EUROPEAN COMMISSION
Date: 14/11/2006
DIRECTORATE-GENERAL JRC
JOINT RESEARCH CENTRE
Institute for Transuranium Elements
Unit: Nuclear Fuells
Action No.: 3112
Copy No.:
Title
RANGE AND STOPPING POWER CALCULATIONS IN
NUCLEONICA
Author(s)
M.Ç. TUFAN, J. GALY and J. MAGILL
Report Nr.:
JRC-ITU-TN-2006/67
Classification:
unclassified
Type of report:
Technical Report
Name
reviewed by the
project coordinator /
or action leader
M.Ç. Tufan
approved by the
project leader
J. Magill
approved by the
head of unit
D. Haas
released by the
director
T. Fanghänel
Date
Signature
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Distribution List
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ITU
x1
J. –F. Babelot (QM Office)
ITU
x1
D. Haas
ITU
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J. Magill
ITU
x1
J. Galy
ITU
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M.Ç. Tufan
ITU
x1
G. Weber
ITU
x2
2
CONTENTS
1.
INTRODUCTION ___________________________________________________ 2
2.
BETHE THEORY OF STOPPING _____________________________________ 2
3.
CALCULATIONS OF STOPPING POWER AND RANGE FOR HEAVY IONS 3
3.1.
Stopping Power Calculations______________________________________________3
a)
b)
3.2.
Nuclear Stopping _____________________________________________________________ 3
Electronic Stopping ___________________________________________________________ 4
Range Calculations ______________________________________________________6
4.
STOPPING POWER CALCULATIONS FOR ELECTRONS AND POSITRONS 7
5.
STOPPING POWER CALCULATIONS FOR MUONS____________________ 10
6.
ACCURACY OF THE RANGE MODULE ______________________________ 11
6.1.
Test Results for Protons _________________________________________________11
a)
b)
c)
6.2.
Protons in H (gas) ___________________________________________________________ 11
Protons in Pb (solid) _________________________________________________________ 13
Protons in water (liquid) ______________________________________________________ 15
Test Results for Alphas _________________________________________________17
a)
b)
c)
6.3.
Alphas in H (gas) ____________________________________________________________ 17
Alphas in Pb (solid) __________________________________________________________ 19
Alphas in water (liquid) _______________________________________________________ 20
Test Results for Electrons _______________________________________________22
a)
b)
c)
6.4.
Electrons in H (gas) __________________________________________________________ 22
Electrons in Pb (solid) ________________________________________________________ 23
Electrons in water (liquid) _____________________________________________________ 24
Test Results for Positrons _______________________________________________26
a)
b)
d)
6.5.
Positrons in air (gas) _________________________________________________________ 26
Positrons in Pb (solid) ________________________________________________________ 27
Positrons in water (liquid) _____________________________________________________ 29
Test Results for Muons _________________________________________________30
a)
c)
d)
Muons in H (gas) ____________________________________________________________ 30
Muons in Pb (solid) __________________________________________________________ 32
Muons in water (liquid) _______________________________________________________ 33
7.
RANGE MODULE IN NUCLEONICA _________________________________ 35
8.
CONCLUSIONS ___________________________________________________ 43
9.
REFERENCES ____________________________________________________ 45
1. INTRODUCTION
In the interaction between charged particles and matter, the stopping power or the average energy
loss per unit path length plays an important role in many fields such as impurity atom implantation in
producing semiconductor devices, structure analysis of solid target by Rutherford backscattering
spectroscopy (RBS), and plasma-first wall interactions in a nuclear-fusion reactor. Stopping power can
be considered in two part: first is the interaction of incident particle with target electrons (called
electronic stopping power), and second is the interaction with target nuclei (called nuclear stopping
power).
Studies of the electronic stopping power started at the beginning of the 20th century. The first
(classical) calculation of the energy loss of energetic particles was made by Bohr [1]. The first quantum
mechanical treatment was by Bethe [2]. This latter theory of stopping power is particularly accurate
when the projectile’s velocity is sufficiently high.
Another important quantity is the range of the charged particle in matter. The range is defined as
the mean path length of the particle in the target matter before coming to rest. Generally, analytic
transport theory and Monte Carlo calculations are used for the range calculations.
In Nucleonica, the RANGE module is included to calculate the stopping powers and ranges of
ions in matter. The stopping power and ranges of almost all ions and, in addition electrons, muons,
positrons, protons and alphas in all elemental targets up to uranium, and compounds can be calculated.
User defined compounds can be included in the calculation. In the RANGE module, SRIM (Stopping
and Range of Ions in Matter) is used to calculate the stopping powers and ranges of ions, protons and
alphas in matter [3]. For electrons, muons and positrons, RANGE uses different formulations.
In this document, we describe:
I. the calculational methodology for stopping power and range,
II. the accuracy of the results,
III. how the RANGE module can be used.
2. BETHE THEORY OF STOPPING
In 1930, Bethe calculated quantum mechanically the electronic energy loss by using the first
Born approximation [4]. According to Bethe, the stopping power is given by
2
Se 
 2mv 2 
4Z12 e 4


