90
Nuclear Instruments and Methods in Physics Research 812 (1985) 90-94
North-Holland, Amsterdam
PROTON STOPPING CROSS SECTIONS FOR CARBON, ALUMINIUM AND GOLD: NEW
EXPERIMENTAL DATA AND CRITICAL ANALYSIS OF THE VALIDITY OF EMPIRICAL FIT
FORMULAS
F. SCHULZ
Gesellschaft ftir
Strahfen - und Vmweltforschung mbH, Physika fisch -Technische Abteilung, D - 8042 Neuherberg, Fed. Rep. Germany
J. SHCHUCHINSKY
Radiation and Solid State Laboratory, New York University, 4 Washington Place, New York, NY 10003, USA
Received 22 January 1985 and in revised form 18 February 1985
We report new experimental data for the stopping cross section S of protons in carbon, aluminium and gold at energies between 8
and 300 keV. The data are found to exhibit largely the same energy dependence as previously reported data. Using empirical fit
functions we have analysed available experimental data in terms of the ratios SC,,,&, from which we derive scaling factors b
applicable to various sets of experimental data. These scaling factors, which typically range from 0.9 to 1.1, are attributed to
deviations of the true foil thickness from the assumed one. Using b-corrections all sets of experimental data can be adjusted to fall on
the same target specific stopping power curve (maximum deviation f 3%). While the adjusted data are well described by the respective
fit function (Al: Andersen and Ziegler) or tabulation (Au: Janni) the empirical data for carbon apparently deserve further
improvements.
1. Introduction
The stopping cross sections for swift ions in matter
are still not known with sufficient accuracy. Inspection
of stopping power compilations for hydrogen fl] and
helium [2] in all elements shows that available experimental data frequently differ by up to 50%. New experiments at low velocities [3-61 indicate that the empirical
fit formulas proposed by Andersen and Ziegler [l] and
Ziegler 121deserve further improvement.
More recently Janni [7] presented proton stopping
power and range tables which, for energies between 20
keV and 1 MeV, were obtained by statistically evaluating the accuracy of the available experimental information (up to June 1979) and then performing least square
curve fits. Almost at the same time Montenegro et al. [S]
presented a “universal” equation for the electronic
stopping of ions in solids. The model is based upon the
effective charge concept. Although this approach approach to be questionable for protons, the agreement
obtained in ref. [S] between the proposed analytical fit
function and selected experimental data looks convincing (at first sight).
We have carried out measurements of the stopping
cross sections for protons in carbon, alu~nium
and
gold at (incident) energies between 8 and 300 keV. The
original motivation of this work was to determine pro0168-583X/85/$03.30
0 Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
ton stopping powers under the same conditions as those
employed in recent measurements of He, N, Ne and Ar
stopping in the same or similar foils [lo]. Proton and
heavy ion stopping powers derived under identical condition constitute the ultimate experimental data base for
a comparison of experimental and theoretical [ll] effective charge fractions of heavy ions.
Another purpose of this study was to assess the
validity of empirical stopping power formulas or tables
[1,7,8]. Using the present and previously reported data
we have evaluated systematic deviations of the energy
dependence of empirical stopping powers from the experimentally observed trends. This procedure allows the
derivation of “adjusted” stopping cross sections which
appear to be accurate to within k 3% or better.
The measurements were performed at NYU. The
experimental set-up was described briefly in ref. [lo]. A
detailed description will be given elsewhere [12]. A
noteworthy improvement of the vacuum system was
accomplished by the incorporation of cryopumps in
place of the diffusion pumps originally in use [10,12].
The velocity loss spectra were measured with a 90”
magnetic spectrometer (momentum resolution Ap/p =
F. Sckult, f. Shchuchimky / Proton sropping crms sections for C, Al and Au
5 x 10e4). In order to detect the analysed protons with
high sensitivity, even at energies of a few keV, the solid
state detectors alloyed
for the heavy ion work [IO&?]
were replaced by channeltrons. The areal density of the
target foils (amorphous
carbon,
polycrystalline
aluminium and gold) was determined with an accuracy
of approximately k 10% as described in ref. [lo] (carbon:
10.6 and 13.6 &g/cm’; alumitium: 26 pg/cm’; gold:
130 and 140 y g/cm')_ Among other methods, thickness
calibration involved measurements of the energy loss of
300 keV He and the use of the respective stopping cross
sections according to Ziegler’s empirical fit [2] (at 300
keV, the stopping powers in C, Al and Au, reported by
different groups, agree to within about + 10% [2]).
