A paramagnetic salt sits on a metal sample holder inside a magnet
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 paramagnetic salt sits on a metal sample holder inside a magnet. The salt carries N non-interacting electron moments on fixed sites, each with two allowed orientations of energy ∓μmB in the applied field B (aligned and anti-aligned); the field energy is not counted in U. Salt and holder are both at T=290K, and B=0.50T. Take μm=μB=9.27×10−24JT−1.
((a)) From the orientation energies and the partition function, derive the total moment M(B,T) along the field, then expand it for μmB≪kBT to obtain Curie's law. State the condition for the expansion and check that it holds here, then find the fractional alignment M/Nμm.
((b)) The holder is modelled as an Einstein solid of Na atoms, each vibrating in three directions with a single frequency ν, one oscillator having levels 0,hν,2hν,… with the lowest at zero. Derive CV from the single-oscillator partition function, give its high- and low-temperature limits, and evaluate CV/3NakB at 290K for θE=hν/kB=200K.