Dynamics of nuclear fluid. II. Normal sound, spin sound, isospin sound, and spin-isospin sound

Starting with the time-dependent Hartree-Fock equation in density matrix form, we investigate the macroscopic description of the dynamics of the nuclear fluid involving the spin and isospin degrees of freedom, in conjunction with the presence of only central exchange interactions. The time-dependent Hartree-Fock equation can be cast into a set of conservation laws of the classical type coupling spin and isospin densities. With simple zero-range interactions, we obtain the normal modes of density propagations and the corresponding speeds of sound waves. In addition to the normal sound waves in which the total density varies with space and time, there are the spin sound waves in which the difference of the spin-up and the spin-down densities varies with space and time, the isospin sound waves in which the difference of neutron and proton densities varies with space and time, and finally, the spin-isospin sound waves in which the difference of the "parallel" spin and isospin densities and the "antiparallel" spin and isospin densities varies with space and time. It is found that for a zero-range interaction whose density dependence is of the type $t_3(1+{\mathcal{P}}^B)\frac{n\delta ({\overrightarrow{\mathrm{r}}}_1-{\overrightarrow{\mathrm{r}}}_2)}{6}$, the speeds of spin sound $a_2$, isospin sound $a_3$ and spin-isospin sound $a_4$ satisfy ${a_2}^2+{a_3}^2=2\mathrm{}{a_4}^2$. With the parameters of Golin and Zamick and Vautherin and Brink, we have in addition $a_3>a_4>a_2$, the numerical values of $a_2$, $a_3$, and $a_4$ being in the range of 0.17 to 0.27c.