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 pellet contains N=2.5×1022 magnetic moments on fixed sites, each of magnitude μm=9.27×10−24JT−1, with two allowed orientations in a field B=1.5T: energy −μmB aligned and +μmB anti-aligned. Immediately after a sudden reversal of the field, only 35 per cent of the moments are aligned with the field and 65 per cent are anti-aligned. The field energy is excluded from U; the moments are weakly coupled to their surroundings. Recall from Chapter 11 that Ω=N!/(N↑!N↓!) and S=kBlnΩ, and that q quanta on Nosc oscillators can be arranged in Ω=(q+Nosc−1)!/[q!(Nosc−1)!] ways.
((a)) Find the entropy of the set of moments from the multiplicity with Stirling's approximation, per moment in units of kB and in total.
((b)) Find the temperature that fits these populations through the Boltzmann factor, and recompute the entropy from F=−NkBTlnZ1 to show the two routes agree. If the pellet is now placed in thermal contact with a copper block at 300K, which way does energy flow?
((c)) A crystal is modelled as an Einstein solid with θE=400K; Nosc=6.0×1022 of its oscillators share q=2.0×1022 quanta. Using U=qhν and 1/T=(∂S/∂U)N with the Chapter 11 count, find its temperature; then obtain the mean energy per oscillator from this chapter's partition function at that temperature and compare it with qhν/Nosc.