Binary theory of electronic stopping, a powerful update to
Bohr’s classic from 1913
Peter Sigmund∗ and Andreas Schinner†
∗
Physics Department, University of Southern Denmark, DK-5230 Odense M, Denmark
†
Institut für Experimentalphysik, Johannes-Kepler-Universität, A-4040 Linz, Austria
INTRODUCTION
In 1913, just before his work on the structure of atoms
and molecules, Niels Bohr published a seminal paper
on the theory of charged-particle penetration in matter,
based solely on classical physics [1]. This work laid the
ground for Bethe’s famous quantum theory of particle
stopping [2], but attention has since then mostly – with
notable exceptions – been paid to its historic and didactic values rather than as a tool in active research. This
despite the well-established fact that the regimes of validity of the Bohr and the Bethe theory in terms of ion
species and energy are roughly complementary [3].
The Born approximation underlying Bethe’s theory
breaks down in the regime of strong interaction, i.e. for
heavy ions at intermediate energies, in particular around
the stopping maximum. It has been known for half a
century that a proper classical theory must be superior
to the Bethe theory in this regime. However, such a
theory was unavailable since also Bohr’s theory is in part
perturbational.
A major obstacle toward establishing a valid theory of
heavy-ion stopping was the Barkas effect. Identified initially as a minute dependence of the stopping force on
the sign of the projectile charge [4], this effect is now
known to cause a major difference between proton and
antiproton stopping [5]. If treated as a second-order perturbation in the projectile charge, the effect would blow
up in case of heavy ions to unreasonable magnitudes [6].
BINARY THEORY
tron binding causes the effective ion-electron interaction
to rapidly drop to zero around the adiabatic radius v/ω,
where v is the projectile speed and ω a resonance frequency. This feature is common to classical and quantal stopping theory. Truncation of the Coulomb interaction at the adiabatic radius can alternatively be achieved
by replacing the Coulomb potential by a screened potential with the adiabatic radius serving as the screening
radius. It was found that rigorous mapping is achieved by
adoption of a Yukawa potential. Additional screening by
bound projectile electrons is easily incorporated.
With this, the elementary stopping event is reduced to
a binary collision which can be treated by classical scattering theory. This implies that the energy transfer depends on the sign of the interaction, i.e., a Barkas effect is
inherent in the theory. Extensive comparisons with rigorous expansions have been performed especially near the
perturbation limit.
In order to establish the scheme as a practical tool it
was necessary to incorporate a number of effects that
were either unknown to Bohr or unimportant for MeV αparticles, the projectiles mainly addressed in that work,
•
•
•
A solution to this problem has been found in the binary
theory of electronic stopping [7, 8]. This theory is classical from the outset but nonperturbational, and it reproduces Bohr’s predictions for distant interactions and
Rutherford’s law for close collisions. This is achieved
by mapping the binding force acting on target electrons
into the ion-electron interaction potential. Indeed, elec-
•
•
Shell corrections, accounting for the intrinsic motion of target electrons, can be allowed for via kinetic theory [9]. This scheme represents a nonrelativistic transformation between moving coordinate
frames which is rigorous for binary scattering.
Energy loss due to projectile excitation can be
added by switching the roles of target and projectile, taking due care of the charge state,
A smooth transition into the Bethe regime is
achieved by addition of an inverse-Bloch correction
[10],
Relativistic terms [11, 12] can be allowed for when
necessary and
Charge exchange can be accounted for [13].
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
98
A central point in Bohr’s paper is the connection between
the stopping force on a charged particle and the optical
properties of the penetrated material. This feature was
confirmed in quantitative terms in the Bethe theory [2],
but most practical evaluations of stopping parameters
do not make use of this connection. In applications of
the binary theory, available oscillator strengths extracted
from measurements are used wherever available [14, 15].
RESULTS
1. We emphasize that adjustable parameters are not involved in these comparisons.
Increased understanding has been achieved of the
material dependence of the stopping cross section, in
particular Z 2 -structure, deviations from Bragg additivity in compounds, and insulator-metal differences [20].
The importance of these features is governed by several
trends which partially go against each other,
•
•
Stopping cross sections
10
•
2
-dE/dx [MeV cm /mg]
O - Al
1
0.1
0.001
0.1
10
1000
80
Si
60
40
20
Since these effects hinge on the contribution to stopping from outer target shells, their magnitude must
increase with decreasing projectile speed as innershell excitation channels get closed.
For stripped ions these effects hinge on the resonance frequency ωv of the outermost shell via the
Bohr logarithm ln(mv 3 /Z 1 e2 ωv ). This causes increased sensitivity to ωv , i.e., increased Z 2 -structure
with increasing atomic number Z 1 of the projectile,
in accordance with measurements [21].
For screened ions the effective interaction range and
hence the magnitude of the stopping cross section is
determined by the competition between screening
radius and adiabatic radius. The screening radius
depends on the projectile and its velocity but is insensitive to the target material. Hence, Z 2 -structure
and related effects tend to decrease with increasing
screening, i.e., increasing atomic number and decreasing velocity.
