Deceleration of electrons for the PAP

The following comes from J.-L. Pouchou and F. Pichoir, “Quantitative analysis of homogeneous or stratified microvolumes applying the model ‘PAP,’” in Electron probe quantitation, Springer, 1991, pp. 35–36.

Bethe Law (used for ZAF)(good for high energy ~ > 30keV):

Get bethe paper to confirm eq - from wiki:

Combinining the constants and adjusting units to get keV/cm:

Where:

$C\mathrm{\left[\frac{keV^2\cdot cm^{2}}{mol}\right]} = \frac{4\pi N_A\mathrm{[mol^{-1}]}(e\mathrm{[C]})^4} {32\pi^2 (\epsilon_0\mathrm{[C^2V^{-1}m^{-1}]})^2} \cdot \left(\frac{(100\mathrm{[cm/m])^2}} {(e\mathrm{[V/eV]})^2(1,000\mathrm{[eV/keV])^2}}\right)$ $=78454\mathrm{\left[\frac{keV^2\cdot cm^{2}}{mol}\right]}$

Which turns into:

$J_i$ is the mean ionization potential of each constituent

At low energies eq.\ref{bethe} produces lower results then experimental penetrations.

At very low energies it is also desirable to have the log term be able to change sign.

So we represent the energy term as $1/f(V)$ where $V=E/J)$:

$\frac{dE}{d \rho s} \mathrm{\left[\frac{keV \cdot cm^2}{g}\right]} = - \frac{M}{J}\cdot \frac{1}{f(V)} \tag{5}\label{eq5}$

Where:

and

The expression for $J_i$ was obtained from empirical fist of experiment see refs:

with: