Two-state paramagnet: the Purcell & Pound experiment with lithium

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.21.

As another example of the formulas we obtained for the two-state paramagnet we can look at the 1951 experiment by Purcell and Pound described in Schroeder on page 102. The dipoles here are provided by lithium nuclei, which is a real-life paramagnet with four spin states, although for the purposes of this problem, we’ll pretend it has only two states. The magnetization is given by

$\displaystyle M=N\mu\tanh\frac{\mu B}{kT} \ \ \ \ \ (1)$

The values in the experiment are

 $\displaystyle \mu$ $\displaystyle =$ $\displaystyle 5\times10^{-8}\mbox{ eV T}^{-1}=8.01\times10^{-27}\mbox{ J T}^{-1}\ \ \ \ \ (2)$ $\displaystyle B$ $\displaystyle =$ $\displaystyle 0.63\mbox{ T}\ \ \ \ \ (3)$ $\displaystyle T$ $\displaystyle =$ $\displaystyle 300\mbox{ K} \ \ \ \ \ (4)$

The magnetization per particle is

 $\displaystyle \frac{M}{N}$ $\displaystyle =$ $\displaystyle \left(8.01\times10^{-27}\right)\tanh\frac{\left(8.01\times10^{-27}\right)\left(0.63\right)}{\left(1.38\times10^{-23}\right)\left(300\right)}\ \ \ \ \ (5)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 9.76\times10^{-33}\mbox{ J T}^{-1} \ \ \ \ \ (6)$

The energy difference between the parallel and antiparallel dipole alignments is ${\Delta U=2\mu B}$, so in this experiment, the energy of a photon required to perform a flip is

 $\displaystyle E$ $\displaystyle =$ $\displaystyle 2\mu B\ \ \ \ \ (7)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 10^{-26}\mbox{ J} \ \ \ \ \ (8)$

This corresponds to a wavelength which can be calculated from Planck’s formula

 $\displaystyle E$ $\displaystyle =$ $\displaystyle h\nu=\frac{hc}{\lambda}\ \ \ \ \ (9)$ $\displaystyle \lambda$ $\displaystyle =$ $\displaystyle \frac{hc}{E}\ \ \ \ \ (10)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\left(6.626\times10^{-34}\right)\left(3\times10^{8}\right)}{\left(10^{-26}\right)}\ \ \ \ \ (11)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 19.9\mbox{ m} \ \ \ \ \ (12)$

With a wavelength of around 20 metres, the photon is in the radio wave region of the spectrum.