Muon Stopping Power and Range
Table of Contents
- 1. Electronic Stopping Power
- 1.1 The Density Effect
- 2. Radiative Stopping Power
- 2.1 Bremsstrahlung
- 2.2 Pair Production
- 2.3 Photonuclear Interactions
- 3. Stopping Power for Compounds and Mixtures
- 3.1 Mean Ionization Energy
- 3.2 Density Effect Parameters
- 4. Range and the CSDA approximation
- 5. References
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Electronic Stopping Power
The electronic stopping power is calculated using the Bethe-Bloch equation for muons [1],
$$ \left(-\frac{dE}{dx}\right)_{electronic}=K\frac{Z}{A}\frac{1}{\beta^2}\left[\frac{1}{2}\ln{\frac{2m_ec^2\beta^2\gamma^2Q_{max}}{I^2}}-\beta^2-\frac{\delta}{2}+\frac{1}{8}\frac{Q^2_{max}}{(\gamma M_{\mu}c^2)^2}\right]+\Delta \left|\frac{dE}{dx}\right| $$
where \(\beta=v/c\), \(\gamma = 1/\sqrt{1-\beta^2}\), \(m_e\) is the electron mass, \(M_{\mu}\) is the muon mass, \(Z\) and \(A\) are the atomic number and mass of the medium, \(N_a\) is Avogadro's number, \(r_e=e^2/4\pi\epsilon_0m_ec^2\) is the classical electron radius, \(I\) is the mean excitation energy, \(\delta\) is the density effect correction to energy loss due to the dielectric polarization of the medium, \(K=4\pi N_A r^2_e c^2\), and,
$$ Q_{max}=\frac{2m_ec^2\beta^2\gamma^2}{1+2\gamma m_e/M_\mu+(m_e/M_\mu)^2} $$
is the kinematic maximum possible electron recoil kinetic energy.
The last term in the Bethe-Bloch equation is the energy loss due to bremsstrahlung from atomic electrons,
$$\Delta \left|\frac{dE}{dx}\right|=\frac{K}{4\pi}\frac{Z}{A}\alpha\left[\ln{\frac{2E}{M_\mu c^2}}-\frac{1}{3}\ln{\frac{2Q_{max}}{m_ec^2}}\right]\ln^2{\frac{2Q_{max}}{m_ec^2}}$$
where \(E=\gamma M_\mu c^2\) is the muon incident energy, and \(\alpha=e^2/4\pi\epsilon_0\hbar c\) is the fine structure constant.
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The Density Effect
The density effect correction is [2],
\begin{equation} \delta= \begin{cases} 0 & \textrm{if} \ x \lt x_0 \ \textrm{(non-conductor)} \\ \delta_0 10^{2(x-x_0)} & \textrm{if} \ x \lt x_0 \ \textrm{(conductor)} \\ 2(ln{10})x-\tilde{C} + a(x_1-x)^k & \textrm{if} \ x_0 \le x \lt x_1 \\ 2(ln{10})x-\tilde{C} & \textrm{if} \ x \ge x_1 \\ \end{cases} \end{equation}
where,
$$ x=\log_{10}\left({p/M_\mu}\right)=\log_{10}{\beta \gamma} $$
(\(p\) is the momentum of the incident particle), and,
$$ \tilde{C}=2\ln\left({I/\hbar\omega_p}\right)+1 $$
where,
$$ \hbar\omega_p=\sqrt{4\pi N_er_e^3} \ m_ec^2/\alpha $$
is the plasma energy of the electrons of the medium considered as free electrons [3] and \(N_e\) is the electron density. In terms of the medium density,
$$ \hbar\omega_p=28.816\sqrt{\rho Z/A} \ \ \textrm{eV} $$
for \(\rho\) in g cm\(^3\).
The parameters \(a, k, x_0, x_1 \textrm{and} \ \delta_0\) are adjusted to give a best fit to numerical calculations, and given in tables for many elements, chemical compounds, and mixtures [2]. These tables also give the mean excitation energy \(I\) in eV.
