To describe the inelastic scattering with the dielectric function, the
relation between the cross section and the mean free path
is required:
1/ is the inverse mean free path and N is the atomic density. To
get the total inverse mean free path
, the inverse mean
free paths of all specific events has to be summed up:
To obtain the probability distribution of the deflection angle or the
energy loss associated with a scattering process, one requires the
differential inverse mean free path. In terms of the dielectric
function, the differential cross section for inelastic scattering of
an electron with kinetic energy E, an energy loss of
and momentum transfer of
is given by:
The term is called
the function.
By integrating equation A.13 over both q and one
obtains the total inelastic mean free path for electrons with incident
energy E. Integrating over
or q leads to the
differential inelastic mean free path for energy loss and angular
deflection, respectively.
A method to compute the energy loss function by use of
i.e.
, extending the function
into the
plane, thus permitting the
calculation of the double differential inelastic mean free path using
optical dielectric data, has been proposed by Penn [Pen87] and
is applied in SESAME.