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Level: Intermediate, Advanced


Contents

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 targets by Rutherford backscattering spectroscopy (RBS), plasma-first wall interactions in a nuclear fusion reactor, and in many others.

The energy loss is generally written with a minus sign in front:

 S_{lin}=-dE/dx,

so that the value of the linear stopping power becomes positive.

Generally, there are two contributions to stopping power: the interaction of incident particles with target electrons (the electronic stopping power, mostly at high energy), and the interaction with target nuclei (the so-called "nuclear" stopping power, mostly at low energy). For electron and positron projectiles, there is no "nuclear" stopping, but the energy loss due to bremsstrahlung is also quite important.

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, 1913. The first quantum mechanical treatment was by Bethe, 1930. In its relativistic version (Bethe, 1932), 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 target matter before coming to rest. The projected range is the distance of travel projected onto the axis of the original direction. Generally (but not here), analytic transport theory and Monte Carlo calculations are used for the range calculations.

In Nucleonica, the Range & Stopping Power++ (R&SP++) application is included to calculate the stopping powers and projected ranges of almost all ions and, in addition, of electrons, muons and positrons in all elemental targets up to uranium, and in pre-defined compounds. In addition, own compounds can be defined. In the R&SP++ application, The Stopping/Range part of Ziegler's program SRIM (Stopping and Range of Ions in Matter) is used to calculate the stopping powers of ions in matter, and the projected ion ranges are calculated using Biersack's program PRAL (Projected Range Algorithm Biersack, 1981). For electrons, muons and positrons, R&SP++ uses different formulations.

In this document, we describe:

Bethe Theory of Stopping, and Mass Stopping Power

In 1930, Bethe calculated quantum mechanically the electronic energy loss by using the first Born approximation (Born, 1926). According to Bethe, the stopping cross section, in c.g.s. units, is given by (ICRU Report 73)

S_e = \frac{4 \pi Z_1^2 e^4}{m v^2} Z_2 \ln( \frac{2 m v^2}{I})

where

v is the projectile’s speed,

Z_1 e is the nuclear charge of the projectile,

Z_2 e is the nuclear charge of the target,

m and e are mass and charge of the electron, and

I is the mean excitation energy of the target.


When relativistic effects are considered, the above equation becomes (Bethe 1932):

S_e = \frac{4 \pi Z_1^2 e^4}{m v^2} Z_2 [\ln( \frac{2 m v^2}{I})-\ln(1-\beta)-\beta^2]

where \beta=\frac{v}{c} and c is the speed of light. Other corrections, such as shell, Z_1^3 and Z_1^4 corrections can be added to the above equation for better agreement with experimental results.

The linear electronic stopping power is related to the electronic stopping cross section as follows (ICRU Report 73):

S_{lin}=-dE/dx=nS_e

where n is the number of target atoms (or molecules) per volume. Evidently, if similar targets differ in state of aggregation (solid vs. gaseous), the linear stopping powers will differ greatly, because of the factor n. It is therefore customary to tabulate mass stopping powers

S_{lin}/\rho=-dE/\rho dx =S_e/M_2

where M2 is the mass of a target atom (or molecule). The R&SP application calculates mass stopping powers.

To obtain electronic mass stopping powers in practical units, according to the Bethe equation, one can use (ICRU Report 49):

S_{lin}/\rho=\frac{4 \pi r_e^2mc^2}{\beta^2} \frac{1}{u} \frac{Z_2}{A_2} Z_1^2 [\ln( \frac{2 m v^2}{I})-\ln(1-\beta)-\beta^2]
</p><p>=(0.307075 MeVcm^2/g) \frac{Z_2}{A_2} Z_1^2 [\ln( \frac{2 m v^2}{I})-\ln(1-\beta)-\beta^2]

where r_e=\frac{e^2}{mc^2} is the classical electron radius, u is the atomic mass unit, A2 is the relative isotopic mass of the target atom, and standard values of the various atomic constants have been used.

Calculation of Stopping Power and Range for 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, 1985, and in the more recent book SRIM - The Stopping and Range of Ions in Matter by J.F. Ziegler, J.P. Biersack and M.D. Ziegler, 2008.

