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

Various NMR properties of solid H2 and D2 are studied, and the following results are obtained. The leading terms in the high-temperature expansion of the second moment M2(T) for H2 are M2(T)=M2(∞)+125/3 d2x(1−x) (βΓ)4 (1−2βΓ−415/64 βΓx), where M2(∞) is the Van Vleck term, Γ is the electric quadrupolequadrupole coupling constant, β≡1/kT, and x is the concentration of ( J=1) molecules. For H2, this expression fits the data qualitatively for T≥5°K. For D2, 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 H2, a reasonable fit to the fourth moment M4 is obtained by the relation M4(T)−M4(∞)≈16/3 M2(∞) [M2(T)−M2(∞)], which is derived by decoupling certain averages required in the otherwise rigorous moment calculation at high temperatures. The spin-lattice relaxation time T1 is calculated by extending the Gaussian approximation for the spectral functions to finite temperatures. The high-temperature result is T1=0.780 (Γ/hc0) x1/2(1−9/ 14 βΓ−1857/1792 βΓx)1/2 for H2, and T1=5.12 (Γ/hc0)×x1/2(1−9/14 βΓ−1857/1792 βΓx)1/2 for ( J=1) molecules in D2. At low concentrations we modify the results of Sung and find T1=2.53 x5/3Γ−1 for H2, and T1=18.7 x5/3Γ−1 for ( J=1) molecules in D2, if T1 is in seconds and Γ in cm−1. These formulas reproduce the concentration dependence of T1 in H2 and D2 very consistently over the entire concentration range x≥0.005. For a quantitative fit to experiment one must take Γ/Γ0 between 0.6 and 0.65 for both H2 and D2, values which are slightly smaller than obtained from other experiments. Here Γ0 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 T1 in the ordered phases of H2 and D2 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) D2 molecules in the ordered phase to be 8.8x kHz. This prediction has recently been confirmed by experiment. Disciplines Physics | Quantum Physics This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/419 PHYSICAL REVIEW B VOLUME 2, NUMB ER 9 NOVEMBER, 1970 Properties of Solid Hydrogen. II. Theory of Nuclear Magnetic Resonance and Relaxation* A. Brooks Harris' Dep«t~«t Of P~ys~~s, Univ«»~yof Pennsylvania, Philadelphia, Pennsylvania 19104 (Received 27 March 1970) Various NMR properties of solid H~ and D& are studied, and the following results are obtained. The leading terms in the high-temperature expansion of the second moment M&(T) for H2 are Mq(T) =Mq{ )+ 3 d x(1 —x) {PI') (1-2pI'-~«pI'x), where M2{ ) is the Van Vleck term, I' is the electric quadrupole-quadrupole coup]jng constaot, P =—1/kT, and x is the concentration of @=1)molecules. For H2, this expression fits the data qualitatively for T 5 K. For D2, 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) moleeules 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&, a reasonable fit to the fourth moment M4 is obtained by the relation M, (T)-M4( )=—,M, ( ) [M,(T)-M, ( )], »6 which is derived by decoupling certain averages required in the otherwise rigorous moment calculation at high temperatures. The spin-lattice relaxation time T» is calculated by extending the Gaussian approximation for the spectral functions to finite temperatures. The hightemperature result is T = 0. 780 (I /hco) x (1—+pI'&&+ pi'x) ~t for H2, and T& = 5. 12 (1"/hco) xr~~t(l-~&4pI'-~I&+ pl'x)~~t for (8=1) molecules in D&. At low concentrations we modify the results of Sung and find T»=2. 53 x5 3I' » for H2, and T» =18.7x5~3I' » for (8=1) molecules in D2, if T» is in seconds and I' in em . These formulas reproduce the concentration dependence of T» in H2 and D2 very consistently over the entire concentration range x ~ 0.005. For a quantitative fit to experiment one must take I'/I'0 between 0. 6 and 0.65 for both H2 and D2, values which are slightly smaller than obtained from other experiments. Here I'0 is the rigid-lattice value of I'. 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» in the ordered phases of H2 and D2 due to libron scattering, making use of the libron density of states calculated by Mertens gt al. At present the data are too scanty for a meaningful comparison with theory. Finally, we calculate the Pake splitting of (J'= 0) D2 molecules in the ordered phase to be 8.Sx kHz. This prediction has recently been. confirmed by experiment.