UBC Theses and Dissertations
Radiative three-nucleon reactions Davis, Ronald Stuart
This thesis seeks simplifications to the formulation of the radiative three-nucleon reactions (i.e. direct capture, and partial and total photodisintegration) by means of group theory and other methods, similar to the simplifications to the three-nucleon Schroedinger equation made by other workers (e.g. Derrick and Blatt (1958)). Some useful results are derived for the S₃ group, the group of permutations of three things. In addition to the usual projection operators (e.g. Eichmann (1963))> operators are formed analogous to the operators J², J₂, and J± of the SU₂ group, and new methods of defining and generating permutation eigenfunctions, which transform according to the irreducible representations of S₃ are developed. Also, properties of the Derrick-Blatt (1958) addition coefficients which simplify their use are demonstrated, expressions are derived for permutations of product functions, a simplification is given for integration of permuted functions, and a Wigner-Eckart theorem for S₃ is derived, A new body-fixed coordinate system is derived which greatly simplifies the formation and use of Euler-angle functions. Three internal coordinate systems, each advantageous for a different problem, are defined, and their interconversions given. For each of them, necessary trigonometric formulae and expressions for integration are given. The Derrick-Blatt (1958) expansion for three-nucleon wave functions in permutation eigenfunctions is modified to take advantage of the new body-fixed coordinates, and relations between functions in the new and old expansions are derived. Expansions of bound states are discussed, and the continuum states of deuteron plus free nucleon and of three free nucleons are expanded in the new basis functions. Previous, erroneous attempts at similar expansions are discussed. The electric dipole and quadrupole and the magnetic-dipole orbital and spin-flip components of the interaction Hamiltonian representing an emitted or absorbed gamma ray are expressed in permutation-eigenfunction operators, using the long-wavelength and first-order-perturbation approximations, whose validity is discussed. Exact, closed, general forms are derived for matrix elements of the Hamiltonian in Euler-angle and spin-isospin variables, and the internal integrations are considerably simplified. Exact expressions, involving integrals over the three internal variables, are thus derived for matrix elements of the overall Hamiltonian between completely general initial and final states. Expressions are given for the cross section in terms of the matrix elements, and further simplifications are given for some of the reactions involved. A few numerical results are given, which show that the cross section depends sensitively on the wave functions used. The application of this work to the study of nuclear structure and interactions is discussed. Three-quark states are also expressed in the new formalism.
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