Liquid drop model fission barrier

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... 

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)... 

In reality, the liquid drop model is oversimplified one should also take into account the stabilisation effects due to shell formation, as well as the presence of an energy barrier in the fission process [711]. 

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. 

It has long been recognized that the liquid-drop model semi-empirical mass equation cannot calculate the correct masses in the vicinity of neutron and proton magic numbers. More recently it was realized that it is less successful also for very deformed nuclei midway between closed nucleon shells. Introduction of magic numbers and deformations in the liquid drop model improved its predictions for deformed nuclei and of fission barrier heights. However, an additional complication with the liquid-drop model arose when isomers were discovered which decayed by spontaneous fission. Between uranium and... 

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.

Height of fission barriers in MeV calculated according to the liquid drop model... 

Fission barriers using the ETFSI (extended Thomas-Fermi plus Strutinsky integral) method have been calculated by Mamdouh et al. (1998, 2001) for a large number of nuclei. Moller et al. (2009) presented calculated fission barriers for heavy elements based on the macroscopic-microscopic finite-range liquid drop model. 

In O Fig. 19.11 the partial fission half-lives of the doubly even isotopes of uranium and beyond are plotted on a logarithmic time scale versus the fissility parameter. In accordance with the expectation from the liquid drop model the dashed line labeled Bld, describing the fission half-life calculated with only the liquid drop barrier Bid, crosses the 2. line at nobeKum. The time Te. is needed for the formation of the electron shell of the atom, the lower time limit considered beyond which a chemical element cannot be formed (Barber et al. 1992). The experimental half-lives follow this general trend. They decrease from uranium to nobelium over more than 20 orders of magnitude, from the age of the solar system down to fractions of seconds. The structure of the isotopic chains of elements from curium to nobelium is caused by a subshell closure at M = 152. 

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]. 

The nuclear models that resulted in the prediction of an island of superheavy nuclei have evolved in response to experimental measurements of the decay properties of the heaviest elements. While the prediction of a spherical magic N = 184 is robust and persists across the models [8], the shell closure associated with Z — 114 is weaker, and different models place it at higher atomic numbers, from Z = 120 to 126 [60-69] or even higher [70] (see Nuclear Structure of Superheavy Elements ). Interpretation of the decay properties of the heaviest elements may support this [71, 72], but the most part decay and reaction data do not conclusively establish the location of the closed proton shell. Because of this, the domain of the superheavy elements can be considered to start at approximately Z = 106 (seaborgium), the point at which the liquid-drop fission barrier has vanished [9]. For our purposes, the transactinide elements (Z > 103) will be considered to be superheavy (see Nuclear Structure of Superheavy Elements ). 

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