# p847ps4 - dimensions μ T k B T ln[1 e-μ T/k B T =...

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Physics 847: Problem Set 4 Due Thursday, April 29 at 11:59 P. M. Each problem is worth 10 pts., unless otherwise specifed. 1. Pathria, Problem 8.8. Silver, lead, and aluminum have 1, 4, and 3 conduction electrons per atom, respectively. 2. Pathria, Problem (8.18). Ignore the last sentence oF the problem. 3. (20 pts.) Free electron gas in two dimensions. (a). What is the relation between the electron density n and the ±ermi wave vector k F in two dimensions? (b). Defne the length r s in two dimensions as the radius oF a circle which contains one electron, so that πr 2 s = 1 /n . ±ind the relation between r s and k F in two dimensions. (c). Show that the density oF states For a Free electron gas is n ( E ) = C (1) For E > 0 and n ( E ) = 0 (2) For E < 0, and fnd the constant C. (This was almost done in problem 1 oF problem set 2.) (d). The electron density and the density oF states are related by n = i -∞ n ( E ) f ( E ) dE, (3) where f ( E ) = [ exp [( E μ ) /k B T ] + 1] - 1 . Show From this that in two

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Unformatted text preview: dimensions μ ( T ) + k B T ln[1 + e-μ ( T ) /k B T ] = μ (0) . (4) (e). Estimate From this result the amount by which μ ( T ) di²ers From μ (0) at a temperature k B T ≪ μ (0). 1 4. The specifc heat at constant volume satisfes the thermodynamic iden-tities C V = p ∂U ∂T P N,V = T p ∂S ∂T P N,V . (5) Show, using these identities, the Nernst postulate (S → 0 as T → 0), and the relations f ( E ) = 1 exp[ β ( E − μ )] + 1 U = V 4 π 3 i d 3 kǫ ( k ) f ( ǫ ( k )) , (6) , that S = − V 4 π 3 k B i d 3 k [ f ln f + (1 − f ) ln(1 − f )] . (7) Here ǫ ( k ) is the energy oF an electron in state k . 5. Calculate the Pauli paramagnetic susceptibility, in the low temperature limit, oF an ideal gas oF Fermions with intrinsic magnetic moment μ * and spin angular momentum J ¯ h , with J = 3 / 2. 2...
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p847ps4 - dimensions μ T k B T ln[1 e-μ T/k B T =...

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