Mean free path of a suddenly created fast electron moving in a degenerate electron gas

A lower bound on the mean free path ($l$) of a swift electron moving in a degenerate electron gas is calculated by implementing a standard theoretical framework for the collision rate, $1/\tau$, with a scattering amplitude characterized by the matrix element of a hole-screened interaction potential taken between plane-wave states. The instantaneous hole around a system's electron is considered at the Hartree-Fock level for the ground-state wave function of the degenerate electron gas. The real transitions in the many-body system are considered by following Galitskii's treatment on an almost perfect Fermi gas of neutral atomic constituents. The analytical results show minima both in $\tau$ and $l$, and they appear at $(E/E_F)\simeq 4.6$ and $(E/E_F)\simeq 2.5$, respectively, where $E$ is the kinetic energy of the fast electron and $E_F$ is the Fermi energy of the host. Comparison with mean free path data obtained recently for Cu is made and a reasonable agreement is found.

Figure 1

The $G_{\tau }\left(x\right)$ (solid curve) and the $G_l\left(x\right)$ (dashed curve) functions in atomic units, for the range $x\in [1.1,10]$ of the dimensionless variable $x=E/E_F$. See the text for further details.

Figure 2

The inelastic mean free path $l\left(E\right)$ as function of the photoelectron energy $\Delta E\equiv (E-E_F)$. The curve with error bars shows the experimental[19] quantity. The dashed curve is based on the present theory. The dotted curve refers to a theoretical description that rests on an optical dielectric function for copper. See the text for further details.