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, eal02, eal00]. 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 [eal94, eal96, LS96]. 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
where the inelastic (in) contribution is calculated as follows
being
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
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
where \(\theta\) is calculated according to ref. [JM72], \(\Phi\) is defined as
and \(\)
represents the average charge of the projectile, which is
determined according to the parameterization given by Pierce and Blann
[PB68]:
The density correction \(\delta\) is described according to the formulation given by Sternheimer [SP71b].
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
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
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
where \(y_r\) is a function of the projectile velocity \(v\)
or
if the projectile velocity is lower than the target Fermi velocity.
\(\Lambda\) is the screening length defined as
and \(C_{1}\) is expressed in the form
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. (45) is obtained according to
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
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
where \(\chi\) is a correction obtained from the Lindhard and Soerensen theory [eal96, LS96] and \(< q >\) is calculated according to eq. (46).
Bibliography
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W.H. Barkas and M.J. Berger. Nasa report sp-3013. Technical Report, NASA, USA, 1964.
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C. Scheidenberger et al. Phys. Rev. Lett., 73(50):, 1994.
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C. Scheidenberger et al. Phys. Rev. Lett., 77(3987):, 1996.
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H. Geissel et al. Nucl. Instr. and Meth. B, 195(3):, 2002.
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H. Weick et al. Nucl. Instr. and Meth. B, 164-165(168):, 2000.
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H. Geissel and C. Scheidenberger. Nucl. Instr. and Meth. B, 136(114):, 1998.
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J.B. Biersack J.F. Ziegler and U. Littmark. The Stopping and Range of Ions in Solids. Pergamon Press, first edition, 1985.
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J.D. Jackson and R.L. McCarthy. Phys. Rev. B, 6(4131):, 1972.
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J. Lindhard and A.H. Soerensen. Phys. Rev. A, 53(2443):, 1996.
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T.E. Pierce and M. Blann. Phys. Rev., 173(390):, 1968.
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C. Scheidenberger and H. Geissel. Nucl. Instr. and Meth. B, 135(25):, 1998.
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R.M. Sternheimer and R.F. Peierls. Phys. Rev. B, 3(3681):, 1971.
- SP71b
R.M. Sternheimer and R.F. Peierls. General expression for the density effect for the ionization loss of charged particles. Physical Review B, 3(11):3681–3692, jun 1971. URL: https://doi.org/10.1103/PhysRevB.3.3681, doi:10.1103/physrevb.3.3681.