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Prob.set7 - Physics 361 Problem Set 7 due P7.1 Anisotropic...

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Physics 361 Problem Set 7 due November 25, 2009 P7.1 Anisotropic mass (Cf. Problem 12.1) As we have seen, the maxima and minima of an energy band need not be isotropic. Thus the general form of the energy near a minimum is ε ( k ) = constant + ¯ h 2 2 ( k - k 0 ) · ˆ M - 1 ( k - k 0 ) , where ˆ M is a 3 × 3 (symmetric) matrix called the mass matrix, and k 0 is the position of the minimum. If k 0 is an energy maximum, the second term is negative. For some metals and most semiconductors all the occupied or empty states are close enough to maxima or minima that ε ( k ) is well described by this mass-matrix formula. a) For such a material find the density of states g ( ε ) and find the specific heat in terms of ˆ M . That is, find the specific heat effective mass m * of p. 48in terms of ˆ M . b) For such materials the crystal momentum k is proportional to the velocity v in the sense v = ˆ A k for some matrix ˆ A . Using Hamilton’s equations ( i.e., the semiclassical formalism of Chapter 12), find the proportionality matrix ˆ A in terms of ˆ M .
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