Fission barriers

Vieira A and Fiolhais C 1998 Shell effeots on fission barriers of metallio olusters a systematio desoription Phys. Rev. B 57 7352... 

In order to determine the maximum atomic mass produced in the r process, we must find the point when induced (destructive) fission enters into competition with (constructive) neutron capture on the path followed by the process across the (N, Z) map of the isotopes. This question requires calculation of the fission barrier far from the region of known nuclei, which is no simple matter. The possibility of producing mythical, superheavy, transuranium nuclei (around Z = 114 and = 184) has not yet been demonstrated. 

We need to find the neutron separation energy and the fission barrier for this nucleus in order to evaluate the ratio ... 

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)... 

Figure 11.2 Qualitative features of the fission barriers for actinide nuclei. (From H. C. Britt, Fission Properties of the Actinides in Actinides in Perspective, N. Edelstein, Ed. Copyright 1982 Pergamon Press, Ltd. Reprinted by permission of H. C. Britt.)...

Nuclei can be trapped in the secondary minimum of the fission barrier. Such trapped nuclei will experience a significant hindrance of their y-ray decay back to the ground state (because of the large shape change involved) and an enhancement of their decay by spontaneous fission (due to the thinner barrier they would have to penetrate.) Such nuclei are called spontaneously fissioning isomers, and they were first observed in 1962 and are discussed below. They are members of a general class of nuclei, called superdeformed nuclei, that have shapes with axes ratios of 2 1. These nuclei are all trapped in a pocket in the potential energy surface due to a shell effect at this deformation. 

It is clear from these basic facts and our picture of fission that spontaneous fission is a barrier penetration phenomenon similar to a or proton decay. The nucleus tunnels from its ground state through the fission barrier to the scission point. Therefore, we would expect the spontaneous fission (SF) half-life to have the form... 

Figure 11.3 Simple parabolic fission barrier. (From R. Vandenbosch and J. R. Huizenga, Nuclear Fission. Copyright 1973 Academic Press. Reprinted by permission of Elsevier.)...

As an exercise, let us compare the spontaneous fission half-lives of two nuclei with barrier heights of 5 and 6 MeV, respectively, and barrier curvatures of 0.5 MeV. One quickly calculates that the spontaneous fission half-lives of these two nuclei differ by a factor of 3 x 105. The barrier heights and curvatures in this example are relevant for the actinides and illustrate the difficulty that a 1-MeV uncertainty in the fission barrier height corresponds to a factor of 105 in the spontaneous fission half-life. 

In our previous discussion, we showed that the fission barrier heights depend on Z2/A and thus so should the spontaneous fission half-lives. In Figure 11.4, we show the dependence of the known spontaneous fission half-lives on x, the fissionability parameter. There is an overall decrease in spontaneous fission half-life with increasing x, but clearly the spontaneous fission half-life does not depend only on Z2/A. One also observes that the odd A nuclei have abnormally long half-fives relative to the even-even nuclei. Also the spontaneous fission half-fives of the heaviest nuclei (Z > 104) are roughly similar with values of milliseconds. 

Fig. 11.8). The resonances associated with fission appear to cluster in bunches. Not all resonances in the compound nucleus lead to fission. We can understand this situation with the help of Figure 11.9. The normal resonances correspond to excitation of levels in the compound nucleus, which are levels in the first minimum in Figure 11.9. When one of these metastable levels exactly corresponds to a level in the second minimum, then there will be an enhanced tunneling through the fission barrier and an enhanced fission cross section. 

Sketch the fission excitation function for the reaction of 232Th with neutrons. The fission barrier is 6.5 MeV, and the binding energies of the last neutron in 232Th and 233Th are 6.90 and 4.93 MeV, respectively. 

Fission, A(n, f), which is most likely at thermal energies but occurs at all energies where the neutron binding energy exceeds the fission barrier height for fissile nuclei. 

It gives reasonable fission barriers, fission isomer energies, and fission isomer lifetimes [KUM86]. Agreement with the fission lifetimes is also quite reasonable (see Fig. 7). This did require the introduction of two additional parameters (the strength and the A A -dependence) for the nuclear part of the fragment-fragment interaction. 

We see that many problems still need to be solved in order to obtain accurate results in Hauser-Feshbach calculations. Some examples are the energy dependence of rotational enhancement of levels in deformed nuclei, the energy and mass dependence of Ml gamma-ray transitions, the importance of E2 transitions, and better estimates of fission barriers. Work in each of these areas will benefit greatly from a better understanding of the discrete levels, particularly in nuclei away from stability. 

KRU81]). The B strength function for nuclei along the decay back paths [coupled with neutron separation energies (Sn), fission barrier heights (Bf) and B"decay Q-values (Qg)] determines the amount of B delayed fission and neutron emission that occurs during the cascade back to the B stability line. 

A recent analysis by Thielemann et al. [THI83] of the effects of B" delayed processes on the progenitors of the Th-U-Pu chronometers showed that these processes (delayed fission in particular) did indeed significantly influence the final abundances of the chronometer progenitors. This leads to a long age for the Galaxy. In view of the importance of this result, it is useful to re-examine the calculation with a nuclear model that includes the effects of nuclear deformation on the B decay rates, fission barriers, and neutron separation energies self-consistently. 

Meyer, B.S., Moller, P., Howard, W.M., Mathews, G.J. Fission Barriers for r-Process Nuclei and Implications for Astrophysics . In Proceedings of the Conference 50 Years with Nuclear Fission , Gaithersburg, Maryland, 25-28 April 1989, pp. 587-591. 

With increasing atomic numbers spontaneous fission begins to compete with a decay and prevails for some radionuclides with Z > 96. However, due to high fission barriers, a decay is still the dominating mode of decay for many heavy nuclides with Z> 105. 

Partial half-lives of spontaneous fission and the number of neutrons set free are listed in Table 5.4. The partial half-lives are calculated by use of eq. (4.41) in section 4.8 (branching decay). They vary between the order of nanoseconds and about lO y. Although the fission barrier for nuclides such as (s 6MeV) is small compared with the total binding energy of the nucleons ( 1800 MeV), spontaneous fission of has a low probability compared with ot decay. 

Significant advances with respect to the quantitative theoretical description of spontaneous fission were achieved by the so-called shell-correction approach (Stru-tinsky, 1967) in which single-particle effects are combined with liquid-drop properties. This approach led to the prediction of a double potential barrier (Fig. 5.19) for some regions of Z and A. From the distorted state II the nuclei may pass much more easily over the fission barrier than from the ground state I. 

The double-humped fission barrier in Fig. 5.19 also makes it possible to explain the very short half-lives, of the order of nano- to microseconds, observed for some spontaneously fissioning nuclear isomers (e.g. "Am). By meas-... 

In section 5.6 it has been shown that nuclides with mass numbers T > 100 are energetically unstable with respect to fission. The fact that fission is not observed is due to the fission barrier. However, if enough excitation energy is transferred to heavy nuclides, the fission barrier can be surmounted and the nuclides undergo fission. 

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