Photoabsorption Ionisation Model

Cross Section for Ionising Collisions

The Photoabsorption Ionisation (PAI) model describes the ionisation energy loss of a relativistic charged particle in matter. For such a particle, the differential cross section \(d\sigma_i/d\omega\) for ionising collisions with energy transfer \(\omega\) can be expressed most generally by the following equations [VSVAael82]:

(42)\[\frac{d\sigma_i}{d\omega} = \frac{2\pi Ze^4}{mv^2} \left\{ \frac{f(\omega)}{\omega\left|\varepsilon(\omega)\right|^2} \left[ \ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} - \frac{\varepsilon_1 - \beta^2\left|\varepsilon\right|^2}{\varepsilon_2} \arg(1-\beta^2\varepsilon^*) \right] + \frac{\tilde{F}(\omega)}{\omega^2} \right\} ,\]

where

\[\begin{split}\tilde{F}(\omega) &= \int_{0}^{\omega}\frac{f(\omega')} {\left|\varepsilon(\omega')\right|^2}d\omega' , \\ f(\omega) &= \frac{m\omega\varepsilon_2(\omega)}{2\pi^2ZN\hbar^2} .\end{split}\]

Here \(m\) and \(e\) are the electron mass and charge, \(\hbar\) is Planck’s constant, \(\beta = v/c\) is the ratio of the particle’s velocity \(v\) to the speed of light \(c\), \(Z\) is the effective atomic number, \(N\) is the number of atoms (or molecules) per unit volume, and \(\varepsilon = \varepsilon_1 + i\varepsilon_2\) is the complex dielectric constant of the medium. In an isotropic non-magnetic medium the dielectric constant can be expressed in terms of a complex index of refraction, \(n(\omega) = n_1 + in_2\), \(\varepsilon(\omega) = n^2(\omega)\). In the energy range above the first ionisation potential \(I_1\) for all cases of practical interest, and in particular for all gases, \(n_1 \sim 1\). Therefore the imaginary part of the dielectric constant can be expressed in terms of the photoabsorption cross section \(\sigma_{\gamma}(\omega)\):

\[\varepsilon_2(\omega) = 2n_1n_2 \sim 2n_2 = \frac{N\hbar c}{\omega} \sigma_{\gamma}(\omega) .\]

The real part of the dielectric constant is calculated in turn from the dispersion relation

\[\varepsilon_1(\omega) - 1 = \frac{2N\hbar c}{\pi}V.p.\int_{0}^{\infty} \frac{\sigma_{\gamma}(\omega')}{\omega'^2 - \omega^2}d\omega' ,\]

where the integral of the pole expression is considered in terms of the principal value. In practice it is convenient to calculate the contribution from the continuous part of the spectrum only. In this case the normalized photoabsorption cross section

\[\tilde{\sigma}_{\gamma}(\omega) = \frac{2\pi^2\hbar e^2Z}{mc} \sigma_{\gamma}(\omega) \left[ \int_{I_1}^{\omega_{max}}\sigma_{\gamma}(\omega')d\omega' \right]^{-1}, \ \omega_{max} \sim 100 \ \mbox{keV}\]

is used, which satisfies the quantum mechanical sum rule [FUJW68]:

\[\int_{I_1}^{\omega_{max}}\tilde{\sigma}_{\gamma}(\omega')d\omega' = \frac{2\pi^2\hbar e^2Z}{mc} .\]

The differential cross section for ionising collisions is expressed by the photoabsorption cross section in the continuous spectrum region:

\[\frac{d\sigma_i}{d\omega} = \frac{\alpha}{\pi\beta^2} \left\{ \frac{\tilde{\sigma}_{\gamma}(\omega)} {\omega\left|\varepsilon(\omega)\right|^2} \left[ \ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} - \frac{\varepsilon_1-\beta^2\left|\varepsilon\right|^2}{\varepsilon_2} \arg(1-\beta^2\varepsilon^*) \right] + \frac{1}{\omega^2}\int_{I_1}^{\omega}\frac{\tilde{\sigma}_{\gamma}(\omega')} {\left|\varepsilon(\omega')\right|^2}d\omega' \right\} ,\]

where

\[\begin{split}\varepsilon_2(\omega) &= \frac{N\hbar c}{\omega} \tilde{\sigma}_{\gamma}(\omega) , \\ \varepsilon_1(\omega) - 1 &= \frac{2N\hbar c}{\pi}V.p.\int_{I_1}^{\omega_{max}} \frac{\tilde{\sigma}_{\gamma}(\omega')}{\omega'^2 - \omega^2}d\omega' .\end{split}\]

For practical calculations using Eq.(42) it is convenient to represent the photoabsorption cross section as a polynomial in \(\omega^{-1}\) as was proposed in [BFR90]:

\[\sigma_{\gamma}(\omega) = \sum_{k=1}^{4}a_{k}^{(i)}\omega^{-k} ,\]

