Fermi Energy of Metallic Lithium

Abstract

A boundary condition method is developed for deriving the coefficient $E_{2n}$ in the power series expansion of the energy of an electron of wave number $k$ moving in the lattice of an alkali metal. (The entire calculation proceeds within the framework of the Wigner-Seitz atomic sphere approximation.) If the electron wave function is expanded as ${\psi }_k\left(\mathbf{r}\right)=e^{i^{\mathbf{k}·\mathbf{r}}}(u_0+u_{1}k+u_2{k}^2+\cdots )$ it is shown that the boundary condition $\left[\right(\frac{\partial }{\partial r}\left)\right(s \mathrm{part}\mathrm{of} u_{2n}\left)\right]r=r_s=0\mathrm{}$ leads naturally to an evaluation of $E_{2n}$ in terms of values at $r_s$ of homogeneous solutions of the Schrödinger equation and their derivatives with respect to energy and radius. In this way, a simple expression for $E_4$ is obtained analogous to that derived by Bardeen for $E_2$. For the case of metallic lithium, this expression leads to the value $E_4=-0.031\mathrm{}$, which agrees with that obtained by the more tedious method of evaluating the expectation value of the Hamiltonian using a wave function correct to the second order in $k$.