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

S_n (E) = \frac{8.462 10^{-15} Z_1 Z_2 M_1 S_n (\epsilon)}{(M_1+M_2) (Z_1^{0.23}+Z_2^{0.23})}\qquad eV/atom/cm^2

where \epsilon is the reduced energy and calculated from

\epsilon=\frac {32.53 M_2 E}{Z_1 Z_2 (M_1+M_2) (Z_1^{0.23}+Z_2^{0.23})}

where E is the energy in keV, and M_1, M_2 are the masses of projectile and target atom, respectively. The reduced nuclear stopping is then given by

S_n(\epsilon)=\frac{\ln(1+1.1383 \epsilon)}{2[\epsilon + 0.01321 \epsilon^{0.21226}+0.19593 \epsilon^{0.5}]}\qquad For \quad \epsilon\leqslant30

S_n(\epsilon)=\frac{\ln( \epsilon)}{2 \epsilon}\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad For \quad \epsilon > 30

It should also be noted that Nuclear Stopping in SRIM is based on a scattering into all solid angles; but in a transmission measurement, the angles are usually restricted by the size of the detector. This can be investigated by using the full Monte Carlo TRIM program contained in SRIM, see paper: Nuclear stopping power and its impact on the determination of electronic stopping power, Helmut Paul, 2012

Electronic Stopping

In R&SP, electronic stopping is calculated using Ziegler's program SRIM. The calculation consists of the following steps as described in by Ziegler's books:

1. For H ions, we directly use coefficients from a table to obtain stopping cross sections in each element. For H energies below 25 keV/u, Sev0.9 except for Z2 ≤ 6 where Sev0.5.

2. For He ions, we multiply the stopping for protons at the same velocity by the He effective charge at that velocity

S_e=S_{e,H}(\gamma Z_1)^2

For energies below 1 keV/u, Sev.

3. For heavier ions, we scale proton stopping powers using Brandt-Kitagawa theory. The calculation for Z1 > 2 takes the following steps:

a) The relative velocity of the ion, vr, is calculated, depending only on the ion velocity, v1, and the target Fermi velocity, vF:

for v_1(=\sqrt{E/25}/v_0)\geqslant v_F

v_r=\frac{v_1}{Z_1^{2/3}}(1+\frac{v_F^2}{5v_1^2}),

and for v_1 < v_F

v_r=0.75v_F\left[1+\left(\frac{2v_1^2}{3v_F^2}\right)-\frac{1}{15}\left(\frac{v_1^2}{v_F^2}\right)^4 \right],

where v0 is the Bohr velocity.

b) The fractional ionization of the ion, q, is calculated as follows:

q=1-exp(0.803y_r^{0.3}-1.3167y_r^{0.6}-0.38157y_r-0.008983y_r^2),

where

y_r = \frac{v_{r}}{v_0Z_1^{1/3}}.

c) The screening length of the ion, \Lambda, is calculated as a function of the ion's charge state

\Lambda=\frac{2a_0(1-q)^{2/3}}{Z_1^{1/3}(1-\frac{1-q}{7})}

d) The relative effective charge of the ion is then calculated using

\gamma=q+0.5(1-q)\left(1/v_F^2\right)\ln\lbrack1+\left(2\Lambda v_F/a_0\right)^2\rbrack

where v_F is the Fermi velocity of target in units of the Bohr velocity, (v_0), and a0 is the Bohr radius.


e) The final stopping power is then found from the effective charge and the equivalent proton stopping power:

S_e=S_{e,H}(\gamma Z_1)^2

4. For very low velocity ions, v_1 < \frac{v_F}{{Z_1}^{2/3}}, we use velocity proportional stopping except for Z_1 < 19 in semiconductor band-gap targets (Z2 = 6, 14, or 32) where we use S_e \propto v^{0.75}

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. Since the stopping power is the energy loss of the projectile per unit path, CSDA range is calculated by

R(E)=\int_{E_{abs}}^{E} \frac{dE'}{S(E')}

where E_{abs} is the energy where particle is effectively absorbed. The CSDA range is the path length travelled by the particle. Since the particle's path is usually not straight due to scattering, the CSDA range is always higher than projected range (R_p), which is the distance between the point where the particle enters the stopping medium and the point where the particle is absorbed (or comes to rest), projected onto the original direction of travel.

