Free Electron Fermi Gas
Phys463.nb 33 6 Free Electron Fermi Gas 6.1. Electrons in a metal 6.1.1. Electrons in one atom One electron in an atom (a hydrogen-like atom): the nucleon has charge +Z e, where Z is the atomic number, and there is one electron moving around this nucleon Four quantum number: n, l and lz, sz. Energy levels En with n = 1, 2, 3 ... Z2 m e4 1 En = - (6.1) 2 2 2 2 32 p e0 Ñ n where m = me mN me + mN » me where me is the mass of an electron and mN is the mass of the nucleon. 13.6 eV Z2 En = - H L (6.2) n2 H L For each n, the angular momentum quantum number [L2 y = l l + 1 y] can take the values of l = 0, 1, 2, 3, 4 … , n - 1. These states are known as the s, p, d, f , g,… states For each l, the quantum number for Lz can be any integer betweenH -Ll and +l lz = -l, -l + 1,…, l - 1, l . For fixed n, l and lz, the spin quantum number sz can be +1/2 or -1/2 (up or down). H L At each n, there are 2 n2 quantum states. In a real atom In a hydrogen-like atom, for the same n, all the 2 n2 quantum number has the same energy. In a real atom, the energy level splits according to the total angular momentum quantum number l. Typically, the state with lower l has lower energy (not always true).
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