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Radiative Stopping Power
The radiative stopping power is expressed as a sum of three contributions; bremsstrahlung, direct pair production and photonuclear interactions. The formulae in this section uses the convention \(c=1\).
Bremsstrahlung
The average energy loss due to bremsstrahlung is calculated by numerical integration,
$$ -\frac{1}{E} \frac{dE}{dx} \Bigg|_{brems}=\frac{N_a}{A} \int_0^1{\nu \frac{d\sigma}{d\nu}\ d\nu}$$
where \(\nu\) is the fraction of the muon energy transferred to the photon, and \(d\sigma/d\nu\) is the differential cross section.
For bremsstrahlung from a screened nucleus [1],
$$ \frac{d\sigma}{d\nu} \Bigg|_{brems,nucl}=\alpha \left( 2Z\frac{m_e}{M_\nu} r_e \right)^2 \left( \frac{4}{3}-\frac{4}{3}\nu+\nu^2 \right) \ \frac{\Phi(\delta)}{\nu} $$
and,
$$ \Phi(\delta)=\ln{\left(\frac{BM_\mu Z^{-\frac{1}{3}}/m_e}{1+\delta\sqrt{e}BZ^{-\frac{1}{3}}/m_e}\right)} - \Delta_n(\delta)$$
where,
\begin{equation} B=\begin{cases} 182.7 & \textrm{for all elements except hydrogen} \\ 202.4 & \textrm{for hydrogen} \\ \end{cases} \end{equation}
\(e=2.7181...\), \(\delta=M_\mu^2\nu/2E(1-\nu)\), and,
$$ \Delta_n=\ln{\left(\frac{D_n}{1+\delta(D_n\sqrt{e}-2)/M_\mu} \right)} $$
where \(D_n=1.54A^{0.27}\).
For bremsstrahlung losses on atomic electrons [1],
$$ \frac{d\sigma}{d\nu} \Bigg|_{brems,elec}=\alpha Z \left(2\frac{m_e}{M_\mu} r_e \right)^2 \left(\frac{4}{3}-\frac{4}{3}\nu+\nu^2 \right) \ \frac{\Phi_{in}(\delta)}{\nu} $$
and,
$$ \Phi_{in}(\delta)= \ln{\left(\frac{M_\mu/\delta}{M_\mu\delta/m_2^2+\sqrt{e}} \right)} - \ln{\left(1+\frac{m_e}{\delta B Z^{-\frac{2}{3}}\sqrt{e}} \right)} $$
where,
\begin{equation} B=\begin{cases} 1429 & \textrm{for all elements except hydrogen} \\ 446 & \textrm{for hydrogen} \\ \end{cases} \end{equation}
The numerical integration of differential cross sections is performed using the algorithms in QUADPACK [4], as implemented by the GNU Scientific Library.
In the specific case of bremsstrahlung, the adaptive integration QAG algorithm with 31 point Gauss-Kronrod rules is used to integrate the differential cross sections.
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Pair Production
The average energy loss due to direct pair production from a screened nucleus is calculated by numerical integration,
$$ -\frac{1}{E} \frac{dE}{dx} \Bigg|_{pair,nucl}=\frac{N_a}{A} \int_0^1{\nu \frac{d\sigma}{d\nu}\ d\nu}$$
where,
$$ \frac{d\sigma}{d\nu} \Bigg|_{pair,nucl}=\frac{\left(2\alpha r_e Z\right)^2}{\pi} \frac{\left(1-\nu \right)}{\nu} \left[\left(f_1+\theta f_3\right) \ln{\left( \frac{2\epsilon}{m_e} \right)} + \phi_2+\theta \phi_4+I \right] $$
and \(\theta=m_e^2/M_\mu^2\). The explicit formulae for the terms \((f_1+\theta f_3)\), \((\phi_2+\theta \phi_4)\) and \(I\) are quite long and given in Appendix B of Ref. [1].
In this specific case, the adaptive integration with singularities QAGS algorithm is used to integrate the differential cross section.