Z
ln
2
mv 2
 I 
(1)
where
v is the projectile’s speed,
Z1e is the nuclear charge of projectile,
Z2e is the nuclear charge of target,
m1 is the projectile mass,
m2 is the target mass,
m is the electron mass, and
I is the mean excitation energy of target.
When relativistic effects are considered, the above equation become
Se 
  2mv 2
4Z 12 e 4

Z
2 ln 
mv 2
  I


  ln 1      2 


(2)
where   v / c and c is the speed of light. Other corrections, such as shell and Z13 corrections can be
added in the above equation for the better agreement with experimental results.
3. CALCULATIONS OF STOPPING POWER AND RANGE FOR HEAVY IONS
SRIM (Stopping and Range of Ions in Matter) is a widely used computer program developed by
J.F. Ziegler and J.P. Biersack for the calculations of stopping power and range. We give here only a
short description of program, further information can be found in detail in The Stopping and Range of
Ions in Solids by J.F. Ziegler, J.P. Biersack and U. Littmark [5].
3.1. Stopping Power Calculations
a) Nuclear Stopping
Nuclear Stopping is important for low energy heavy particles. When the projectile energy
becomes high, nuclear stopping is not important, and can be neglected in the calculations.
For practical calculations, nuclear stopping is given by
3
S n (E) 
8.462  10 15 Z1 Z 2 M 1 S n ( )
( M 1  M 2 )( Z10.23  Z 20.23 )
eV/atom/cm2
(3)
where  is the reduced energy and calculated from

32.53M 2 E
Z1 Z 2 ( M 1  M 2 )( Z10.23  Z 20.23 )
(4)
where E is in keV. The reduced nuclear stopping is then given by
S n ( ) 
ln( 1  1.1383 )
2   0.01321 0.21226  0.19593 0.5
S n ( ) 
ln(  )
2


For  ≤ 30
(5)
For  > 30
(6)
b) Electronic Stopping
In SRIM, electronic stopping is considered in three different region: i) low energy region
(E< 25 keV/amu), ii) high energy region (E > 200 keV/amu) iii) intermediate energy region
(25 keV/amu< E< 200 keV/amu). To calculate the stopping power for ions except H, SRIM uses the
proton scaling rule (see Eq.13).
Firstly, a fitted function is used to calculate proton stopping power in each element in SRIM. For
protons of energy E (keV/amu), the fitting functions are
S e, LOW  A1 E A2  A3 E A4
(7)
and
S e, HIGH
A

A5 ln  6  A7 E 
 E
.

A8
E
(8)
For intermediate energies (25 keV  E  10 MeV) stopping power can be calculated from
1
S e, INT

1
S e, LOW

1
S e, HIGH
,
(9)
and for energies higher than 10 MeV,
2
ln( E )
E
 ln( E ) 
S e  A9  A10
 A11 
  A12
E
ln( E )
 E 
(10)
where coefficients Ai are fitting parameters and available for each Z2 in the program. For lower energies
4
(E < 25 keV/amu), velocity proportional stopping used:
 E
S e  S e, INT 
 E0



