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.

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