Blundell Thermal Physics · Problem 20.5

The partition function

Physics — Statistical Mechanics · spin-half-paramagnet-thermodynamics

Problem

Show that for

N

non-interacting spin-

\frac{1}{2}

particles in a magnetic field

B

the energy

U

is given by

U = -N\mu_B B \tanh\left(\frac{\mu_B B}{k_B T}\right), \tag{20.52}

the heat capacity is given by

\frac{C}{Nk_B} = \left(\frac{\mu_B B}{k_B T}\right)^2 \mathrm{sech}^2\left(\frac{\mu_B B}{k_B T}\right), \tag{20.53}

and the entropy is given by

\frac{S}{Nk_B} = \ln\left[2\cosh\left(\frac{\mu_B B}{k_B T}\right)\right] - \frac{\mu_B B}{k_B T}\tanh\left(\frac{\mu_B B}{k_B T}\right). \tag{20.54}