Time-Dependent Hartree-Fock Theory of Nuclear Collective Oscillations

A theory of nuclear collective oscillations is presented which does not involve introducing extra variables and subsidiary conditions. This time-dependent self-consistent field method is applied to the breathing mode of a spherically symmetric nucleus and yields a value for the frequency of oscillation which is more accurate than that from a previous treatment in terms of one-nucleon excitation, but which becomes identical to the latter in the case of weak nucleon-nucleon interaction. In cases where nucleon exchange can be neglected, the new estimate reduces to the frequency derived from the simple classical theory of a compressible fluid. By means of an electric monopole sum rule which is derived for $T=0\mathrm{}\to T=0\mathrm{}$ transitions, it is shown that in general the classical formula overestimates the breathing mode frequency. From the sum rule it also follows that the 6.06-Mev $0^{+}$ state in ${\mathrm{O}}^{16}$ is related only indirectly to the breathing mode, which must itself be at a higher excitation energy.