## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

11-1-1970

#### Publication Source

Physical Review B

#### Volume

2

#### Issue

9

#### Start Page

3495

#### Last Page

3525

#### DOI

10.1103/PhysRevB.2.3495

#### Abstract

Various NMR properties of solid H_{2} and D_{2} are studied, and the following results are obtained. The leading terms in the high-temperature expansion of the second moment M_{2}(T) for H_{2} are M_{2}(T)=M_{2}(∞)+125/3 d_{2}x(1−x) (βΓ)^{4} (1−2βΓ−415/64 βΓx), where M_{2}(∞) is the Van Vleck term, Γ is the electric quadrupole-quadrupole coupling constant, β≡1/kT, and x is the concentration of (J=1) molecules. For H_{2}, this expression fits the data qualitatively for T≥5°K. For 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) 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 H_{2}, a reasonable fit to the fourth moment M_{4} is obtained by the relation

M_{4}(T)−M_{4}(∞)≈16/3 M_{2}(∞) [M_{2}(T)−M_{2}(∞)],

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 (Γ/hc_{0}) x^{1/2}(1−9/14 βΓ−1857/1792 βΓx)^{1/2} for H_{2}, and T_{1}=5.12 (Γ/hc_{0})×x^{1/2}(1−9/14 βΓ−1857/1792 βΓx)^{1/2} for (J=1) molecules in D_{2}. At low concentrations we modify the results of Sung and find T_{1}=2.53 x^{5/3}Γ^{−1} for H_{2}, and T_{1}=18.7 x^{5/3}Γ^{−1} for (J=1) molecules in D_{2}, if T_{1} is in seconds and Γ in cm^{−1}. These formulas reproduce the concentration dependence of T_{1} in H_{2} and D_{2} very consistently over the entire concentration range x≥0.005. For a quantitative fit to experiment one must take Γ/Γ_{0} between 0.6 and 0.65 for both H_{2} and D_{2}, values which are slightly smaller than obtained from other experiments. Here Γ_{0} is the rigid-lattice value of Γ. 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 H_{2} and 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) D_{2} molecules in the ordered phase to be 8.8x kHz. This prediction has recently been confirmed by experiment.

#### Recommended Citation

Harris, A.
(1970).
Properties of Solid Hydrogen. II. Theory of Nuclear Magnetic Resonance and Relaxation.
*Physical Review B,*
*2*
(9),
3495-3525.
http://dx.doi.org/10.1103/PhysRevB.2.3495

**Date Posted:** 12 August 2015

This document has been peer reviewed.