SRIM uses PRAL (Projected Range ALgorithm) equations for calculating projected range. To second order it involves iterating the difference equation

R_p(E_0+\Delta E_0)=R_p(E_0)+\left[\frac{4E^2-(2E\mu S_n+\mu Q_n)R_p(E_0)}{4ES_t-2\mu Q_n}\right]\frac{\Delta E_0}{E}

where \mu=\frac{M_2}{M_1} , E_0 is the initial energy of the projectile and Q_n is the second moment of the nuclear energy loss and given in units of (eV)2/Ǻ by

Q_n=\frac{4M_1M_2}{(M_1+M_2)^2(4+0.197\epsilon^{-1.691}+6.584\epsilon^{-1.0494})}\left (\frac{f_s}{f_{\epsilon}}\right)

where

f_s=3.14159A^2\frac{4\mu}{(1+\mu)^2}\frac{\rho}{f_{\epsilon}}

and

f_{\epsilon}=\frac{A \mu}{(1+\mu)Z_1 Z_2 14.4}

with A=0.4685/(Z_1^{0.23}+Z_2^{0.23}) and \rho is the atomic density of target in units of Atoms/\overset{\circ}{A}^3.

Stopping Power Calculations for Electrons and Positrons

The Stopping Power calculation for electrons traversing matter is similar to that for heavy charged particles. The interaction of incident electrons with atomic electrons, leading to excitation and ionization, can be calculated from Bethe’s theory, and it is called the “Collisional Stopping Power”. In addition to that, electrons are accelerated in the Coulomb field of nuclei, and this leads to electromagnetic radiation, the so-called "Bremsstrahlung". The corresponding stopping power is the “Radiative Stopping Power”.

In the R&SP application, we have used the effective charge approximation for collisional stopping power. The formulation which is used is given by Gümüs, 2005. The collisional stopping power is calculated by

S_{col}=\frac{4\pi e^4 z^*^2}{mv^2}\left(\frac{N_A}{A}\right)Z_2^*\left[\ln(E/I^*)-F(\tau)/2\right]

where

F(\tau)=1-\beta^2+\left[\frac{\tau^2}{8}-(2\tau+1)\ln(2)\right]/(1+\tau)^2

\tau=E/(m_ec^2)

N_A 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}=\sigma_0\frac{N_A Z^2}{A}(E+ m_e c^2)\overline{B_r}

\overline{B_r} is a function of Z and E. Its value is 16/3 for E << 0.5 MeV, 6 for E = 1 MeV, 12 for E= 10 MeV and 15 for E = 100 MeV (Attix, 1986).

Strength of Bremsstrahlung depends on the target's atomic number (Z), and it is proportional to Z^2 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


\frac{S_{rad}}{S_{col}}=\frac{ZE}{800}

at high energies ( more than 10 MeV). Where, 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.

In Tufan & Gümüş, 2011 the radiative stopping power is given by:

S_{rad} \approx E/X_0

where \frac{1}{X_0}=4\alpha r_e^2 \frac{N_A}{A} \Bigl\{ Z^2\bigl[L_{rad} -f(Z)\bigl]+ZL_{rad}' \Bigl\}

with f(Z) = a^2 \bigl[ (1+a^2)^{-1} + 0.20206 - 0.0369a^2 + 0.0083a^4 - 0.002a^6 \bigl]

and a=\alpha Z, \alpha = 1/137 is the fine structure constant and r_e = 2.818 fm is the classical electron radius. L_{rad} and L_{rad}' are tabulated values.

The R&SP++ application 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 sections Test Results for Electrons and Test Results for Positrons.

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 R&SP++ application, 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_{col}=\frac{4\pi e^4 z^*^2}{mv^2}\left(\frac{N_A}{A}\right)Z_2^*\left[\ln(E/I^*)+\frac{1}{2}\ln(1+\tau/2)+F(\tau)/2\right]

where

F(\tau)=2\ln(2)-\frac{\beta^2}{12}\left[23+\frac{14}{\tau+2}+\frac{10}{(\tau+2)^2}+\frac{4}{(\tau+2)^3}\right]

others are same as for electrons ( i.e. \tau, \beta, z*). The formulation which is used in the R&SP++ application is given in (Gümüs, 2005).

Projected Range vs. CSDA Range for Electrons and Beta particles

When a particle traverses a medium, the particle's path is usually not a straight due to scattering. The CSDA range is a very close approximation to the average path length traveled by a charged particle as it slows down to rest, calculated in the continuous-slowing-down approximation (CSDA). In this approximation, the rate of energy loss at every point along the track is assumed to be equal to the total stopping power. Energy-loss fluctuations are neglected. The CSDA range is obtained by integrating the reciprocal of the total stopping power with respect to energy.

