Theory of Ortho-Para Conversion and its Effect on the NMR Spectrum of Ordered Solid Ortho-Hydrogen

The NMR spectrum of a system of nuclear spins in thermal equilibrium is directly proportional to $\frac{\ensuremath{\mu}{H}_{0}}{kT}$, where $\ensuremath{\mu}{H}_{0}$ is the difference in Zeeman energy between adjacent nuclear magnetic states and $T$ is the temperature. The metastable system, solid ortho-hydrogen, is far from thermal equilibrium because of the large rotational energy of the ($J=1$) molecules. Thus the populations of the three magnetic states of the ($I=1$) total-nuclear-spin wave functions are affected not only by the magnetic field and the temperature but also by the rate of ortho-para conversion from each of the three states. In this paper we calculate the difference $D$ between the ortho-para conversion rates from the ${m}_{I}=0$ and the ${m}_{I}=\ifmmode\pm\else\textpm\fi{}1$ states for a crystal of ortho-hydrogen in the ordered state. It is found that $D$ depends on ($3{cos}^{2}\ensuremath{\beta}\ensuremath{-}1$), where $\ensuremath{\beta}$ is the angle between the magnetic field and the symmetry axis of the molecular wave function. We then compute the steady-state populations of the nuclear-spin states as a function of $\frac{\ensuremath{\mu}{H}_{0}}{kT}$, $D$, and the nuclear spin-lattice relaxation time ${T}_{1}$. These are used to calculate the shape of the NMR spectrum of a powder sample for values of ${T}_{1}$ which are appropriate to the ordered state. The result is that the usual Pake line shape is distorted by an enhancement which is linear in frequency shift and proportional to ${T}_{1}D$. An expression is also derived for the average ortho-para conversion rate as a function of molar volume and the Debye energy which shows that the conversion rate, which we have calculated for the two-phonon process, is negligible below 20 ${\mathrm{cm}}^{3}$/mole. By contrast, experiments show that at this molar volume the rate increases sharply with $\frac{1}{V}$. Our conclusion is that the increasing rate is due to a one-phonon process which is only effective for $V$ less than about 22 ${\mathrm{cm}}^{3}$/mole.

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