0.45
 E
S e  S e, INT 
 E0



0.35
for Z2 > 7
(11)
for Z2 = 3-6
(12)
The proton scaling rule is employed to calculate electronic stopping power for ions which are
heavier than protons. The electronic stopping power of heavy ions is proportional to that of protons at
same velocity and target, so one can calculate the stopping power by using
S e  S e, H ( Z1 ) 2
(13)
where  Z1 is the effective charge of the projectile and  is the fractional effective charge of the
projectile.
For 4He ions,
 2  1  exp( 0.2865  0.1266B  0.001429B 2  0.02402B 3  0.01135B 4  0.001475B 5 )c 2
(14)
where B  ln( E ) and c is a correction factor which corrects for the polarization of target atoms by the
projectile (the Z13 effect):

c  1  (0.007  0.00005Z 2 ) exp  (7.6  ln( E )) 2

(15)
Consequently, stopping power of He ions is calculated from
S e, He  S e, H 4 2
S e, He  S e, H 4 2
E
E0
for E > 1keV/amu
(16)
for E ≤ E0 (=1 keV/amu)
(17)
For ions with Z1 > 2,
  q  0.5(1  q)(1 / vF2 ) ln 1  (4vF / 1.919) 2  C
(18)
where vF, is Fermi velocity of target in units of Bohr velocity (v0) and q is the degree of ionization and
given by
q  1  exp( 0.803 y r0.3  1.3167 y r0.6  0.38157 y r  0.008983 y r2 ) .
(19)
C stands for the Z13 correction and is given by
C  1

1
(0.08  0.0015Z 2 ) exp  (7.6  ln( E )) 2
2
Z1
5

(20)
and ,

2a 0 (1  q ) 2 / 3
 1 q 
Z 11 / 3 1 

7 

(21)
is the screening function. The last term, which is called effective ion velocity yr, is given by
yr 
v1
v F2
(
1

),
Z12 / 3
5v12
for v1( 
E / 25 / v0 ) ≥ vF
(22)
and
0.75v F   2v12
1  
yr 
Z12 / 3   3v F2

 1  v1
  
 15  v F



4

.

for v1 < vF
(23)
By using the above equations, stopping power of heavier ions (Z1 > 2) can be calculated from
proton scaling rule for energies higher then 25 keV/amu. In low energy region, stopping power is given
by
 E
S e  S e, H ( Z 1) 
 E0



0.55
 E
S e  S e, H ( Z 1) 
 E0



0.375
 E
S e  S e, H ( Z 1) 
 E0



0.375
 E
S e  S e, H ( Z 1) 
 E0



0.47
2
2
2
2
for Z1 = 3,
(24)
for Z2 < 7,
(25)
for Z1 < 18 and Z2 = 18 or 32,
(26)
and other situations,
(27)
where E0 is the minimum energy used in the above equations. This minimum energy E0 is determined
by the Fermi velocity of target.
In addition, SRIM uses another correction, called Fermi velocity correction, which is applied for
energies below the 10 MeV/amu. Further information can be found in the program package.
3.2. Range Calculations
Most of the transport calculations and Monte Carlo simulations for the calculation of Range are
based on the so-called Continuous Slowing Down Approximation (CSDA). In this approximation, it is
assumed that the particle loses its energy in a continuous way and at a rate equal to the stopping power.
6
Since the stopping power is the energy loss of projectile per unit path, CSDA range (or Bethe range) is
calculated by
dE 
 S ( E )
Eabs
E
R( E ) 
(28)
where Eabs is the energy where particle is effectively absorbed. The CSDA range is the path length
travelled by the particle and since energy-loss fluctuations are not considered, the CSDA range is
always higher than projected range (Rp) which is the distance between the point where particle enters
the stopping medium and the point where particle is absorbed (or comes to rest). It becomes important
when the projectile’s energy is low enough.
SRIM uses PRAL (Projected Range ALgorithm) [6] equations for calculating projected range.
To second order it involves iterating the difference equation
 4 E 2  2 ES n  Qn R p ( E0 )  E0
R p ( E0  E0 )  R p ( E0 )  

4 ES t  2Qn

 E
where  
(29)
M2
, E0 is the initial energy of the projectile and Qn is the second moment of the nuclear
M1
energy loss and given in units of (eV)2/Ǻ by
Qn 
 fs 
4M 1 M 2
 
