Ashcroft Mermin eng · Problem 10

Superconductivity

Physics — Statistical Mechanics · paramagnet-specific-heat-adiabatic-demagnetization

Problem

Show that for an ideal paramagnet whose free energy has the form (31.52), the specific heat at constant field is simply related to the susceptibility by

c_H = T\left(\frac{\partial S}{\partial T}\right)_H = \frac{H^2 \chi}{T}, \tag{31.90}

or, in the regime where Curie's law is valid,

c_H = \frac{1}{3}\frac{N}{V}k_B J(J+1)\left(\frac{g\mu_B H}{k_B T}\right)^2. \tag{31.91}

Estimating the lattice vibrational contribution to the specific heat by Eq. (23.27), show that the lattice contribution drops below that of the spins at a temperature

T_0

of order

T_0 \approx \left(\frac{N}{N_i}\right)^{1/5}\left(\frac{g\mu_B H}{k_B \Theta_D}\right)^{2/5}\Theta_D. \tag{31.92}

(Here

N_i

is the total number of ions, and

N

is the number of paramagnetic ions.) What is a typical size for

g\mu_B H / k_B \Theta_D

in a field of

10^4

gauss?

Related theory

Heat Capacity Specific Heat

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