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 μH0k T, where μH0 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 mI=0 and the mI=±1 states for a crystal of ortho-hydrogen in the ordered state. It is found that D depends on (3cos2β-1), where β 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 μH0k T, D, and the nuclear spin-lattice relaxation time T1. These are used to calculate the shape of the NMR spectrum of a powder sample for values of T1 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 T1D. 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 cm3/mole. By contrast, experiments show that at this molar volume the rate increases sharply with 1V. Our conclusion is that the increasing rate is due to a one-phonon process which is only effective for V less than about 22 cm3/mole.