Semiclassical Aspect of Collective Motion in a Layer-Structured Many Particle System

Abstract

Semiclassical argument is presented for the layer vibration in the many particle system with a layer structure, which is expected to be realized in a dense nuclear matter. Within a toy model, the presence of this collective mode is shown to result in a small reduction of system's energy. The alternating-layer-spin-structure (ALS) model of dense nuclear matter proposed by Takatsuka et al.l) is characterized by three points; 1. two-dimensional or layered structure of nucleonic Fermi liquid, 2. one-dimensional lattice structure along the direction (which we define as z-direction) perpendicular to surfaces of laminated layers with alternating values of «363), and 3. the pion condensation accompanied by spin-isospin ordering of nucleons. Nucleon part of the ground state wave function of this model can be approximated by the Slater determinant of the single particle wave function which represents the above characteristics. Particularly, for the motion in z-direction, the Gaussian form has been shown practically sufficient as an approxima­ tion of exact Wannier function. This implies that individual particle's motions in z-direction are regarded as the zero-point vibrations of uncorrelated harmonic oscil­ lators. However, it is also possible that such degrees of freedom are absorbed into collective motion as vibration of layers just like the surface vibrations of nuclei. 2 ) Like zero sound,3) this type of excitation would have dispersion relation which is linear at small wavevectors. A reduction of system's energy will therefore occur by absorption of independent oscillator-degrees of freedom into such a collective mode. A systematic method to separate collective modes was developed by Bohm and Pines 4 ) and was applied to quantization of plasma-oscillation in electron gas. This method called random phase approximation is also applicable to develop a quantum theory of layer vibration in the ALS-model of nuclear matter. However, a semiclas­ sical argument is sufficient to gain an insight into a fundamental aspect of the effect of collective motion. In this paper, on the presumption of the existence of layer structure, we find the dispersion relation of collective mode of layer vibration to estimate the correlation energy on the basis of classical mechanics of a very simplified model. The longitudinal mode of layer vibration was discussed by Ruivo 5 ) on the basis of the generator coordinate method, while our argument will include the most general class of collective motion. We consider a system of laminated layers of area Q.l., each of which consists of N/ particles. The model Hamiltonian is