Phys560Notes-3 - Review Drude model (free electron...

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Review Drude model – (free electron approximation) + (independent electron approximation) + (Maxwell Boltzmann statistics) Sommerfeld model – (free electron approximation) + (independent electron approximation) + (Fermi Dirac statistics) Next: Band structure – (crystal potential) + (independent electron approximation) + (Fermi Dirac statistics) Crystal potential arises from the crystal structure made of atoms -- important for understanding the differences among metals, semiconductors, and insulators. Crystal Structure Crystal structure is evident from naturally formed facets of many mineral samples. Based on x-ray diffraction: Crystal structure = periodic, (b) infinite (a) repetition of structural units (c) (a) Infinite: 23 10 N  , bulk properties are independent of surfaces. (b) Periodic: translational symmetry Define Bravais lattice with lattice vector 11 2 2 33 nn n  Ra a a ; 1,2,3 n = integers 1,2,3 a primitive lattice vectors (noncoplanar); they generate the lattice. 2d example: (c) Structural unit: basis = a set of atoms associated with each lattice point Summarize : crystal structure = Bravais lattice + basis Theorem : A crystal looks the same from every Bravais lattice point. 1 TH T H (system is invariant under translation) a 2 a 1
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Theorem : Primitive vectors are not unique; there are ways of choosing them. 2d example: Definition: primitive unit cell = a region in space, when translated using all lattice vectors R , fills neatly the whole space (w/o overlapping regions or voids) 2d examples: 2 possible choices out of a 1 a 2 ' a 2 " a 2 3d example: (obvious choice) Theorem : volume of primitive cell  12 3 c V N  aaa ; V = volume of crystal, N = # of lattice points One unit cell contains one lattice point (on average). Definition: Wigner-Seitz unit cell = the region of space that is closer to a selected lattice point (origin O) than to any other lattice points; it is a primitive unit cell. Method of construction: 1. Draw lines from O to all nearby lattice points. 2. Draw bisectors (planes in 3-d). 3. Find the smallest volume enclosed.
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This note was uploaded on 03/21/2012 for the course PHYS 560 taught by Professor Flynn during the Spring '08 term at University of Illinois, Urbana Champaign.

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Phys560Notes-3 - Review Drude model (free electron...

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