The total reaction cross section is an essential quantity in particle and heavy-ion transport codes when determining the mean free path of a transported particle. Many transport codes determine the distance a particle is transported before it collides with the target or is stopped in the target material, with the Monte Carlo (MC) method using semiempirical parametrization models for the total reaction cross sections. In order to improve the well-known Kox and Shen models of total reaction cross sections and allow the models to be used at energies below 30 MeV/nucleon, we propose a modified parametrization of the transparency parameter. We also report that the Kox and Shen models have a projectile-target asymmetry and should be used so that the lighter nucleus is always treated as the projectile

Kohama, A.

Lantz, M.

Sihver, Lembit

Chalmers Research

Improved parametrization of the transparency parameter in Kox and Shen models of total reaction cross sections

Chalmers Publication Library

PHYSICAL REVIEW C 89, 067602 (2014)
Improved parametrization of the transparency parameter in Kox
and Shen models of total reaction cross sections
L. Sihver,1,* M. Lantz,2,3 and A. Kohama3
1Nuclear Engineering, Applied Physics, Chalmers University of Technology, Gothenburg, Sweden
2Applied Nuclear Physics, Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden
3RIKEN Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan
(Received 17 March 2014; revised manuscript received 19 May 2014; published 30 June 2014)
The total reaction cross section is an essential quantity in particle and heavy-ion transport codes when
determining the mean free path of a transported particle. Many transport codes determine the distance a particle
is transported before it collides with the target or is stopped in the target material, with the Monte Carlo (MC)
method using semiempirical parametrization models for the total reaction cross sections. In order to improve the
well-known Kox and Shen models of total reaction cross sections and allow the models to be used at energies
below 30 MeV/nucleon, we propose a modified parametrization of the transparency parameter. We also report
that the Kox and Shen models have a projectile-target asymmetry and should be used so that the lighter nucleus
is always treated as the projectile.
DOI: 10.1103/PhysRevC.89.067602 PACS number(s): 24.10.Lx, 25.40.−h, 25.70.−z, 25.75.−q
Introduction. Accurate knowledge of the nucleon-nucleus
and nucleus-nucleus total reaction cross sections is of impor-
tance for many different fields including fundamental nuclear
physics, simulations of radiation damage to equipment used on
space vehicles, and transport of the heavy ions in the galactic
cosmic rays (GCR) through the interstellar medium, as well
as to estimate the biological effects of space radiation. During
the past decades, new applications have also arisen within
transmutation, reactor science, and medicine, especially in
radiotherapy with protons and ions. Several medical accel-
erator facilities are operating or are planned for construction.
In order to be able to calculate complex geometries, including
production and transport of protons, neutrons, α particles, and
heavy ions in different materials, including human tissues and
organs, three-dimensional Monte Carlo (MC)–based transport
codes must be used. The total reaction cross section and the
decay lifetime of the particle are essential quantities when
determining the mean free path of a transported particle.
Many particle and heavy-ion transport codes choose the next
collision point using the MC method and the calculated total
reaction cross section. Due to calculation speed, simplicity,
and accuracy, semiempirical parametrization models for the
total reaction cross section are most often used in these codes.
During the 1980s, Kox and coworkers performed a series
of experiments where the total reaction cross sections were
measured over a wide energy range for 12C and 20Ne on
different target materials [1–4]. Besides generally supporting
the development of microscopic models for the prediction
of total reaction cross sections, they also developed their
own general parametrization, commonly referred to as the
Kox model, that was valid for many different projectiles
at energies above 30 MeV/nucleon [4]. The Kox model
follows the semiempirical Bradt-Peters form [5]. A few
years later the model was improved by Shen et al. [6], by
