Alternative models for impact ionisation by hadrons and PIXE
Early developments of proton and \(\alpha\) particle impact
ionisation cross sections in Geant4 are reviewed in a detailed paper
devoted to PIXE simulation with Geant4 [eal09].
This article also presents alternative developments for PIXE
simulation, their validation with respect to experimental data and the
first Geant4-based simulation involving PIXE in a concrete experimental
use case: the optimization of the graded shielding of the X-ray
detectors of the eROSITA [eal07] mission. The new
developments described in [eal09] are released in
Geant4 in the pii package (in source/processes/electromagnetic/pii).
The developments for PIXE simulation described in [eal09] provide
a variety of proton and \(\alpha\) particle cross sections for the
ionisation of K, L and M shells:
- theoretical calculations based on the ECPSSR
[BL81] model and its variants (with
Hartree-Slater corrections [Lap05], with the
“united atom” approximation [Cip07b] and
specialized for high energies [Lap08]),
- theoretical calculations based on plane wave Born approximation
(PWBA),
- empirical models based on fits to experimental data collected by Paul
and Sacher [PS89] (for protons, K shell),
Paul and Bolik [PB93] (for \(\alpha\), K
shell), Kahoul et al. [KND08]) (for protons, K,
shell), Miyagawa et al. [MNM88], Orlic et al.
[OST94] and Sow et al.
[SOLT93] for L shell.
The cross section models available in Geant4 are listed in Table 11.
Table 11 Cross Section Models in Geant4
Particle, shell |
Model |
Z range |
Protons, K shell |
|
ECPSSR |
6-92 |
ECPSSR High Energy |
6-92 |
ECPSSR Hartree-Slater |
6-92 |
ECPSSR United Atom |
6-92 |
ECPSSR reference [PS89] |
6-92 |
PWBA |
6-92 |
Paul and Sacher |
6-92 |
Kahoul et al. |
6-92 |
Protons, L shell |
|
ECPSSR |
6-92 |
ECPSSR United Atom |
6-92 |
PWBA |
6-92 |
Miyagawa et al. |
40-92 |
Orlic et al. |
43-92 |
Sow et al. |
43-92 |
Protons, M shell |
|
ECPSSR |
6-92 |
PWBA |
6-92 |
\(\alpha\), K shell |
|
ECPSSR |
6-92 |
ECPSSR Hartree-Slater |
6-92 |
ECPSSR reference [PB93] |
6-92 |
PWBA |
6-92 |
\(\alpha\), L and M shell |
|
ECPSSR |
6-92 |
PWBA |
6-92 |
The calculation of cross sections in the course of the simulation is
based on the interpolation of tabulated values, which are collected in a
data library. The tabulations corresponding to theoretical calculations
span the energy range between 10 keV and 10 GeV; empirical models are
tabulated consistently with the energy range of validity documented by
their authors, that corresponds to the range of the data used in the
empirical fits and varies along with the atomic number and sub-shell.
ECPSSR tabulations have been produced using the ISICS software
[LC96][Cip07a], 2006 version; an extended version,
kindly provided by ISICS author S. Cipolla
[Cip08], has been exploited to produce tabulations
associated with recent high energy modelling developments
[Lap08].
An example of the characteristics of different cross section models is
illustrated in Fig. 22. Fig. 23 shows various
cross section models for the ionisation of carbon K shell by proton,
compared to experimental data reported in
[PS89].
The implemented cross section models have been subject to rigorous
statistical analysis to evaluate their compatibility with experimental
measurements reported in [PS89],
[OSaSMT94], [SC84] and to compare
the relative accuracy of the various modelling options.
The validation process involved two stages: first goodness-of-fit
analysis based on the \(\chi^2\) test to evaluate the hypothesis of
compatibility with experimental data, then categorical analysis
exploiting contingency tables to determine whether the various modelling
options differ significantly in accuracy. Contingency tables were
analyzed with the \(\chi^2\) test and with Fisher’s exact test.
The complete set of validation results is documented in
[eal09]. Only the main ones are summarized here;
Geant4 users interested in detailed results, like the accuracy of
different cross section models for specific target elements, should
should refer to [eal09] for detailed information.
Regarding the K shell, the statistical analysis identified the ECPSSR
model with Hartree-Slater correction as the most accurate in the energy
range up to approximately 10 MeV; at higher energies the ECPSSR model in
its plain formulation or the empirical Paul and Sacher one (within its
range of applicability) exhibit the best performance. The scarceness of
high energy data prevents a definitive appraisal of the ECPSSR
specialization for high energies.
Regarding the L shell, the ECPSSR model with “united atom” approximation
exhibits the best accuracy among the various implemented models; its
compatibility with experimental measurements at 95% confidence level
ranges from approximately 90% of the test cases for the L\(_3\)
sub-shell to approximately 65% for the L\(_1\) sub-shell.
According to the results of the categorical analysis, the ECPSSR model
in its original formulation can be considered an equivalently accurate
alternative. The Orlic et al. model exhibits the worst accuracy with
respect to experimental data; its accuracy is significantly different
from the one of the ECPSSR model in the “united atom” variant.
The implementation of these models for the hadron impact ionisation process
is included in the G4hImpactIonisation
class, which is largely based on
the original G4hLowEnergyIonisation
process.
Bibliography
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