Why does the maximum in the stopping cross section for
protons occur at approximately 100 keV most of the time?
Remigio Cabrera-Trujillo1, Peter Apell2 , Jens Oddershede1 3, and John R. Sabin1 3
1 Department of Physics, University of Florida, Gainesville, FL 32611, USA
2 University Outreach, Kristianstad University, 291 88 Kristianstad, Sweden
3 Department
of Chemistry, University of Southern Denmark, 5230 Odense M, Denmark
Abstract. Using simple arguments based on the dominant contributions of the valence electrons to the stopping power
at the maximum we are able to explain the general observation that the Bragg peak in the proton stopping cross section
occurs at a projectile energy of the order of 100 keV/amu for most targets. Furthermore, theoretical justification of the
observed trend is provided using a Harmonic Oscillator model.
INTRODUCTION
DISCUSSION
Consideration of the experimental and/or theoretical
stopping cross section for swift protons impinging on a
variety of targets, shows that the maximum in the stopping curve generally occurs at a proton energy of approximately Emax 100 keV/amu, corresponding to a velocity
of about 2v0 where v0 is the Bohr velocity. Although Emax
varies from a low of about 50 keV to 200 keV depending
upon the composition, state of aggregation of the target,
and projectile charge state, it is remarkable that Emax is
generally stable in this region.
As velocity, rather than energy, is the important quantity in energy deposition, it becomes clear that the presence of the ubiquitous 100 keV peak implies that there
has to be a velocity scale set by the ingredients in the
problem. In this case, the velocity scale is set by that of
the responding charges, and is either the Bohr velocity
or the Fermi velocity. Both occur at approximately the
same value, yielding the immediate result that the natural
velocity scale for the problem is of the order of v0 , corresponding to an energy of 25 keV. In this case, the peak
lies between the low velocity regime, frequently treated
by DFT, and the high velocity region, where the first Born
approximation applies and Bethe theory is appropriate.
Obviously, the velocity scale in the former region is set
by the Fermi velocity and the latter by the Bohr velocity.
Since on general grounds one can argue that the stopping
cross section increases with velocity in the first region
and decreases in the second region, one might expect a
maximum in between.
On the other hand, there has been previous attempts to
describe the maximum of S v , either by a free parameter
model (3) or by the LPA approximation (13).
In simplest terms, the stopping cross section of a target
for a swift ion projectile at some given velocity v, comes
from the integral of the velocity dependent differential
collision cross
section for the particle with the target elec
energy
trons, dσ v dΩ, times the velocity dependent
transfer from the projectile to the electrons, ∆E v :
S v dσ v dΩ ∆E v dΩ
(1)
For a given target, the collision cross section is largest
for collisions of the projectile with the outermost, or valence, electrons, as they have the largest orbital radii, r .
The collision cross section will get smaller as one considers collisions with progressively deeper lying electrons.
When a collision does occur, the energy transfer cross
section will be greatest when the projectile velocity approaches that of the outermost (lowest energy) electron
in the target, since equality of the two velocities gives the
optimal condition for energy transfer. Perhaps the easiest way to see this is to take Bohr’s atom (2) as a model
for the target. If one pushes the model further, and considers the target as a harmonically bound electron, then
resonance will occur when the angular frequency of the
electron and of the projectile are the same. Then from
simple classical uniform circular motion considerations
one would expect resonance when the electron and projectile velocities are equal.
Thus, one expects a maximum in the stopping cross
section as a function of projectile velocity when both collision cross section and energy transfer are maximized
as a function of projectile velocity, namely for the case
when the projectile has velocity similar to that of the out-
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
86
5.00
Table 1. Valence shell orbital contribution to
proton stopping cross section at Smax .
4.00
3.50
IP1
1/2
(eV
1/2
)
4.50
3.00
2.50
2.00
0
5
10
15
20
25
30
35
40
atomic number
FIGURE 1. Experimental values (9) of IP vs. atomic number
for the first 36 elements.
ermost target electrons. As expressed by Fano (note 24 in
his 1963 Annual Reviews article (5)), “Roughly speaking,
one may say that the electrons of the inner shells, K, L, ...
cease to contribute to the stopping power in succession as
the incident particle’s velocity decreases. However, the
cutoff of each shell is gradual.” (vide infra, Table 1).
At what projectile energy, or velocity, might the maximum be expected to occur?
Atomic
Target
Shells
vmax
(au)
Contribution
to Stopping (%)
He
Li
N
Ne
Na
Ar
K
Fe
Ni
Zn
Ge
Kr
1s
2s
2s, 2p
2s, 2p
3s
3s, 3p
4s
4s
4s
4s, 4p
4s
4s, 4p
1.9
0.5
1.3
2.9
0.5
1.9
0.4
0.7
0.6
0.7
1.0
1.7
100.0
98.7
96.0
99.2
96.8
97.2
93.2
93.1
96.1
93.8
94.8
94.7
proton stopping cross section at the maximum for atomic
targets, as calculated using the methods of Ref. 6. It thus
appears that the assertion that stopping at the maximum
is primarily due to the valence electrons is correct. Note
that for the transition metals, there can be at most a few
percent contribution to the stopping from the 3d shell.
