Some problems in the theory of nuclear structure being a study of weak collective effects in the nuclear shell model

Eigen-states generated by considering particles moving independently in a spherically symmetric potential well are, according to the shell modal of Mayer and Jensen, good approximations for calculating the ground state and low energy properties of nuclei, particularly near closed shells. But even for nuclei differing by only one particle from a closed shell, weak collective effects have been found that can only be explained by calling into play most or all of the particles in the closed shell core. Hitherto, effects of this type have generally been interpreted in terms of weak collective coupling between the odd particle and the surface of the core.
In Chapter 1 the shell model of Mayer and Jensen and the collective model of Bohr and Mottelson are briefly discussed and an alternative approach, more nearly in the spirit of the shell model, is proposed for the explanation of the weak collective effects in which the residual two-body interaction between the odd particle and the core is used as a perturbation to mix in excited core configurations.
The formalism for this type of configuration mixing is developed in Chapter 2 for collective contributions to the expectation value of the electric quadrupole operator. These will come from admixtures of excitations of the core to 2+ states differing from the zero order state by only one particle. For an L-S closed shell these will result from excitatios of a particle to a state two shells away in the oscillator sense and from higher excitations. For a j-j closed shell there may also be contributions from excitations to the other member of the spin-orbit doublet. In either case, by virtue of the energy stability of the closed shall core, the threshold for these excitations will be high. is some cases, as for example O17 with harmonic oscillator wave functions, stringent selection rules will operate to give non-vanishing contributions from only a few excitations, and in this case the configuration mixing sum can be evaluated exactly.
In other cases, however, the number of possible excitations is very large and a complete evaluation of the sum is prohibitive, but progress can still be made by replacing the energy denominators in the configuration mixing sum by a mean energy, and doing closure. If we assume that the combined effect of the large energy denominator, the selection of states by the quadrupole operator, and the potential coupling, all contribute to cutting off the actual configuration mixing sum quickly, then the mean excitation energy will be fairly independent of the odd particle states, and can be assigned as a property of the core. If one can do this, one obtains a one-parameter fit of the weak collective quadruple effects in a given nucleus or for a given closed shall core. Situations in which the closure assumption that is the assumption that the moan excitation energy is a property of the core, independent of the odd particle state, is justified are discussed and the formalism for reducing the matrix elements in the closure calculation to a sun over particle states in the core is presented in chapter 2.
In C17 the low lying positive parity levels and the ground state magnetic moment are very well understood on the pure single particle model. But the decay of the 2s1/2 first excited state to the ground state is E2 with a lifetime of the order of a single proton lifetime. Furthermore the d5/2 ground state is usually taken to have a very small negative quadrupole moment. Recent work, however, indicates that a good deal of doubt still exists about the value of the moment. Both of these electric effects are forbidden by a pure single particle model for an odd neutron. In Appendix 1 the effect of centre of mass recoil is considered and it is shown that even when large departures from proper internal states are considered, the recoil contribution to the ground state quadrupole moment is extremely small.
Chapter 3 is demoted to a discussion of these weak collective offsets from the point of view of configuration mixing. Firstly the closure formalism is applied using an interaction with a Gaussian radial dependence and a Rosenfeld exchange mixture. The wave functions are taken as determinants of harmonic oscillator single particle eigen-functions. The matrix elements are then evaluated by expending the interaction in spherical harmonics. A discussion of the evaluation of radial integrals is presented in Appendices 2, 3 and 4. We find that the mean excitation energy needed to fit the old value for the ground state moment is about 7 times larger than the one needed to fit the E2. This result is independent of the exchange mixture and very insensitive to the range of the force. In fact since for oxygen the range of the force is of the order of the nuclear radius, reasonable variations in the form of the force or of the wave functions have almost no effect on the relative values of the calculated collective effects.
If one neglects spin-orbit splittings, the closure assumption is exact for O17, since for harmonic oscillator functions the selection rules on matrix elements of the quadrupole operator only allow excitations from the core of 1s to 1d, 1p to 2p, and 1p to 1f, all of have have an energy of twice too oscillator splitting. A detailed calculation without closure of the separate contribution of each of the excitations in j-j coupling using the same interaction and wave functions as in the closure calculation is presented. We find that the 1p to 2p excitations contribute very little, and that the 1p3/2 to 1f7/2 gives the largest contribution, but in general the calculation yields no groat surprises. One still finds that no reasonable values of the parameters will remove the factor of seven between the fitting of the E2 and of the ground state moment. Using semi-empirical values for the excitation energies, one finds that one can fit the transition probability quite well, but this still leaves the ground state quadrupole moment a factor of seven too large.
The failure of other attempts to fit these two collective effects is also discussed and a brief discussion of the possibility of obtaining agreement with experiment by using a zero order state with an equilibrium deformation is presented.
In Chapter 4 configuration mixing with closure is to the weak collective effects in the region of the lead doubly closed shell. To evaluate the matrix elements we use a nuclear density distribution which reproduces the main features of the nuclear charge density obtained from the high energy electron scattering experiments. (In Appendix 5 we discuss the terms neglected in doing this.) For the odd particles we use eigen-states in an infinite potential well. The interaction is treated in a range expansion going to one term beyond zero range and its exchange dependence is taken as a Rosenfeld mixture. The matrix elements ere evaluated in a straightforward manner, the radial integrals being done graphically. A fit of experiment can be made treating the mean excitation energy and the density edge thickness as parameters.
The formalism is applied to two E2 transitions in Pb207, an E2 transition in Pb206, and the collective part of the ground state quadrupole moment of Bi209. In Pb206 a wave function is used that allows for configuration mixing between the two neutron holes. One finds that for a value of the edge thickness consistent with the electron scattering data, and only for such an edge thickness, the mean excitation energies for the two transitions in Pb207 and for the ground state moment of Bi209 all agree very well. For Pb206 the mean excitation energy consistent with the data is somewhat smaller, but this can be accounted for in terms of the terms discussed in Appendix 5, as well as in terms of an expected "softness" of Pb206 due to the fact that it differs from a closed shall by two rather than only one particle.
A discussion of the corroboration of the closure assumption is given. Finally the weak surface coupling formalism is compared with the configuration mixing approach with closure, and it is seen that the latter contains the former, but also contains more, essentially through the relaxation of the requirement of incompressibility of the core.