Ionization

Method

The class $G4MuIonisation$ provides the continuous energy loss due to ionization and simulates the 'discrete' part of the ionization, that is delta rays produced by muons. The approach described in Section 7.1 is used. The value of the maximum energy transferable to a free electron $T_{max}$ is given by the following relation:

$$T_{max} =\frac{2mc^2(\gamma^2 -1)}{1+2\gamma (m/M)+(m/M)^2 } .$$ (9.1)

Here $m$ is the electron mass and $M$ the muon mass. The method of calculation of the continuous energy loss and the total cross section are explained below.

Continuous Energy Loss

The integration of 7.1 leads to the Bethe-Bloch restricted energy loss formula [1] :

$$\left. \frac{dE}{dx} \right]_{T < T_{cut}} = 2 \pi r_e^2 mc... ...c{T_{up}}{T_{max}} \right) - \delta - \frac{2C_e}{Z} \right ]$$ (9.2)

where

$$\begin{array}{ll} r_e & \mbox{classical electron radius:} ... ...unction} \\ C_e & \mbox{shell correction function} \end{array}$$

In a single element the electron density is

$$n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A}$$

( $\mathcal{N}_{av}$: Avogadro number, $\rho$: density of the material, $A$: mass of a mole). In a compound material

$$n_{el} = \sum_i Z_i \: n_{ati} = \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} .$$

$w_i$ is the proportion by mass of the $i^{th}$ element, with molar mass $A_i$.

The mean excitation energy, $I$, for all elements is tabulated according to the ICRU recommended values [2].

Density Correction

$\delta$ is a correction term which takes into account the reduction in energy loss due to the so-called density effect. This becomes important at high energy because media have a tendency to become polarised as the incident particle velocity increases. As a consequence, the atoms in a medium can no longer be considered as isolated. To correct for this effect the formulation of Sternheimer [3] is used:
$x$ is a kinetic variable of the particle : $x = \log_{10}(\gamma \beta) = \ln(\gamma^{2} \beta^{2})/4.606$,
and $\delta(x)$ is defined by

$$\begin{array}{rll} \mbox{for} & x < x_0 : & \delta(x) = 0 \\... ...\ \mbox{for} & x > x_1 : & \delta(x) = 4.606 x - C \end{array}$$ (9.3)

where the matter-dependent constants are calculated as follows:

$$\begin{array}{lcl} h\nu_p & = & \mbox{ plasma energy of the ... ... = & 4.606(x_a - x_0)/(x_1 - x_0)^m \\ m & = & 3 . \end{array}$$ (9.4)

For condensed media

$$\begin{array}{ll} I < 100 \: \mbox{eV} & \left \{ \begin{arr... ... x_0 = 0.326 C - 1.5 & x_1 = 3 \end{array} \right . \end{array}$$

and for gaseous media

$$\begin{array}{rlll} \mbox{for} & C < 10. & x_0 = 1.6 & x_1 =... ...r} & C \geq 13.804 & x_0 = 0.326 C -2.5 & x_1 = 5 . \end{array}$$

Shell Correction

$2C_e/Z$ is the so-called shell correction term which accounts for the fact that, at low energies for light elements and at all energies for heavy ones, the probability of collision with the electrons of the inner atomic shells (K, L, etc.) is negligible. The semi-empirical formula used in GEANT4, applicable to all materials, is due to Barkas [4]:

$$C_e(I, \beta\gamma) = \frac{a(I)}{(\beta\gamma)^2} +\frac{b(I)}{(\beta\gamma)^4} +\frac{c(I)}{(\beta\gamma)^6} .$$ (9.5)

The functions a(I), b(I), c(I) can be found in the source code. This formula breaks down at low energies, and is valid only when $\beta\gamma > 0.13$ ($T > 7.9$ MeV for a proton). For $\beta\gamma \leq 0.13$ the shell correction term is calculated as:

$$\left . C_{e}(I,\beta\gamma) \rule{0mm}{5mm} \right \vert _{... ...ta\gamma=0.13)\frac{\ln(T/T_{2l})}{\ln(7.9 \: \rm MeV/T_{2l})}$$ (9.6)

i.e. the correction is switched off logarithmically from $T=7.9$ MeV to $T=T_{2l}=2$ MeV.

Parameterization

The mean energy loss can be described by the Bethe-Bloch formula (9.2) only if the projectile velocity is larger than that of the orbital electrons. In the low energy region this is not the case, and the parameterization from the ICRU'49 report [5] is used in the $G4BraggModel$ class. The Bethe-Bloch model is applied to muons of higher kinetic energies

$$T > 2 * M_{\mu}/M_{proton} MeV.$$ (9.7)

The details of the low energy parameterization are described in Section 11.10.

Total Cross Section per Atom and Mean Free Path

For $T \gg I$ the differential cross section can be written as [1]

$$\frac{d\sigma }{dT} = 2\pi r_e^2 mc^2 Z \frac{z_p^2}{\beta^2... ...t[ 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} \right] .$$ (9.8)

In GEANT4 $T_{cut} \geq 1$ keV. Integrating from $T_{cut}$ to $T_{max}$ gives the total cross-section per atom :

$$\sigma ( Z,E,T_{cut} ) = \frac {2\pi r_e^2 Z z_p^2}{\beta^2}mc^2 \times$$ (9.9)

$$\left[ \left( \frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) - \fr... ...T_{max}} \ln \frac{T_{max}}{T_{cut}} + \frac{T_{max} - T_{cut}}{2E^2} \right] .$$

In a given material the mean free path is

$$\begin{array}{lll} \lambda = (n_{at} \cdot \sigma)^{-1} & o... ...\left( \sum_i n_{ati} \cdot \sigma_i \right)^{-1} . \end{array}$$ (9.10)

The mean free path is tabulated during initialization as a function of the material and of the energy of the incident muon.

Simulating Delta-ray Production

A short overview of the sampling method is given in Chapter 2. Apart from the normalization, the cross section 9.8 can be factorized :

$$\frac{d\sigma}{dT}=f(T) g(T) with T \in [T_{cut}, \ T_{max}]$$ (9.11)

where

$$f(T) = \left(\frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) \frac{1}{T^2}$$ (9.12)

$$g(T) = 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} .$$ (9.13)

The energy $T$ is chosen by

1. sampling $T$ from $f(T)$
2. calculating the rejection function $g(T)$ and accepting the sampled $T$ with a probability of $g(T)$.

After successful sampling of the energy, the direction of the scattered electron is generated with respect to the direction of the incident muon. The azimuthal angle $\phi$ is generated isotropically. The polar angle $\theta$ is calculated from energy-momentum conservation. This information is used to calculate the energy and momentum of both scattered particles and to transform them into the global coordinate system.

Status of this document

09.10.98 created by L. Urbán.
14.12.01 revised by M.Maire
30.11.02 re-worded by D.H. Wright
01.12.03 revised by V. Ivanchenko

Bibliography

1. Particle Data Group. Rev. of Particle Properties. Eur. Phys. J. C15. (2000) 1. http://pdg.lbl.gov
2. ICRU Report No. 37 (1984)
3. R.M.Sternheimer. Phys.Rev. B3 (1971) 3681.
4. W.H. Barkas. Technical Report 10292,UCRL, August 1962.
5. ICRU (A. Allisy et al), Stopping Powers and Ranges for Protons and Alpha Particles, ICRU Report 49, 1993.