Comparing Fig. 1 to the variation of the stopping
power maximum for protons over the same range of elements, calculated by via the kinetic theory of stopping
(14) and shown in Fig. 2, it is clear that the structure
of the two curves is very similar. In addition, the magnitudes are qualitatively, although not precisely, correct.
Considering the range of values for IP shown in Fig. 1,
they correspond to valence electron velocities of the order
of, but less than 100 keV, leading to values of Emax (that
energy at which Smax occurs) of the same order.
Note that the figures and table display data for atomic
targets only. However, similar arguments should apply
to molecular, condensed, and cluster targets as well. In
fact, the measured maxima in the proton stopping cross
sections for substances ranging from CCl4 , through stainless steel, to human muscle, all show (6) maxima in the
projectile energy range of 100 50 keV/amu.
One expects little deviation from atomic results for the
gaseous diatomic elements, but one might expect some
deviation in metals, as the electronic structure is band-like
rather than atomic-like (12). In this case the appropriate
velocity to compare with is the Fermi velocity [see Fig. 3
in Ref. (12)]. This causes some changes in the numbers,
but does not vitiate the argument. As can be seen in Fig.
3, where the Fermi velocities for metals are plotted (15),
most of them lie in the range around 1 au which, again,
corresponds to a proton energy of order 25 keV.
Phenomenology
There are several ways to determine the appropriate
velocity in the problem. To lowest order, the orbital energy, εv , of the outermost valence electron, in its HartreeFock incarnation where the electron moves in an average
local potential, is related to its kinetic energy, Tv , via an
orbital version of the virial theorem as εv Tv Vv 2Tv ,
where the orbital kinetic energy (in Hartree atomic units)
is Tv 12 v2v . For most systems, we can invoke Koopmans’
theorem. In this case, εv IP, the first ionization potential of the target, so that vv IP. In Fig. 1 is plotted the
square root of the first ionization potential (9) for the first
36 atoms vs. atomic number.
Instead of IP, one might as well plot the atomic
velocity parameters calculated from Hartree-Fock wavefunctions by Ponce (11). Comparing the Ponce results to
Fig. 1, similar behavior as a function of atomic number is
apparent. In addition, one finds that the average value of
the valence electron velocity hovers in the range 0 5 2 0
au; namely spanning the 10 - 100 keV/amu energy range
for projectile velocity [E p keV 25v2 (au)] mentioned
above. Perhaps the IP formulation is more useful, however as the arguments made here should apply to molecular and condensed phase targets as well as atoms, and for
those systems, IP’s are more readily available than are
calculated velocity parameters.
Consider proton projectiles. Table 1 gives the relative
contribution from the valence shell electrons to the total
87
250
and, in the first Born approximation, Bethe version (1), it
is the mean excitation energy that governs the stopping.
Note that this is not fundamentally different from the explanation above, as, for an electron gas, the basis of this
explanation, the Fermi velocity is a function of the density. Thus, if one plots the orbital mean excitation energy
of the atoms [e.g. as in those reported in Ref. (12)] vs.
atomic number, one sees behavior very much like Fig. 1
(vide supra).
Similarly, if one extends the Bethe (1) formulation
to
lower energy, one finds that for each orbital, vmax Ii e 2me , where Ii is the orbital mean excitation energy
(10) for orbital i and e is the base of Naperian logarithm.
Most of the valence orbital mean excitation energies lie
in the range around 15 eV (with the glaring exception of
Ne), again, consistently, leading to E p ’s of the order of 25
keV/amu.
Perhaps these estimates could be revised upward
somewhat, as, in an experiment, there will be some contribution from deeper lying electrons with larger IP’s and
I0 ’s. Collision cross sections for these more energetic
(smaller values of r ) electrons will have a smaller collision cross section, and consequent lower probability for
energy transfer. Similarly, in an experiment, a proton
beam may also have very small components of both for a
low energy beam in a target of finite thickness.
Emax (keV)
200
150
100
50
0
0
10
20
30
40
atomic number
FIGURE 2. Proton projectile energy at the stopping power
maximum as a function of atomic number as calculated by the
kinetic theory for atomic targets.
A similar argument could be made for other projectiles, provided care is taken in terms of the projectile
charge. For the case of He2 which is (statistically)
nearly fully stripped near the stopping maximum, this
would lead to the expectation that Smax should occur in
the range Emax 100 400 keV. Examination of Vol. 4 of
Ziegler’s Stopping Powers and Ranges series (15) shows
this to be the case.
