Paramagnetism — Question 1

Question 1

Two thermal populations produce a bounded paramagnetic magnetization

Independent moments have field-axis projections $+\mu$ and $-\mu$, energies $-\mu B$ and $+\mu B$, and number density n. Let $x=\mu B/(k_BT)$ and assign Boltzmann weights to the two states.

1. Normalize the populations and derive M as a function of x.

2. Find x and the population ratio when M is one-half of the saturation magnetization.

Question 1 - Solution

Two thermal populations produce a bounded paramagnetic magnetization

(a) Method and calculation

$$P_+=\frac{e^x}{e^x+e^{-x}},\quad P_-=\frac{e^{-x}}{e^x+e^{-x}}.$$$$\boxed{M=n\mu(P_+-P_-)=n\mu\tanh x}.$$

(b) Interpretation and further calculation

$$M/(n\mu)=1/2\Rightarrow\frac{e^{2x}-1}{e^{2x}+1}=\frac12\Rightarrow e^{2x}=3.$$$$\boxed{x=\tfrac12\ln3,\quad N_+/N_-=3}.$$

Physical check

Saturation is nmu, because it counts the largest allowed field-axis projection of every moment.