adjusting the parameters and extending its validity to energies
*sihver@chalmers.se
below 30 MeV/nucleon. However, the energy-dependent
transparency parameter, C(E), is not given in numerical or
functional form in the article but is only shown in a figure.
Furthermore, the Shen model gives different results for some
of the systems where the projectile and target nuclei have
different masses, if run in the inverse kinematics. Nevertheless,
the models by Kox and Shen are still available as options in
several Monte Carlo particle and transport codes.
In this paper a modified and improved parametrization
of the transparency parameter in the Kox and Shen models
is therefore presented. The main improvement with our
parametrization compared to the original ones is that it results
in a smoother energy dependence of the total reaction cross
sections and extends the models to be used for projectile
energies below 30 MeV/nucleon. We also report that the Kox
and Shen models have a projectile-target asymmetry and we
therefore give the recommendation that the models should
be used so that the lighter nucleus is always treated as the
projectile, no matter what the actual case might be.
It should be pointed out that there are several parametriza-
tions of later date that in general give a more accurate
prediction of the total reaction cross sections, e.g., the
model developed by Tripathi et al. [7–9] and the empiri-
cally modified version [10] of the Tripathi model used in
the three-dimensional particle and heavy-ion transport code
FLUKA [11,12]. Other promising models used in different parti-
cle and heavy ion transport codes are the Sihver model [13,14]
and the models developed by Takechi et al. [15] and Iida
et al. [16]. However, it should be pointed out that there is no
current model which agrees well with the experimental data for
all possible nucleon-nucleus and nucleus-nucleus collisions in
the energy range 10–104 MeV/nucleon.
The transparency parameter. The Kox and Shen models
are based on the strong absorption model and have the same
general form:
σR = πR2
[
1 − B
ECM
]
, (1)
0556-2813/2014/89(6)/067602(4) 067602-1 ©2014 American Physical Society
BRIEF REPORTS PHYSICAL REVIEW C 89, 067602 (2014)
where R is the interaction radius and ECM is the center-of-mass
energy of the collision. The term B, which describes the effects
of the Coulomb barrier, has a slightly different form in the two
models:
BKox = ZpZte
2
1.3
(
A
1/3
p + A1/3t
) , (2)
BShen = 1.44ZpZte
2
Rp + Rt + 3.2 −
RpRt
Rp + Rt , (3)
Ri = 1.12A1/3i − 0.94A−1/3i (i = p,t),
whereAp,At,Zp,Zt,Rp, andRt are the mass numbers, charges,
and radii of the projectile (p) and target (t), respectively. For
simplicity, the values of all fixed parameters are here included
in the equations. The complete expressions can be found in the
original articles [4,6]. The forms for the interaction radii, R,
are the same, except that the Shen model has one extra term,
RKox = 1.1
(
A1/3p + A1/3t + 1.85
A
1/3
p A
1/3
t
A
1/3
p + A1/3t
− C(E)
)
+ 5(At − 2Zt)Zp
ApAt
, (4)
RShen = RKox + 0.176E−1/3CM
A
1/3
p A
1/3
t
A
1/3
p + A1/3t
. (5)
It should be noted that the last term in Eq. (4), the so-
called neutron excess term, is only used at energies below
200 MeV/nucleon in the Kox model, with the motivation
that at higher energies the n-p total cross section becomes
equal to the p-n and p-p total cross sections. In the Shen
model the neutron excess term is used also at energies
above 200 MeV/nucleon. C(E) is the energy-dependent trans-
parency term and E is the energy per nucleon in the laboratory
system. In the Shen model the transparency parameter is
denoted C ′(E), in the original paper [6], because it is extended
to energies below 30 MeV/nucleon. It is, however, called C(E)
throughout this paper.
As seen from Eqs. (1)–(4), it is, besides the Coulomb cor-
rection term, B, for low energies, the transparency parameter
that gives the energy dependence for the total reaction cross
section in the Kox model. In the Shen model there is also a
slight contribution to the energy dependence from Eq. (5).
For the Kox model a number of values are given for the
transparency parameter, as function of different energies in
the energy range 30–2100 MeV/nucleon. Figure 1 shows the
values for C(E) given by Kox et al. as black dots. The simplest
way to determine the transparency parameter at other energies
is by straight line interpolation between the given values. In
Fig. 1 this is shown as a dashed black line.
