Paramagnetism — Question 3

Question 3

A saturation bound diagnoses failure of a Curie extrapolation

A two-state paramagnet has $M=n\mu\tanh x$, with $x=\mu B/(k_BT)$. The linear Curie approximation would give $M_{\mathrm{lin}}=n\mu x$.

1. At x=2 compare the exact and linear magnetizations as fractions of saturation and find the relative overestimate.

2. Use $\tanh x\simeq x-x^3/3$ to estimate a small-x criterion for a linear overestimate below 1 percent.

Question 3 - Solution

A saturation bound diagnoses failure of a Curie extrapolation

(a) Method and calculation

$$M/(n\mu)=\tanh2=0.964028,\quad M_{\mathrm{lin}}/(n\mu)=2.$$$$\boxed{\frac{M_{\mathrm{lin}}-M}{M}=\frac{2}{\tanh2}-1=1.07463}\quad(107.46\%).$$

(b) Interpretation and further calculation

$$\frac{x-\tanh x}{\tanh x}\simeq\frac{x^2}{3}\quad(x\ll1).$$$$\boxed{x\lesssim\sqrt{0.03}=0.173}\quad\text{as a leading-order estimate}.$$

Physical check

An extrapolation exceeding nmu violates the microscopic saturation bound even before a detailed error calculation.