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Radiation Effects
1976, VOI.27, pp. 179-181
Printed in Great Britain
STOPPING CROSS-SECTIONS AND ATOMIC POTENTIALS
J. C. ECKARDT, W. MECKBACH and R. A. BARAGIOLA
Centro Atbrnico Bariloche, Cornisibn Nacional d e Enerpa Atornica,
Instituto de Fisica “Dr. Jose A. Balseiro ”, Universidad Nacional de Cuyo, Bariloche, Argentina
(Received May 16, I 9 7 3
A strong correlation is shown to exist throughout the periodic table between the 22 dependence of the electronic
stopping cross-sections for 4He ions o f a given velocity and the Z dependence of the atomic potentials at a certain
radius. An empirical formula is presented which allows the extrapolation o f stopping cross-section data for any ion
at a given velocity in some elements, to elements for which experimental data are not available. This study covers the
energy interval 100 < E/m < 1000 keV/amu.
In the last few years, several authors have presented
measurements and theories about the oscillating behaviour of the stopping cross-section S either as a
function of target atomic number Zz for a given projectile or as a function of projectile atomic number
Z , for a fixed target.’-’’
Ziegler and Chu’ have presented semiempirical
values of S(Zz)for 400-4000 keV ‘He ions on 92
target materials. These values were derived from experimental data and from the Lindhard-Winther theory for
electronic stopping.
Chu and Powers4 have proposed that a Hartree-FockSlater (HFS) potential could be used to account for the
structure of S ( Z z ) they observed for He ions in the
region 22 G Zz Q 29. We have found that it is possible
to fit the semiempirical values of Ziegler and Chug at
each energy by the expression S = CVr, where V is the
HFS potential of the target at a certain radius r from
the nucleus and Cis a constant scale factor. For a given
4He+energy, a fit is obtained using constant values for
r a n d C, the oscillatory dependence with Z2 being
accounted for only by the variation of V with Z z .
For each data set corresponding to a given He’
energy C-and r were computed to obtain a best fit of
the semiempirical S ( Z z ) of Ziegler and Chu and the
HFS potentials as calculated by Lu et aLz’ This was
done using values for Z 2 20. The data for elements
with Z2 < 20 were not included since the accuracy of
the fits is less at these low Z z values. After establishing
Cand r, S(Zz)was also calculated for these elements.
Table I shows the values of C and r for different
projectile energies. Figure 1 shows S = CVr curves at
three different energies together with the semiempirical
values of Ziegler and Chu.’ Also shown for 400 keV/
179
amu are the theoretical results of the same authors. It
is seen that our curve has the same shape as the theoretical curve but it agrees better with the semiempirical
data.
Since the Z2 dependence of S is similar for ions other
than 4He3i’i6722there is a possibility to obtain values
of S ( Z z ) for other projectiles for which S is known
for some elements. We propose that
w,,zz ,
u) = W
l
, u ) T(u)
w ,zz 1
where r is given by Table I as a function of the velocity
of the projectile u and C(Z,, u ) is a quantity which
depends only on Z1 and on u .
TABLE I
Radial distances r, and scale factors C for 4He ions, at different
.
.
ion velocities
. .
Elm
U
r
keV/amu
au
a,
C
6.9445. 10-9 cm/at
100
150
200
250
300
350
400
450
500
600
700
800
900
1000
2.00
2.45
2.83
3.16
3.46
3.74
4.00
4.24
4.47
4.90
5.29
5.66
6.00
6.32
1.549
1.367
1.284
1.133
1.133
1.064
1.ooo
0.939
0.882
0.829
0.779
0.732
0.687
0.687
35.9
32.8
30.7
25.6
24.9
22.1
19.5
17.3
15.4
13.2
11.5
10.1
8.9
8.5
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180
J. C. ECKARDT, W. MECKBACH AND R. A. BARAGIOLA
TABLE I1
Scale factor C for 35CI ions at different energies
Elm
keV/amu
C
6.9445.10* cm/at
r
a.
~~
~~
343
457
571
714
857
1.064
0.939
0.8 29
0.779
0.687
266
256
235
232
208
-
other elements. Figure 2 shows the values so derived
for 3 5 C ions
~ in Uranium as a function of Elm, together
with values tabulated by Northcliffe and Schilling.24
It is interesting to note that the discrepancy between
both curves is of the same magnitude as the discrepancy
of the semiempirical values of Ziegler and Chu’ and
the tabulations of Northcliffe and Schilling24 for 4He
on U.
r
SOOl
m
FIGURE 1 Stopping cross-sections for 4He ions as a function
of Z 2 . Full drawn line: semiempirical data of Ziegler and Chug;
dotted line: C-r . Y; and dashed line: theoretical values from
Ref. 9.
a) Elm= 100 keV/amu
r = 1.549 a,
C = 35.9.6.9445.10-9 cm/at.
b) Elm = 400 keV/amu
r = 1.000 a,
C = 19.5.6.9445. lo-’ cm/at.
c) E/m = 1000 keV/amu
r = 0.687 a,
C = 8.5 .6.9445 . 10-9 cm/at.
1
I
I
I
I
Elm
I . . .
m
I
YXMO
[keVlamu)
FIGURE 2 Stopping cross-sections for 3 s C l ions in U as a
function of energy per atomic mass unit. Full drawn line:
extrapolated from experimental data of Ward ef aLZ3;dashed
line: from Northcliffe and Schilling.24
The same calculations were made for I6O ions as
projectiles, using experimental data of Porat and
Ramavataram2’ and Ward ef U I ! , , ~ for fixed values of
the ion velocities. It was again found that C did not
depend on Z,.
ACKNOWLEDGEMENTS
This formula was tested by calculating C ( Z , , u ) for
35 C1 projectiles of different velocities in the targets
Ni, Ge, Y,Ag and Au, using experimental S values of
Ward et 0L2’ and the values of r(u) given in Table I.
At each velocity, we obtain the same factor C for
all the measured elements within 5%, the quoted
errors in the experimental data being 4%. This, together with the fact that the elements used cover a
wide Z2 range, some of them lying at maxima and
minima of the oscillatory S(Z2)curve, gives a good
proof that C i s independent of Z 2 .
The calculated C values, listed in Table 11, were
then used to extrapolate the data of Ward ef aL23 to
We wish to thank Lic. V. H. Ponce and Lic. N. Arista for
stimulating discussions. Mrs. Bibiana de Espincl is gratefully
acknowledged for making the computer program.
This work has been supported partially by the Organization
of American States, Multinational Program in Physics.
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STOPPING CROSS-SECTIONS AND ATOMIC POTENTIALS
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