Material dependence of electronic stopping 1
Peter Sigmund and André Fettouhi
Physics Department, University of Southern Denmark, DK-5230 Odense M, Denmark
Andreas Schinner
Institut für Experimentalphysik, Johannes-Kepler-Universität, A-4040 Linz, Austria
Abstract
The dependence of the electronic stopping cross section on the atomic number of the target
material has been studied on the basis of binary stopping theory over a wide range of beam
energies. A distinct dependence on ion type and energy is found for the predicted Z 2 -structure
which becomes significantly less pronounced for heavy ions than for protons and antiprotons.
This behavior is caused primarily by the interplay between projectile screening and closing
of inner target shells as a function of the beam velocity and introduces a pronounced Z 2 structure in the effective-charge ratio. Related effects are found for stopping in three different
phases of carbon and in lithium fluoride.
Key words: Stopping; stopping power; stopping force; Z 2 -structure; Bragg rule; phase
effect; LiF; diamond
1 Introduction
Electronic stopping of charged particles in matter is known to be rather insensitive to specific
properties of the stopping material [1]. If the stopping force is expressed as −d E/d x = N S,
the dominating material dependence is contained in the density N [atoms/volume], whereas all
remaining dependence is summarized in the stopping cross section S. Dependent on the required
accuracy, S can be approximated as an atomic parameter varying smoothly with the atomic
number Z 2 of the stopping material independent of the state of aggregation.
Limitations of this simple picture exist and are commonly classified as follows,
• Z 2 -structure, i.e., nonmonotonic dependence of the stopping cross section on the atomic
number of the material,
1
Email [email protected]
Preprint submitted to Nuclear Instruments and Methods B
17 May 2002
• Phase effects such as gas-solid differences, insulator-metal differences, and deviations from
the Bragg additivity rule of stopping cross sections for compounds and alloys.
While the error introduced by ignoring such effects has barely been found to exceed the 20-30 %
level, numerous applications require higher accuracy in stopping forces. More important, little
systematic knowledge is available, in particular with regard to the dependence on type and energy
of the penetrating beam.
The present work represents an attempt to address the effects outlined above on the basis of the
binary theory of electronic stopping [2]. Predictions of this theory have been verified successfully
for several elemental target materials and for numerous projectiles ranging from antiprotons to
argon over an energy range from 1 keV/u to 1 GeV/u [3–5]. The theory is particularly well suited
for the present purpose because its main input consists of oscillator-strength spectra which are
known for a large number of elemental and compound materials.
2 Qualitative survey
With the stopping cross section expressed in standard form
4π Z 12 Z 2 e4
S=
L,
mv 2
(1)
where Z 1 and v are the atomic number and speed of the projectile and −e and m the electron
charge and mass, the pertinent physics becomes condensed in the dimensionless stopping number
L which can be expressed in the form
X
L=
fn L n ,
(2)
n
where f n is the dipole oscillator strength of the n’th target resonance ωn , normalized according
P
to n f n = 1. Eq. (2) dates in essence back to Bohr [6] but was quantified by Bethe [7]. At
high speed L n approaches the Bethe logarithm L n ∼ ln(2mv 2 / ωn ). In that limit all material
P
dependence of L reduces to the ‘I -value’ defined by ln I = n f n ln( ωn ). We recall that the
logarithmic dependence of L n on ωn originates in an integration over all impact parameters from
zero to the adiabatic radius v/ωn [6].
Z 2 -structure and phase effects hinge on the contribution of outer, incompletely-filled target shells
to the stopping cross section. These effects are small in the high-velocity regime because the
contribution of outer target shells to stopping is small except for very light target materials.
With decreasing projectile speed a number of effects become significant which may either
strengthen or weaken Z 2 -structure and related phenomena,
(1) The relative contribution of outer target shells to the stopping force increases with decreasing speed because of closing inner-shell excitation channels. This gives rise to enhanced
2
(2)
(3)
(4)
(5)
Z 2 -structure. At the same time the effect of the orbital motion of target electrons (shell
correction) enhances the dependence of L n on ωn beyond the logarithmic dependence.
In case of significant projectile screening due to electrons bound to the projectile, the
screening radius gradually replaces the adiabatic radius as the effective range of interaction.
Since the screening radius is only weakly dependent on the medium, this effect tends to
weaken and eventually remove this type of Z 2 -structure with decreasing velocity.