-.
-
-- - -
, ( /
)
,$,,)
,
. !
Jonni
AndersensZiegler
Montenegro et al.
ti: 1
B
01
0‘7
91
(
/
,
1
,,,(
,
# ,
k
,,,J
IQ2
IO
ENERGY/ MASS
lo3
IkeVfu)
Fig. 2. As fig. 1 but for aluminiurn.
3. Results
The measured proton velocity loss spectra exhibited
u&modal distributions. For carbon and alurninium the
loss peaks were almost Gaussian in shape_ In the case of
gold the loss peaks exhibited a tail OR the high-energy
side which is attributed to channeling [lO,f3]. This tail
could be removed by offsetting the angle of observation
by l-3” from the direction of beam propagation [IO].
Because of the absence of a channeled loss component,
the mean random energy loss n = E, - EoU, and the
corresponding stopping cross sections could be de
termined unambiguously and rather precisely ( E, is the
incident ion energy and Ena, the mean ion energy after
passage through the foil).
The experimental results of this work are presented
in figs. 1-3 (full circles). The measured cross sections S
are plotted versus the appropriate
“mean”
ion
energy/mass, E,/M, in the foil. If the stopping cross
section can be described, in the interval i& c E < E,
H--Au
e
this work
-.----
Jonni
Andenen & Ziegfef
Montenegro et al.
$00
lo3
1
ENERGYi MASS
lkeVlut
Fig. 3. As fig. 1 but for gold.
by S(E) = kEP, the correct value for E, is [12]
;
I
E I&
0
0
$
9
5
s
5 IOL
%
m
8 5%
(3
In this study the error intr~uced
the power p is less than 1%.
H-C
by ~ncert~nties
of
* this work
4. Discussion
- ,----
t
E
Oo
!7J 1
Jonni
Andersen 8 Ziegter
Montenegro et al
10
ENERGY/MASS
lo2
(keV/uI
lO>
Fig. 1. Proton stopping cross section for carbon versus the
appropriate mean energy of the protons in the target foil (see
text). Previously reported empirical fit functions [1,7,8] are
shown for comparison.
In figs. 1-3 the results of this study are compared
with the fit formulas suggested in refs. [1,7,8]. The
extent to which the experimental data and the fit functions agree depends on the target material. The agreement is seen to be best for aluminium (fig. 2) and less
satisfactory for carbon (fig. 1) and gold (fig. 3). Even in
the latter case, however, the maximum relative deviation
does not exceed 20%.
Considering the fit formulas one will note that the
curves of Andersen and Ziegler [l] and Janni [7] generally agree rather well with each other (except for gold
STOPPfNC POWER WORKSHOP
F. Schulz, J. Shchuchinsky / Proton stopping cross sections for C, Al and Au
92
around the maximum of the stopping cross section, cf.
fig. 3). The maximum in the fit functions of refs. [1,7] is
located at almost the same energy, i? = E( S = S_).
This emperical number 8,, [1,7] is in accordance with
observed in this study. The agreement with i? of
R,
Montenegro et al. [8] is not as good, notably in the case
of carbon and gold where the calculated
,6 is apparently too low.
As pointed out in sect. 2 the thickness of the foils
used in this study was known with an estimated accuracy of &-lo%. Accordingly
part of the discrepancy
between our experimental
data and the fit formulas can
be attributed to deviations between the assumed and the
true foil thickness. If we multiply our data for aluminium
by a factor
of 1.05 the agreement
with
the
Andersen-Ziegler
fit is almost perfect. A similar agreement cannot be achieved for carbon and gold, i.e., in
these cases the observed energy dependence of the stopping cross section deviates significantly
from the empirical fits (cf. figs. 1 and 3).
