Paramagnetism — Question 2

Question 2

Continuous orientations introduce a factor of one-third in the Curie response

Compare classical freely oriented moments of magnitude mu with a two-state projection model. For the classical model, set $u=\cos\theta$ and use $Z=2\pi\int_{-1}^{1}e^{xu}du$, where $x=\mu B/(k_BT)$. Assume weak magnetization, so B is approximately mu0H.

1. Derive the classical mean field-axis moment using the logarithmic derivative of Z.

2. Expand at small x and compare classical susceptibility with the two-state result from tanh x.

Question 2 - Solution

Continuous orientations introduce a factor of one-third in the Curie response

(a) Method and calculation

$$Z=4\pi\frac{\sinh x}{x},\quad\langle\mu_z\rangle=\mu\partial_x\ln Z.$$$$\boxed{\langle\mu_z\rangle=\mu(\coth x-1/x)}.$$

(b) Interpretation and further calculation

$$\coth x-1/x=x/3+O(x^3)\Rightarrow\boxed{\chi_{\mathrm{classical}}=\frac{\mu_0n\mu^2}{3k_BT}}.$$$$\tanh x=x+O(x^3)\Rightarrow\boxed{\chi_{\mathrm{two-state}}=\frac{\mu_0n\mu^2}{k_BT}=3\chi_{\mathrm{classical}}}.$$

Physical check

The two models have different allowed orientations; the factor of three is not a unit conversion or an error in the Boltzmann weights.