Properties of Solid Hydrogen. II. Theory of Nuclear Magnetic Resonance and Relaxation

Various NMR properties of solid ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ are studied, and the following results are obtained. The leading terms in the high-temperature expansion of the second moment $M_2\mathrm{}\left(T\right)\mathrm{}$ for ${\mathrm{H}}_2$ are $M_2\mathrm{}\left(T\right)=M_2\left(\infty \right)+\frac{125}{3}{d}^{2}x(1-x)\mathrm{} {\left(\beta \Gamma \right)}^4 (1-2\mathrm{}\beta \Gamma -\frac{415}{64}\beta \Gamma x)$, where $M_2\left(\infty \right)$ is the Van Vleck term, $\Gamma$ is the electric quadrupole-quadrupole coupling constant, $\beta \equiv \frac{1}{\mathrm{kT}}$, and $x$ is the concentration of ($J=1\mathrm{}$) molecules. For ${\mathrm{H}}_2$, this expression fits the data qualitatively for $T\ge 5$°K. For ${\mathrm{D}}_2$, the observed second moment agrees with our calculations only for very small or very large values of $x$. For intermediate values of $x$, the observed second moment is much smaller than expected, which leads us to propose that the resonance of the ($J=1\mathrm{}$) molecules is too broad to be observable. Under this assumption, we find a temperature-dependent contribution at 5°K about 100 times smaller than that given above, in rough agreement with experiment. For ${\mathrm{H}}_2$, a reasonable fit to the fourth moment $M_4$ is obtained by the relation $M_4\mathrm{}\left(T\right)-M_4\left(\infty \right)\approx \frac{16}{3}{M}_2\left(\infty \right) \left[M_2\mathrm{}\right(T)-M_2(\infty \left)\right],$ which is derived by decoupling certain averages required in the otherwise rigorous moment calculation at high temperatures. The spin-lattice relaxation time $T_1$ is calculated by extending the Gaussian approximation for the spectral functions to finite temperatures. The high-temperature result is $T_1=0.780\mathrm{} \left(\frac{\Gamma }{h{c}_0}\right) x^{\frac{1}{2}}{(1-\frac{9}{14}\beta \Gamma -\frac{1857}{1792}\beta \Gamma x)}^{\frac{1}{2}}$ for $H_2$, and $T_1=5.12\mathrm{} \left(\frac{\Gamma }{h{c}_0}\right)\times x^{\frac{1}{2}}{(1-\frac{9}{14}\beta \Gamma -\frac{1857}{1792}\beta \Gamma x)}^{\frac{1}{2}}$ for ($J=1\mathrm{}$) molecules in $D_2$. At low concentrations we modify the results of Sung and find $T_1=2.53\mathrm{} x^{\frac{5}{3}}{\Gamma }^{-1\mathrm{}}$ for ${\mathrm{H}}_2$, and $T_1=18.7\mathrm{} x^{\frac{5}{3}}{\Gamma }^{-1\mathrm{}}$ for ($J=1\mathrm{}$) molecules in ${\mathrm{D}}_2$, if $T_1$ is in seconds and $\Gamma$ in ${\mathrm{cm}}^{-1\mathrm{}}$. These formulas reproduce the concentration dependence of $T_1$ in ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ very consistently over the entire concentration range $x\ge 0.005$. For a quantitative fit to experiment one must take $\frac{\Gamma }{{\Gamma }_0}$ between 0.6 and 0.65 for both ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$, values which are slightly smaller than obtained from other experiments. Here ${\Gamma }_0$ is the rigid-lattice value of $\Gamma$. Both the resonance and the relaxation data tend to confirm that in the solid all interactions must be renormalized to take account of lattice vibrations. We also obtain explicit analytic results for $T_1$ in the ordered phases of ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ due to libron scattering, making use of the libron density of states calculated by Mertens et al. At present the data are too scanty for a meaningful comparison with theory. Finally, we calculate the Pake splitting of ($J=0\mathrm{}$) ${\mathrm{D}}_2$ molecules in the ordered phase to be $8.8x$ kHz. This prediction has recently been confirmed by experiment.