In order to check the quality of our data with respect
to their energy dependence
we have carried out a detailed comparison
with previously reported data. We
made the rather reasonable assumption that in all measurements the largest error is introduced
by uncertainties in the determination
of the foil thickness. In order
to assess likely errors in foil thickness we derived ratios
of stopping cross sections, S,,,/Sril.
Visual inspection
of these ratios revealed a common energy dependence.
Almost perfect agreement between the various sets of
data could be achieved by multiplication
with an individual factor b, where b usually ranged from 0.9 to 1.1.
The procedure is illustrated in fig. 4 which shows
scaled ratios of stopping cross sections b&.,/S,,
using
the fit function according to Andersen and Ziegler [l],
Sri, = S,. Two observations
are evident. (i) The energy
dependence of S observed in this study agrees quite well
with previously reported data. (ii) The common trend in
the energy dependence of SeXPis not described properly
by the suggested fit functions. In all cases b&,/S,,
=
f(E) rather than bS,,,/S,,
= const (ideally b&,/S,,
=
1.0). Accordingly
we conclude that presently available
fit functions usually deserve an energy dependent
correction in order to describe experimental
data properly,
i.e. with an accuracy of better than f 3%.
It should be pointed out that the selection of the
respective set of b-values for a given target was not
completely arbitrary since, for each target, we tried to
arrive at mean value b between 0.97 and 1.03. The use
of this “boundary
condition” was based upon the idea
that the error in foil thickness determination
should
roughly cancel by averaging over data from different
laboratories.
Although the choice of the data sets may be considered somewhat arbitrary we based our decision on some
practical aspects. (i) The measured stopping cross sections had to cover a sufficiently large range in energy,
preferably around the maximum (since this is the region
where the employed procedure is most sensitive to systematic differences between different sets of data). (ii)
Experimental
results of those authors who provided
data for more than one target material (out of the three
investigated by us) were given preference to those who
studied one material only. (iii) Data which revealed
systematic energy-dependent
deviations from the common trend were not taken into account in the final
compilation
of stopping cross sections. As discussed
5
%
=
3
this work 1x1.091
o vws D 62 1.0.881
A 0 8 0 63 1x0.98)
l
!j --jl
m
1
10
ENERGY/
MASS
v
0
q
SoZ 65 1x1.103
Ms K 80 (x0.971
MsK 82 1x0.98)
102
(keV/ u)
10-
g
l
ti
IO’
?I
&
”
B
5-0
c
o
A
v
0
q
this work
VW L D 62
Oe D 63
Ss 2 65
Ma K 80
Ms K 82
(x1.091
Ix0.88)
(x0.981
1x110)
1x0.97)
ix0.981
Fig. 4. Scaled ratio of the proton stopping cross section
is the fit function suggested by
b&,, L&z for carbon (S,
Andersen and Ziegler [l]). The factor b for the respective set of
data is specified in parentheses. The mean value of b is 1.00.
For comparison S,.,,/S,
and SJanni/SAz are also plotted.
References: M. et al. [8], Janni [7], VW & D 62 [14], 0 & D 63
[El, S St Z 65 [16], M & K 80 [17], M & K 82 [6].
Fig. 5. Adjusted proton stopping cross section for carbon.
Experimental data as in fig. 4. The tabulated stopping cross
section due to Janni [7] is shown for comparison.
F. Schulz, J. Shchuchinsb / Proton stopping cross sections for C, AI and Au
93
5-7. Also shown is that fit function out of refs. [1,7,8]
which describes the new set of data best. Whereas for
alumiuium (fig. 6) and gold (fig. 7) the fit functions due
this work
(x1.05)
o
M&K
(xO.8L)
q
MsK
82 (x0.83)
OMD65/H,D(x1.03)
l
AO
0’
80
I
10
ENERGY/
MASS
I
,,I,,
10'
102
(keV/uI
Fig. 6. Adjusted stopping cross section for protons (H) and
deuterons (D) in ahnninium. The scaling factors b are specified
in parentheses. The mean value of b is 1.00 (counting M & K
80; 82 only once). The empirical fit function due to Andersen
and Ziegler (A & Z) [l] is shown for comparison. References:
Bii 36 [18], Ma 53 [19], OMD 65 [20], M & K 80 [17], M & K
82 [6].
below the employed procedure may in fact be used to
identify these deviations.
