momentum L and the impact
parameter b are related by
L = bpi. (3)
After the interaction, the particle moves in a direction
that makes an angle ϑ with the polar axis. Using polar
coordinates (r, φ) in the scattering plane, we have φ(t =
−∞) = pi and φ(t =∞) = ϑ. Notice that the scattering
angle θ is the angle between the initial and final directions
of the projectile, and it can only take values in the range
[0, pi]. A small detector at an angle θ may receive particles
which have circled the centre of force and emerge with
angles ϑ = ±θ plus integer multiples of 2pi.
To obtain the geometrical equation of the trajectory,
we consider the constants of motion
L = Mr2φ˙ and E =
p2
2M
+ V (r) = p2i . (4)
Combining these equalities with the expression of the ve-
locity in polar coordinates we obtain [2]
r˙ = ± 1
M
√
p2 − L
2
r2
. (5)
The condition r˙ = 0 determines the distance r0 of clos-
est approach, i.e., the turning point of the radial motion.
Collisions of charged particles with atoms Marc Barroso Mancha
The sign on the right-hand side of Eq. (5) is − while
the projectile is approaching the scattering centre, and
+ when the particle has passed the point of closest ap-
proach. Using the identity φ˙ = L/(Mr2), we can write
dφ =
φ˙
r˙
dr = ± L/r
2√
p2(r)− L2/r2 dr. (6)
Considering that the trajectory is symmetric with respect
to the point of closes approach, we obtain [2]
ϑ(L) = pi − 2
∫ ∞
r0
L/r2√
p2(r)− L2/r2 dr. (7)
Although the deflection angle ϑ(L) varies continuously
with L, the “inverse” function L(θ) may be multivalued,
see Fig. 1.
Considering that the number of projectile particles
scattered per unit time to directions with polar angle
between θ and θ + dθ is equal to the number of inciding
particles with impact parameters between b and b + db
per unit time, we can conclude that scattering differential
cross section (DCS) is given by [2]
dσ
dΩ
=
1
2p2i sin θ
∑
j
∣∣∣∣dL2dθ
∣∣∣∣
L=Lj
, (8)
where the summation is over the angular momenta L that
yield deflection angles ϑ corresponding to the scattering
angle θ.
A. Scattering by a screened Coulomb potential
In the case of the atomic potential (1) we have
ϑ(L) = pi−2
∫ ∞
r0
L/r2√
p2i − 2MZ1Ze2Φ(r)/r − L2/r2
. (9)
To facilitate the numerical calculations, we change the
integration variable to u = (1− r0/r)1/2 and write
ϑ(L) = 2 arctan
(
MZ1Ze
2Φ(r0)
Lpi
)
+
+ 4
∫ 1
0
{
1√
CΦ(r0) + 2− u2
−
− 1√
CΦ(r0) + 2− u2 − Cg(u)
}
(10)
with
C =
2MZ1Ze
2r0
L2
, g(u) = r0
Φ(r)− Φ(r0)
r − r0 . (11)
My Fortran program computes this integral for a dense
grid of L values by using a 20-point adaptive Gauss-
Legendre quadrature [3], and represents the function
ϑ(L) by means of the natural cubic spline that inter-
polates the numerical table. The derivatives in Eq. (8)
are obtained by computing dθ/dL using the interpolating
spline.
B. A second look at ϑ(L)
The scattering characteristics are embedded in the
function ϑ(L) [4]. Interesting phenomena arise when the
function has either maxima or minima, and when L(θ) is
multivalued. A maximum or minimum of ϑ(L) causes a
sharp peak in the DCS, a feature called rainbow scatter-
ing. When ϑ(L) passes smoothly through a point where
sinϑ = 0 (that is, through ϑ = 0, or ±npi) the DCS be-
comes infinite and the situation is called glory scattering.
If the energy of the projectile is close to a maximum of
the effective radial potential, the radial component of the
velocity is small and we may have orbiting trajectories.
All these effects are related to each other, and some of
them imply the existence of others. We will see examples
of them in the Results section.
−300
−250
−200
−150
−100
−50
0
−10 −5 0 5 10
ϑ
(d
eg
)
logL2
E = 500 eV
E = 1000 eV
E = 2000 eV
E = 5000 eV
FIG. 1: Classical deflection angle as a function of the angular
momentum for scattering of electrons of the indicated energies
by gold atoms.
