Phys560Notes-12 - Lattice Specific Heat Classical:...

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Lattice Specific Heat Classical: Equi-partition theorem: 22 11 1 2 xx kx m v kT  6 degrees of freedom each atom – 3 each from kinetic energy and potential energy. Internal energy/volume 3 kT u V  N ; N = total number of atoms (including basis) Volume specific heat 3 V Ck V N Dulong-Petit law Well obeyed by solids at high T (~RT and above), but not at low T where the classical model fails. (Ask yourself – why?) Quantum: i E i Ze partition function State index   s in k specified by phonon numbers in each mode   0 exp exp 1 exp 1 2 exp 2 1e x p s s s ss ns n s s sn s s Zn n n             k k k kk k k k k  111 2 ks s s un Z n VV k k  , where 1 1 s s n e k k s n k = average number of phonons in mode s k (Bose-Einstein distribution with = 0) 1 1 s s Vs u Cg d g d TVT T e T e e  
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Phys560Notes-12 - Lattice Specific Heat Classical:...

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