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Physical Review C

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

Cheuk-Yin Wong, Theodore A. Welton, J. A. Maruhn

Published Apr 1, 1977• 17 citations
Publisher

Abstract

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 ${t}_{3}(1+{\mathcal{P}}^{B})\frac{n\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{1}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{2})}{6}$, the speeds of spin sound ${a}_{2}$, isospin sound ${a}_{3}$ and spin-isospin sound ${a}_{4}$ satisfy ${{a}_{2}}^{2}+{{a}_{3}}^{2}=2{{a}_{4}}^{2}$. 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.NUCLEAR STRUCTURE Dynamics of nuclear fluid, normal sound wave, spin sound wave, isospin sound wave, and spin-isospin sound wave. Time-dependent Hartree-Fock theory. Estimate speeds of different sound waves.

DOI
10.1103/physrevc.15.1558

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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 t3(1+PB)n\ensuremathδ(r\ensuremath→1\ensuremath−r\ensuremath→2)6{t}_{3}(1+{\mathcal{P}}^{B})\frac{n\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{1}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{2})}{6}t3​(1+PB)6n\ensuremathδ(r\ensuremath→1​\ensuremath−r\ensuremath→2​)​, the speeds of spin sound a2{a}_{2}a2​, isospin sound a3{a}_{3}a3​ and spin-isospin sound a4{a}_{4}a4​ satisfy a22+a32=2a42{{a}_{2}}^{2}+{{a}_{3}}^{2}=2{{a}_{4}}^{2}a2​2+a3​2=2a4​2. With the parameters of Golin and Zamick and Vautherin and Brink, we have in addition a3>a4>a2{a}_{3}>{a}_{4}>{a}_{2}a3​>a4​>a2​, the numerical values of a2{a}_{2}a2​, a3{a}_{3}a3​, and a4{a}_{4}a4​ being in the range of 0.17 to 0.27c.NUCLEAR STRUCTURE Dynamics of nuclear fluid, normal sound wave, spin sound wave, isospin sound wave, and spin-isospin sound wave. Time-dependent Hartree-Fock theory. Estimate speeds of different sound waves.
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Working within a time-dependent Hartree-Fock framework, one develops a semiclassical approximation appropriate for large systems. It is demonstrated that the standard semiclassical approach, the Thomas-Fermi approximation, is inconsistent with Hartree-Fock theory when the basic two-body interaction is short-ranged (as in nuclear systems, for example). However, by introducing a simple extension of the Thomas-Fermi approximation, one overcomes this problem. One also discusses the infinite nuclear matter problem and point out that time-dependent Hartree-Fock theory yields collective modes of the zero sound variety instead of ordinary hydrodynamic (first) sound. One thus emphasizes that one should be extremely circumspect when attempting to cast the equations of motion of time-dependent Hartree-Fock theory into a hydrodynamic-like form.
At low temperatures, elementary excitations of a one-dimensional quantum liquid form a gas that can move as a whole with respect to the center of mass of the system. This internal motion attenuates at exponentially long time scales. As a result, in a broad range of frequencies, the liquid is described by two-fluid hydrodynamics, and the system supports two sound modes. The physical nature of the two sounds depends on whether the particles forming the quantum liquid have a spin degree of freedom. For particles with spin, the modes are analogous to the first and second sound modes in superfluid 4He^{4}\mathrm{He}4He, which are the waves of density and entropy, respectively. When dissipative processes are taken into account, we find that at low frequencies, the second sound is transformed into heat diffusion, while the first sound mode remains weakly damped and becomes the ordinary sound. In a spinless liquid, the entropy and density oscillations are strongly coupled and the resulting sound modes are hybrids of the first and second sound. As the frequency is lowered and dissipation processes become important, the crossover to a single-fluid regime occurs in two steps. First the hybrid modes transform into predominantly density and entropy waves, similar to the first and second sound, and then the density wave transforms to the ordinary sound and the entropy wave becomes a heat-diffusion mode. Finally, we account for the dissipation due to viscosity and intrinsic thermal conductivity of the gas of excitations, which controls attenuation of the sound modes at high frequencies.
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