( M 1  M 2 ) 2 (4  0.197 1.691  6.584 1.0494 )  f  
(30)
where
f s  3.14159 A 2
4

2
(1   ) f 
(31)
and
f 
A
(1   ) Z 1 Z 2 14.4
(32)
with A  0.4685 /( Z10.23  Z 20.23 ) and  is the atomic density of target in units of atoms/Ǻ3.
4. STOPPING POWER CALCULATIONS FOR ELECTRONS AND POSITRONS
The Stopping Power calculation for electrons which are traversing through matter is similar to
that of heavy charged particles. The interaction of incident electrons with target electrons can be
7
calculated from Bethe’s theory, and this gives rise to the “Collisional Stopping Power”. The interaction
between incident electrons and target nucleus results in Bremsstrahlung, and this gives rise to the
“Radiative Stopping Power”.
In the RANGE Module, we have used the effective charge approximation for collisional stopping
power. The formulation which is used in the RANGE module is given in Ref.[7]. The collisional
stopping power is calculated by
S coll
4 e 4 z *

mv 2
2

 NA  *  E 

 Z 2 ln  *   F ( ) / 2
 A   I 

(33)
where
2
F ( )  1    [
2
8
 (2  1) ln( 2)] /(1   ) 2
  E /( me c 2 ) ,
NA is Avogadro’s number,
Z 2* is the effective charge of the target,
A is the mass number of the target,
I* is the effective mean excitation energy of target,
z* is the effective charge of electron.
When charged particles are accelerated or decelerated, they radiate and the energy of this
radiation can be any value from 0 to the energy of incident particles. This is the source of the radiative
stopping power or Bremsstrahlung. This is more important especially fast electrons, since the mass of
electron is much lower than that of nucleus it is accelerated more rapidly when it is in the coulomb
field of nucleus. The radiative stopping power is given by
S rad   0
N AZ 2
( E  me c 2 ) Br
A
(34)
where
0 
1  e2 


137  me c 2 
Br is a function of Z and E. Its value is [8]
8
Br  16 / 3
6
 12
for E  0.5MeV
for E  1MeV
for E  10MeV
.
 15
for E  100MeV
Strength of Bremsstrahlung depends on the target's atomic number (Z), and it is proportional to Z2
and also proportional to energies of incident electrons. On the other hand, collisional stopping power is
proportional to Z. So, the ratio of the radiative stopping power to the collisional stopping power is
approximately given by
S rad ZE

S coll 800
(35)
at high energies ( more than 10 MeV), and E is the energy of the incident electrons in units of MeV. At
high energies, this ratio can be used to calculate the radiative stopping power. The RANGE module
uses this ratio to calculate radiative stopping power.
Radiative energy-loss becomes more important for energies above 10 MeV. The rate increases
logarithmically and at high energies it becomes the predominant mechanism of energy loss. This can be
seen in the figures and tables given in Chapter 6.3.
The positron has the same mass and a charge opposite that of the electron, and the structure of a
positron track in matter is frequently assumed to be similar to that of an electron. Like electrons, for
positron stopping there are two mechanisms: Collisional Stopping Power and Radiative Stopping
Power.
In the RANGE Module, calculations of stopping power and range for positrons are almost same.
Radiative stopping power and CSDA Range calculations are the same as for electrons, but collisional
stopping power is calculated by
S coll
4 e 4 z *

mv 2
2

 NA  *  E  1   


 Z 2 ln  *   ln 1    F ( ) / 2
2
 A   I  2 

(36)
where
14
10
4 
(37)


23 
.
2
12 
  2   2   23 
others are same as for electrons ( i.e. , , z*). The formulation which is used in the RANGE module is
given in Ref.[9].
F  ( )  2 ln( 2) 
 
9
5. STOPPING POWER CALCULATIONS FOR MUONS
The muon is an elementary particle whose charge (-1 e) and spin (1/2) are equal to that of the
electron. It is sometimes regarded as a "heavy" electron, because its mass is 207 times the electron
mass and its interactions with matter are very similar to those of electrons. Muon interactions with
matter differ significantly from electron interactions purely as a result of its much greater mass. For
example the stopping power for electrons, particularly in the high energy regime, is dominated by the
bremsstrahlung process, which is not the case for muons unless the energies are in the multi-GeV range.
On the other hand, in this multi-GeV regime radiative processes are more pronounced than for other
heavy charged particles and ions.
For muons, the stopping power can be calculated by [10]
Se  K
Z 1
A 2
2
 1 2me c 2  2 2Qmax

