Paramagnetism — Question 4

Question 4

Ideal adiabatic demagnetization preserves the spin entropy parameter

For N independent two-state moments, the spin entropy is $S=Nk_B[\ln(2\cosh x)-x\tanh x]$, where $x=\mu B/(k_BT)>0$. The sample is thermally isolated and the field is lowered reversibly. Neglect all non-spin heat capacity and interactions.

1. Show that constant entropy requires constant x, hence a relation between B and T.

2. If initially B=2.0 T and T=1.5 K, find T at B=0.20 T. Explain why the ideal formula alone cannot establish an attainable zero-temperature state.

Question 4 - Solution

Ideal adiabatic demagnetization preserves the spin entropy parameter

(a) Method and calculation

$$\frac{dS}{dx}=-Nk_Bx\operatorname{sech}^2x<0\quad(x>0).$$$$dS=0\Rightarrow dx=0\Rightarrow\boxed{B/T=\text{constant}}.$$

(b) Interpretation and further calculation

$$\boxed{T_f=T_iB_f/B_i=(1.5)(0.20/2.0)=0.150\,\mathrm K}.$$$$B\to0\Rightarrow T\to0\ \text{only within the ideal spin model}.$$

Physical check

Residual interactions and non-spin heat capacities invalidate the ideal extrapolation at sufficiently small fields.