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8. Curie susceptibility: consider N non-interacting quantized spins in a magnetic field B = B2, and at a temperature T. The work done by the field is given by BM,, with a magnetization M, =µ, m,. For each spin, m, takes only the 2s+1 values -s, -s+ 1,...,s-1, s. (a) Calculate the Gibbs partition function 2Z(T, B). (Note that the ensemble corresponding to the macrostate (T, B) includes magnetic work.) (b) Calculate the Gibbs free energy G(T, B), and show that for small B, Nu?s(s+1)B G(B) = G(0) – +0(B*). 6k T (c) Calculate the zero field susceptibility x= aM,jƏB\Ro, and show that it satisfies Curie's law x= c/T. (d) Show that C- Cy = cB /T², where Cg and C are heat capacities at constant B and M, respectively.

8. Curie susceptibility: consider N non-interacting quantized spins in a magnetic field B = B2, and at a temperature T. The work done by the field is given by BM,, with a magnetization M, =µ, m,. For each spin, m, takes only the 2s+1 values -s, -s+ 1,...,s-1, s. (a) Calculate the Gibbs partition function 2Z(T, B). (Note that the ensemble corresponding to the macrostate (T, B) includes magnetic work.) (b) Calculate the Gibbs free energy G(T, B), and show that for small B, Nu?s(s+1)B G(B) = G(0) – +0(B*). 6k T (c) Calculate the zero field susceptibility x= aM,jƏB\Ro, and show that it satisfies Curie's law x= c/T. (d) Show that C- Cy = cB /T², where Cg and C are heat capacities at constant B and M, respectively.

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8. Curie susceptibility: consider N non-interacting quantized spins in a magnetic field B = B2, and at a temperature T. The work done by the field is given by BM,, with a magnetization M, = µE m,. For each spin, m, takes only the 2s+1 values -s, -s+ 1,...,s-1, s. (a) Calculate the Gibbs partition function 2(T, B). (Note that the ensemble corresponding to the macrostate (T, B) includes magnetic work.) (b) Calculate the Gibbs free energy G(T, B), and show that for small B, Nµ²s(s+1)B 6k T G(B) = G(0) – +0(B*). (c) Calculate the zero field susceptibility x = OM,/aB|-0, and show that it satisfies %3D B=0 Curie's law x = c/T. (d) Show that C,-CM =cB/T², where Cg and CM are heat capacities at constant B and M, respectively.
Transcribed Image Text:8. Curie susceptibility: consider N non-interacting quantized spins in a magnetic field B = B2, and at a temperature T. The work done by the field is given by BM,, with a magnetization M, = µE m,. For each spin, m, takes only the 2s+1 values -s, -s+ 1,...,s-1, s. (a) Calculate the Gibbs partition function 2(T, B). (Note that the ensemble corresponding to the macrostate (T, B) includes magnetic work.) (b) Calculate the Gibbs free energy G(T, B), and show that for small B, Nµ²s(s+1)B 6k T G(B) = G(0) – +0(B*). (c) Calculate the zero field susceptibility x = OM,/aB|-0, and show that it satisfies %3D B=0 Curie's law x = c/T. (d) Show that C,-CM =cB/T², where Cg and CM are heat capacities at constant B and M, respectively.
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