 1 Qmax
dE
2
ln






2
2 
I
2 8 (Mc ) 
dx
2
(38)
where
K / A=0.307075 MeV/g/cm2 for A=1g/mol,
M is the muon mass,
 is the density factor,
Qmax

2me c 2  2 2

,
1  2 me / M  (me / M ) 2
dE
is the bremstrahlung from atomic collisions and is given by
dx
dE
K Z  2 E 1 2Qmax


 ln
 ln
dx 4 A  Mc 2 3 me c 2
 2Qmax
 ln
2
 me c
2

 ,

and
 =1/137.035999 is the fine structure constant.
Radiative stopping power results from the interaction of muons with the coulomb field of the nucleus.
This is important only at extremely high energies, i.e. more than 100 GeV for Uranium and more than
2.5 TeV for hydrogen. In the RANGE module we have not included radiative stopping power for
muons.
10
6. ACCURACY OF THE RANGE MODULE
The RANGE module uses SRIM for the calculation of the stopping powers and ranges for the
projectiles Z=1-92 in targets Z=1-92, various predefined and also user created compounds. SRIM
results agree with the experimental data within less than 5 %. On the other hand, SRIM has not been
developed specifically for protons and alphas, but it is used the RANGE module. We tested the
RANGE module for protons and alphas in gas, solid and liquid targets, and have compared results with
PSTAR and ASTAR. PSTAR and ASTAR [11] have been developed especially for protons and alphas,
respectively. In this chapter we compare the results.
The RANGE module has also been tested for electrons, positrons and muons in gas, solid and
liquid targets. For electrons we compared results with ESTAR [11] which is the part of STAR program
groups together with PSTAR and ASTAR, while for positrons we compared our results with ICRU
Report 37 [12].
In comparison, we calculated the difference by using
Referance Value - Our Result
Referance Value
 100 .
(39)
Mean errors are calculated as simple arithmetic average value from above equation.
6.1. Test Results for Protons
We calculated the stopping powers and ranges of H (gas), Pb (solid) and water (liquid) for
protons and compared the results with PSTAR.
a) Protons in H (gas)
As can be seen in fig.1, overall agreement with PSTAR is quite good. Comparing the RANGE
module's results with PSTAR, the overall mean error in energy range from 1 keV to 1 GeV is 0.8 %,
mean error is 1.8 % in energies below 400 keV and mean error in energies below 10 keV is 2.5 %.
11
Figure 1. The stopping power results for protons in H (gas).
Figure 2 shows the range results of protons in H (Gas). As can be seen, agreement with PSTAR
is quite good. Mean errors are 2.4 %, 5.8 % and 11.2 % in the energies between 1 keV an 1 GeV, below
400 keV and 10 keV, respectively.
12
Figure 2. The range results for protons in H (gas).
b) Protons in Pb (solid)
To test the range module for solids targets, we have chosen lead as a target. Obtained results for
the stopping power are compared with PSTAR and these results are plotted in fig.3 for comparison.
The agreement is quite good in the high energy region, but in low and intermediate energy regions, as
can bee seen from the fig.3, there is some deviation from the PSTAR results. The mean errors in the
energy ranges between 1 keV and 1 GeV, below 400 keV and below 10 keV are 3.4 %, 8.0 % and
15.8 %, respectively.
For range calculations, obtained results are given in fig. 4. It can be seen that range results how
better agreement than the stopping power results. The mean errors are 2.5 %, 5.7 % and 7.7 % for the
energies between 1 keV and 1 GeV, below 400 keV and below 10 keV, respectively.
13
Figure 3. The stopping power results for protons in Pb (solid)
14
Figure 4. The range results for protons in Pb (solid)
c) Protons in water (liquid)
We have carried out the calculation of stopping power and ranges for protons in water (liquid) to
test the range module in liquid targets. Obtained results for stopping power are shown in fig.5 together
with the results of PSTAR. Agreements with the PSTAR are 4.4 % for energies between 1 keV and 1
GeV, 8.0 % for energies below 400 keV and 8.5 % for energies below 10 keV.
15
Figure 5. The stopping power results for protons in water (liquid)
For range calculations, we have obtained the results shown in fig. 6. The mean errors are 3.0 %,
5.1 % and 4.3 % for the energies between 1 keV and 1 GeV, below 400 keV and below 10 keV,
respectively.
16
Figure 6. The range results for protons in water (liquid).
6.2. Test Results for Alphas
In this section, we give the results of stopping power and ranges for alphas in H (gas), Pb (solid)
