ATIMA energy-loss model

ATIMA (ATomic Interaction with MAtter) classes, G4AtimaEnergyLossModel and G4AtimaFluctuations, implemented in Geant4 since version 10.5 predicts the energy loss and energy-loss straggling of ions penetrating matter for kinetic energies ranging from 1 keV/u to 450 GeV/u. The model is developed at GSI Helmholtz Center for Heavy Ion Research GmbH since 1994. In the last two decades the model has been widely validated for ions using experimental data obtained from experiments carried out at the fragment separator FRS [GS98][SG98]. Basically, the model is based on the Bethe formula but including corrections from the theory developed by Lindhard and Soerensen [LS96], which make this model a powerful tool to predict the energy loss and energy-loss straggling of medium and heavy ions accelerated at relativistic energies [eal94][eal96][eal00][eal02]. This section is devoted to explain the main ingredients and equations of ATIMA model in Geant4.

Continuous energy loss

Above kinetic energies of 30 MeV/u, the stopping power is obtained from the theory developed by Lindhard and Soerensen (LS) [LS96] including the following corrections: the shell effects [BB64], a Barkas term [JM72][Lin76] and the Fermi-density effect [SP71a]. Nuclear size effect of projectiles, which are important for ions moving at high relativistic velocities, comes also from the LS theory. The mean charge of the projectiles is parametrized according to ref. [PB68]. For ions with medium and high atomic numbers the LS theory differs substantially from the Bethe formula because the later is based on the first-order Born approximation, while the LS theory is the exact solution for two-body free electron [LS96][eal94][eal96]. In addition, energy transfer in elastic collisions with the whole target atom is also included.

In ATIMA the continuous energy loss per unit of path length is calculated according to

(43)\[\frac{dE}{dx} = \left(\frac{dE}{dx}\right)_{in} + \left(\frac{dE}{dX}\right)_{elastic}\]

where the inelastic (in) contribution is calculated as follows

\[\left(\frac{dE}{dx}\right)_{in} = 2 \pi r_e^2 mc^2 n_{el} \frac{<q>^2 Z_{T}}{A_{T} \beta^2} \left( \left [ \ln \left( \frac{2mc^2 \beta^2 \gamma^2 } {I^2}\right) - \beta^2 - \frac{C}{Z_{T}} \right] B + LS - \delta/2 \right)\]

being

\[\begin{split}r_e &= \mbox{classical electron radius } = e^2/(4 \pi \epsilon_0 mc^2 ) \\ mc^2 &= \mbox{mass-energy of the electron} \\ n_{el} &= \mbox{electron density in the material} \\ I &= \mbox{mean excitation energy of the material} \\ Z_{T} &= \mbox{atomic number of the material} \\ <q> &= \mbox{average charge of the hadron or ion in units of the electron charge} \\ \gamma &= E/mc^2 \\ \beta^2 &= 1-(1/\gamma^2) \\ C &= \mbox{Shell correction function} \\ B &= \mbox{Barkas term} \\ LS &= \mbox{LS term including nuclear size effects for ions at relativistic velocities} \\ \delta &= \mbox{density effect function}\end{split}\]

The LS term accounts for nuclear size and scattering corrections to the Bethe formula [eal94]. The values of LS are interpolated between pre-calculated tables obtained by an analysis of partial waves each contributing with different phase shift [LS96]. These partial waves were calculated with a model developed by Soerensen and they were then summed up for the tables used in ATIMA.

The shell correction term C accounts for the fact that at projectile velocities comparable or even smaller than the orbital velocities of the bound target electrons the energy transfer is less effective. This correction is considered only at low energies \(\gamma \beta < 0.13\) and is expressed in the form

\[ \begin{align}\begin{aligned}C = 10^{-9} [(422.377\eta^{-2}+30.4043\eta^{-4}-0.38106\eta^{-6}) I^{2} +(3.858019\eta^{-2} -0.1667989\eta^{-4}\\ +0.00157955\eta^{-6}) I^{3} ]\end{aligned}\end{align} \]

where \(\eta = \gamma \beta\).

