Weizsacker formula

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 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 stability valley of nudei. The vertical projection of the bottom of the valley on the Z x A plane follows the line of stable nuclei (cf. Fig. 2.7 in Chap. 2) at an "altitude" expressed by the ratio of the nudidic mass (M) and the mass number (A). The horizontal projection of the bottom of the valley onto the (M/A) x A plane is essentially the same as the graph labeled "experimental" in Fig. 2.3 in Chap. 2, except that it is upside down. The isobaric cross sections of the valley are parabolic as a consequence of the Weizsacker formula (2.3) in Chap. 2. (See also the legend of Fig. 7.2 in this chapter, as well as Fig. 14.5 in Chap. 14, Vol. 2.) The local dips marked are related to magic numbers (Ca N=20 Fe-Ni A/=28, Z= 28 Y A(=50 Ba Af=82). Beta decay occurs all along the valley. On the near side of the valley decay is predominant, on the far side decay and electron capture (EC) compete with each other as indicated on the parabola. Alpha decay occurs for A > 60 mainly on the far side of the valley... 

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... 

In order to calculate the binding energy E ) of the nuclei, Weizsacker developed a semi-empirical formula based on the drop model ... 

One finds that the LDA overestimates the calculated binding energy. At the BHF minimum the difference is about 85 MeV for the total nucleus, which is close to the surface correction contribution in the Bethe-Weizsacker mass formula for (as 118MeV). This suggests that the LDA approximation misses the major contribution to the surface tension and also the major contribution of the surface effects to the compressibility of a finite nucleus. This is supported by the observation that the LDA result for as a function of density is only slightly above the energy versus density curve of nuclear matter. 

The Weizsacker semiempirical mass formula ( Eq. (2.3)) correctly predicts that a emission can be expected mainly for the heavy elements. Nevertheless, the shell effects, which were not taken into account in the formula, also play an important role in a decay. 

Unfortunately, adding the Dirac exchange formula to the TF model does not improve the quality of the calculated electron density. The TFD density suffers from the same undesirable characteristics as does the TF density. A major enhancement of these two overly simplified models was made through addition of an inhomogeneity the electron density correction to the kinetic energy density functional. This was first investigated by von Weizsacker, who derived a correction that depends upon the gradient of the density, namely. 

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