and water (liquid). We have compared the results for RANGE module with those from ASTAR.
Obtained results are shown in the figs.7-12 for these targets.
a) Alphas in H (gas)
Calculated results are shown in fig. 7 for stopping power and in fig .8 for range. We have also given
the mean errors in tables (see figs.7-8) for stopping power and for range, respectively.
17
Figure 7. The stopping power results for alphas in H (gas)
Figure 8. The range results for alphas in H (gas).
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b) Alphas in Pb (solid)
The calculations of stopping power and range for the alphas were carried out in Pb to test solid
targets. Obtained results are shown in figs. 9 and 10 for stopping power and range, respectively. As can
be seen from the figures, obtained results are quite agree with the results of ASTAR.
Figure 9. The stopping power results for alphas in Pb (solid).
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Figure 10. The range results for alphas in Pb (solid).
c) Alphas in water (liquid)
Calculated results for the alphas in water(liquid) are shown in figs. 11-12 for the stopping power
and range, respectively. The results agree with the ASTAR results very well.
20
Figure 11. The stopping power results for alphas in water (liquid).
Figure 12. The range results for alphas in water (liquid).
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6.3. Test Results for Electrons
The calculation of the stopping powers and ranges for electrons in matter is different from the
calculation of other ions in matter. Calculation procedure used in RANGE module is explained in
chapter 4. In this section, we have given the results of the stopping power and ranges for electrons in H
(gas), Pb (solid) and water (liquid).
a) Electrons in H (gas)
Obtained results for electrons in H (gas) are shown in figs 13-14 for the stopping power and range,
respectively. In these calculations, we have compared our results with the results of ESTAR. In the
figures, we have shown separately the total, electronic and radiative stopping power.
Figure 13. The stopping powers for electrons in H (gas).
22
Figure 14. The range of electrons in H (gas).
b) Electrons in Pb (solid)
Again, we chose Pb to test the stopping powers and range for the electrons in solids targets.
Obtained results are shown figs. 15-16.
23
Figure 15. The stopping powers for electrons in Pb (solid).
Figure 16. The range of electrons in Pb (solid).
c) Electrons in water (liquid)
For the liquid targets, we have chosen the water as a test material. Calculated results are shown in
24
figs. 17-18 for stopping powers and ranges, respectively.
Figure 17. The stopping powers for electrons in water (liquid).
Figure 18. The range of electrons in water (Liquid).
25
6.4. Test Results for Positrons
The calculation of the stopping powers and ranges for positrons in matter is similar to that of
electrons. Calculation procedure used in RANGE module is explained in chapter 4. In this section, we
have given the results of the stopping power and ranges for positrons in air (gas), Pb (solid) and water
(liquid) to predict how good the results are in three kinds of matter. We compared our results with
ICRU Report 37 [12].
a) Positrons in air (gas)
Obtained results for positrons in air (gas) are shown in figs 19-20 for the stopping power and range,
respectively. In the figures, we have shown separately the total, electronic and radiative stopping power.
Figure 19. The stopping powers for positrons in air (gas).
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Figure 20. The range of positrons in air (gas).
b) Positrons in Pb (solid)
Again, we chose Pb to test the stopping powers and range for the positrons in solids targets.
Obtained results are shown figs. 21-22.
27
Figure 21. The stopping powers for positrons in Pb (solid).
Figure 22. The range of positrons in Pb (solid).
28
d) Positrons in water (liquid)
For the liquid targets, we have chosen the water as a test material. Calculated results are shown in
figs. 23-24 for stopping powers and ranges, respectively.
Figure 23. The stopping powers for positrons in water (liquid).
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Figure 24. The range of positrons in water (liquid).
6.5. Test Results for Muons
We calculate stopping power and range for muons in H (gas), Pb (solid) and water (liquid). Our
calculation procedure is based on that of Groom [10], but we have not calculated radiative stopping
power.
a) Muons in H (gas)