The average energy loss due to electronic pair production is calculated directly,
$$ -\frac{1}{E} \frac{dE}{dx} \Bigg|_{pair,elec}=\frac{Z}{A}\left(0.073 \ln{\left(\frac{2E/M_\mu}{1+gZ^{\frac{2}{3}}E/M_\mu} \right)}-0.31 \right) \ \times 10^{-6} \ \textrm{cm}^2/\textrm{g} $$
where,
\begin{equation} g=\begin{cases} 1.95\times 10^{-5} & \textrm{for all materials except hydrogen} \\ 4.4 \times 10^{-5} & \textrm{for hydrogen} \\ \end{cases} \end{equation}
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Photonuclear Interactions
The average energy loss due to photonuclear interactions is calculated by numerical integration,
$$ -\frac{1}{E} \frac{dE}{dx} \Bigg|_{photonucl}=\frac{N_a}{A} \int_0^1{\nu \frac{d\sigma}{d\nu}\ d\nu}$$
where,
$$ \frac{d\sigma}{d\nu} \Bigg|_{photonucl}=\frac{\alpha}{2\pi}A\sigma_{\gamma N}(\epsilon)\nu \left(0.75 \ G(x)\left[\kappa\ln{\left(1+\frac{m_1^2}{t}\right)}-\frac{\kappa m_1^2}{m_1^2+t}-\frac{2M_\mu^2}{t}\right] + 0.25\left[\kappa \ln{\left(1+\frac{m_2^2}{t} \right)}-\frac{2M_\mu^2}{t} \right]+ \\ \frac{M_\mu^2}{2t}\left[0.75 \ G(x)\frac{m_1^2}{m_1^2+t}+0.25 \ \frac{m_2^2}{t} \ \ln{\left(1+\frac{t}{m_2^2}\right)} \right] \right) $$
where,
\(\epsilon\) is the energy loss of the muon, \(\sigma_{\gamma N}(\epsilon)\) is the photoabsorption cross section, \(\nu=\epsilon/E\),
$$ G(x)=\frac{3}{x^3}\left(\frac{x^2}{2}-1+e^{-x}(1+x) \right) \textrm{ ,} $$
$$ \kappa=1-\frac{2}{\nu}+\frac{2}{\nu^2} \textrm{ ,} $$
$$ t=\frac{M_\mu^2 \nu^2}{1-\nu} \textrm{ ,} $$
$$ x= 0.00282 A^{1/3} \sigma_{\gamma N} \textrm{ ,} $$
\(m_1^2=0.54 \textrm{ GeV}^2\), and \(m_2^2=1.8 \textrm{ GeV}^2\).
The cross section for photon-nucleon interactions for \(\epsilon > 5\) GeV is,
$$ \sigma_{\gamma N}=114.3+1.647\ln^2{\left(0.0213\epsilon\right)} \ \textrm{in \(\mu\)b} $$
For \(\epsilon \le 5\) GeV the data in Ref. [5,6] is interpolated.
In this specific case, the adaptive integration with singularities QAGS algorithm is used to integrate the differential cross section.
