Chapter 12: Solid state physics6312.3Crystal vibrations12.3.1A lattice with one type of atomsIn this model for crystal vibrations only nearest-neighbour interactions are taken into account. The force onatomswith massMcan then be written as:Fs=Md2usdt2=C(us+1-us) +C(us-1-us)Assuming that all solutions have the same time-dependenceexp(-iωt)this results in:-Mω2us=C(us+1+us-1-2us)Further it is postulated that:us±1=uexp(isKa) exp(±iKa).This gives:us= exp(iKsa). Substituting the later two equations in the fist results in a system of linearequations, which has only a solution if their determinant is 0. This gives:ω2=4CMsin2(12Ka)Only vibrations with a wavelength within the first Brillouin Zone have a physical significance. This requiresthat-π< Ka≤π.The group velocity of these vibrations is given by:vg=dωdK=Ca2Mcos(12Ka).and is 0 on the edge of a Brillouin Zone. Here, there is a standing wave.
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