The Barkas effect, i.e. a contribution dependent on the sign of the projectile charge, enhances
the effect of the shell correction for a negatively-charged projectile such as an antiproton, but
opposes it for a positively-charged projectile, where it hence tends to weaken Z 2 -structure.
Quantitative details are also affected by the Bloch term at low velocities, i.e. in the regime
where Bohr’s classical theory is superior to Bethe’s theory that is based on the Born approximation.
At velocities where energy loss to projectile excitation becomes significant the relative
magnitude of Z 2 -structure tends to decrease.
Similar considerations apply to phase effects. However, an important additional point of consideration here is the gas-solid difference in charge state [8] which produces a gas-solid difference
in the stopping force.
From these considerations one would expect the most pronounced Z 2 -structures to be observed
for antiprotons where screening effects are absent. For positive ions these effects should be
most visible at velocities where screening is unimportant but shell corrections important, i.e.
2/3
2/3
v Z 1 v0 and v Z 2 v0 , respectively, where v0 is the Bohr velocity.
3 Theoretical scheme
The essentials of the binary theory have been described in a recent summary [3].
In brief, the stopping number of an individual oscillator is evaluated from a nonperturbative
extension of Bohr’s classical theory [6], and an inverse-Bloch correction is applied that extends
the range of validity of the theory into the Born regime. Shell corrections are incorporated via
kinetic theory [9]. Static projectile screening is allowed for.
Bundled oscillator strengths f n are employed to characterize the response of the target, extracted
from spectra given in refs. [10, 11] that are averaged logarithmically over pertinent frequency
intervals as described in [4] and [3]. Velocity spectra governing shell corrections are either
computed from atomic wave functions or, for conduction electrons, from a Fermi distribution.
4
Z 2 -structure
Figure 1 shows stopping forces evaluated from the binary theory for argon and helium ions
in charge equilibrium and for bare protons over six decades in energy and for elements up
3
10
1
4
L = (mv /4πZ1Z2e ) S
Ar - Z2
-1
10
10
2
mmmm`
mmmm`
µµµµ
2
10
10
10
-3
2
1
1
10
10
10
10
3
-1
-2
-3
-5
0
10
20
30
40
Z2
10
1
10
4
L = (mv /4πZ1Z2e ) S
He - Z2
10
10
3
2
1
1
2
mmmm`
mmmm`
µµµµ
2
10
-1
10
10
10
10
-3
10
-1
-2
-3
-5
0
10
20
30
40
Z2
10
1
4
L = (mv /4πZ1Z2e ) S
H - Z2
-1
10
2
10
10
2
10
3
2
1
1
10
-3
10
10
10
10
-1
-2
-3
-5
0
10
20
30
40
Z2
Fig. 1. Z 2 -structure according to the binary theory of stopping for argon and helium ions in charge
equlilibrium (upper and middle graph) and bare protons (lower graph). Projectile excitation taken into
account for argon but ignored for helium. Beam energy from 10 3 MeV/u (top curve) to 10 −3 MeV/u
(bottom curve).
4
mmmm`
mmmm`
to copper. The quantity shown is the stopping number L. In all cases, Z 2 -structure is found for
energies up to 100 keV/u and very little structure from 1 MeV/u upward. However, the amplitude
increases significantly from argon to proton bombardment. This reflects primarily the influence
of screening 2 .
500
Pb - Z2
300
-15
200
S/Z2 [10
mmmm`
mmmm`
2
eVcm ]
400
0.5 MeV/u
100
1 MeV/u
0
20
40
60
80
100
Z2
5
He - Z2
3
0.5 MeV/u
2
1 MeV/u
S/Z2 [10
-15
2
eVcm ]
4
1
0
20
40
60
80
100
Z2
Fig. 2. Measured stopping cross sections per target electron according to [12]. Upper graph: Pb ions;
lower graph: He ions.
Experimental evidence may be found from data by Geissel [12] shown in figure 2 comparing
stopping forces on Pb and He ions in charge equilibrium at 0.5 and 1.0 MeV/u. While a detailed
comparison of the observed structure is not feasible because of different systems and different
coverage with data, it is evident that Z 2 -structure is less pronounced for Pb than for helium.
Stopping tables for heavy ions [13–17] make use of the effective-charge concept to scale stopping
forces on hydrogen or helium ions up to heavier ions. The underlying assumption is that the
2
Projectile excitation, which has been taken into account for argon and ignored for helium, also tends
to smoothen Z 2 -structure, but that effect accounts for less than 25 % of the total stopping force, and only
at the lowest energy.