Townsend and Wilson [17] have proposed an energy-
dependent parametrization of the transparency parameter in
the following way:
C(E) = 1.91 − 16.0e−0.7274E0.3493 cos(0.0849E0.5904), (6)
where E is the projectile kinetic energy given in MeV/nucleon
in the laboratory system. This parametrization is based on the
FIG. 1. (Color online) The C(E) parameter as given by Kox
(black dots), and a simple straight line interpolation between the
dots (dashed black line). The dotted (red) line is the parametrization
by Townsend and Wilson according to Eq. (6), and the dash-dotted
(blue) line is the Shen model as it is used in GEANT4, using Eqs. (7)
and (8). The proposed modified parametrization, using a combination
of Eqs. (6) and (9) (see the text for details), is given as a solid
black line. The line for the Townsend and Wilson model has been
extrapolated somewhat below the lower limit of its given validity in
order to show its behavior for lower energies.
C(E) values by Kox [4] and gives good agreement with the
Kox data above 30 MeV/nucleon, while it is not defined below
this energy range. At 30 MeV/nucleon it agrees within 10%,
and at higher energies within 5%. In Fig. 1 the parametrization
by Townsend and Wilson is shown as a dotted (red) line.
The simple form of the parametrization by Townsend and
Wilson, and the resulting smooth shape, makes it attractive for
use in simulation codes where parametrized total reaction cross
sections are required. As mentioned above, it has the drawback
of not being defined at energies below 30 MeV/nucleon. In
Fig. 1 the curve is drawn to somewhat lower energies in order
to show the trend if used below 30 MeV/nucleon.
Besides the extra term in Eq. (5), and a few modified
parameters which increased the validity of the model to more
projectile-target systems, the main improvement of the Shen
model was the extension to energies below 30 MeV/nucleon;
see Fig. 2 in Ref. [6]. But the extension of C(E) is not given in
any numerical or functional form. Attempts to extract values
from the figure should be discouraged for two reasons:
(i) The figure seems to be drawn by hand, and close
inspection shows that the curve above 30 MeV/nucleon
does not agree fully with the values of C(E) given by
Kox et al. [4]. Therefore there is no reason to believe
that the suggested curve below 30 MeV/nucleon can
be extracted in a way that agrees with the intention of
the authors.
(ii) The energy scale is inconsistent. Most likely it is
the marker for the energy 103 MeV/nucleon that
has been accidentally placed at the position for
2×103 MeV/nucleon.
In the Monte Carlo simulation toolkit GEANT4 [18–20] the
Kox and Shen models can be selected among other models.
067602-2
BRIEF REPORTS PHYSICAL REVIEW C 89, 067602 (2014)
FIG. 2. (Color online) Comparison of the energy dependence of
total reaction cross sections for 12C on C calculated with the Kox and
Shen models. The dashed black line is the Kox model, and the dotted
(red) line is the result of the Kox model by using the Townsend and
Wilson parametrization of C(E). The dash-dotted (blue) line is the
Shen model with the approach from GEANT4, and the solid black line
is the modified parametrization presented in this work. Experimental
data, shown as open circles, are taken from Ref. [21].
The authors of the code have defined C(E) as
C(E) =
(
− 10
1.55
+ 2.0
)(
x
1.5
)3
for x < 1.5, (7)
C(E) = −10
x5
+ 2.0 for x  1.5, (8)
where x = log10(E). The shape of C(E) is shown in Fig. 1 as a
dash-dotted (blue) line. As seen it does not reproduce the peak
values at about 200 MeV/nucleon, and it gives a rather sharp
corner where the two parts merge at about 31.6 MeV/nucleon.
We therefore propose a new parametrization where we use
a modified version of Eq. (7) from the GEANT4 model, that
gives a smooth overlap with the Townsend and Wilson model,
Eq. (6), at E = 45 MeV/nucleon, i.e.,
C(E) =
(
− 10
1.55
+ 2.0
)(
x
1.38
)3
+ 0.0006E, (9)
where E is the projectile energy per nucleon in the laboratory
system. In Fig. 1 this version is shown as a solid black line for
energies below 45 MeV/nucleon.
Results. The different models discussed previously have
been checked for a large number of cases together with avail-
able experimental data. Figure 2 shows the total reaction cross
section for 12C ions on C targets, together with experimental
data from Ref. [21]. Figures 3 and 4 show the same situation
for the systems 12C+27Al and 12C+Fe, respectively. As seen