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Stopping Power for Compounds and Mixtures
Compounds and mixtures are conceived to be composed of thin layers of pure elements or compounds, whose data is available, in a proportion consistent with Bragg additivity [7]. If \(n_j\) is the number of the \(j\)th kind of atoms, and \(w_j\) its weight fraction,
$$ w_j=\frac{n_j A_j}{\sum_{i} {n_i A_i}} $$
then,
$$ \biggr \langle \frac{dE}{dx} \biggr \rangle = \sum_{j} {w_j} \frac{dE}{dx} \Bigg|_j $$
The \(Z\)-dependent terms in the Bethe-Bloch equation for a compound or mixture is equivalent to a single material with,
$$ \biggr \langle \frac{Z}{A} \biggr \rangle = \sum_{j} {w_j} \frac{Z_j}{A_j} $$
$$ \ln{\langle I \rangle} = \frac{\sum_{j} {w_j} \frac{Z_j}{A_j}\ln{I_j}}{\sum_{j} {w_j} \frac{Z_j}{A_j}} $$
Mean Ionization Energy
Due to molecular binding effects, electrons in compounds and mixtures are more tightly bound than in the constituent elements. Therefore, the calculated effective \(\langle I \rangle\) is generally underestimated by the Bragg additivity assumption. To improve accuracy, Berger and Seltzer [8,9] devised a simple scheme for assigning temporary \(I\) values to constituent elements depending on the type of compound or mixture, and physical state:
(a) For gases,
| Constituent | \(I\) [eV] |
|---|---|
| H | 19.2 |
| C | 70.0 |
| N | 82.0 |
| O | 97.0 |
(b) for solids and liquids,
| Constituent | \(I\) [eV] |
|---|---|
| H | 19.2 |
| C | 81.0 |
| N | 82.0 |
| O | 106.0 |
| F | 112.0 |
| Cl | 180.0 |
(c) for all other constituents the values are multiplied by 1.13.
The scheme is superseded if an experimental value for the compound or mixture is available.
There are other schemes that take instead into account the type of chemical bonds in the compound, with less degrees of accuracy. We therefore ignore these and adopt the scheme by Berger and Seltzer described above.
Density Effect Parameters
Bragg additivity has no meaning for the calculation of the density effect parameters of a compound or mixture. The following recipe by Sternheimer and Peierls [10] is used:
- \(k\) is always taken as 3.00.
- \(\tilde{C}=2\ln\left({I/\hbar\omega_p}\right)+1\), with \(\hbar\omega_p=28.816\sqrt{\rho Z/A}\) eV for \(\rho\) in g cm\(^3\) as before.
- For solids and liquids,
\begin{equation} x_1= \begin{cases} 2.0 & \textrm{if} \ I \lt 100 \ \textrm{eV, and} \\ & x_0= \begin{cases} 0.2 & \textrm{if} \ \tilde{C} \lt 3.681 \\ 0.326\tilde{C}-1.0 & \textrm{otherwise} \\ \end{cases} \\ 3.0 & \textrm{if} \ I \ge 100 \ \textrm{eV, and} \\ & x_0= \begin{cases} 0.2 & \textrm{if} \ \tilde{C} \lt 5.215 \\ 0.326\tilde{C}-1.5 & \textrm{otherwise} \\ \end{cases} \\ \end{cases} \end{equation} - For gases, \begin{equation} x_0= \begin{cases} 1.6 \ \textrm{and} \ x_1=4.0 & \textrm{if} \ \tilde{C} \lt 10.0 \\ 1.7 \ \textrm{and} \ x_1=4.0 & \textrm{if} \ 10.00 \le \tilde{C} \lt 10.50 \\ 1.8 \ \textrm{and} \ x_1=4.0 & \textrm{if} \ 10.50 \le \tilde{C} \lt 11.00 \\ 1.9 \ \textrm{and} \ x_1=4.0 & \textrm{if} \ 11.00 \le \tilde{C} \lt 11.50 \\ 2.0 \ \textrm{and} \ x_1=4.0 & \textrm{if} \ 11.50 \le \tilde{C} \lt 12.25 \\ 2.0 \ \textrm{and} \ x_1=5.0 & \textrm{if} \ 12.25 \le \tilde{C} \lt 13.804 \\ 0.326\tilde{C}-1.5 \ \textrm{and} \ x_1=5.0 & \textrm{if} \ \tilde{C} \ge 13.804 \\ \end{cases} \end{equation}
- \begin{equation} a=\frac{\tilde{C}-2(\ln{10})x_0}{(x_1-x_0)^3} \end{equation}
If a gas is not at the normal density \(\rho_0\) (1 atmosphere and 20° C), Sternheimer and Peierls [10] noted that at the same temperature and momentum \(p\),
$$ \delta_r(p) = \delta(p\sqrt{r}) $$
where \(r=\rho/\rho_0\). For the new density condition represented by the subscript \(r\), this implies,
$$ \tilde{C}_r = \tilde{C}-\ln{r} $$
$$ x_{0r} = x_0 -\frac{1}{2}\log_{10}{r} $$
$$ x_{1r} = x_1 -\frac{1}{2}\log_{10}{r} $$
while \(a\) and \(k\) remain unchanged under the transformation.
