A crystal carries electron magnetic moments on fixed sites
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Electricity, Magnetism & Electromagnetism · Statistical foundations and the equilibrium ensembles
Thermal Physics, Ch Two-level systems, paramagnets and the Einstein solid
Problem
A crystal carries N=4.0×1022 electron magnetic moments on fixed sites, each of magnitude μm=9.27×10−24JT−1, with two allowed orientations in an applied field B=2.0T: energy −μmB aligned and +μmB anti-aligned. A measurement finds 70 per cent of the moments aligned with the field and 30 per cent anti-aligned. The moments are weakly coupled to the lattice, and the field energy is excluded from U. Recall from Chapter 11 that the multiplicity of such a set is Ω=N!/(N↑!N↓!), where N↑ and N↓ are the numbers aligned and anti-aligned, and that S=kBlnΩ; recall from Chapter 11 that q quanta shared among Nosc oscillators can be arranged in Ω=(q+Nosc−1)!/[q!(Nosc−1)!] ways.
((a)) Use the multiplicity with Stirling's approximation to find the entropy of the set of moments, per moment in units of kB and in total.
((b)) Find the temperature that fits these populations through the Boltzmann factor, then compute the entropy again from the free energy F=−NkBTlnZ1 of this chapter, and show the two routes agree. State the sign of T and say why.
((c)) A copper crystal is modelled as an Einstein solid with θE=240K; Nosc=3.0×1022 of its oscillators share q=7.5×1021 quanta. From the Chapter 11 count, with U=qhν and 1/T=(∂S/∂U)N, find the temperature of the crystal. Then compute the mean energy per oscillator from this chapter's partition function at that temperature and show it equals qhν/Nosc.