our approach gives a smooth overlap with the Townsend and
Wilson model and thus a smooth energy dependence over the
full energy range.
It should be noted that our approach does not change
the predictions at very low energies, and no modification
has been attempted in order to give a better agreement with
experimental data. This would require a modification of the
Coulomb repulsion term in Eq. (1) and leads to a completely
new parametrization. We consider that to be beyond the scope
of this study. Regarding the difference in results at the peak
FIG. 3. (Color online) The energy dependence of total reaction
cross sections, predicted with the different versions of the Kox and
Shen models, for 12C+27Al. Experimental data, shown as open circles,
are taken from Ref. [21].
value around a few tens of MeV/nucleon, our approach looks
smoother and more physically correct considering the general
energy dependence of nucleus-nucleus total reaction cross
sections in that energy region.
Projectile-target asymmetry. We also point out that the Kox
and Shen models have a projectile-target asymmetry, i.e.,
σR(p + t) = σR(t + p) for Ai = 2Zi (i = p,t), (10)
where p and t denotes the projectile and target, respectively.
This is due to the last term in Eq. (4), which was introduced
by Kox et al. as an option for heavy target nuclei. Shen et al.
made it permanent in their model, but it is not obvious from
Ref. [6] that this anomaly exists, and it may therefore cause
unexpected results for the user. For systems with both a heavy
projectile and a heavy target the difference is small, but for
systems where either the projectile or target is a proton or
3He the difference is significant. Figure 5 shows an example
for protons on 28Si. The lines show the result of the Shen
model with our modified parametrization: the solid black line
for the case that the proton is the projectile and the dashed
(red) line for 28Si as the projectile. As can be seen, the model
works much better for the case with the proton as projectile.
In some cases, such as for protons on 12C, both options give
FIG. 4. (Color online) The energy dependence of total reaction
cross sections, predicted with the different versions of the Kox and
Shen models, for 12C+Fe. Experimental data, shown as open circles,
are taken from Ref. [21].
067602-3
BRIEF REPORTS PHYSICAL REVIEW C 89, 067602 (2014)
FIG. 5. (Color online) Total reaction cross sections for the sys-
tem p+28Si predicted with the Shen model using the modified
parametrization of this work. The black line shows the proton as
projectile while the dashed (red) line shows the case when the proton
is treated as the target. Experimental data are taken from Refs. [22,23].
somewhat ambiguous results. But based on clear cases such as
the one in Fig. 5, our advice is that the lighter particle always
should be treated as the projectile, when using the Kox or
Shen models. Such a function is already implemented in the
PHITS code [24].
Conclusions. We have proposed a modified parametrization
of the transparency parameter in the Kox and Shen models of
total reaction cross sections. Our modification is based on the
parametrizations used by Townsend and Wilson [17] and by
the GEANT4 [18–20] collaboration, and improves the usage
of the Kox [4] and Shen [6] model by giving a smoother
energy dependence of the cross sections. Furthermore it
extends the models to be used for projectile energies below
30 MeV/nucleon. We also give the recommendation that
due to an inherent projectile-target asymmetry the mod-
els should be used so that the lighter nucleus is always
treated as the projectile, no matter what the actual case
might be.
More experimental data are needed to benchmark and
improve the existing total reaction cross-section models. From
the available experimental data it is not possible to determine
which of the models displayed in the present work give the
most accurate prediction of total reaction cross sections over
the entire energy range.
Acknowledgments. Part of this work is supported by the
RIKEN iTHES project. L.S. greatly acknowledges A.K. for
his kind support that made it possible for Sihver to stay two
months at RIKEN, Japan. M.L. work is partly funded by a
grant from JSPS.
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067602-4


Improved parametrization of the transparency parameter in Kox and Shen models of total reaction cross sections

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