Path Lengths of 19.6 keV electrons in air at atmospheric pressure

The projected range (R_p) is the distance between the point where the particle enters the stopping medium and the point where the particle is absorbed (or comes to rest), projected onto the original direction of travel. The CSDA range is always higher than projected range (R_p). It is usually the projected range which is measured in experiments. In the various models used to calculate the range, it is usually the CSDA range which is calculated. The difference between the two quantities is especially large for electrons, since they are strongly scattered due to their small mass. It is therefore of interest to know how large are the differences betwen these two quantities.

An example can be seen in the image on the left (taken from ICRU Report 16) which is based on measurements by Williams (1931) for 19.6 keV electrons in oxygen at 0° C and 1 atm, measured in a Wilson cloud chamber. The figure shows the integral distributions of CSDA path lengths (S) and of projected ranges (R). Clearly, the CSDA ranges are larger than the projected ranges. The ordinate shows the fraction of electrons that have R (or S) > ℓ, the abscissa value. On differentiating the curves, one obtaines gaussians with maxima at about \overline{R} and \overline{S}, i.e. the mean R is about 0.32 cm, and the mean S is about 0.63 cm.

Comparison of the projected range using the Flammersfeld relation and the CSDA range for electrons calculated using nucleonica

Projected ranges for beta particles have been measured decades ago, using a Geiger counter and absorber sheets. The figure to the left compares the measured projected range for beta decays in aluminum (calculated using the Flammersfeld formula below), which is similar to the projected range of monoenergetic electrons, with the CSDA range for monoenergetic electrons calculated using Nucleonica. Evidently, the CSDA range is again larger than the projected range, by about 100 % at low energy and about 30% at high energy.

The quantitative relationship between maximum beta energy and projected range is given by the following experimentally determined empirical equation (Flammersfeld, 1946; Knop and Paul, 1971):


R = 0.11 ( \sqrt {1+22.4E^2}-1) \quad \quad \quad  0 < E < 3MeV


where R = projected range (g/cm2) and E = maximum beta energy (MeV). In the R&SP++ application, this relation has been used to calculate the Projected ranges of electrons in materials.


Following Cember (2009) the figure below shows the experimentally determined relationship between the projected range and the maximum energies of various beta emitters using different absorbers. The figure shows clearly that the required thickness of absorber for any given beta energy decreases as the density of the absorber increases. Experimental data further show that ability to absorb energy from beta particles depends mainly on the number of absorbing electrons in the path of the beta i.e. on the areal density (electrons/cm2) of electrons in the absorber, and, to a much lesser degree, on the atomic number of the absorber. For this reason, in the calculation of shielding thickness against beta particles, the effect of atomic number is neglected (beta shields are almost always made from low-atomic-numbered materials.) Areal density of electrons is approximately proportional to the product of the density of the absorber material and the linear thickness of the absorber, thus giving rise to the unit of thickness called the density or mass thickness. If mass stopping power were plotted instead of linear stopping power, the curves would be much closer to each other.

Range–energy curves for beta particles in various substances. (Adapted from Radiological Health Handbook. Washington, DC: Office of Technical Services; 1960.)





References

  • ICRU Report 16: Linear Energy Transfer, Intern. Comm. on Radiation Units and Measurements, Washington, D.C., 1970
  • A. Flammersfeld, Naturw, 33 (1946) 280. pdf



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 Groom, 2001

S_e=K\frac{Z}{A}\frac{1}{\beta^2}\left [ \frac{1}{2} \ln \left(\frac {2m_e c^2 \beta^2 \gamma^2 Q_{max}}{I^2}\right)-\beta^2-\frac{\delta}{2}+\frac{1}{8}\frac{Q_{max}^2}{\left(\gamma M c^2\right)^2}\right]+\Delta \left|\frac{dE}{dx}\right|

where

K/A = 0.307075 MeV/g/cm2 for A=1g/mol,

M is the muon mass,

\delta is the density factor,

\Delta \left|\frac{dE}{dx}\right| is the bremstrahlung from atomic collisions and is given by


\Delta \left|\frac{dE}{dx}\right|=\frac{K}{4\pi}\frac{Z}{A}\alpha\left[\ln \left(\frac {2E }{M c^2}\right)-\frac{1}{3}\ln \left(\frac {2Q_{max} }{m_e c^2}\right)\right]\left(\ln \frac {2Q_{max} }{m_e c^2}\right)^2

and

\alpha =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 our calculations, since we have not considered such a high energy we have not included radiative stopping power for muons.

Accuracy of the Range and Stopping Power (R&SP) Application

Comparison with experimental data for positive ions

Note: The stopping power part of SRIM has not been changed since SRIM 2003.

