Energy Loss Fluctuations

The total continuous energy loss of charged particles is a stochastic quantity with a distribution described in terms of a straggling function. The straggling is partially taken into account in the simulation of energy loss by the production of \(\delta\)-electrons with energy \(T > T_{cut}\) ((30)). However, continuous energy loss ((29)) also has fluctuations. Hence in the current Geant4 implementation different models of fluctuations implementing the \(G4VEmFluctuationModel\) interface:

• G4UniversalFluctuation;

• G4BohrFluctuations;

• G4IonFluctuations;

• G4AtimaFluctuations;

• G4PAIModel;

• G4PAIPhotModel.

The first model is the default one used in main Physics List and will be described below. Other models have limited applicability and will be described in chapters for Hadron and Ion Ionisation, PAI (Photoabsorption Ionisation Model) and ATIMA (ATIMA energy-loss model) models.

Fluctuations in Thick Absorbers

The total continuous energy loss of charged particles is a stochastic quantity with a distribution described in terms of a straggling function. The straggling is partially taken into account in the simulation of energy loss by the production of \(\delta\)-electrons with energy \(T > T_c\). However, continuous energy loss also has fluctuations. Hence in the current Geant4 implementation two different models of fluctuations are applied depending on the value of the parameter \(\kappa\) which is the lower limit of the number of interactions of the particle in a step. The default value chosen is \(\kappa = 10\). In the case of a high range cut (i.e. energy loss without delta ray production) for thick absorbers the following condition should be fulfilled:

\[\Delta E > \kappa \ T_{max}\]

where \(\Delta E\) is the mean continuous energy loss in a track segment of length \(s\), and \(T_{max}\) is the maximum kinetic energy that can be transferred to the atomic electron. If this condition holds the fluctuation of the total (unrestricted) energy loss follows a Gaussian distribution. It is worth noting that this condition can be true only for heavy particles, because for electrons, \(T_{max}=T/2\), and for positrons, \(T_{max}=T\), where \(T\) is the kinetic energy of the particle. In order to simulate the fluctuation of the continuous (restricted) energy loss, the condition should be modified. After a study, the following conditions have been chosen:

(34)\[\Delta E > \kappa \ T_c\]

(35)\[T_{max} \leq 2 \ T_c\]

where \(T_c\) is the cut kinetic energy of \(\delta\)-electrons. For thick absorbers the straggling function approaches the Gaussian distribution with Bohr’s variance [eal93]:

(36)\[\Omega^2 = 2\pi r^2_e m_e c^2 N_{el}\frac{Z_h^2}{\beta^2} T_c s \left(1 - \frac{\beta^2}{2} \right),\]

where \(r_e\) is the classical electron radius, \(N_{el}\) is the electron density of the medium, \(Z_{h}\) is the charge of the incident particle in units of positron charge, and \(\beta\) is the relativistic velocity.

Fluctuations in Thin Absorbers

If the conditions (34) and (35) are not satisfied the model of energy fluctuations in thin absorbers is applied. The formulas used to compute the energy loss fluctuation (straggling) are based on a very simple physics model of the atom. It is assumed that the atoms have only two energy levels with binding energies \(E_1\) and \(E_2\). The particle-atom interaction can be an excitation with energy loss \(E_1\) or \(E_2\), or ionisation with energy loss distributed according to a function \(g(E) \sim 1/E^2\) :

(37)\[\int_{E_0}^{T_{up}} g(E)\ dE = 1 \Longrightarrow g(E) = \frac{E_0 T_{up}}{T_{up}-E_0} \frac{1}{E^2} .\]

The macroscopic cross section for excitation \((i=1,2)\) is

(38)\[\Sigma_i = C \frac{f_i}{E_i} \frac{\ln[2mc^2 \ (\beta\gamma)^2/E_i]-\beta^2} {\ln[2mc^2 \ (\beta\gamma)^2/I]-\beta^2}\ (1-r)\]

and the ionisation cross section is

(39)\[\Sigma_3 = C \frac{T_{up}-E_0} {E_0 T_{up}\ln(\frac{T_{up}}{E_0})}\ r\]

where \(E_0\) denotes the ionisation energy of the atom, \(I\) is the mean ionisation energy, \(T_{up}\) is the production threshold for delta ray production (or the maximum energy transfer if this value smaller than the production threshold), \(E_i\) and \(f_i\) are the energy levels and corresponding oscillator strengths of the atom, and \(C\) and \(r\) are model parameters.