C. Validity of the classical theory
It is pertinent to ask when the classical theory is valid,
that is, under what circumstances the classical DCSs
agrees with the predictions of the quantum formulation?
Bohr [5] considered this question on the basis of a simple
argument. He regarded the incident beam of projectiles
as an incoming wave that diffracts through a small cir-
cular hole with permeable edges. Then he estimated the
minimal angular aperture of the transmitted beam by
adding the apertures caused by diffraction and by the
lateral variation of the potential within the hole, and
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Collisions of charged particles with atoms Marc Barroso Mancha
found that
(∆θ)min =
√
~
∣∣∣∣ dθdL
∣∣∣∣. (12)
Consequently, the classical method should be valid as
long as (∆θ)min is much less than the deflection θ of the
particles due to the field. In the case of atomic potentials,
Bohr’s argument implies that the classical calculation is
valid for angles larger than
θclass ' ~
pia0Z1/3
, (13)
where a0Z
1/3 is an estimate of the “atomic radius”. The
correctness of Bohr’s condition will be confirmed numer-
ically below in the Results Section.
III. QUANTUM COLLISION THEORY
In the quantum theory the scattering of particles by
central potentials is described by the solution of the time-
independent Schro¨dinger equation(
− ~
2M
∇2 + V (r)
)
ψ(~r) = Eψ(~r) =
p2i
2M
ψ(~r) (14)
with the following asymptotic behaviour [6],
ψ~ki(~r) ∼r→∞ ei
~ki·~r +
eikir
r
f(θ, φ). (15)
The first term on the right-hand side is a plane wave
with wave vector ~ki (≡ ~pi/~), which describes the inci-
dent electron beam, and the second term is a spherical
wave modulated by the scattering amplitude f(θ, φ). The
results from this stationary-state solution are equivalent
to those from a more rigorous formulation that considers
the time evolution of wave packets. The potentials con-
sidered in the present work are of finite range, that is,
V (r) vanishes faster than r−1 as r →∞.
Without loosing generality, we choose a reference
frame with its origin at the centre of force and the z
axis in the direction of the initial momentum, so that
~k = kzˆ and ~r · ~ki = rki cos θ, where θ is the polar scat-
tering angle. It is important to notice that, for central
potentials, the scattering wave (15) is independent of the
azimuthal angle φ, and the same holds for the scattering
amplitude, f(θ, φ) = f(θ). It can be shown that, under
these conditions, the DCS is given by
dσ
dΩ
= |f(θ)|2 (16)
A. Partial-wave expansion
The scattering wave (15) can be expanded as
ψ~k(~r) = (2pi)
−3/2 1
kr
∑
`
(2`+ 1)i`
× exp(iδ`)PE`(r)P`(cos θ), (17)
where P`(cos θ) are the Legendre polynomials, and
PE`(r) are the solutions of the radial equation with the
asymptotic boundary condition
PE`(r) ∼r→∞ sin
[
kr − `pi
2
+ δ`
]
. (18)
The quantities δ` are the phase shifts, which represent
the effect of the potential on the large-r behaviour of the
radial wave function. Notice that the expansion (17) is
also valid for the plane waves, which are solutions of Eq.
(14) with V ≡ 0, in which case δ` = 0 for all `.
From the analysis of the asymptotic behaviour of the
series (17) we obtain the following partial-wave series for
the scattering amplitude
f(θ) =
1
ki
∑
`
(2`+ 1) exp(iδ`) sin(δ`)P`(cos θ). (19)
Inserting this expansion into Eq. (16), with the aid of the
orthogonality of the Legendre polynomials, we have
σ =
4pi
k2i
∑
`
(2`+ 1) sin2(δ`). (20)
From the partial-wave expansion we see that
σ =
4pi
ki
Imf(0). (21)
This equality is known as the optical theorem, it ex-
presses the fact that the scattering process conserves the
number of particles.
Because the calculation of each phase shifts requires
solving the corresponding radial equation, which is a deli-
cate and lengthy work, we consider approximate methods
to compute the phase shifts.
