Magnetism and Electrons — Question 10

Question 10

A thermal-energy comparison tests whether spin alignment can be strong

For a noninteracting electron spin take moment projections $\pm\mu_B$ in field B=1.00 T at T=300 K. Use $\mu_B=9.274\times10^{-24}$ J/T and $k_B=1.381\times10^{-23}$ J/K. Level populations obey Boltzmann weights.

1. Find the ratio of the Zeeman gap to thermal energy and the lower-to-upper population ratio.

2. Find the temperature at which the gap equals kBT for the same field; explain what this comparison does and does not imply about alignment.

Question 10 - Solution

A thermal-energy comparison tests whether spin alignment can be strong

(a) Method and calculation

$$\frac{\Delta E}{k_BT}=\frac{2\mu_BB}{k_BT}=\boxed{4.47695\times10^{-3}}.$$$$\boxed{N_{\mathrm{lower}}/N_{\mathrm{upper}}=e^{\Delta E/(k_BT)}=1.004487}.$$

(b) Interpretation and further calculation

$$\Delta E=k_BT_*\Rightarrow\boxed{T_*=2\mu_BB/k_B=1.34308\,\mathrm K}.$$$$T=T_*\Rightarrow N_{\mathrm{lower}}/N_{\mathrm{upper}}=e,\quad\boxed{P_{\mathrm{lower}}=e/(1+e)\simeq0.731}.$$

Physical check

An energy gap comparable to thermal energy gives a bias, not complete polarization; room-temperature populations here are nearly equal.