Protons and alpha particles

To judge the accuracy of the SRIM program used in R&SP, it is useful to compare to experimental data. Such comparisons can be done either graphically [16] or by statistical analysis. Here we show the statistical results from the paper by Paul [17] which is based on the use of program “Judge” [18]. This program calculates the normalized differences


 \quad \delta_{SRIM} = (S_{exp}- S_{SRIM})/S_{exp}} \quad \quad \quad (1)


for every data point. Here, Sexp is the experimental value from [16], and SSRIM the corresponding SRIM value for the same ion, same target and same energy. Then, in every range of specific energy (i.e., energy per nucleon), it determines the average normalized difference


\Delta = <\delta_{SRIM}{>} \quad \quad \quad (2)


and its standard deviation


\sigma = \sqrt{<\delta^2_{SRIM}{>}- <\delta_{SRIM}{>}^2} \quad \quad \quad (3)


The averages designated by < > are unweighted, except that experimental data that appear to be in conflict with the majority have been rejected [16]. The calculated average normalized difference (and its standard deviation) is then a reliable indicator of how well SRIM reproduces the experimental data. A small Δ usually signifies good agreement between table and experimental data; in such a case, σ is related to the mean experimental accuracy, and σ may be taken as a measure of the accuracy of the table, as determined from experiment.

Table 1 shows the results for H ions in 17 solid elements, as a function of E/A1 , where E is the energy and A1 is the mass number of the ion. SRIM represents the data well, and better than other programs [17]. The number of measured points gives an indication of the reliability of results. Since measurements at low energy are more difficult than at high energy, σ always increases with decreasing energy.


Table 1. Mean normalized deviations Δ ± σ (in %) for H ions in the 17 solid elements covered by the PSTAR Table, using eq. (1).

Table 1. Mean normalized deviations Δ ± σ (in %) for H ions in the 17 solid elements covered by the PSTAR Table, using eq. (1).


Table 2 shows the results for gases. Again, SRIM represents the data well. The results for gases are better than for solids.

Table 2. Mean normalized difference Δ ± σ (in %) for H ions in all elemental gases except F, Cl, Rn, using eq. (1).


Table 3 shows the results for He ions in 16 elemental solids (adding results for 39 more elemental targets would not change the values Δ and σ much), and Table 4 for elemental gases (adding the data for Cl and for gaseous Br would change the values Δ and σ very little) except F, Cl, Rn. Again, the results for gases are much better than for solids.

Table 3. Mean normalized deviations Δ ± σ (in %) for He ions in 16 solid elements covered by the ASTAR Table, using eq. (1)


Table 4. Mean normalized difference Δ ± σ (in %) for He ions in all elemental gases except F, Cl, Rn, using eq. (1)


Similar results for protons and alphas in compounds are shown in the same reference [17].

Ions from 3Li to 18Ar

Table 5 shows a comparison with SRIM stopping for ions from 3Li to 18Ar on solid elemental targets; similarly, Table 6 shows results for gases. Comparisons for compounds are also shown in [17].

Table 5. Mean normalized deviations Δ ± σ (in %) for ions from 3Li to 18Ar in all the elemental solids covered by MSTAR, using eq. (1)


Table 6. Mean normalized deviations Δ ± σ (in %) for ions from 3Li to 18Ar in all (elemental and compound) gases covered by MSTAR for which there are data, using eq. (1).


Ions from 19K to 92U

Table 7 shows comparisons with SRIM of data for ions from 19K to 92U [ii]. Here, Δ increases with energy: at the highest energy, only the non-perturbational Lindhard-Sørensen theory (calculated using ATIMA) is correct [17]. Table 8 shows results for gases. Results for compounds are also given in [17].

Table 7. Mean normalized deviations Δ ± σ (in %) for SRIM of experimental data for 31 ions from 19K to 92U in all 54 solid elemental targets for which there are data in [17], using eq. (1)



Table 8. Mean normalized deviations Δ ± σ (in %) for SRIM, for ions from 19K to 92U in all elemental gas targets for which there are experimental data in [17], using eq. (1)


Comparison with other tables

The R&SP application 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. Here, we test the R&SP application module for protons and alphas in gas, solid and liquid targets, comparing results with programs PSTAR and ASTAR [12] that have been developed especially for protons and alphas, respectively.