The oscillator strengths \(f_i\) and energy levels \(E_i\) should satisfy the constraints

(40)\[f_1 + f_2 = 1\]

(41)\[f_1 \cdotp \ln E_1 + f_2 \cdotp \ln E_2 = \ln I .\]

The cross section formulas (38),(39) and the sum rule equations (40),(41) can be found e.g.in Ref.[Bic88]. The model parameter \(C\) can be defined in the following way. The numbers of collisions (\(n_i\), \(i=1,2\) for excitation and \(3\) for ionisation) follow the Poisson distribution with a mean value \(\langle n_i \rangle\). In a step of length \(\Delta x\) the mean number of collisions is given by

\[\langle n_i \rangle = \Delta x \ \Sigma_i\]

The mean energy loss in a step is the sum of the excitation and ionisation contributions and can be written as

\[\frac{dE}{dx} \cdotp \Delta x = \left \{ \Sigma_1 E_1 + \Sigma_2 E_2 + \int_{E_0}^{T_{up}} E g(E) dE \right \} \Delta x .\]

From this, using Eq. (38) - (41), one can see that

\[C = dE/dx .\]

The other parameters in the fluctuation model have been chosen in the following way. \(Z \cdotp f_1\) and \(Z \cdotp f_2\) represent in the model the number of loosely/tightly bound electrons

\[f_2 = 0 \mbox{ for } Z = 1\]

\[f_2 = 2/Z \mbox{ for } Z \geq 2\]

\[E_2 = 10 \mbox{ eV } Z^2\]

\[E_0 = 10 \mbox{ eV } .\]

Using these parameter values, \(E_2\) corresponds approximately to the K-shell energy of the atoms (and \(Z f_2 = 2 \) is the number of K-shell electrons). The parameters \(f_1\) and \(E_1\) can be obtained from Eqs.(40) and (41). The parameter \(r\) is the only variable in the model which can be tuned. This parameter determines the relative contribution of ionisation and excitation to the energy loss. Based on comparisons of simulated energy loss distributions to experimental data, its value has been fixed as

\[r = 0.55.\]

Width Correction Algorithm

This simple parametrization and sampling in the model give good values for the most probable energy loss in thin layers. The width of the energy loss distribution (Full Width at Half Maximum, FWHM) in most of the cases is too small. In order to get good FWHM values a relatively simple width correction algorithm has been applied. This algorithm rescales the energy levels \(E_1\), \(E_2\) and the number of excitations \(n_1\), \(n_2\) in such a way that the mean energy loss remains the same. Using this width correction scheme the model gives not only good most probable energy loss, but good FWHM value too.

Width correction algorithm is in the model since version 9.2. The updated version in the model (in version 9.4) causes an important change in the behaviour of the model: the results become much more stable, i.e. the results do not change practically when the cuts and/or the stepsizes are changing. Another important change: the (unphysical) second peak or shoulder in the energy loss distribution which can be seen in some cases (energy loss in thin gas layers) in older versions of the model disappeared. Limit of validity of the model for thin targets: the model gives good (reliable) energy loss distribution if the mean energy loss in the target is \(\ge (few \hspace{2 mm} times) * I_{exc}\), where \(I_{exc}\) is the mean excitation energy of the target material.

This simple model of energy loss fluctuations is rather fast and can be used for any thickness of material. This has been verified by performing many simulations and comparing the results with experimental data, such as those in Ref. [LPUrban95]. As the limit of validity of Landau’s theory is approached, the loss distribution approaches the Landau form smoothly.

Sampling of Energy Loss

If the mean energy loss and step are in the range of validity of the Gaussian approximation of the fluctuation (34) and (35), the Gaussian sampling is used to compute the actual energy loss (36). For smaller steps the energy loss is computed in the model under the assumption that the step length (or relative energy loss) is small and, in consequence, the cross section can be considered constant along the step. The loss due to the excitation is

\[\Delta E_{exc} = n_1 E_1 + n_2 E_2\]

where \(n_1\) and \(n_2\) are sampled from a Poisson distribution. The energy loss due to ionisation can be generated from the distribution \(g(E)\) by the inverse transformation method:

\[u = F(E) = \int_{E_0}^E g(x) dx\]

\[E = F^{-1}(u) = \frac {E_0}{1-u \frac{T_{up}-E_0}{T_{up}}}\]

where \(u\) is a uniformly distributed random number \(\in [0,\ 1]\). The contribution coming from the ionisation will then be

\[\Delta E_{ion} = \sum_{j = 1}^{n3} \frac {E_0} {1-u_j \frac{T_{up}-E_0}{T_{up}}}\]

where \(n_3\) is the number of ionisations sampled from the Poisson distribution. The total energy loss in a step will be \(\Delta E = \Delta E_{exc} + \Delta E_{ion}\) and the energy loss fluctuation comes from fluctuations in the number of collisions \(n_i\) and from the sampling of the ionisation loss.

Bibliography

Bic88

H. Bichsel. Rev. Mod. Phys., 60:663, 1988.

eal93

M.J. Berger et al. Report 49. Journal of the International Commission on Radiation Units and Measurements, os25(2):NP–NP, may 1993. ICRU Report 49. URL: https://doi.org/10.1093/jicru/os25.2.Report49, doi:10.1093/jicru/os25.2.report49.

LPUrban95

K. Lassila-Perini and L. Urbán. Nucl. Inst. Meth. A, 362:416, 1995.