Nuclear drop theory

During recent decades a great amount of knowledge about the properties of atomic nuclei has been gathered. An extensive theory of nucleonic interactions and nuclear structure [liquid-drop theory (7), shell theory (2, 3), unified theory (4), cluster theory (5—7)] has been developed... 

The earliest theory of nuclear structure was the liquid drop theory, developed around 1940. It pictured the nucleus as a structureless drop of liquid in which repulsions between positive charges are opposed by a kind... 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

The close-packed-spheron theory of nuclear structure may be described as a refinement of the shell model and the liquid-drop model in which the geometric consequences of the effectively constant volumes of nucleons (aggregated into spherons) are taken into consideration. The spherons are assigned to concentric layers (mantle, outer core, inner core, innermost core) with use of a packing equation (Eq. I), and the assignment is related to the principal quantum number of the shell model. The theory has been applied in the discussion of the sequence of subsubshells, magic numbers, the proton-neutron ratio, prolate deformation of nuclei, and symmetric and asymmetric fission. 

The attractive potential exerted on the electrons due to the nuclei - the expectation value of the second operator VNe in equation (1-4) - is also often termed the external potential, Vext, in density functional theory, even though the external potential is not necessarily limited to the nuclear field but may include external magnetic or electric fields etc. From now on we will only consider the electronic problem of equations (1 -4) - (1 -6) and the subscript elec will be dropped. 

C. F. von Weizsacker developed a crude theory of nuclear masses in 1935. The theory takes as its basis the idea that nuclei behave like incompressible uniformly charged liquid drops. How can we account for the variation of nuclear masses We begin by stating that... 

A mathematical model of the processes leading to steam explosions has been developed for the contact of hot liquids or molten solids dropping into water [1]. A review of detonation theory leads to an attempt to apply it to melt/water combinations, with particular reference to molten oxides in nuclear reactor melt-down. Alumina /water is more violent than urania/water [2]. 

The mechanisms and data of the fission process have been reviewed recently by Leachman (70). Several different approaches have been used in an effort to explain the asymmetry of the fission process as well as other fission parameters. These approaches include developments of the liquid drop model (50, 51,102), calculations based on dependence of fission barrier penetration on asymmetry (34), the effect of nuclear shells (52, 79, 81), the determinations of the fission mode by level population of the fragments (18, 33, 84), and finally the consideration of quantum states of the fission nucleus at the saddle point (15, 108). All these approaches require a mass formula whereby the masses of the fission fragments far removed from stability may be determined. The lack of an adequate mass formula has hindered the development of a satisfactory theory of fission. The fission process is highly complex and it is not surprising that the present theories fall short of a full explanation. 

But if we are concerned with more complex aspects of nuclear structure, the liquid drop model of the nucleus won t do. Suppose we are interested, for example, in the pattern of stability and instability that governs the collection of nuclear isotopes. Why is there a line of stability about which the stable nuclei are concentrated, with deviation from that line, which is plotted with numbers of protons and numbers of neutrons as axes, indicating the likelihood that the nucleus in question will be unstable Much insight can be gained from a model that treats the nucleons in the nucleus as moving on orbits in an overall potential field. Here, the nucleons are treated as if they were like the electrons in their orbits that surround the nucleus in the atom. Numbers are assigned that are parallels to the familiar quantum numbers of atomic electron theory, and orbits for the nucleons in the nucleus characterized by these quantum numbers are posited. Just... 

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