In a magnetic-cooling stage a paramagnetic salt holds identical magnetic moments on…
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
In a magnetic-cooling stage a paramagnetic salt holds N identical magnetic moments on fixed lattice sites. Each moment has two allowed orientations in the applied field B, of energy −μmB when aligned with the field and +μmB when anti-aligned. The moments do not interact with one another, the field energy itself is not counted in U, and the whole sample is in equilibrium at temperature T.
((a)) Starting from the two orientation energies, write the single-moment partition function Z1, form ZN and the free energy F, and hence derive the total moment along the field, M=−(∂F/∂B)T. Check the saturation limit. Then evaluate M/Nμm and M for one mole of electron moments, μm=μB=9.27×10−24JT−1, at T=2.0K in B=3.0T.
((b)) The lattice of the same salt is modelled as an Einstein solid: each of its Na atoms vibrates in three directions with one frequency ν, one oscillator having levels 0,hν,2hν,… with the lowest set at zero. Derive CV from the single-oscillator partition function, state the high- and low-temperature limits, and evaluate CV/3NakB at 2.0K for a lattice with θE=hν/kB=150K.