Theory of Librational Motion in the α Phase of Nitrogen

Abstract

<p>Librational motion in the α phase of solid nitrogen is studied theoretically, assuming the crystal structure to belong to the space group Pa3 and ignoring the translational motions entirely. A general expansion is written down for an arbitrary nonsingular potential function of the positions of two nitrogen molecules, and methods are presented for obtaining the expansion coefficients for some particular potential model. The crystal Hamiltonian is written down and the mean field approximation is briefly discussed. The eigenstates of the mean-field Hamiltonian, the oblate spheroidal wave functions, are shown to correspond formally to the eigenstates of the two-dimensional isotropic harmonic oscillator, and this correspondence is exploited to define boson creation and annihilation operators for excitations of a single molecule. The crystal Hamiltonian is second-quantized in terms of these operators, and the bilinear terms which result are diagonalized by an RPA treatment which is an extension of one given by Raich and Etters. The uniqueness of the present diagonalization scheme is established and explicit expressions are developed for the three-operator terms in the Hamiltonian, which are essential for the later consideration of spin-lattice relaxation. Numerical calculations of the Raman frequencies show that the potential models proposed by Kohin and by Raich and Mills do not exhibit the correct anisotropic behaviour, the computed Raman frequencies being 57% to 103% too high in the first case and 28% to 53% too high in the second. Without taking account of libron-libron interactions, it is possible to obtain temperature-dependent Raman frequencies, but this dependence is negligibly weak. The temperature dependence of the nuclear quadrupole resonance frequency is not well reproduced by the theory, although a potential which is constructed to fit the Raman frequencies is more successful than an a priori potential. Using an approach due originally to Van Kranendonk and Walker, an a priori theory of quadrupolar spin-lattice relaxation is developed. The direct and first-order Raman processes, and a process involving a three-libron vertex, are shown not to contribute to relaxation. The anharmonic Raman process is considered, using first-order perturbation theory and the expressions for the three-operator terms. The relaxation time is found to be in fair agreement with experiment above about 10 K, and it is argued that a direct process involving phonons would be the dominant relaxation mechanism at low temperatures.</p>