A salt contains independent electron magnetic moments of magnitude on fixed sites

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids

Thermal Physics, Ch Two-level systems, paramagnets and the Einstein solid

Problem

A salt contains N=5.0×1023N = 5.0\times10^{23} independent electron magnetic moments of magnitude μm=9.27×10−24 J T−1\mu_{\rm m} = 9.27\times10^{-24}\,\mathrm{J\,T^{-1}} on fixed sites, each with two allowed orientations in the applied field. It is held at T=300 KT = 300\,\mathrm{K} in a field B=0.50 TB = 0.50\,\mathrm{T}. ((a)) Show that the weak-field condition μmB≪kBT\mu_{\rm m}B \ll k_BT holds, and use Curie's law to find the total moment MM along the field, in J T−1\mathrm{J\,T^{-1}}. ((b)) The field is now doubled to 1.00 T1.00\,\mathrm{T} and the temperature halved to 150 K150\,\mathrm{K}. By what factor does MM change, what is its new value, and is Curie's law still a good approximation? ((c)) In the same cryostat sits a block of a solid whose Einstein temperature is θE=240 K\theta_E = 240\,\mathrm{K}; at 300 K300\,\mathrm{K} its heat capacity is 0.95×3NkB0.95 \times 3Nk_B, on the shoulder of the Dulong-Petit plateau. A second solid has θE=480 K\theta_E = 480\,\mathrm{K}. At what temperature does the second solid reach the same fraction 0.950.95 of 3NkB3Nk_B? Justify the step you use.

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