The photoabsorption ionization model

An expression for the distribution of energy loss can be derived considering the energy loss as the sum of the energy transfers in the electromagnetic interactions between the particle and the atom. As the interaction is small (i.e. the energy transfer is small compared to the energy of the passing particle), Born approximation can be used in the perturbation theory. In the derivation, the atomic transition current is considered as a sum of the transition currents of its electrons.

$\epsilon$ is the complex dielectric constant of a medium which describes the electromagnetic properties of the medium and thus the effect of the field of an atom on the energy loss of the particle.

The complex dielectric constant can be written $\epsilon =\epsilon$₁+ i ε₂ where $\epsilon$₁ describes the polarization and $\epsilon$₂ the absorptive properties of the medium. $\epsilon$₂ can be expressed with the help of the oscillator strength function $f(k,\omega )$ which describes the coupling of the electrons to the field of the atom. $\epsilon$₂= {2 π²N e²m ω}f(k,ω)

m is the mass of the electron and N is the electron density. In a simplified model, the photoabsorption cross-section $\sigma$_γ(ω) can be used for description of $f(k,\omega )$ : $f(k,\omega )\; =\; \{mc2\pi 2e2Z\}\sigma$_γ(ω)

Z being the atomic number of the medium. The real part of $\epsilon$ can be expressed as an integral of the imaginary part according to the Kronig's and Kramers' dispersion relation[]:

$\epsilon$1- 1	$=$	$\{2\pi \}P\; \int$0&inf;{x ε2(x) dxx2- ω2}
	$=$	$\{2\; N\; c\pi Z\}P\; \int$0&inf;{σγ(x) dxx2- ω2}

where P indicates the Cauchy principal value. Using these assumptions for the interaction between the projectile and the atom, the following form is obtained for the collision cross-section []:

$\{d\sigma dE\}$	$=$	$-\; \{e2\pi 2c2n$a} Im[ ( {1β2ε}- 1 ) ln{2mv2E(1-β2ε)}]
		$+\; \{2\pi Ze4mv2E2\}\int$0&inf;{f(E')ε(E')2}dE'

where $n$_a is the number of atoms per $cm3$ , v velocity of the particle, and $\beta =v/c$ . The number of collisions per distance x and the energy E is then $\{N2dx\; dE\}=\; n$_a{dσdE}

For the simulation purposes, the number of primary collisions in a unit length with energy loss greater than a certain E is computed and tabulated for several values of the Lorentz factor. This is done by integrating

$(\; \{dNdx\})$>E

$=$

$\int$EEmaxna{dσdE}dE