The R&SP application has also been tested for electrons, positrons and muons in gas, solid and liquid targets. For electrons we compared results with ESTAR [12] which is the part of STAR program groups together with PSTAR and ASTAR, while for positrons we compared our results with ICRU Report 37 [13].

In comparison, we calculated the difference by using

\frac{|Reference Value - Our Value|}{Reference Value}x100

Mean errors are calculated as simple arithmetic average value from above equation. Please note that the accuracy of the calculated mean errors depends on the location of the points chosen. The data points for the chosen energies are listed in the Table tab, and they are denoted as small circles in the various graphs.

Test Results for Protons: Comparisons with PSTAR

We calculated the stopping powers and ranges of H (Gas), Pb (solid) and water (Liquid) for protons and compared the results with PSTAR.

Protons on H (Gas)

As can be seen in the figure below, overall agreement with PSTAR is quite good. Comparing the R&SP++ application 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 %.

The stopping power results for protons in H (Gas).


The figure below 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.

The range results of protons in H (Gas).


Protons on 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 shown in the figure below for comparison. The agreement is quite good in the high energy region, but in low and intermediate energy regions, as can bee seen, 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.

The stopping power results for protons in Pb (Solid).


For range calculations, obtained results are given in the figure below. 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.

The range results of protons in Pb (Solid).


Protons on Water (Liquid)

We have carried out the calculation of stopping power and ranges for protons in water (liquid) to test the R&SP++ applicaiton in liquid targets. Obtained results for stopping power are shown in the figures below together with the results using PSTAR 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.

The stopping power results for protons in Water (Liquid).


For range calculations, we have obtained the results shown in the figure below. 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.

The range results of protons in Water (Liquid).


Test Results for Alphas: Comparisons with ASTAR

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 R&SP++ application with those from ASTAR. Obtained results are shown in the figures below for these targets.

Alphas in H (Gas)

Calculated results are shown in the figures below for stopping power range. We have also given the mean errors in tables for stopping power and for range, respectively.

The stopping power results for alphas on H (Gas).


The range results of alphas on H (Gas).


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 figures below for stopping power and range, respectively. As can be seen from the figures, obtained results are quite agree with the results of ASTAR.

The stopping power results for alphas on Pb (Solid).


The range results of alphas on Pb (Solid).


Alphas in Water (Liquid)

Calculated results for the alphas in water(liquid) are shown below for the stopping power and range, respectively. The results agree with the ASTAR results very well.

The stopping power results for alphas on Water (Liquid).


The range results of alphas on Water (Liquid).


Test Results for Heavier Ions

In a recent paper by Paul and Sánchez-Parcerisa (see reference below), the authors present a critical overview of stopping power programs of heavy ions in solid elements and compare results with a large collection of experimental data to test their validity (see typical results in the figure below).

In the figure below, Prof. Helmut Paul from the Johannes Kepler University in Linz, Austria, has calculated the stopping power for carbon ions in carbon. In addition to the codes SRIM and MSTAR, results form a number of programs currently available are shown. The authors conclude that although interesting new approaches have been used in some of the programs and codes, both SRIM (as used in Nucleonica's Range & Stopping Power Application) and MSTAR still seem to be slightly better at reproducing the available data.

The stopping power for carbon ions in carbon. Courtesy Helmut Paul, 2014. See references to the curves and data below.


References to the curves and data:

  • ICRU Report 73, Stopping of ions heavier than Helium, J. ICRU 5 (1) (2005) 1.
  • A. Javanainen, Nucl. Instr. Meth. B 285 (2012) 158.


The stopping power for carbon ions in carbon, calculated with Nucleonica's Range & Stopping Power application which is based on SRIM.


References

- Helmut Paul, Daniel Sánchez-Parcerisa, A critical overview of recent stopping power programs for positive ions in solid elements, Nuclear Instruments and Methods B 312 (2013) 110

- H. Paul, MSTAR version 3, 2003. Available from http://www.exphys.jku.at/stopping/MSTARWWW/MSTAR312.zip.

- Stopping Power: Graphs and Data, Dr. Helmut Paul



Test Results for Electrons: Comparisons with Beta particles and with ESTAR

The calculation of the stopping powers and ranges for electrons in matter is different from the calculation for heavier particles. The calculation procedure used in the R&SP++ application is explained in Stopping Power Calculations for Electrons and Positrons. In this section, we first compare with beta particles and we then give the results of the stopping power and ranges for electrons in H (gas), Pb (solid) and water (liquid).

