homework7 - Physics 481: Condensed Matter Physics - Test...

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Physics 481: Condensed Matter Physics - Test prep Homework 7 due date: Friday, March 5, 2011 Problem 1: Tetragonal lattice (15 points) A hypothetical monoatomic substance crystallizes in a centered tetragonal structure. The con- ventional unit cell can described by primitive vectors ( a, 0 , 0) , (0 ,a, 0) , (0 , 0 ,c ) with c = 3 a/ 2 and a basis consisting of two atoms at positions (0 , 0 , 0) and ( a/ 2 ,a/ 2 ,c/ 2). The lattice constant is a = 4 . 2 ˚ A. a) Calculate the maximum space filling for this lattice. b) Find the primitive vectors of the reciprocal lattice. c) A powder specimen of the substance is analyzed by X-ray diffraction using the Debye-Scherrer method. The wavelength of the X-rays is 1.5 ˚ A. Calculate the angles of the first four diffraction rings. Problem 2: Phonon dispersion in a chain with long-range interactions (Ashcroft-Mermin 22.1 , 15 points) Consider a one-dimensional chain of identical atoms of mass M . The springs are not only between nearest neighbors but between all pairs of atoms. Thus, the elastic energy reads
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