5
1
10
LAr/LHe
mmmm`
mmmm`
Eff charge
10
10
0.1
3
2
1
1
10
10
10
-1
-2
-3
0.01
0
10
20
30
40
Z2
1
Eff. charge
LAr / LH
10
10
0.1
10
3
2
1
1
10
10
10
-1
-2
-3
0.01
0
10
20
30
40
Z2
Fig. 3. Effective-charge ratio for argon ions evaluated from data in figure 1. Upper graph: Reference to
Helium in charge equilibrium. Lower graph: Reference to bare protons. Projectile excitation neglected
everywhere. Beam energy from 10 3 MeV/u (top curve) to 10 −3 MeV/u (bottom curve).
effective-charge ratio
γ2 =
L(Z 1 , Z 2 )
,
L(Z 1,ref , Z 2 )
(3)
with Z ref = 1 or 2, is independent of or only weakly dependent on Z 2 .
Figure 3 shows calculated effective-charge ratios for argon ions with helium and hydrogen as
reference ions, respectively, evaluated from the data in figure 1. It is seen that γ 2 is only weakly
2/3
dependent on Z 2 from 1 MeV/u upward. Note that the screening limit v ' Z 1 v0 lies at 1.2
MeV/u for argon ions. Conversely, pronounced Z 2 -structure is found at lower energies. This
structure is evidently caused by the reference ion and is particularly pronounced in case of bare
protons. Note that the structure is inverted compared to that in the stopping cross section (figure
1, upper graph), i.e. the effective charge of argon has maxima for noble-gas targets, and that
the oscillation amplitude is enhanced compared to that of the stopping number (and hence the
stopping force).
6
N - Z2 at v = v0
Binary theory
N-Z2: Land et al
N-Z2: Santry et al
N-Z2: Ward et al
N-N2: Ormrod
10
N-H2: Weyl
S/Z2 [10
-15
2
eVcm ]
20
N-Z2: Price et al
5
0
10
20
30
40
Z2
Fig. 4. Stopping forces on nitrogen ions at v = v 0 , measurements from [18–23] and calculations from
binary theory.
These results discourage the use of effective-charge scaling below ∼1 MeV/u. Unfortunately,
the coverage with experimental data is not sufficient to unambiguously support or reject this
conclusion. The most comprehensive set of data in the velocity range v ∼ v 0 is found for nitrogen
ions. Figure 4 shows measured stopping forces on nitrogen ions compared to calculated values
from the binary theory. Reasonable agreement is found in absolute magnitude and, by and large, in
the oscillatory structure. The most pronounced discrepancy is with the noble-gas data of Price et
al. [19] which fall significantly below both the calculated values, neighboring experimental data
for solids, and gas data (N2 ). Apart from those noble-gas data, surprisingly little Z 2 -structure is
found for this comparatively light ion at a comparatively low projectile speed. Clearly, if stopping
forces are to be determined by interpolation of experimental data, interpolating between stopping
forces is to be preferred to interpolation between effective charges.
5 Amorphous carbon, diamond and graphite
As an example of a phase effect we consider the stopping cross section of three forms of solid
carbon. One set of oscillator strengths was utilized to model amorphous carbon and graphite,
and another one to model diamond [11]. Shell corrections were modeled on the basis of atomic
velocity distributions for amorphous carbon and diamond, and with a Fermi distribution for
graphite. While this may seem a bit schematic, it should be adequate for an estimate of the
magnitude of the differences to be expected between the three materials.
7
20
Graphite
mmmm`
mmmm`
2
-dE/dx [MeVcm /mg]
Ar - C
10
Amorphous
5
Diamond
Projectile exc
2
1
0.01
0.1
1
10
100
E/A1 [MeV]
0.5
2
-dE/dx [MeVcm /mg]
-
p -C
0.2
Graphite
Amorphous
Diamond
0.1
0.05
0.001
0.01
0.1
1
10
E [MeV]
Fig. 5. Stopping in carbon. Upper graph: Argon ions, target excitation only, but contribution due to
projectile excitation shown separately. Lower graph: Antiprotons.
