Paramagnetism — Question 6

Question 6

Moment fluctuations control the differential paramagnetic response

A two-state moment has $\mu_z=\pm\mu$ and mean $\langle\mu_z\rangle=\mu\tanh x$, with $x=\mu B/(k_BT)$. The number density n and temperature are fixed.

1. Find the variance of the moment projection.

2. Differentiate M with respect to B and relate the result to the variance; find its limits at zero and very large field.

Question 6 - Solution

Moment fluctuations control the differential paramagnetic response

(a) Method and calculation

$$\langle\mu_z^2\rangle=\mu^2,\quad\operatorname{Var}(\mu_z)=\mu^2-\mu^2\tanh^2x.$$$$\boxed{\operatorname{Var}(\mu_z)=\mu^2\operatorname{sech}^2x}.$$

(b) Interpretation and further calculation

$$\boxed{\frac{\partial M}{\partial B}=\frac{n\mu^2}{k_BT}\operatorname{sech}^2x=\frac{n\operatorname{Var}(\mu_z)}{k_BT}}.$$$$B\to0:\ \partial_BM\to n\mu^2/(k_BT);\quad B\to\infty:\ \boxed{\partial_BM\to0}.$$

Physical check

A saturated sample has a large magnetization but a small differential response because its projection fluctuations are suppressed.