The Barkas correction term B accounts for close and distant collisions and is introduced as a polarisation effect. This term is parameterized in the form

\[B = 1 + 2 <q> \frac{\theta}{\sqrt(Z_{T})\Phi}\]

where \(\theta\) is calculated according to ref. [JM72], \(\Phi\) is defined as

\[\Phi = \frac{\gamma^2 \beta^2}{Z_{T}\alpha}\]

and \(\) represents the average charge of the projectile, which is determined according to the parameterization given by Pierce and Blann [PB68]:

(44)\[<q> = Z_{P}(1-e^{(-0.95 \frac{\beta}{\alpha Z_{P}})})\]

The density correction \(\delta \) is described according to the formulation given by Sternheimer [SP71].

Below 10 MeV/u ATIMA uses an older version of Ziegler’s SRIM code [JFZL85], in which the continuous energy loss per unit of path length is calculated according to

\[\left(\frac{dE}{dx}\right)_{in} = Se (\gamma_1 Z_{P})^2\]

where Se represents the stopping power per unit of path length of a proton passing through the same material. The effective charge \(\gamma_{1} \) is parameterized as

\[\gamma_1 = q_1 +0.5(1 - q_1)\left( \frac{v_0}{v_F} \right)^2 ln\left[ 1 + \left( \frac{2\Lambda v_F}{a_0 v_0} \right)^2 \right]C_1\]

where \(v_0 \) is the Bohr velocity, \(a_0 \) is the Bohr radius and \(v_F \) is the target Fermi velocity that depends on \(Z_{T} \). Here \(q_1 \) is defined according to

\[q_1 = 1 - exp( 0.803y_r^{0.3} -1.3167y_r^{0.6} - 0.38157y_r -0.008983y_r^{2} )\]

where \(y_r \) is a function of the projectile velocity \(v\)

\[y_r = v \left( 1 + \frac{v_F^2}{5v_1^2} \right)\]

or

\[y_r = \frac{0.75 v_F}{v_0 Z_{P}^{2/3}} \left( 1 + \frac{2v^2}{3v_F^2} - \frac{v^4}{15v_F^4} \right)\]

if the projectile velocity is lower than the target Fermi velocity.

\(\Lambda \) is the screening length defined as

\[\Lambda = \frac{2a_0(1-q_1)^{2/3}}{Z_{P}^{1/3}(1-(1-q_1)/7)}\]

and \(C_{1} \) is expressed in the form

\[C_1 = 1 + \frac{1}{Z_{P}^2} \left( 0.18 + 0.0015 Z_{T} \right) exp\left( -15.2 + 2ln(E_{P}) \right)\]

where \(E_{P} \) is the projectile kinetic energy in units of keV/u.

In the intermediate energy range \(10 < E_{P} < 30 MeV/u \) ATIMA interpolates between the two parameterizations.

Finally, the elastic contribution in eq. (43) is obtained according to

\[\left(\frac{dE}{dx}\right)_{elastic} = \frac{846.21 \cdot 10^{-23} Z_{P} Z_{T} A_{P} A_{v} \chi} {A_{T} A_{sum} Z_{pow} }\]

where \(Z_{P} \) is the atomic number of the projectile, \(A_{sum} = A_{T} + A_{P} \), \(Z_{pow} = Z_{T}^{0.23} + Z_{P}^{0.23} \) and

\[\begin{split}\begin{array}{rllll} \chi = & \frac{ln(\epsilon)}{2\epsilon} & : & \mbox{for} & \epsilon > 30 \\ \chi = & 0.5 \frac{ln(1+1.01323\epsilon)}{\epsilon + 0.01321\epsilon^{0.21226}+ 0.19593\epsilon^{0.5}} & : & \mbox{for} & \epsilon \leq 30 \end{array}\end{split}\]

where \(\epsilon \) is defined as: \(\epsilon = 32530 A_{T} A_{P} E_{P}/(Z_{T} Z_{P} A_{sum} Z_{pow}) \) with \(E_{P} \) in units of keV/u.

Fluctuations of energy loss

ATIMA also accounts for the determination of fluctuations of the energy lost by ions penetrating matter. Here, the energy-loss straggling comes also from the Lindhard and Soerensen theory [eal96]. The variance is defined a

\[\Omega^2 = 4 \pi N_a m_e c^2 r_e^2 \gamma^2 \chi <q>^2 \frac{Z_{T}}{A_{T}}\]

where \(\chi \) is a correction obtained from the Lindhard and Soerensen theory [LS96][eal96] and \(< q > \) is calculated according to eq. (44).

Bibliography

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