# Sets the zero of energy and may be taken to be

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sets the zero of energy and may be taken to be constant. Also, h is Planck’s constant and ( ) 1/ B k T β = . The function ( ) g ν is the density of frequencies, i.e. ( ) d g ν ν is the number of frequencies within [ ] , d ν ν ν + . Einstein assumed that all 3 N oscillators vibrate at the same frequency, E ν . This implies that ( ) ( ) 3 E g N ν δ ν ν = . Note that ( ) x δ is the standard Dirac delta function. (a) (14 pts.) Derive an expression for the entropy, ( , ) S S N T = , of a crystal of N atoms in the Einstein approximation. Express the answer in terms of the dimensionless parameter / E T Θ , where ( ) E E B h k ν Θ = is the Einstein temperature of the crystal. (b) (14 pts.) Derive the high- and low- T asymptotic forms of S . (c) (14 pts.) Prove that rd 0 as 0 (3 Law) . S T (d) (14 pts.) Derive an expression for the heat capacity, ( , ) C C N T = , of a crystal of N atoms in the Einstein approximation. Express the answer in terms of the dimensionless parameter / E T Θ . (e) (14 pts.) Derive the high- and low- T asymptotic forms of C . Prove that 0 as 0. C T 3 4 12 5 pm D T C R π = Θ ( ) 1 2 0 ( , ) ( ) ( ) ln 1 e . h o B A A N T U N k T d g v h β ν ν β ν = = + +

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- 2 - ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 2 3 2 3 2 2 3 2 3 4 1 1 1 1 1 1 2 3 4 1 1 1 1 2! 3! 1 1 1 ln 1 2 3 4 x dx x x dx x x f x f x x x x x x x x x e x x x x x x x x δ δ = = = + + + = + + + = + + + + + = + + " " " "
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