next up previous contents
Next: Exchange and correlation functions Up: Development of an electron Previous: Static properties of the   Contents

Dynamic properties of the uniform electron gas

We study electron excitations at finite temperature by giving the electron positions and sizes initial random velocities, and propagating the resulting dynamics at constant energy and volume. We use as a test case a NaCl lattice of 64 electrons with $ r_{s}$ = 2 bohr. At low temperatures, the electrons make small excursions about their equilibrium positions, but at higher temperatures they mix more freely (Figure 6.8).

Figure 6.8: Electron trajectories in a uniform electron gas ($ r_{s}$ = 2 bohr) at low and high temperature.

We can measure the heat capacity of the electron liquid by plotting the total energy as a function of temperature (T = 50 to 500 K, Figure 6.10). In a classical solid, the heat capacity at low temperatures is given by the Dulong and Petit expression $ C_{v} = 3 k_{B} N$, since there is an equipartition of energy among all the degrees of freedom in the solid. In metals, the heat capacity at temperatures below the Fermi temperature ($ T_{F}$ = 140,000 K for $ r_{s}$ = 2 bohr) scales as $ C_{v} \propto T/T_{F}$; the heat capacity is much lower than would be expected from a classical solid, because only states near the Fermi level can be excited [16] (Figure 6.9).

Figure 6.9: At low temperatures, the heat capacity of a metal goes to zero because only electrons near the Fermi level are excited.

The eFF electron gas has a heat capacity that matches classical, not quantum, statistics (Figure 6.10), suggesting that all of the available modes are being excited uniformly, a contention further supported by the spectrum of phonon excitations (Figure 6.11) derived by computing the Fourier transform of the velocity autocorrelation function [18].

Figure 6.10: eFF uniform electron gas has the heat capacity of a solid crystal, not a metal with Fermi-Dirac statistics. The heat capacity in this figure is given by the slope, since we are plotting total energy versus temperature.

Figure 6.11: Plasma oscillations are excited uniformly over the range of temperatures considered.

Since we are simulating electrons as classical particles interacting via effective potentials, it is not surprising that we reproduce classical and not quantum statistics. Other researchers who have created quasiclassical models of the uniform electron gas have reproduced the correct Fermi-Dirac distribution of momenta using momentum dependent potentials, which spread out the particles in momentum phase space. For example, in 1987, Dorso and Randrup [9] applied a Pauli potential of the form

$\displaystyle V(p, q) = V_{0} (\hbar/p_{0} q_{0})^{D} \exp(-p_{ij}^{2}/p_{0}^{2} + q_{ij}^{2}/q_{0}^{2})

to a periodic system of point particles with positions $ q$ and momenta $ p$, and found that a Metropolis simulation produced a proper Fermi-Dirac distribution of momenta. In their system, the repulsion is greater when two nearby particles have different momenta; investigation by Cordero and Hernandez [19] has shown that this kind of potential causes nearby particles to ``lock momenta'' and move collectively. In 1997, Ortner et al. [10] used Dorso's potential to simulate some dynamic properties of the uniform electron gas, including plasma oscillations.

In the future we will try adding a momentum-dependent repulsion function to eFF to reproduce Fermi-Dirac distributions of momenta. However, we will need to proceed carefully to ensure (1) that energy remains conserved, and that there is no energy loss via hysteresis effects and (2) that we still obtain correct dynamics when electrons are associated with nuclei.

next up previous contents
Next: Exchange and correlation functions Up: Development of an electron Previous: Static properties of the   Contents
Julius 2008-04-29