Comparison of the maximum projected range of beta particles using the Flammersfeld relation and the CSDA range for electrons calculated using Nucleonica


Electrons in H (Gas)

Obtained results for electrons in H (gas) are shown in the figures below 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.

The stopping power results for electrons on H (Gas).


The range results of electrons on H (Gas).


Electrons in Pb (Solid)

Again, Pb was chosen as target material to test the stopping powers and range for the electrons in solids targets. The results are shown in the figures below.

The stopping power results for electrons on Pb (Solid).


The range results of electrons on Pb (Solid).


Electrons in Water (Liquid)

As a liquid target, we have chosen water as test material. Calculated results are shown in the figures below for stopping powers and ranges, respectively.

The stopping power results for electrons on Water (Liquid).


The range results of electrons on Water (Liquid).


Test Results for Positrons

The calculation of the stopping powers and ranges for positrons in matter is similar to that of electrons. The calculation procedures used in R&SP++ application is explained in the section Stopping Power Calculations for Electrons and Positrons. In this section, we give the results of the stopping power and ranges for positrons in three different states of matter: in air (gas), Pb (solid) and water (liquid). The results are compared with those given in ICRU Report 37 [ICRU 13, 1084].


Positrons in air (gas)

Obtained results for positrons in air (gas) are shown in the figures below for the stopping power and range, respectively. In the figures, we have shown separately the total, electronic and radiative stopping power.

The stopping power results for positrons in H (Gas).


The range results of positrons in H (Gas).


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 figures below.

The stopping powers for positrons in Pb (solid).


The range of positrons in Pb (solid).


Positrons in water (liquid)

For the liquid targets, we have chosen the water as a test material. Calculated results are shown below for stopping powers and ranges, respectively.

The stopping powers for positrons in water (liquid).


The range of positrons in water (liquid).


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 et al. [2011], but we have not calculated radiative stopping power.

Muons in H (gas)

Calculated results for muons in H (gas) are shown in the figure belows for stopping power and for the 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 figures below, 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%.

The stopping power for muons in H (gas).


The CSDA range of muons in H (gas).


Muons in Pb (solid)

For Pb (solid) target, we have shown the calculated results in the figures below 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%.

The stopping power for muons in Pb (solid).


The CSDA range of muons in Pb (solid).


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%. Results are shown in the figures below.

The stopping power for muons in water (liquid).


The CSDA range of muons in water (liquid).


Using Range&Stopping Power++ (R&SP++) Application

Range & Stopping Power

The online version of the Range & Stopping Power++ (R&SP++) application is shown in figure below. All inputs are included in a single page. In this page, there are two panels: the first is for projectile (left panel) and the second is for target. One can choose the projectile and target composition by using these boxes. On pressing the Start button, the results are shown in a small summary table (which can be expanded to show details) and in graphical form (showing the Stopping Power, Range and the Bragg Curves).

Main Page of the Range & Stopping Power++ (R&SP++) application in Nucleonica.


Defining the Projectile

Defining the projectile: to choose a projectile, the first combo box in the main page is used. When one clicks on this combo box (see figure below), 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).

Types of projectiles available in the Range & Stopping Power++ (R&SP++) application.


If the user selects "other ions", two new combo boxes appear as shown below. The first allows choosing an element from Z=1 to 92 and second allows defining the isotopes.

Choosing ions as projectiles.


Defining the energy of projectile: in the "Projectile ion" section, different energy units and values can be selected as shown below.

Energy units in R&SP++ application.


In R&SP++ application, the relativistic energy is considered when converting the units. The formula is given by

E=m_0 c^2 (\gamma-1)

where \gamma=1/(1-\beta^2)^(0.5), \beta=v/c , m_0 is the rest mass of the projectile and c is the speed of light.


Defining the Target

Elements

Defining the target: in the R&SP application one can choose an element from Z=1 to 92 (see image below) 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 figure). 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.

Defining a target in R&SP++ application.


Predefined Compounds

If the user selects "Predefined compound", the page shown below appears. 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.

Target: Predefined compound.


User defined Compounds

If the user selects "User defined compound", the page shown below appears. 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.

Target: User-defined compound.


SRIM can make significant corrections to the stopping of ions in compounds, based on the chemical binding of the material.

For information on the compound correction factor see:

SRIM documentation: Examples of Compound Correction.

The Stopping and Range in Compounds


Table

In the Table tab, shown below, results are given in tabular form. In the header part of the page, general information is given on the SRIM version used for the calculations, the date of the calculation, and a summary of the projectile and target properties.