On the basis of the scaling procedure illustrated in
fig. 4 we have derived “adjusted”
stopping cross sections, using both our own and previously reported experimental data. The results, Sadj = b&r, together with
the employed scaling factors b are presented
in figs.
H.D -
Au
to Andersen and Ziegler [l] and Janni [7], respectively,
describe the “adjusted” stopping cross sections very
well, a sufficient accurate fit is not yet available for
carbon. The closest agreement is achieved using Janni’s
fit (fig. 5).
Whilst it is evident from figs. 5-7 that the scaled
experimental data generally agree to within better than
f3% the procedure illustrated in fig. 4 may also serve
to identify “unusual” data. For example, the results
recently reported by Mertens and Krist [6] for aluminium
exhibit a rather rapid drop at energies above 200 keV
(cf. fig. 6). This drop is at variance with the present and
all previously reported data. A similar, though not as
pronounced fall-off is seen in their results for carbon
and gold [18] (figs. 4, 5 and 7; clearly evident for gold in
a data analysis corresponding to fig. 4). The origin of
this unusual energy dependence is not easy to assess
without a knowledge of all details of the experimental
approach as well as of the method employed for deriving stopping cross sections from measured energy loss
spectra.
As another example we like to mention that the
stopping cross sections for gold reported by Bader [24]
(scaling factor b = 0.98) exhibit a shift towards higher
energies by about 10 to 20%. This shift raises some
doubt whether the terminal voltage of the accelerator
employed was calibrated properly. In view of this uncertainty Bader’s data are not included in fig. 7.
Last but not least, a comment on the quality of our
own thickness determination is in order. According to
figs. 4-7 the true foil thickness was 5 to 11% smaller
than anticipated. Since recent heavy ion stopping power
measurements at NYU were carried out using the same
and similar foils (and the same procedure for thickness
calibration) we suspect that the data reported in ref. [lo]
have to be corrected by the scaling factors given in figs.
4-7.
Janni
l
this
work
(xl.lll
.ri”-, ;;F;iZ;,,,,j
10
ENERGY/MASS
lo2
(k&‘/u)
10’
Fig. 7. Adjusted stopping cross section for protons (H) and
deuterons (D) in gold. The scaling factor b is specified in
parentheses. The mean value of b is 1.03. The tabulated cross
section due to Janni [7] is shown for comparison. References:
Ba 36 [18], vWMB 71 [21], T & P 80; 81 [22,5], M & K 80; 82
[17,6], BSG 84 [23].
5. Conclusion
Based upon a critical analysis of new and previously
reported proton stopping cross sections for carbon,
aluminium and gold we have derived sets of “adjusted”
experimental data which appear to be accurate to within
*3% or better. The new data provide means for a
critical test of previously suggested fit functions or
tabulated data. Corrections to the empirical stopping
cross sections are desirable in the case of carbon.
The improvements accomplished by determining and
critically analysing ratios of stopping cross sections,
Se,+/&, are evident. Therefore we suggest that this
procedure should be employed in evaluations of the
STOPPING POWER WORKSHOP
94
F. Schulz, J. Shchuchinsky / Proton stopping cross sections for C, Al and Au
quality of new experimental data as well as in attempts
to derive optimum sets of stopping cross sections from
experimental data of different sources.
This work was initiated by the late Prof. W. Brandt
to whom we are grateful for his enthusiastic encouragement and advice in course of the measurements. Thanks
are due to C. Peterson (NYU) for technical support
during the experiments and to K. Wittmaack (GSF) for
the numerous discussions and assistance during the
preparation of this paper. Part of this work was supported by the US Department of Energy.
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and J.F. Ziegler,
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stopping
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End of the Proceedings of the Workshop
on Stopping Power for Low Energy Ions