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midsemester_test - Physics 481 Condensed Matter Physics...

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Physics 481: Condensed Matter Physics - Midsemester test Friday, March 4, 2011 Problem 1: Structure determination (70 points) Debye-Scherrer X-ray diffraction is used to study a powder specimen of a monoatomic substance that is known to crystallize in a cubic Bravais lattice structure with primitive vectors ~a 1 = ( a, 0 , 0) , ~a 2 = (0 , a, 0) and ~a 3 = (0 , 0 , a ). The wavelength of the X-rays is 1.4 ˚ A. a) Find the primitive vectors of the reciprocal lattice. (15 points) b) Find the four shortest possible lengths of reciprocal vectors. (20 points) c) The first diffraction ring is at an angle of ϑ = 17 . 9 from the incident direction. Determine the lattice constant a . (20 points) d) Find the angles of the next three diffraction rings. (15 points) Problem 2: One-dimensional Morse solid (80 points points) Consider N identical atoms of mass M whose motion is restricted to the x -axis. Nearest neighbor atoms are coupled by the so-called Morse potential V M ( r ) = D 1 - e - α ( r - r 0 ) 2 - D where r is the distance between them and D , α , and r 0 are positive constants.
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