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phys documents (dragged) 69 - Chapter 12: Solid state...

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Unformatted text preview: Chapter 12: Solid state physics 63 12.3 Crystal vibrations 12.3.1 A lattice with one type of atoms In this model for crystal vibrations only nearest-neighbour interactions are taken into account. The force on atom s with mass M can then be written as: F s = M d 2 u s dt 2 = C ( u s +1- u s ) + C ( u s- 1- u s ) Assuming that all solutions have the same time-dependence exp(- i t ) this results in:- M 2 u s = C ( u s +1 + u s- 1- 2 u s ) Further it is postulated that: u s 1 = u exp( isKa ) exp( iKa ) . This gives: u s = exp( iKsa ) . Substituting the later two equations in the st results in a system of linear equations, which has only a solution if their determinant is 0. This gives: 2 = 4 C M sin 2 ( 1 2 Ka ) Only vibrations with a wavelength within the rst Brillouin Zone have a physical signicance. This requires that- < Ka . The group velocity of these vibrations is given by: v g = d dK = Ca 2 M cos( 1 2 Ka ) ....
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