Nuclear liquid-drop model

Strutinsky did not address the question of how to calculate microscopically the smooth part E (which necessarily entails specifying the smooth density p). Instead he circumvented this question by substituting for E the empirical energies, ldm of the nuclear liquid-drop model, namely he suggested that... 

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... 

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

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... 

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,... 

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. 

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

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). 

It is the general consensus among nuclear physicists38) that the unsaturated behaviour of nuclei with A below 12 in Eq.(6) is atypical, in sofar the liquid drop model becomes a reasonable approximation in heavier nuclei. It may be noted that the smallest drop consisting of A identical spheres, where at least one particle is not in the surface, occurs for (both cuboctahedral and icosahedral) A = 13, related to the fact that 4 7r = 12.56637... This model was proposed by C.F. von Weiszacker in 1935, and one example 39) of the parametrization of the total atomic weight in the unit of 0.001 chemical unit is... 

A similar type of research took place also in nuclear physics during the thirties with a systematic characterization of different properties for a number of atomic nuclei [24]. As an example can be mentioned the studies of the neutron cross sections as a function of the number of neutrons or protons in the nuclei, which showed systematic variations with very small values at certain numbers corresponding to nuclei with 20, 50, 82 and 126 neutrons. This discovered periodicity was rather different compared with the periodicity of atomic properties as the first ionization potential and electron affinity for alkali and noble gas atoms. Speaking at a meeting of the Chemical Society on April 19, 1934, the centenary of the birth of Mendeleev, Rutherford concluded, /< may be that a Mendeleev of the future may address the Fellows of this Society on the Natural Order of Atomic Nuclei and history may repeat itself [25]. Measurements of for example nuclear spins for a number of isotopes also showed a similar type of periodicity as found in neutron cross sections. This kind of periodicity could not at that time be understood from the commonly used liquid drop model [26] but based on the single particle model formulated by Mayer, Haxel, Jensen and Suess in 1949 [27]. 

A simple relation between any length parameter and the nuclear mass number A follows directly from geometrical considerations and the assumption of constant nuclear (mass) density ( liquid drop model or homogeneous model ), e.g. for the rms radius... 

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. 

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.

The fission process can be described in terms of the liquid drop model. The drop is capable of undergoing various deformations considered as vibrational modes. As the drop becomes more distorted, it eventually breaks into two primary fragments. The two fragments are usually unequal in mass, with a mass distribution that depends on the manner in which the drop splits. In nuclear fission, absorption of a neutron induces similar oscillations in the target radionuclide that distort its shape until it splits into the two primary FF. A wide array of possible mass combinations exists for the FF. Shown below are three such possibilities for... 

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... 

Detailed studies of the vast array of fission products occupied a small army of chemists at the Metallurgical Laboratory in Chicago (Coryell and Sugarman 1951, see also Siegel 1946). The individual nuclides had to be identified, their nuclear properties determined, and their yields in fission measured. One of the important general results was the realization that, under slow-neutron bombardment, and Pu split asymmetrically into two fragments of unequal mass, whereas the liquid-drop model predicts a symmetric mass split. This behavior remained an intriguing puzzle for 3 decades. 

On the other hand, the liquid drop model can correctly reproduce the binding energies of nuclei (see, e.g.,0 Eq. (2.3)), nuclear masses, and the threshold potential of nuclear fission, but it cannot describe the shell effects, ground state spins, and many other quantum characteristics. 

The liquid drop model can also explain many nuclear phenomena successfully. The most important ones are as follows the nuclear volume is proportional to the mass number (A) (O Eq. (2.15)) the binding energy per nucleon is approximately constant in a wide mass-number region (O Fig. 2.3) the nuclear masses can be rather well described by the Weizsacker formula (O Eq. (2.3)) nuclear fission (see Chap. 3 in this Volume) Hofstadter s electron scattering experiments show that the nuclear volume is filled up with nucleons rather uniformly. However, the liquid drop model also has its weak points, e.g., it cannot give account of the shell effects. 

The viewpoint of the shell model is very different from that of the liquid drop model. The nucleons are looked at as standing waves occupying well-defined quantum states in a potential that is formed by the other nucleons. The chemist knows such a system fi-om the behavior of the electrons in the atomic shell. Whereas in the atomic shell the potential and the forces acting are known, this is not the case for nuclei. The form of the nuclear potential well is not known exactly. The form of the nuclear potential well can be parameterized by a harmonic oscillator, by a square well potential or - more realistically - by a Woods-Saxon potential (see Fig. 2.2 in Chap. 2). 

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... 

Fermi irradiated uranium with slow neutrons, and observed a variety of radioactivities that he tentatively identified as being transuranium elements [1]. We now know that these radioactive species were the products of the fission of the in the sample. Study of the chemical properties of these new nuclides led to the subsequent discovery of fission in 1939 [2, 3], Explanation of the fission process was closely connected to the creation of the liquid-drop model [4—6], in which the nucleus is treated like an incompressible charged fluid with surface tension. See Nuclear Structure of Superheavy Elements for more information on nuclear structure and the stability of the heaviest nuclides. 

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]. 

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