Dielectric function tools

New in version 0.7.

Number of effective electrons

New in version 0.7.

The Bethe f-sum rule gives rise to two definitions of the effective number (see [Egerton2011]):

n_{\mathrm{eff1}}\left(-\Im\left(\epsilon^{-1}\right)\right)=\frac{2\epsilon_{0}m_{0}}{\pi\hbar^{2}e^{2}n_{a}}\int_{0}^{E}E'\Im\left(\frac{-1}{\epsilon}\right)dE'

n_{\mathrm{eff2}}\left(\epsilon_{2}\right)=\frac{2\epsilon_{0}m_{0}}{\pi\hbar^{2}e^{2}n_{a}}\int_{0}^{E}E'\epsilon_{2}\left(E'\right)dE'

where n_a is the number of atoms (or molecules) per unit volume of the sample, \epsilon_0 is the vacuum permittivity, m_0 is the elecron mass and e is the electron charge.

The get_number_of_effective_electrons() method computes both.

Compute the electron energy-loss signal

New in version 0.7.

The get_electron_energy_loss_spectrum() “naively” computes the single-scattering electron-energy loss spectrum from the dielectric function given the zero-loss peak (or its integral) and the sample thickness using:

S\left(E\right)=\frac{2I_{0}t}{\pi
a_{0}m_{0}v^{2}}\ln\left[1+\left(\frac{\beta}{\theta(E)}\right)^{2}\right]\Im\left[\frac{-1}{\epsilon\left(E\right)}\right]

where I_0 is the zero-loss peak integral, t the sample thickness, \beta the collection semi-angle and \theta(E) the characteristic scattering angle.