Figure 5 shows calculated stopping forces on antiprotons and argon ions spanning four orders
of magnitude in projectile energy. Antiprotons are considered in order find a maximum phase
effect. For argon, the stopping force due to target and projectile excitation, respectively, are given
separately.
The difference in sensitivity to target parameters is striking. At energies around and below the
stopping maximum, variations of ∼ 30 % are found for antiprotons while corresponding effects
for argon remain below ∼ 10 % except for v < v0 where the theoretical description is known to
become less reliable, as is evidenced by the inability of the scheme to predict Z 1 -oscillations for
carbon [3]. The contribution from projectile excitation tends to further decrease these variations
on a relative scale.
6 Lithium fluoride
As an example for deviations from the Bragg additivity rule we study the case of lithium fluoride,
where a comparatively large effect is to be expected because outer-shell electrons constitute a
8
large fraction of all target electrons, and because of a major rearrangement of the lithium 2s
electron. Bundled oscillator strengths for this system have been reported recently [5].
100
-15
2
eV cm ]
O in LiF
S [10
Bragg
LiF
F
Li
10
10
-3
10
-1
10
1
10
3
E/A1 [MeV]
-
p in LiF
Bragg
LiF
F
Li
1
S [10
-15
2
eV cm ]
10
0.1
10
-3
10
-1
10
1
10
3
E/A1 [MeV]
Fig. 6. Stopping cross section of lithium, fluorine, and lithium fluoride for oxygen ions i charge equilibrium
(upper graph) and antiprotons (lower graph), calculated from binary theory. Also included is the sum of
the elemental stopping cross sections (Bragg rule).
Figure 6 shows stopping cross sections for oxygen ions in charge equilibrium and for antiprotons. It is seen that Bragg additivity is accurately fulfilled above the stopping maximum while
discrepancies are found at lower velocities which amount to approximately a factor of two for
antiprotons. This is clearly due to the contribution from the 2s electron of metallic lithium. For
oxygen ions the same effects are much less pronounced as a consequence of projectile screening.
Projectile excitation has been ignored.
The stopping cross sections of Li and F for antiprotons show a crossover, while for oxygen ions,
fluorine shows a larger stopping cross section at all velocities. This feature is asserted to be due
to the Barkas effect which increases with Z 2 at constant velocity and depends on the sign of the
projectile charge.
9
7 Conclusions
We emphasize that details of the predicted Z 2 -structure hinge on available oscillator-strength
spectra which differ significantly in quality from element to element. However, the same spectra
have been employed in the evaluation of stopping forces on different projectiles, and the same
theoretical procedure was employed over the entire range of beam energies. Therefore, predicted
variations from projectile to projectile deserve some confidence.
In particular, the prediction that the magnitude of Z 2 -structure should decrease from antiprotons
over protons and helium ions to heavier ions is well supported by theoretical arguments. Similar
conclusions emerge from our estimates of stopping in three phases of carbon, and particularly
from estimated deviations from the Bragg additivity rule which are far more pronounced for
antiprotons than for oxygen ions.
The significance of Z 2 -structure hinges primarily on a competition between two opposite tendencies: Enhanced structure is expected with decreasing beam velocity because of increased
sensitivity to outer, incompletely-filled target shells. Conversely, diminished structure is expected with increasing projectile screening. Thus, pronounced structural effects are expected in
a velocity range where stopping is primarily due to outer electrons and the primary Coulomb interaction is only weakly screened. This energy interval is rather wide for antiprotons and protons
and shrinks rapidly with increasing atomic number of the projectile.
Our findings have implications on the effective-charge concept which is supposed to account
for projectile screening and has generally been employed in the past to predict stopping forces
on heavy ions. It has been shown previously that the overall variation of the effective charge
has little relation to projectile screening [24]. Figure 3 confirms that γ 2 is only a meaningful
quantity, i.e., independent of Z 2 , in the velocity range where projectile screening is insignificant.
Conversely, at beam energies below 1 MeV for argon, the effective charge is predicted to undergo
Z 2 -oscillations that are more pronounced than those inherent in the stopping force. These oscillations are due to the reference ion and are hence more pronounced when protons are chosen
as a reference than for helium. We therefore conclude that any use of the effective charge is
4/3
discouraged below the screening limit Z 1 ×0.025 MeV/u.
Acknowledgements
Extensive discussions with R. Bimbot, H. Geissel and H. Paul are gratefully acknowledged. This
work has been supported by the Danish Natural Science Research council (SNF).
10
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11