In the main table, as a function of the projectile ion energy, the electronic and nuclear stopping powers are given as well as the range, and longitudinal and lateral straggling. The projectile energies can range from 1 keV to 1 GeV with the results suitably paged. The table results can be downloaded as an Excel spreadsheet or as a csv (comma separated variable) file.

At the bottom of the page the stopping power results in the table can be expreseed in alternative stopping power units.

Stopping power and range results in tabular form as a function of the projectile ion energy.


When the projectile ion is an electron or positron, the tabulated results shown below are slightly different to those for light and heavy ions. As a function of electron or positron energy, the collisional, radiative and total stopping powers are given together with the projected range, expressed as a mass thickness, and the radiation yield.

Stopping power and range results in tabular form as a function of the projectile electron energy.


Options

In the Options tab, the minimum and maximum energies used for the calculations can be set. The minum and maximum energies are 1 keV and 1 GeV respectively.

In the Options tab, the minimum and maximum energies used for the calculations can be set.


Compound Details

When the target material is a compound, either predefined user defined, the Compound Details tab shows information on the selected compound. In this tab, a new compound can be created or an existing compound can be edited by adding or deleting components.

Compound details create/edit page.


Conclusions

Overall agreement between the R&SP++ application 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 [ICRU 49, 1993], 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%)

For electrons, we have coded the new formulation [Gümüs et al., 2005] 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 the section Test Results for Electrons. 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., 2005] 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., 2001]. 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, the R&SP++ application does not use any database in the calculations. On the other hand SRIM, which is used by the R&SP++ application, uses its own database which was created by Ziegler and Biersack. This database 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 [Ziegler et al., 2008]. In addition, for the R&SP++ application 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.

References

  1. N.Bohr, Phil. Mag. 25, 10 (1913).see pdf
  2. H.A. Bethe, Ann. Phys. (Leipzig) 5, 325 (1930).see pdf
  3. H.A. Bethe, Z. Phys. 76, 293 (1932)
  4. http://www.srim.org
  5. M. Born, Z. Phys. 38, 803 (1926).
  6. J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985.
  7. J.F. Ziegler, J.P. Biersack, M.D. Ziegler, SRIM - The Stopping and Range of Ions in Matter, SRIM Co., 2008
  8. J.P. Biersack, Nucl. Instrum. Methods B 182-183, 199 (1981).
  9. H. Gümüs, Radiat. Phys. Chem. 72, 7 (2005).
  10. F. H. Attix, Introduction to radiological physics and radiation dosimetry, Wiley&Son, New York, 1986.
  11. M.Ç. Tufan, H. Gümüş, Journal of Nuc. Mat. 412 (2011) 308-314.
  12. H.Gümüs, Ö.Kabadayi and M.C.Tufan, Calculation of the Stopping Power for Intermediate Energy Positrons, Chinese J. Phys 44 (4), 290 (2006)
  13. D.E. Groom, N.V. Mokhov and S.I. Striganov, MUON STOPPING POWER AND RANGE TABLES: 10 MeV - 100 TeV, At. Data Nucl. Data Tables 78, No 2, 183 (2001).
  14. http://physics.nist.gov/PhysRefData/Star/Text/contents.html
  15. Stopping Powers for Electrons and Positrons. ICRU Report 37, ICRU, Bethesda, Md, 1984.
  16. Stopping Powers and Ranges for Protons and Alpha Particles ICRU Report 49, ICRU, Bethesda, MD, 1993.
  17. H. Paul, Stopping Power for Light Ions, http://www.exphys.jku.at/stopping/
  18. H. Paul, The stopping power of matter for positive ions, in: G. Natanasabathi (Eds.): Modern practices in radiation therapy, chapter 7, Page(s) 113-132, InTech Open Access Company, 2012, http://www.intechopen.com/books/modern-practices-in-radiation-therapy
  19. H. Paul, A. Schinner. An empirical approach to the stopping power of solids and gases for ions from 3Li to 18Ar, Nucl. Instrum. Methods B 179 (2001) 299-315

External Links

Stopping Power: Graphs and Data, Dr. Helmut Paul

Particle Interactions in Detectors, Dr. Peter Hobson

Passage of particles through Matter, 2002

Cloud chamber researches in nuclear physics and cosmic radiation, P. M. S. Blackett, Nobel lecture, 1948

On the Stopping Power of Fast Particles and on the Creation of Positive electrons, H. Bethe, W. Heitler, 1934

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