Electron Gas Model
Another way of attacking the problem is to start with
Lindhard’s local plasma approximation (8). Here the
mean excitation energy, I0 , is related to the local electron
density (ρ) and thus to the Fermi velocity (v f ) through the
plasma frequency (ω p ) by
I0
h̄ω p
4πe2 ρ
me
2eme v3f
1 2
3πh̄ Harmonic Oscillator Model
In a more formal manner, one can understand the previous results by modeling the bound electron in the harmonic oscillator model. There are two important parameters describing the stopping. The first is the mean excitation energy, I0 , which describes the contribution from
excitation in the target electronic structure to the stopping cross section. The second is the projectile charge
state, which depends on the electron capture and loss
cross sections for the collision partners, and thus, in a
target is a function of the projectile velocity. Therefore,
in a simplistic quantitative description of the maximum
in the stopping cross section, these must be taken into account. Now, in order to obtain the maximum in the stopping cross section, we would like to express it in analytical form. Using the analytical expression for the stopping
cross section from the harmonic oscillator model (4) for
the valence shell with associated mean excitation energy
I0 , one obtains the maximum (dS dv 0) when equation
(3) is satisfied
(2)
1.20
1.00
vf (au)
0.80
0.60
0.40
0.20
0.00
0
5
10
15
20
25
30
35
40
atomic number
FIGURE 3. Fermi velocities of metals as a function of atomic
number (theoretical, continuous entries in Ref. 11).
88
E1 x 1 1 dZ1
Z1 dE p xE0 x 0
(3)
collision with the electron and projectile matching velocities.
90
80
Ep,max (keV)
70
60
CONCLUSION
50
40
It appears that the outermost electrons in many targets
of different chemical composition, state of aggregation,
and physical state, have quite similar characteristic velocities, leading to similar energy absorbing properties. Thus
they have similar values of leading to the 100 keV peak.
A similar argument also reproduces the well-known variation of the atomic stopping power maximum with target
atomic number.
30
20
10
0
0
5
10
15
Target atomic number Z2
20
FIGURE 4. Proton kinetic energy for which the maximum in
the stopping occurs, as given by the solution to Eq. (3). Also,
for comparison, we show the energy for Smax as given by the
program SRIM (15) ( ! ).
ACKNOWLEDGMENTS
Thanks are due to Prof. Raul Baragiola for helpful discussion leading to this contribution. This work was supported in part by grants from the Office of Naval Research
(#N00014-96-1-0707 to JRS) and from the National Science Foundation (CHE-9974385 to JRS).
Here Ei x is the exponential integral function of order i, x M p I0 4me E p , and Z1 is the projectile effective
charge. For a proton projectile with an effective charge
1 3
described by Z1 Z1 " 1 exp v v0 Z1 $# (see e. g.
(15)), equation (3) can be solved numerically (see Fig.
4).
We note that the maximum of the stopping cross section is proportional to the target mean excitation energy;
I0 2xme . Thus for exthat is, E p M p I0 4xme or v ample, for a hydrogen target with I0 19 eV we obtain
x 0 125, which gives Emax 55 keV consistent with the
experimental value.
In Fig. 4, we present the solution to Eq. (3) for the first
18 elements, as well as the results for H, He, Li, and Ne
obtained by the SRIM program (15). As we note from
the previous results, the maximum in S occurs between
25-100 keV, as previously discussed.
REFERENCES
1. Bethe, H. A., Ann. Phys. (Leipzig) 5, (1930) 325.
2. Bohr, N., Philos. Mag. 25, (1913) 10.
3. Brandt, W., Atomic Collisions in Solids 1, (1975) 261.
4. Cabrera-Trujillo, R., Phys. Rev. A 60, (1999) 3044.
5. Fano, U., Ann. Rev. Nucl. Sci. 13, (1963) 1.
6. Janni, J. F., At. Data Nucl. Data Tables 27, (1982) 341.
7. Kumakhov, M. A., and Komarov, F. F., Energy loss and ion
ranges in solids, Gordon and Breach, Science Publishers,
1981.
8. Lindhard, J., and Scharff, M., Kgl. Dan. Vidensk. Selsk.
Mat. Fys. Medd. 27, (1953) no. 15.
The Bohr model
9. Moore, C., NSRDS-NBS 34, (1970) .
10. Oddershede, J., and Sabin, J. R., At. Data. Nucl. Data Tables 31, (1984) 275.
Finally, to prove that the maximum of S v occurs at
a projectile velocity in resonance with the target electrons velocity, we invoke Bohr’s model. For an electronic
harmonically bound with a frequency w0 , Bohr found
(2) that
the energy absorbed by the target is ∆E b % v &
πe2 ' E b % v% w0 ' 2 me , where E is the electric field produced by the projectile (7). It is interesting to note that
E is evaluated at the resonant frequency of the target, i.e
w w0 . Following Kumakhov’s et al. (7) deduction of
Bohr’s model, it is easy to prove that the maximum in
∆E occurs at w0 v p b with b being the projectile impact parameter, or when v p w0 b ve , i.e for a head-on
11. Ponce, V. H., At. Data Nucl. Data Tables 19, (1977) 63.
12. Sabin, J. R., and Oddershede, J., Nucl. Instr. Meth. B12,
(1985) 80.
13. Semrad, D., J. Phys. B: At. Mol. Phys. 14, (1981) 4527.
14. Sigmund, P., Phys. Rev. A 26, (1982) 2497.
15. Ziegler, J. F., Biersack, J. P., and Littmark, U., The Stopping
and Range of Ions in Solids, New York: Pergamon Press,
1985.
89