Unformatted text preview: H = ¯ hω ( a † a + 1 / 2). c)  n i denotes the normalized eigenstate with energy E n = ¯ hω ( n + 1 / 2). Show that a †  n i = √ n + 1  n + 1 i and a  n i = √ n  n1 i . Problem 3: Lowtemperature speciﬁc heat in d dimensions and for nonlinear dispersion laws (AshcroftMermin problem 23.2, 15 points) Consider small lattice vibrations in a ddimensional crystal in harmonic approximation. a) For the Debye model, i.e. a linear dispersion ω = c  k  of all phonon modes, calculate the phonon density of states and show that it varies as ω d1 . What is the Debye frequency? b) Determine the phonon contribution to lowtemperature speciﬁc heat. c) Investigate what would happen for a nonlinear phonon dispersion ω ∼  k  ν (anomalous sound). Show that the lowtemperature speciﬁc heat would vanish as T d/ν in d dimensions....
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 Spring '11
 ThomasVojta
 Physics, Mass, Work, Fundamental physics concepts, Condensed matter physics, Phonon, lowtemperature specific heat, nonlinear phonon dispersion

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