Nucleus liquid drop model

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 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)].

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. 

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,... 

The liquid drop model, in which nucleons are considered to be packed together in the nucleus like molecules in a liquid... 

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. 

We have already shown how one model for the nuclear structure, the liquid drop model, has helped us to explain a number of nuclear properties, the most important being the shape of the stability valley. But the liquid drop model fails to explain other important properties. In this chapter we shall try to arrive at a nuclear model which takes into account the quantum mechanical properties of the nucleus. 

In Chapter 3 we observed that the binding energy per nucleon is almost constant for the stable nuclei (Fig. 3.3) and that the radius is proportional to the cube root of the mass number. We have interpreted this as reflecting fairly uniform distribution of charge and mass throughout the volume of the nucleus. Other experimental evidence supports this interpretation (Fig. 3.4). This information was used to develop the liquid drop model, which successfully explains the valley of stability (Fig. 3.1). This overall view also supports the assumption of a strong, short range nuclear force. 

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... 

The ground state energy, E, of a nucleus can be regarded as a sum of the liquid drop model energy (including deformation), pairing correction, 6p, and the shell... 

Strutinsky developed an extension of the liquid drop model which satisfactorily explains the fission isomers and asymmetric fission. For such short half-lives the barrier must be only 2-3 MeV. Noting the manner in which the shell model levels vary with deformation ( 11.5, the "Nilsson levels"), Strutinsky added shell corrections to the basic liquid-drop model and obtained the "double-well" potential energy curve in Figure 14.14b. In the first well the nucleus is a spheroid with the major axis about 25 % larger than the minor. In the second well, the deformation is much larger, the axis ratio being about 1.8. A nucleus in the second well is metastable (i.e. in isomeric state) as it is unstable to y-decay to the first well or to fission. Fission from the second well is hindered by a 2 - 3 MeV barrier, while from the first well the barrier is 5 - 6 MeV, accounting for the difference in half-lives. 

Figure 16.5 shows the variation in nuclear deformation calculated for the fission barrier of 298114 Qf particular interest are the small local fluctuations at small deformation. The minimum of 8 MeV at zero deformation constrains the nucleus to a spherical shape. Spontaneous fission is a very slow process in this situation since it involves tunneling through the 8 MeV barrier. These local fluctuations in the potential energy curve in Figure 16.S result from adding corrections for shell effects to a liquid drop model. The resistance to deformation associated with closed shell nuclei produces much longer half-lives to spontaneous fission than would be expected from calculations based on a standard liquid drop model.

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. 

The liquid-drop model made a division of the nucleus seem possible. They sat down on a log. Meitner found a scrap of paper and a pencil in her purse. She drew circles. Couldn t it be this sort of thing ... 

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... 

The main features of this behavior can be understood on the basis of the liquid drop model (von Weizsacker 1935 Bethe and Bacher 1936). According to this model, the nucleus is an incompressible liquid drop, in which the electric charge is distributed uniformly. The binding energy E is described by the Weizsacker formula ... 

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

These observations indicate that fission of metal clusters occurs when the repulsive Coulomb forces due to the accumulation of the excess charges overcome the electronic binding (cohesion) of the cluster. This reminds us immediately of the well-studied nuclear fission phenomenon and the celebrated liquid drop model (LDM) according to which the binding nuclear forces are expressed as a sum of volume and surface terms, and the balance between the Coulomb repulsion and the increase in surface area upon volume-conserving deformations allows for an estimate of the stability and fissility of the nucleus [12, 13]. 

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... 

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