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Collision theory as applied to the calculation of a relaxation time Nielsen, Katherine Stephanie

Abstract

An expression for the spin-lattice relaxation time, T₁, of a dilute monatomic gas can be derived starting from the quantum-mechanical Boltzmann equation. The real difficulty in calculating the relaxation time for a particular system lies in the evaluation of the transition operator which appears in the expression for T₁ˉ¹. In this thesis, the relevant part of the transition operator, t₁, is estimated by a distorted-wave Born approximation (DWBA). The monatomic gas is approximated by a specific model. In this model the collisions described by t₁ are governed by two potentials: one, the isotropic rigid sphere potential, V₀, and the other, the anisotropic dipole-dipole nuclear spin interaction potential, V₁. The latter interaction describes the coupling between the degenerate nuclear spin states of the atoms and the translational degrees of freedom in the gas. The former (isotropic) potential governs the explicit form of the rigid sphere distorted wave. After the DWBA transition operator is substituted into the equation for the relaxation time, the expression for T₁ˉ¹ breaks up into two terms, the "diagonal" and "non-diagonal" contributions. At this stage the explicit expression for T₁ˉ is sufficiently complicated that, in order to finish the calculation, analytical approximations to the diagonal and non-diagonal terms are made. These approximations may be succinctly described by stating that they result in two separate evaluations, a linear and a quadratic one, for the overall relaxation time. The magnitude of a small parameter c² , which appears in the exponential term of T₁ˉ¹ , is used as the basis for neglecting certain contributions to the integrals which arise in estimating T₁ˉ¹. The linear and quadratic approximations yield numerical factors of 3,50 and 2.56 respectively, in the expression for the relaxation time. These values are to be compared with the factor of 2 obtained elsewhere.

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