Density of States and Specific Heat

5 Specific Heat and Density of States The specific heat at constant volume is given by 

Since according to (2.33-25) there are N different values in the range 2Tr/a of the first Brillouin zone which are uniformly distributed, we have 

It is more convenient to transform (2.133b) into an integral over ui. We therefore introduce the frequency distribution g-(wj such that g. (o) is the 

There is a normalizing condition for gj(o)), namely, that the total number of modes of branch j is equal to N  

If the density of states or the dispersion relation is known, c (T) can, in principle, be calculated numerically from the relations (2.139-142) by using a computer. Even in simple cases such as the monoatomic or diatomic chain with nearest-neighbour interactions, for which the dispersion relations are given analytically by (2.44) and (2.14) or (2.36), respectively, the evaluation of the integrals must be performed numerically. Before introducing approximate distribution functions, we discuss the qualitative behaviour of g.(w) for the linear NaCl crystal [(2.36), Fig.2.4]. Since g-(w) is inversely proportional to dwj/dq and dwj/dq = 0 for the acoustic and optic modes at q = ir/a and for the optic mode at q = 0, we obtain singularities in g] (o)) at 0) = CD = (2f/m ) and in 92(w) at w = o) = (2f/m2) and at w = = (2f/y).  

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