Model liquid-drop

Let us begin with a discussion of the probability of fission. For the first approximation to the estimation of the fission barrier, we shall use the liquid drop model (Chapter 2). We can parameterize the small nonequilibrium deformations, that is, elongations, of the nuclear surface as 

it is natural to express the fissionability of nuclei in terms of a parameter a, that is, this energy ratio, and is called the fissionability parameter. Thus 

Coulomb energy of a charged sphere surface energy of the sphere 

We can approximate the Coulomb and surface energies of a uniformly charged sphere by the following expressions  

Note that the parameter (Z2/A)critical is the ratio of two empirical constants related to the strength of the Coulomb and surface (nuclear) forces. If we take the view that the limit to the size of the periodic table is given by the point at which the heaviest nuclei spontaneously fission 

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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 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. 

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. 

Woosley et al. (1984), and the compressible liquid drop model EOS of Lamb et al. (1978) up to nuclear densities. At higher densities we added to the lepton pressure a cold pressure as given by Baron et al. in the... 

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 7.10 Contours of the Q value for the emission of a 12C nucleus as a function of neutron and proton numbers calculated with the liquid drop model mass formula. The contour lines me separated by 10 MeV. The dotted curve indicates the line of B stability [Eq. (2.9)].

Figure 7.11 Locus of neutron and proton driplines as predicted by the liquid drop model.

Figure 11.2 shows some of the basic features of fission barriers. In Figure 11.2, the fission barriers as estimated from the liquid drop model for a range of actinide nuclei are shown. The fission barrier height decreases, and the maximum (saddle point)... 

As we learned in Chapter 2, it is necessary to include shell effects in the liquid drop model if we want to get reasonable values for nuclear masses. Similarly, we must devise a way to include these same shell effects into the liquid drop model description of the effect of deforming nuclei. Strutinsky (1967) proposed such a method to calculate these shell corrections (and also corrections for nuclear pairing) to the liquid drop model. In this method, the total energy of the nucleus is taken as the sum of a liquid drop model (LDM) energy, LDM and the shell (8S) and pairing (8P) corrections to this energy,... 

Models of nuclei have grown in sophistication as new discoveries about subatomic particles have been made. One of the simplest was suggested by Niels Bohr, the Danish scientist who contributed a great deal to our understanding of atomic structure. Bohr compared the nucleus to a drop of liquid. His liquid drop model proposes that nucleons are packed together like the molecules in a liquid. Nucleons at the surface of the... 

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. 

Spontaneous nuclear fission takes place when the natural oscillations of a heavy nucleus cause it to break into two nuclei of similar mass (Fig. 17.21). In terms of the liquid drop model, we can think of the nucleus as distorting into a dumbbell shape and then breaking into two smaller droplets. An example is the disintegration of americium-244 into iodine and molybdenum ... 

Induced nuclear fission is fission caused by bombarding a heavy nucleus with neutrons (Fig. 17.23). In terms of the liquid drop model, the nucleus breaks into two droplets when struck by a projectile. Nuclei that can undergo induced fission are called fissionable. For most nuclei, fission takes place only if the impinging neutrons travel so rapidly that they can smash into the nucleus and drive it apart with the shock of impact uranium-238 undergoes fission in this way. Fissile nuclei, however, are nuclei that can be nudged into breaking apart even by slow neutrons. They include uranium-235, uranium-233, and plutonium-239, the fuels of nuclear power plants. 

This model is similar to the liquid drop model but regards the tropoelastin monomer as fibrillar. It features a broad coil made up of the repeating tetrapeptide units GVPG in which glycine residues occupy the exterior portions exposed to solvent, while proline, valine, and other hydro-phobic residues are buried. Each monomer is fibrillar, made up of alternating sections of crosslink regions and oiled coils (Gray et al., 1973). The... 

Y =1 indicates that the effect of adding an odd neutron is just half of the effect of adding a pair. Such a behaviour would be expected, e.g. by the Liquid Drop Model. The other extreme case (y s 0) is obtained when the charge distribution of the nucleus with even N completely ignores the additional neutron. Generally,... 

Following on from this work two types of mathematical model were developed that do not rely on measuring the contact area. These models are the "liquid-drop" model (Yoneda, 1973) and the elastic membrane model (Cheng, 1987a Feng and Yang, 1973 Lardner and Pujara, 1980). 

The liquid-drop model was used to model the deformation of sea urchin eggs (Yoneda, 1973). This theory assumes that the tensions in the wall during the compression are uniform and isotropic as is stated by Cole (1932) and Yoneda (1964). However, Hiramoto (1963) suggested that the circumferential tensions are actually up to two times greater than the tensions in the meridian direction. This result suggests that the use of the liquid-drop model may not be appropriate to determine material properties of cells. 

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

Several models were adopted to explain the structure of stable and radioactive nuclei. The liquid drop model assumes that protons and neutrons coalesce to form a liquid drop of high density (spherical, or prolate spheroidal, or oblate spheroidal) Weizsacker s20 semiempirical mass formula of 1935... 

However, the liquid-drop model does not account for the relative stability of certain nuclei called "islands of (relative) nuclear stability" (Z and/or N = 2, 8, 20, 28, 50, 82,126,184). 

The primary result of this analysis is that evaporation occurs only at very low levels of excitation (low velocities of impact). Very hot clusters do not evaporate. They shatter into small pieces. The theory does not exhibit an intermediate regime of cluster fission into two (or three,. ..) roughly equally sized subclusters, as in a liquid drop model.The transition from the evaporative to the shattering regime is a quite abrupt function of the velocity of impact. 

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