Liquid drop model binding energy

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.

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. 

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 ... 

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 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]. 

Fig. 1 Binding energy per nucleon in the liquid drop model. Isocontours for binding energies of 8.5, 8, 7.5, 7, and 6.5 MeV are shown, together with the line N = Z and fissility contours X = 7 lA = 18,30,40,50. A fissility jc 40 gives the limit of stability in the LDM. Nuclei beyond this line are stabilised entirely by shell effects...

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. 

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