Liquid drop model of the

The close-packed-spheron model differs from the conventional liquid-drop model of the nucleus in having spherons rather than nucleons as the units. This is a simplification , 4GdiM,ir>4, for example, is described in terms of 45 spherons, rather than 154 nucleons. 

In considering the physical forces acting in fission, use may be made of the Bohr liquid drop model of the nucleus. Here it is assumed that in its uonual energy state, a nucleus is spherical and lias a homogeneously distributed electrical charge. Under the influence of the activation eneigy furnished by the incident nentron, however, oscillations are set up which tend to deform the nucleus. In the ellipsoid form, the distribution of the protons is such that they are concentrated in the areas of the two foci. The electrostatic forces of repulsion between the protons at the opposite ends of the ellipse may then further deform the nucleus into a dumbbell shape. Rrom this condition, there can be no recovery, and fission results. 

Nuclear fission has generally been explained theoretically in terms of die liquid-drop model of the nucleus, In this model, the incident neutron... 

FIGURE 17.21 In spontaneous nuclear fission, the oscillations of the heavy nucleus in effect tear the nucleus apart, thereby forming two or more smaller nuclei of similar mass. This picture is based on the liquid drop model of the nucleus. 

Why was Dr. Noddack s suggestion ignored The reason is that she was ahead of her time. Bohr s liquid-drop model of the nucleus had not yet been formulated, and so there was at hand no accepted way to calculate whether breaking up into several large fragments was energetically allowed. 

If Noddack s physics was avant garde, her chemistry was sound. By 1938 her article was gathering dust on back shelves, but Bohr had promulgated the liquid-drop model of the nucleus and the confused chemistry of uranium increasingly preoccupied Lise Meitner and Otto Hahn. 

Placzek was skeptical. The situation is more confused than ever, he told Bohr. He began then to specify the sources of confusion. He was directly challenging the relevance of Bohr s liquid-drop model of the nucleus. The Danish laureate paid attention. 

Niels Bohr proposes a liquid drop model of the atomic nucleus. 

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... 

An estimate of the fission threshold can be obtained from the energy required to distort the nucleus into an extreme shape which results in complete separation into fragments. It has been shown that this calculation can be based on the liquid-drop model of the nucleus. The two principal contributions to the distortion energy of the nucleus are the surface-tension effect from the nuclear forces between the constituent... 

Vibrational Spectroscopy.—The liquid-drop model of the nucleus has been used as the basis of calculations of the low-energy excitation spectra of polynuclear... 

R. H. Stuewer, "The origin of the liquid-drop model and the interpretation of nuclear fission, Perspectives on Science 2 (1994) 76-129. 

Not all properties of the nuclei can be explained by the shell model. For calculation of binding energies and the description of nuclear reactions, in particular nuclear fission, the drop model of the nucleus has proved to be very useful. In this model it is assiuned that the nucleus behaves hke a drop of a liquid, in which the nucleons correspond to the molecules. Characteristic properties of such a drop are cohesive forces, surface tension, and the tendency to split if the drop becomes too big. 

Figure 13. The cluster size dependence of the calculated binding energies per atom for a He) cluster (N = 6.5 x 103 to 1.88 x lO ) of radius R without a bubble (marked as cluster) and for a cluster with a bubble at the equilibrium electron bubble radius Rf, (marked as cluster + bubble). The experimental binding energy per atom in the bulk [232, 248], E /N = —0.616 meV (R, N = cxd), is presented (marked as bulk). Previous computational results for the lower size domain N = 128-728 [51-54, 106, 128, 129] are also included. The calculated data for the large (N = 10 —10 ) clusters (A = 6.5 x 1Q3 to 1.88 x 10 ), as well as the bulk value of Ec/N without a bubble, follow a linear dependence versus 1 /R and are represented by the liquid drop model, with the cluster size equation [Eq. (58)] (solid line). The dashed curve connecting the E /N data with a bubble was drawn to guide the eye. The calculated data for the smaller clusters (N = 128) manifest systematic positive deviations from the liquid drop model, caused by the curvature term, which was neglected.

Both the liquid-drop model and the single-particle model assume that the mass and charge of the nucleus are spherically symmetric. This is true only for nuclei close to the magic numbers other nuclei have distorted shapes. The most common assumption about the distortion of the nuclide shape is that it is ellipsoidal, i.e. a cross-section of the nucleus is an ellipse. Figure 11.6 shows the oblate (flying-saucer-like) and prolate (egg-shaped) ellipsoidally distorted nuclei the prolate shape is the more common. Deviation from the spherical shape is given by... 

K. K. Nanda, S. N. Sahu, and S. N. Behera. Liquid-drop model for the size-dependent melting of low-dimensional systems. Phys. Rev. A, 66 013208-1-8, 2002... 

The Combination of the Liquid Drop Model and the Shell Model According to... 

The true behavior of a fissioning nucleus can be described only by combining the liquid drop model and the shell model. 

Mercury electrodeposition is a model system for experimental studies of electrochemical phase formation. On the one hand, the product obtained is a liquid drop, corresponding very well with the liquid drop model of classical nucleation theory. Besides, electron transfer is fast [61] and therefore the growth of nuclei is controlled by mass transport to the electrode surface [44]. On the other hand, the properties of the mercuryjaqueous solution interface have been the object of study for over a century and hence are fairly well understood. The high overpotential for proton reduction onto both mercury and vitreous carbon favor the study of the process over a wide range of overpotentials. In spite of the complications introduced by the equilibrium between the Hg +, Hg2 " ", and Hg species, this system offers an excellent opportunity to verily the fundamental postulates of the electrochemical nucleation theory. In fact, the dependence of the nucleation rate on the oxidation state of the electrodepositing species is fiiUy consistent with theory critical nuclei appear with similar sizes and onto similar number densities of active sites... 

The liquid-drop model was very successful in reproducing the beta-stable nuclei at a given atomic mass (A) as a function of atomic number (Z) and neutron number (AO, and the global behavior of nuclear masses and binding energies. Early versions of the liquid-drop model predicted that the nucleus would lose its stability to even small changes in nuclear shape when zVa > 39, around element 100 for beta-stable nuclei [6, 7]. At this point, the electrostatic repulsion between the protons in the nucleus overcomes the nuclear cohesive forces, the barrier to fission vanishes, and the lifetime for decay by spontaneous fission drops below lO" " s [8]. Later versions of the model revised the liquid-drop limit of the Periodic Table to Z = 104 or 105 [9]. 

The close-packed-spheron theory8 incorporates some of the features of the shell model, the alpha-particle model, and the liquid-drop model. Nuclei are considered to be close-packed aggregates of spherons (helicons, tritons, and dineutrons), arranged in spherical or ellipsoidal layers, which are called the mantle, the outer core, and the inner core. The assignment of spherons, and hence nucleons, to the layers is made in a straightforward way on... 

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. 

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