Fission barrier heights and lifetimes for heavy and superheavy nuclei

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$M. Winiarski, Prezentacje multimedialne w LATEX-u$

TABLE 1. avoi =-15.4920 MeV asur= 16.9707 MeV acur = 3.8602 MeV To = 1.21725 fm Parameters entering the LSD model K-voi= 1.8601 Ksui= 2.2938 Kcur = -2.3764 C4 =0.9181 MeV FISSION BARRIER HEIGHTS AND a-DECAY LIFETIMES Using a gradient method (see Ref. [13] for details), we have located all physically relevant stationary points for a sample of 18 actinide nuclei. Fig. 2 displays the fissionbarrier heights for these nuclei calculated as the difference between the total ground state and the saddle-point energy. In the lower part of the figure the difference of the experimental and theoretical barrier heights is given. We conclude that our macroscopic- > ^ CD > 2 m X CO 7.5 7 6.5 6 5.5 5 4.5 4 2 1 0 -1 - -3 n 23ep^ n 232T-„ n 234y + 2S*U + 232Th + ^*PU , D D D 23Bp^ 236^ + 2S*Th + 23Bp„ B D + ^ 238^ 238.. 240^ni H 2lJgn 240y + 2*"U 140 142 144 146 148 150 , N + »«cm + »*Pu , n 24ep^ B gSgm * 2Sg[" + Pu , n 260(,m + 26"cm , J - - •: - -, i - 152 154 156 FIGURE 2. Theoretical barrier heights Sth of actinide nuclei from ^^^Th to ^'"Cf (upper) and difference between experimental [20] and theoretical barrier heights (lower part) as function of neutron number microscopic approach with the LSD macroscopic energy, the Strutinsky shell correction and the pairing treatment in terms of the BCS approach with the uniform-gap method for the determination of the pairing strength is able to reproduce the fission barrier heights on the average within less than 1 MeV. Since our theoretical estimates for the barrier heights are almost all higher than the experimental values, this discrepancy will still be reduced e.g. when taking the effect of different proton-neutron deformations into account, as has been demonstrated in Ref. [13] where it was shown that such an additional degree of freedom in the macroscopic-microscopic approach can lead to an energy gain up to about 1 MeV. Note that in our calculations no zero-point vibration energy is included since the macroscopic-microscopic model or the Hartree- Fock-Bogoliubov type approach are based on a variational method which by definition yields an upper bound of the nuclear ground-state energy and no further corrections are therefore needed. The obtained accuracy is practically of the same quality as the barrier 236
heights estimates of Ref [20]. Keep in mind, however, that the parameters of the finite range LDM approach [21] were re-adjusted to reproduce experimental barriers in the best possible way. The parameters of the LSD model used in our approch were, on the contrary, adjusted only to experimentally known gound-state masses. In this respect it is quite gratifying to notice the good capacity of our method to reproduce fission barriers, which seems to indicate that the essential physics is correctly contained in our approach and this, as pointed out before, with a minimal number of adjustable parameters. 801 6 801 8 802 0 802 2 802 4 802 6 802 8 803 0 803 2 803 4 803 6 803 8 804 0 804 2 804 4 804 6 804 8 805 0 805 2 805 4 805 6 805 8 806 0 FIGURE 3. Energy landscape of the nucleus Pu at a deformation corresponding to the second saddle point as function of the left-right asymmetry parameter a and the non-axiality parameter 77. It is also quite interesting to investigate the importance of taking left-right asymmetry and non axiality into account when calculating fission barrier heights. For that purpose we have calculated the energy gain obtained including, in addition to the elongation and neck parameters c and /i, also the left-right asymmetry and non-axiality parameters a and 77 • It is generally admitted that non-axiality plays a role in the vicinity of the first barrier and that left-right asymmetry is of mayor importance at the second barrier. It turns out, however, in our study that both these degrees of freedom are important as well at the first as at the second barrier and that the commulative effect of both is of the order of 1-2 MeV at the first, but reaches up to 7 MeV at the second barrier. We would like to attract the reader's attention to the fact that even if at a given deformation point, as at the second saddle, the minimal energy turns out to be axially symmetric, the energy landscape in our 4-dimensional deformation space might show another local minimum at almost the same energy which exhibits a substantial non-axiality, as shown in Fig. 3 for the nucleus ^'^''Pu at a deformation corresponding to the second saddle. This second local minimum found here at fixed values of c and h is believed to belong to a different fission valley and that one would need to carry out a dynamical calculation taking mass parameters into account to make any reasonable predictions about a fission path or fission lifetime. Motivated by our quite encouraging results on fission-barrier heights caused, as we believe, by a correct description of nuclear deformation energies, we can make some predictions on a-decay half lives. For such an estimate the Viola-Seaborg expression [22] is often used. Recently an even simpler expression [23, 24] has been proposed which for even-even nuclei reads log,oTa{Z,N) aZ VQa{Z,N) -hZ- (13) 237
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Fission barrier heights and lifetimes for heavy and superheavy nuclei

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