where the coefficients, \(a_{k}^{(i)}\) result from a separate least-squares fit to experimental data in each energy interval \(i\). As a rule the interval borders are equal to the corresponding photoabsorption edges. The dielectric constant can now be calculated analytically with elementary functions for all \(\omega\), except near the photoabsorption edges where there are breaks in the photoabsorption cross section and the integral for the real part is not defined in the sense of the principal value. The third term in Eq.(42), which can only be integrated numerically, results in a complex calculation of \(d\sigma_i/d\omega\). However, this term is dominant for energy transfers \(\omega > 10\) keV, where the function \(\left|\varepsilon(\omega)\right|^2 \sim 1\). This is clear from physical reasons, because the third term represents the Rutherford cross section on atomic electrons which can be considered as quasifree for a given energy transfer [AWWMJ80]. In addition, for high energy transfers, \(\varepsilon(\omega) = 1 - \omega_{p}^{2}/\omega^2 \sim 1\), where \(\omega_{p}\) is the plasma energy of the material. Therefore the factor \(\left|\varepsilon(\omega)\right|^{-2}\) can be removed from under the integral and the differential cross section of ionising collisions can be expressed as:

\[\frac{d\sigma_i}{d\omega} = \frac{\alpha} {\pi\beta^2\left|\varepsilon(\omega)\right|^2} \left\{ \frac{\tilde{\sigma}_{\gamma}(\omega)}{\omega} \left[ \ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} - - \frac{\varepsilon_1-\beta^2\left|\varepsilon\right|^2}{\varepsilon_2} \arg(1-\beta^2\varepsilon^*) \right] + \frac{1}{\omega^2}\int_{I_1}^{\omega}\tilde{\sigma}_{\gamma}(\omega')d\omega' \right\} .\]

This is especially simple in gases when \(\left|\varepsilon(\omega)\right|^{-2} \sim 1\) for all \(\omega > I_1\) [AWWMJ80].

Energy Loss Simulation

For a given track length the number of ionising collisions is simulated by a Poisson distribution whose mean is proportional to the total cross section of ionising collisions:

\[\sigma_i = \int_{I_1}^{\omega_{max}}\frac{d\sigma(\omega')}{d\omega'}d\omega' .\]

The energy transfer in each collision is simulated according to a distribution proportional to

\[\sigma_i(>\omega) = \int_{\omega}^{\omega_{max}} \frac{d\sigma(\omega')}{d\omega'}d\omega' .\]

The sum of the energy transfers is equal to the energy loss. PAI ionisation is implemented according to the model approach (class G4PAIModel) allowing a user to select specific models in different regions. Here is an example physics list:

const G4RegionStore* theRegionStore = G4RegionStore::GetInstance();
G4Region* gas = theRegionStore->GetRegion("VertexDetector");
...
if (particleName == "e-")
{
  ...
  G4eIonisation* eion = new G4eIonisation();
  G4PAIModel*     pai = new G4PAIModel(particle,"PAIModel");

  // here 0 is the highest priority in region 'gas'
  eion->AddEmModel(0,pai,pai,gas);
  ...
}
...

It shows how to select the G4PAIModel to be the preferred ionisation model for electrons in a G4Region named VertexDetector. The first argument in AddEmModel is 0 which means highest priority.

The class G4PAIPhotonModel generates both \(\delta\)-electrons and photons as secondaries and can be used for more detailed descriptions of ionisation space distribution around the particle trajectory.

Photoabsorption Cross Section at Low Energies

The photoabsorption cross section, \(\sigma_{\gamma}(\omega)\), where \(\omega\) is the photon energy, is used in Geant4 for the description of the photo-electric effect, X-ray transportation and ionisation effects in very thin absorbers. As mentioned in the discussion of photoabsorption ionisation (see Photoabsorption Ionisation Model), it is convenient to represent the cross section as a polynomial in \(\omega^{-1}\) [BFR90] :

\[\sigma_{\gamma}(\omega) = \sum_{k=1}^{4}a_{k}^{(i)}\omega^{-k}.\]

Using cross sections from the original Sandia data tables, calculations of primary ionisation and energy loss distributions produced by relativistic charged particles in gaseous detectors show clear disagreement with experimental data, especially for gas mixtures which include xenon. Therefore a special investigation was performed [VMAPeal94] by fitting the coefficients \(a_{k}^{(i)}\) to modern data from synchrotron radiation experiments in the energy range of 10-50 eV. The fits were performed for elements typically used in detector gas mixtures: hydrogen, fluorine, carbon, nitrogen and oxygen. Parameters for these elements were extracted from data on molecular gases such as N₂, O₂, CO₂, CH₄, and CF₄ [eal73][eal77]. Parameters for the noble gases were found using data given in the tables [MW76][WM80].

Bibliography

AWWMJ80(1,2)

Allison W.W.M. and Cobb J. Ann. Rev. Nucl. Part. Sci., 30:253, 1980.

BFR90(1,2)

Biggs F. and Lighthill R. Technical Report SAND 87-0070, Preprint Sandia Laboratory, 1990.

eal73

Lee L.C. et al. J.Q.S.R.T., 13:1023, 1973.

eal77

Lee L.C. et al. Journ. of Chem. Phys., 67:1237, 1977.

FUJW68

Fano U. and Cooper J.W. Rev. Mod. Phys., 40:441, 1968.

MW76

G.V. Marr and J.B. West. Absolute photoionization cross-section tables for helium, neon, argon, and krypton in the vuv spectral regions. Atom. Data Nucl. Data Tabl., 18:497–508, 1976.

VMAPeal94

Grichine V.M., Kostin A.P., and Kotelnikov S.K. et al. Bulletin of the Lebedev Institute, 1994.

VSVAael82

Asoskov V.S., Chechin V.A., and Grichine V.M. at el. Technical Report 140, Lebedev Institute annual report, 1982. p.3.

WM80

J.B. West and J. Morton. Atom. Data Nucl. Data Tabl., 30:253, 1980.