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Phys560Notes-2

# Phys560Notes-2 - Sommerfeld Theory of Metals a quantum...

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Sommerfeld Theory of Metals – a quantum theory of independent free electrons Physical picture: Focus on one electron. Assume all other charges in the system are smeared out into a static neutral background (atomic details, fluctuations, & correlations ignored). All electrons are "equal." A simplified jellium model. Main difference from Drude model – Fermi Dirac statistics. Consider an electron gas in a cubic box of volume V = L 3 . Two commonly used boundary conditions: 1. Infinite well, = 0 outside V ; standing wave solutions; convenient for some problems. 2. Periodic boundary condition (PBC), or Born-von Karman BC:    ,, , , x Lyz xy Lz xyz L xyz   Wave function repeats. 1D example: We use PBC – convenient for mode counting. It is unphysical in 2D and 3D, but bulk properties are independent of V and boundary conditions as V  . Plane wave solution: 1 exp i V  k rk r k = wave vector; quantum number     i    kk k pr r kr  It is a momentum eigenfunction.     22 2 pk H mm  k k rr r k r 2 k m k (dispersion relation) PBC: exp exp xy z x y z ik x L ik y ik ik x ik y ik z        etc.   exp exp exp 1 ik L ik L ik L  2 kn L integers n   A dense grid of points in k space. Volume each k point 3 3 28 LV    Density of points in k space = 3 8 V 3 8 V d k k for V

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Adding spin,  1 exp i V     k rk r 3 , 2 8 s V d k k Ground state at T = 0 (minimum energy) based on Pauli exclusion principle: 22 0,0,0 ,0,0 2 2 0, ,0 0, ... g A LL L L         A = Antisymm operator g = a Slater determinant Total energy is a minimum occupied k values are enclosed in a sphere in k space (Fermi sphere; Fermi surface; radius = Fermi wave vector) 33 332 4 2 88 3 3 F F F F kk VV V Nd k k   k 3 2 3 F Nk n V 1 2 3 3 F kn ~ 1 Å -1 for typical metallic density. (Fermi energy) 2 F F k m ~ 1.5 – 15 eV, typically.
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Phys560Notes-2 - Sommerfeld Theory of Metals a quantum...

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