Asymptotic Models for Studying Kinetics of Formation of Compact Objects with Strong Internal Bonds

Abstract

An asymptotic method has been developed for investigation of kinetics of formation of compact objects with strong internal bonds. The method is based on the uncertainty relation for a coordinate and a momentum in space of sizes of objects (clusters) with strongly pronounced collective quantum properties resulted from exchange interactions of various physical nature determined by spatial scales of the processes under consideration. The proposed phenomenological approach has been developed by analogy with the all-known ideas about coherent states of quantum mechanical oscillator systems for which a product of coordinate and momentum uncertainties (dispersions) accepts the value, which is minimally possible within uncertainty relations. With such an approach the leading processes are oscillations of components that make up objects, mainly: collective nucleon oscillations in a nucleus and phonon excitations in a mesostructure crystal lattice. This allows us to consider formation and growth of subatomic and mesoscopic objects in the context of a single formalism. The proposed models adequately describe characteristics of formation processes of nuclear matter clusters as well as mesoscopic crystals having covalent and quasi-covalent bonds between atoms.

Cluster Model of Formation of Subnuclear and Subatomic Objects

“…The random interaction of these seeds results in their coalescence, and compact nuclear matter clusters are formed in the system. It is assumed that, as in the case of the growth processes of nanocrystals with strong interatomic bonds [6], oscillations of nucleons on the surface of contacting nuclear clusters with temporarily (virtually) broken external bonds can lead to the mutual compensation of these unused bonds. As a result, consolidation of adjacent clusters takes place and larger objects of nuclear medium with different masses are formed.…”

“…for describing the processes under consideration [5,6]. This 1D approach makes it possible to neglect deviations of the object geometric shape from the spherical one.…”

“…A reasonable choice for this scale is the period of high-frequency nucleon oscillations in the nucleus. This parameter can be defined as t 0 = 2r 0 /c 0 ≈ 5 × 10 -23 s. The approximate quantity λ is found by matching the solutions for the first maximum of ϕ, which corresponds to the seeds, and the maximum of ϕ for small clusters [6]. The calculations yield 15 2 λ ≈ .…”

“…We define two mechanisms of growth of nuclear matter clusters: 1) the small flux of seeds, when each of them has time to occupy its place on the surface of the cluster before starting interaction with the next seed; 2) the high flux of seeds, when they affect the cluster almost simultaneously (equivalent to collision of clusters). Linearization of Equation (1) makes it possible to produce analytical expressions for increase in the mean size a of large clusters (a  a 0 ) with time [6]. In case of the first growth mechanism the approximate law of increasing the average size of clusters can takes the following form…”

“…The carried out study of the asymptotic properties of function φ(a, t) showed that in the end of the irreversible aggregation of objects as a result of interaction of large clusters with small seeds in a stochastic conservative system the distribution density of large clusters with a  a 0 (a 0 is a size of a seed) is inversely proportional to their masses: ϕ ∝ m -1 (an "inverse-mass" law) [6]. We define two mechanisms of growth of nuclear matter clusters: 1) the small flux of seeds, when each of them has time to occupy its place on the surface of the cluster before starting interaction with the next seed; 2) the high flux of seeds, when they affect the cluster almost simultaneously (equivalent to collision of clusters).…”