Calculated results for muon in H (gas) are shown in figure 25 for stopping power and in figure 26
for CSDA Range. The agreement with the results of work of Groom et al. is about 0.5% for both the
collisional stopping power and the total stopping power in the energy range from 1 MeV to 1 GeV. In
fact we have calculated only collisional stopping power, but we compare our results both collisional
and total (collisional+radiative) stopping power results. As seen from the figure 25, the results agree
well with the results of work of Groom et al. We also compare our CSDA range results with the Groom
et al.'s results and we see that the agreement is 4.1%.
30
Figure 25. The stopping power for muons in H (gas).
Figure 26. The CSDA range of muons in H (gas).
31
c) Muons in Pb (solid)
For Pb (solid) target, we have shown the calculated results at figures 27 and 28 for stopping
power and range, respectively. The agreement with the results of work of Groom et al. is 7.3% for both
the collisional stopping power and for the total stopping power in the energy range from 1 MeV to 1
GeV. In this case, agreement is not good as before but it is still reasonable. The agreement for CSDA
Results is 9.9%.
Figure 27. The stopping power for muons in Pb (solid).
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Figure 28. The CSDA range of muons in Pb (solid).
d) Muons in water (liquid)
In this case we found that the agreements are 1.6% and 1.5% for the collisional and the total
stopping power, respectively, in the energy range from 1 MeV to 1 GeV. For CSDA Range, the
agreement is 1%. we have shown these results at the figures 29 and 30.
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Figure 29. The stopping power for muons in water (liquid).
Figure 30. The CSDA range of muons in water (liquid).
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7.
RANGE MODULE IN NUCLEONICA
The online version of RANGE module is shown in figure 31.
Figure 31. Main Page of RANGE module in NUCLEONICA.
All inputs are included in a single page. In this page, there are two parts: the first is for projectile
(upper part) and the second is for target. One can choose the projectile and target composition by using
these boxes.
Defining the projectile: to choose projectile, first combo box in the main page is used. When
one clicks on this combo box (see fig.32), one can see various pojectiles in the drop down menu. These
projectiles are electrons, positrons, protons, alphas, muons and other ions ( Z=1 to 92).
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Figure 32. Types of projectiles available in the RANGE module.
If user selects "other ions", two new combo boxes appear as shown in fig.33. The first allows
choosing an element from Z=1 to 92 and second allows defining the isotopes.
Figure 33. Choosing elements and isotopes as a target.
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Defining the energy of projectile: in the "Projectile ion" section, different energy units and
values can be selected as shown in fig.34. In RANGE module, relativistic energy is considered and
when converting the units a relativistic calculation is done. The formula is given by
E  m0 c 2 (  1)
(40)
where   1 1   2 ,   v / c , m0 is the rest mass of the projectile and c is the speed of light.
Figure 34. Energy units in RANGE module
Defining the target: in the RANGE module one can choose an element from Z=1 to 92 (see
fig.35) or a predefined or user defined compound as target. Firstly one must choose a type of target
from the radio buttons under the target combo box (see fig.31). After defining the type of target, the
user can select an actual target from the third combo box if an element or predefined compound has
been selected.
37
Figure 35. Defining a target in RANGE
If the user selects "user defined compound", the page shown in fig.36 appears.
Figure 36. User-defined compound.
38
The user can also define the target phase as a gas or solid and also its density which is predefined
in the program, by using the radio button and text box in target section of the main page.
After defining the projectile-target system, the calculation can be started by clicking the RUN
button. The results are shown ath the bottom of fig. 37a and 37b.
Figure 37a. Input and output for a calculation for 1 MeV protons in hydrogen (gas).
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Figure 37b. Input and output for a calculation for 1 MeV electrons in hydrogen (gas).