B. The plane-wave Born approximation
In the plane-wave Born approximation, the states of
the projectile before and after the interaction are rep-
resented as plane waves with respective linear momenta
~p = ~kzˆ and ~p ′ = ~~k′, and the interaction is treated as a
perturbation to first order. The DCS can then be calcu-
lated by means of the Fermi golden rule. This approach
gives the following scattering amplitude
f (B)(θ) ≡ − M
2pi~2
∫
d~r exp(i~q · ~r)V (r), (22)
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Collisions of charged particles with atoms Marc Barroso Mancha
where ~q ≡ ~p − ~p ′ is the moment transfer. Evidently,
f (B)(θ) is proportional the Fourier transform of the po-
tential. The angular part of the integral can be performed
analytically, to yield a scattering amplitude that is real
and, hence, it does not satisfy the optical theorem.
In the case of the potential (1), the Born scattering
amplitude is
f (B)(θ) = − 2M
~2
Z1Ze
2
∑
i
Ai
1
α2i + q
2
. (23)
Inserting the partial-wave expansion (17) of the plane
wave into Eq. (22) we obtain
f (B)(θ) =
1
2k
∑
`
(2`+ 1)(2δ
(B)
` )P`(cos θ)δ
(B)
` (24)
where the quantities
δ
(B)
` = −
2M
~2
k
∫ ∞
0
j2` (kr
′)V (r′)r′2dr′ (25)
are the Born phase-shifts. As suggested by the similarity
of the series (19) and (24), the Born phase shifts provide
a good approximation to the numerical phase shifts when
the latter are small in magnitude, that is, when |δ`|  1.
The Born phase shifts for the potential (1) are given
by
δ
(B)
` = −
M
~2k
Z1Ze
2
∑
i
AiQ`
(
1 +
α2i
2k2
)
. (26)
where Q`(x) are the Legendre functions of the second
kind. In my program, these functions are calculated from
their recurrence relations and the explicit expressions for
Q0(x) and Q1(x). Notice that this calculation is very fast
and accurate.
C. Wentzel-Kramers-Brilluin approximation
Fairly accurate phase shifts can be calculated by using
the Wentzel-Kramers-Brillouin (WKB) semi-classical ap-
proximation, together with the Langer correction, that
yields the following formula [7]
δ
(WKB)
` =
1
2
(
`+
1
2
)
pi − kr0 +
∫ ∞
r0
[√
F`(r)− k
]
dr,
(27)
where
F`(r) = k
2 − 2M
~2
V (r)− (`+ 1/2)
2
r2
, (28)
and r0 is the largest zero of F`(r).
To simplify the calculation, in my program the in-
tegration variable is changed to x = 1/r, and the re-
sulting integral is computed by using the adaptive 20-
point Gauss-Legendre quadrature method, which yields
the phase shifts to very high accuracy.
D. Practical partial-wave expansion method
The scattering amplitude is calculated from its partial-
wave expansion using the WKB phase shifts for orders
` ≤ L and the Born phase shifts for ` > L, with the
cutoff value L equal to the lowest angular momentum for
which |δ(B)` | < 0.001. To ensure continuity of the phase
shifts with `, we set
δ` =
{
δ
(WKB)
` if ` < L,
C`δ
(B)
` otherwise
(29)
with
C` ≡ 1 +
(
δ
(WKB)
L
δ
(B)
L
− 1
)
exp
(
−a`− L
L
)
(30)
In my Fortran program, all the Born phase shifts larger
than about 10−10 are calculated first, then the cut-off
L is determined, and finally the WKB phase shifts with
` ≤ L are calculated.
Generally, the convergence of the partial-wave series
(19) is slow. It can be accelerated by adding the Born
scattering amplitude and subtracting its partial-wave ex-
pansion. Thus, we have
f(θ) = f (B)(θ) +
∑
`
F`P`(cos θ), (31)
with
F` = 1
2ik
(2`+ 1)
[
exp(2iδ`)− 1− 2iδ(B)`
]
. (32)
IV. RESULTS
The theory and the numerical methods described
above have been implemented in a Fortran program that
calculates the classical and quantum DCSs for collisions
of electrons of a given energy with neutral atoms. The
examples presented here correspond to target gold atoms
(Z = 79). Because the program only computes the clas-
sical DCS for angles satisfying the Bohr condition (13),
we plot the classical DCS only for angles larger than 45◦.
