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8. Curie susceptibility: consider N non-interacting quantized spins in a magnetic field B = Bê, 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, Nu²s(s+1)B G(B) = G(0) - +0(B*).

8. Curie susceptibility: consider N non-interacting quantized spins in a magnetic field B = Bê, 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, Nu²s(s+1)B G(B) = G(0) - +0(B*).

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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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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