In RANGE module, one can see the results in a table or graph for all projectiles. However, the
energy range in the table for ions Z=1 from 92 can be changed only for ions Z=1 from 92 from the
Options menu before clicking on Run (see fig.38). Graph options are also available and the user can
change these options (see fig.39).
Figure 38. Options menu for changing the energy range and the stopping power units.
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Figure 39. Graph options. This will appear at the end of the Results page.
In the RANGE one can see full details of calculation for all projectiles. However, the "details"
menu is different for different type of projectiles. These pages are shown figs.40 and 41.
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Figure 40. Details menu for electrons.
Figure 41. Details menu for projectiles heavier than electrons.
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8.
CONCLUSIONS
Overall agreement between the RANGE Module and PSTAR and ASTAR is quite good. There
is no problem in the calculation of the stopping power and ranges for the protons and alphas. In the
ICRU report 49 [13], it is stated that in the high energy region uncertainties are 1 % to 2 % and for low
energies uncertainties are
•
2% to 5% at 1000 keV, (0.1% to 4%)
•
5% to 10% at 100 keV, (0.3% to 8%)
•
10% to 15% at 10 keV, (0.7% to 11%)
•
20% to 30% at 1 keV. (5% to 22%).
When we compare the results of the RANGE module with the results of STARs, the esults agree
well.
For electrons, we have coded the new formulation [7] for collisional stopping power. For
radiative stopping power we used a simple ratio which is given by Eq.35. We have compared our
results with ESTAR in Chapter 6. Overall agreement is quite good especially for collisional stopping
power and CSDA Range. On the other hand, for radiative stopping power agreement is more than 10%.
The calculation procedure for calculating stopping power and range for positrons in matter is
almost same as for electrons. For collisional stopping power, we used the formulation of Gümüs et al.
[9] and for radiative stopping power we used Eq.35. When we compared our results with ICRU 37
report, It was found that agreement was about 10% in 10keV-1GeV energy range for both stopping
power and range.
For muons, we have calculated only collisional stopping power by using the formulation which is
indicated at the work of Groom et al. [11]. As seen from the figures and calculation results, our results
agree well. But we calculate the stopping power at the energies under 1 GeV. At high energies,
radiative stopping becomes important, so one must calculate also radiative stopping power. On the
other hand, radiative stopping power is important above 100 GeV in almost all every matter. This
shows that our calculation can be also used at the energies above 1 GeV for muons.
Furthermore, RANGE module does not use any database in the calculations. On the other hand
SRIM, which is used by the RANGE module, uses its own database which was created by Ziegler and
Biersack. This data base contains the fitting parameters for calculation of stopping power and range.
These parameters were obtained by using all available experimental results for H and He [3, 5]. In
43
addition, for the RANGE module the new database for mean excitation energy can be prepared and this
database can be used instead of calculating the mean excitation energy in the future.
This technical note has also been released in Nucleonicawiki [14].
44
9.
REFERENCES
1.
N.Bohr, Phil. Mag. 25, 10 (1913).
2.
H.A. Bethe, Ann. Phys. (Leipzig) 5, 325 (1930).
3.
www.srim.org
4.
M. Born, Z. Phys. 38, 803 (1926).
5.
J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon,
New York, 1985.
6.
J.P. Biersack, Nucl. Instrum. Methods B 182-183, 199 (1981).
7.
H. Gümüs, Radiat. Phys. Chem. 72, 7 (2005).
8.
F. H. Attix, Introduction to radiological physics and radiation dosimetry, Wiley&Son, New
York, 1986.
9.
H.Gümüs, Ö.Kabadayi and M.C.Tufan, Chinese J. Phys 44 (4), 290 (2006)
10. D.E. Groom, N.V. Mokhov and S.I. Striganov, At. Data Nucl. Data Tables 78, No 2, 183
(2001).
11. http://physics.nist.gov/PhysRefData/Star/Text/contents.html
12. Stopping Powers for Electrons and Positrons. ICRU Report 37, ICRU, Bethesda, Md, 1984.
13. Stopping Powers and Ranges for Protons and Alpha Particles. ICRU Report 49, ICRU,
Bethesda, MD, 1993.
14. http://www.nucleonica.net:81/wiki/index.php/Help:Range_%26_Stopping_Power
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