Consequently, pronounced structural effects are expected
in the stopping of antiprotons where screening is absent
even at the lowest projectile speeds. Conversely, these effects tend to be less pronounced for heavier ions within
the region of heavy screening, i.e., around the stopping
maximum. An example is shown in figure 2 which shows
very dramatic deviations from Bragg additivity for antiprotons in LiF and a very weak effect for oxygen in the
same material [20].
E/A [MeV]
-dE/dx [keV/µm]
mmmm`
mmmm`
INPUT
Binary, 3 shells
Binary, 5 shells
Straggling
0
10
100
1000
Energy [keV]
FIGURE 1. Upper graph: Equilibrium stopping force on
oxygen in aluminium. Points: Experimental data from 16
sources compiled by Paul [16]. Solid line: Binary theory. Lower
graph: Antiprotons in Si: Calculations for two sets of input
compared with experimental data from ref. [5]. From [17].
In its present form the theory has been applied successfully to quantify stopping parameters for ions ranging in atomic number from antiprotons to argon over six
orders of energy (1 keV/u to 1 GeV/u) for numerous target materials [17, 18, 19]. Examples are shown in figure
99
Binary theory allows predictions on collisional
energy-loss straggling. Figure 3 shows results of the
first systematic study of the interplay between Barkas
effect and shell correction on this phenomenon [22].
The Barkas effect can be turned off by taking averages
between calculations for protons and antiprotons. It
is instructive here to study the contributions from the
principal target shells.
Several of the curves show a shoulder which was first
found by Bethe & Livingston [23] and which is caused
by the shell correction. This type of shoulder is seen in
all curves labelled ‘average’ in the lower graph, most
pronouncedly so for the L shell. However, the upper
mm
mmm
1.0
-
p in LiF
10
+
H - Si
Average
p - Si
W 0/W B
mmmm`
mmmm`
Bragg
LiF
F
Li
1
S [10
-15
2
eV cm ]
No shell correction
L shell
0.5
M shell
0.1
K shell
10
-3
10
-1
10
1
10
0
0.01
3
0.1
E/A1 [MeV]
1
10
E/A1[MeV]
1.0
O in LiF
Shell correction included
S [10
Bragg
LiF
F
Li
10
10
W/W B
100
-15
2
eV cm ]
0.8
-3
10
-1
0.6
+
H - Si
Average
p - Si
L shell
0.4
M shell
0.2
10
1
10
K shell
3
0
0.01
E/A1 [MeV]
FIGURE 2. Calculated stopping force on antiprotons (upper
graph) and oxygen (lower graph) ions in LiF. Solid lines are
based on known oscillator strengths of LiF. From [20].
graph, where no shell corrections were applied, shows
pronounced shoulders for protons caused by the Barkas
effect. No shoulders are found in the average curves in
accordance with common experience.
Shoulders caused by the Barkas effect are seen to occur at rather low velocities. According to kinetic theory
[9] the shell correction reshuffles the stopping cross section according to the relative motion of ion and target
electron. It is seen in the lower graph that this rearrangement wipes out most of the Barkas effect on straggling
for the inner shells which are characterized by high orbital speeds. Conversely, for the M shell, the shoulder
caused by the shell correction is barely visible while that
caused by the Barkas effect is only slightly attenuated by
the shell correction.
Although straggling is caused predominantly by close
collisions we have found a substantial reduction in straggling due to projectile screening [22]. Even weak screening was found to cause a noticeable effect which, however saturates rapidly with increasing neutralization.
100
0.1
1
10
E/A1[MeV]
FIGURE 3. Ratio of energy-loss straggling calculated from
binary theory and Bohr value W B = 4π Z 12 Z 2 e4 for protons and
antiprotons in silicon. Taking the average turns off the Barkas
effect. Contributions separated into those from principal target
shells. Upper graph: No shell correction applied. Lower graph:
shell correction incorporated. From [22].
Outlook
The initial goal of this program was to develop a theoretical tool for heavy-ion stopping that would not depend
on empirical scaling. The incorporation of the inverseBloch correction into the scheme [24] was found to extend the range of validity of the theory very efficiently
into the Bethe regime. This implies that we now have
shell and Barkas corrections even for protons that are
competitive with the results of much more elaborate calculations [19].
All calculations go over impact-parameter-dependent
energy losses as an intermediate step. Hence, binary
theory is a useful tool in the study of bulk and surface
channeling. First results [25] refer to a now-classical
experiment involving channeled heavy ions in ‘frozen
charge states’ [26, 27].
The theory can also predict nonequilibrium stopping
forces as a function of charge state [18]. In conjunction
with a statistical scheme [13] this will eventually allow
computation of energy-loss spectra differentiated into
incident and exit charge state.
ACKNOWLEDGMENTS
This work has been supported by the Danish Natural
Science Research Council (SNF).
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