Range and the CSDA approximation
The Continuous Slowing Down Approximation (CSDA) is an approximation to the average distance traveled by a charged particle in a medium as it slows down to rest. Fluctuations in the energy loss are ignored. The CSDA range is calculated by integrating the reciprocal of the total stopping power with respect to energy,
$$R(E)=\int_{E_0}^E \left(-\frac{dE'}{dx} \right)^{-1} dE' $$
where \(E_0\) is sufficiently small for the result to be insensitive to its actual value.
For muons, the radiative stopping power at low energies is negligible. The choice of \(E_0\) is therefore dictated by the Bethe-Bloch equation, which is accurate for \(T\ge 10\) MeV, and somewhat accurate for \(T\gt 1\) MeV. In actual integrations we therefore use,
$$R(E)=R_0 + \int_{E_0}^E \left(-\frac{dE'}{dx} \right)^{-1} dE' $$
where \(E_0\) is now the energy limit at which the Bethe-Bloch equation can be considered sufficiently accurate, and \(R_0\) is the range for \(E \lt E_0\), estimated by other means.
Since the Bethe-Bloch has a very small projectile mass dependence, introduced by \(Q_{max}\), the stopping power depends only on projectile velocity. This means the stopping power can be scaled by projectile mass,
$$\left(-\frac{dE}{dx}\right)_{M_1,T_1} = \left(-\frac{dE}{dx}\right)_{M_2,T_2} $$
provided \(T_2=(M_2/M_1)T_1\).
To estimate \(R_0\) we use the tabulated electron range in the ICRU Report 37 [11] as implemented by ESTAR [12], and scale accordingly with projectile mass.
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References
- D. E. Groom, N. V. Mokhov, S. I. Striganov, "Muon Stopping Power and Range Tables 10 MeV-100 TeV", Atomic Data and Nuclear Data Tables 78, 183 (2001).
- R. M. Sternheimer, M. J. Berger, and S. M. Seltzer, "Density effect for the ionization loss of charged particles in various substances", Atomic Data and Nuclear Data Tables 30, 261 (1984).
- U. Fano, "Penetration of Protons, Alpha Particles, and Mesons", Ann. Rev. Nucl. Sci. 13, 1, (1963).
- R. Piessens, E. De Doncker-Kapenga and C. W. Überhuber. "QUADPACK: a subroutine package for automatic integration". Springer, ISBN: 3-540-12553-1. 1983.
- T. A. Armstrong et al., "Total Hadronic Cross Section of γ Rays in Hydrogen in the Energy Range 0.265-4.215 GeV", Phys. Rev. D5, 1640 (1972).
- T. A. Armstrong et al., "The total photon deuteron hadronic cross section in the energy range 0.265–4.215 GeV", Nucl. Phys. B41, 445 (1972).
- W. H. Bragg, R. Kleeman, "On the α particles of radium, and their loss of range in passing through various atoms and molecules". Philos. Mag. 10, 318, (1905).
- M. J. Berger, S. M. Seltzer, "Stopping powers and ranges of electrons and positrons", National Bureau of Standards, U.S. Department of Commerce, 82-2550-A, 1982.
- S. M. Seltzer, M. J. Berger, ""Evaluation of the collision stopping power of elements and compounds for electrons and positrons", Int. J. Appl. Radiat. and Isot., 33, 1189, 1982.
- R. M. Sternheimer, R. F. Peierls, "General Expression for the Density Effect for the Ionization Loss of Charged Particles", Phys. Rev. B 3, 3681, 1971.
- ICRU Report 37, "Stopping Powers for Electrons and Positrons", (ICRU, Washington DC, 1984.)
- NIST, "Stopping-Power & Range Tables for Electrons, Protons, and Helium Ions".
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