Paramagnetism — Question 8

Question 8

A paramagnetic gas at constant pressure has an extra temperature factor

A dilute ideal gas of two-state moments has weak-field susceptibility $\chi=\mu_0n\mu^2/(k_BT)$ and equation of state $P=nk_BT$. Moment magnitude mu is temperature independent.

1. Find susceptibility as a function of P and T.

2. Compare the susceptibility ratios when temperature doubles at fixed pressure and at fixed number density. State the magnetization ratio at fixed applied field in each case.

Question 8 - Solution

A paramagnetic gas at constant pressure has an extra temperature factor

(a) Method and calculation

$$n=P/(k_BT)\Rightarrow\boxed{\chi(P,T)=\frac{\mu_0P\mu^2}{k_B^2T^2}}.$$$$P\text{ fixed},\ T_f=2T_i\Rightarrow\boxed{\chi_f/\chi_i=1/4}.$$

(b) Interpretation and further calculation

$$n\text{ fixed},\ T_f=2T_i\Rightarrow\boxed{\chi_f/\chi_i=1/2}.$$$$M=\chi H,\ H\text{ fixed}\Rightarrow\boxed{M_f/M_i=1/4\text{ or }1/2,\text{ respectively}}.$$

Physical check

At fixed pressure the density falls as temperature rises, adding another inverse-temperature factor to the volume susceptibility.