Magnetic Resonance — Question 5

Question 5

Thermal Zeeman polarization is governed by a hyperbolic tangent

Two equally degenerate spin levels have energies $\mp\Delta/2$. Their thermal populations are $N_l,N_u$. Define $p=(N_l-N_u)/(N_l+N_u)$.

1. Derive p as a function of the gap and temperature.

2. Find the high-temperature limit, then evaluate the exact p when $\Delta/(k_BT)=\ln3$.

Question 5 - Solution

Thermal Zeeman polarization is governed by a hyperbolic tangent

(a) Method and calculation

$$N_u/N_l=e^{-\Delta/(k_BT)},\quad p=\frac{1-e^{-\Delta/(k_BT)}}{1+e^{-\Delta/(k_BT)}}.$$$$\boxed{p=\tanh[\Delta/(2k_BT)]}.$$

(b) Interpretation and further calculation

$$\Delta\ll k_BT:\quad\boxed{p\simeq\Delta/(2k_BT)}.$$$$\Delta/(k_BT)=\ln3\Rightarrow\boxed{p=(1-1/3)/(1+1/3)=1/2}.$$

Physical check

The approximation applies only when the level gap is small relative to thermal energy. At positive temperature the lower level is more populated, so equilibrium does not give inversion.