The Free Electron Gas

Abstract

The structure of a solid, its composition of atoms, as well as the electronic, optical and lattice properties are essentially determined by the electrons. However, not all electrons of the atoms constituting the solid are involved in the same way. At the beginning of Chap. 2 we have distinguished between the core electrons and the valence electrons. The former are tightly bound to the nuclei and extend over a distance (much) smaller than the lattice constant, while the wave functions of the latter overlap with those of the neighboring atoms, thus giving rise to the chemical binding. Therefore, we describe the solid as being composed out of ions (nuclei plus closed shell electrons) in equilibrium positions at R ⁰ _n and valence electrons, which are responsible for the stability of the solid and its electronic properties. They are ruled by the Hamiltonian

$$ \mathcal{H} = \sum\limits_{l = 1}^N {\frac{{p_l^2}}{{2m}} + \frac{1}{2}\sum\limits_{\mathop {k,l{\text{ }} = {\text{ }}1}\limits_{k \ne l} }^N {\frac{{{e^2}}}{{4\pi {\varepsilon _0}\left| {{r_k} - {r_l}} \right|}}} } + \sum\limits_{n,l} {v({r_l} - R_n^0) + \sum\limits_{mn} {V(R_n^0 - R_m^0)} } $$

((4.1))

.