Figure 2 displays the calculated classical DCS for elec-
trons with various kinetic energies (the same energies
for which ϑ is represented at Fig. 1). The four exam-
ples show a backward glory, with the DCS diverging at
θ = 180◦, while the three lower energies present a clear
rainbow at an angle that increases with the energy of the
projectile.
Results from the quantum theory are shown in Figure
3, for the same set of kinetic energies as in Fig. 2, for
the sake of comparison. It is worth mentioning that the
present quantum calculations are in fairly good agree-
ment with elaborate Dirac partial-wave calculations and
with available experimental data [8].
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Collisions of charged particles with atoms Marc Barroso Mancha
1.0e-19
1.0e-18
1.0e-17
1.0e-16
1.0e-15
60 80 100 120 140 160 180
d
σ
/
d
Ω
(c
m
2
/
sr
)
θ (deg)
E = 500 eV
E= 1000 eV
E = 2000 eV
E = 5000 eV
FIG. 2: Classical DCS for collisions of electrons with the in-
dicated energies and gold atoms.
10−19
10−18
10−17
10−16
10−15
10−14
0 20 40 60 80 100 120 140 160 180
d
σ
/
d
Ω
(c
m
2
/
sr
)
θ (deg)
E = 500 eV
E= 1000 eV
E = 2000 eV
E = 5000 eV
FIG. 3: Quantal DCS for collisions of electrons with the in-
dicated energies and gold atoms.
Despite the evident differences between the classical
and quantum results, the classical theory still gives DCS
with the correct order of magnitude. Of course, the
diffraction-like structures of the quantum DCS cannot
be reproduced by the classical theory. At higher ener-
gies, however, the quantum DCS varies monotonously
with the scattering angle and we may expect a closer
agreement between the two theories. This is confirmed
by the results for 5 keV electrons shown in Fig. 4, where
the classical and quantum predictions are seen to agree
pretty well at intermediate angles.
10−19
10−18
10−17
10−16
10−15
20 40 60 80 100 120 140 160 180
d
σ
/
d
Ω
(c
m
2
/
sr
)
θ (deg)
Quantum DCS, E = 5000 eV
Classical DCS, E = 5000 eV
FIG. 4: Classical and quantum DCSs for collisions of 5 keV
electrons with gold atoms.
V. CONCLUSIONS
In this work we have studied the classical and quan-
tum theories for the scattering of charged particles by
neutral atoms. We have considered relatively simple cal-
culation schemes, which allow rapid computations, and
I have written a Fortran program that gives the DCS
for collisions of electrons with arbitrary kinetic energies
colliding with atoms of any element. From the numerical
results I have understood the various features of the clas-
sical DCS. I have also verified the correctness of Bohr’s
condition, by verifying that the classical and quantum
theories yield similar results when the condition holds.
Acknowledgments
I would like to thank my advisor, Dr. Francesc Salvat,
for his guidance and help all through this work. I would
also wish to express my gratitude to my parents and my
colleagues.
[1] Salvat, et al. Analytical Dirac- Hartree-Fock-Slater screen-
ing function for atoms (Z = 192), Phys. Rev. A 36,467474,
1987.
[2] Herbert Goldstein, Charles Poole, John Safko Classical
Mechanics, 3rd ed. (Addison-Wesley Publishing Company,
2001)
[3] Course notes of Computational Physics, 2016, UB.
[4] B. H. Bransden Atomic Collision Theory, 1st ed. (W. A.
Benjamin, 1970)
[5] Bohr, N. The penetration of atomic particles through mat-
ter, Vi- densk. Selsk. Mat. Fys. Medd. 18, 1144, 1948.
[6] R. Shankar Principles of Quantum Mechanics, 2nd. ed.
(Plenum Press, 1980)
[7] Charles J. Joachain Quantum Collision Theory, 1st. ed.
(North-Holland Publishing Companym, 1975)
[8] F. Salvat Optical-model potential for electron and
positron elastic scattering by atoms, Phys. Rev. A. 68,
012708, 2003.
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