ERNEST ORLANDO LAWRENCE BERKELEY NATIONAL LABORATORY Fission Transient Times from Fission Probabilities of Neighboring Isotopes

We present a new and straightforward method to esti mate the fission transient time by utilizing the cumulative fission probabilities of neighboring isotopes. The fission prob abilities were determined as the ratio of the measured fis sion cross sections to the Bass Model fusion cross sections. For five neighboring 185- 189 0s compound nuclei produced in 3 Het He-induced reactions on separated isotope W tar gets, the transient time rv is estimated to be smaller than 25 x 10- 21 seconds, and the most likely value of rv is about.

We present a new and straightforward method to estimate the fission transient time by utilizing the cumulative fission probabilities of neighboring isotopes. The fission probabilities were determined as the ratio of the measured fission cross sections to the Bass Model fusion cross sections. For five neighboring 185 -189 0s compound nuclei produced in 3 Het He-induced reactions on separated isotope W targets, the transient time rv is estimated to be smaller than 25 x 10-21 seconds, and the most likely value of rv is about. lOxl0-21 seconds.
The evolution of a fissioning nucleus from an assumed spherical shape towards the fission saddle, and eventually to the scission point, has been studied extensively (1)(2)(3). If the transient time ( TD) that a nucleus takes to evolve from a ground state shape to the saddle point is longer than the characteristic time for compound nucleus decay (TeN), then the fission probability is expected to be suppressed, and additional particles can be emitted as compared to those predicted by the standard theory. If on the other hand the transient time is short compared to TeN, then the stationary Kramers current (4] (i.e., the transition state fission rate) is expected.
Prescission particles can be emitted either before the system reaches the fission saddle, or during the descent from saddle to scission. Therefore, the fission time inferred from prescission particle emission is the sum of the transient time discussed above and of the time required for the nucleus to descend from saddle to scission. It is important to distinguish between presaddle and postsaddle times since postsaddle times do not affect the fission ·probability. Efforts have been made to separate the presaddle and postsaddle time components by examining the differences in the mean kinetic energy of charged particles emitted pre-and postsaddle (17]. The separation of presaddle and postsaddle particle emissions is, however, fraught with difficulties and ambiguities.
The transient time has a strong and direct effect on the fission probability. Consequently, its magnitude may 1 be determined more reliably from fission probabilities {18] rather ·than from indirect methods such as particle/photon emission.
In the following, we illustrate a new method to estimate the transient time, based upon high precision fission probabilities of several neighboring isotopes. This approach is based on the fact that, except for a factor accounting for the transient time effects, the second chance fission probability of a nucleus (Z, A) appears as the first chance fission probability of the neighboring nucleus (Z, A-1), whose second chance fission probability is in turn the first chance fission probability of its neighbor (Z,A-2), and so on. This novel approach, which does not involve any consideration beyond the fission saddle, automatically bypasses the difficulties associated with the separation of the presaddle and postsaddle particle emissions.
Assuming a step function for the transient time effects, the fission decay width can be written as  i, E-L l:l.Ej ), respectively. The inverse of these decay i=1 constants defines the corresponding characteristic times: . Given a transient time TD and assuming a step function for the transie.nt time effects, the number of nuclei (Z, A-i) must satisfy the balance equations:  [19]:
This solution, as written above, also provides the algorithm to follow the decay chain until all the excitation energy is exhausted.
With the solution N;(t) (i=0,1,2,. ··)in hand, the total fission probabilities P} can be simply calculated as where The transition state fission width r>co) = >. . 1 n and the neutron decay width r n = )..,h can be estimated as [18] r<ooJ Ps J r ex e , (10) Pn(E-Bn-E~") ex (11) where B 1 is the fission barrier, l:l.Ec is the pairing condensation energy, and flshell is the ground state shell effect of the daughter nucleus after neutron emission. For an even-even nucleus, l:l.Ec = (1/2)gll6; and for an odd A nucleus, l:l.Ec = (1/2)gllij-1:1 0 , where flo is the gap parameter and g is the doubly-degenerate single particle levels (g = (3/7r 2 )a with a being the level density parameter either at the saddle (aJ ), or at the ground state  (23]. For each excitation function, the contributions from first, second, third, · · · chance fission to the total fit (solid line) are shown. The TD value obtained from this simultaneous fit to seven excitation functions is lOxl0-21 sec, and a1/an is 1.062. 94.0%, 82.5%, 93.8% and 97.3%, respectively. Fission events were identified by detecting both fission fragments in two large area parallel plate avalanche counters. The experimental details are described in ref. [21]. The statistical errors of the measured fission cross sections u 1 are smaller than 2% for the compound nucleus Os isotopes at excitation energies above 50 MeV. Since the fission cross sections for all four isotopes were measured with the same detector setup in a single experiment, the systematic errors are estimated to be small ( "'4%). The fission excitation functions for compound nuclei 186 • 187 • 188 0s produced in 4 He-induced reactions on 182 · 183 • 184 W targets are also available [22]. All these excitation functions cover an excursion in fission cross section from 10-5 mb to 10 mb. The fusion cross sections u 0 of the above reactions can be estimated with theoretical models such as the Bass Model (23] (see Fig. 1).
We extracted a value for rv by fitting simultaneously 3 all the available fission excitation functions for the compound nuclei e 89 0s, 188 0s, 187 0s, 186 0s and 185 0s) with Eqs. 6 & 7. In order to reduce possible correlations between different parameters, we proceeded as follows. To extract the fission barriers B 1, we first fit the low energy ( <70 MeV) portion of the fission excitation functions. In this fit, the· fission barriers B 1 for the nuclei 189,188,187,186,1850s were taken as free parameters. The value of rv was set to zero since first chance fission is expected to dominate at low energies. Setting rv = 0 reduces the formalism to the form with which the first chance fission probability is usually obtained. The ratio a 1 I a" was assumed to be the same for all nuclei, but its value was let free in the fit. a" was assumed to be Al8  Fig. 2. We now let rv free, and fit the complete excitation functions with TD and a 1 I an as the only free parameters, using the fission barriers obtained above as the fixed parameters. In Fig. 1, we show the simultaneous fit for five neighboring osmium compound nuclei among which 186 0s and 187 0s were produced in both 3 He-and 4 Heinduced reactions. All seven fission excitation functions are well reproduced with only two free parameters, and the value obtained for rv from this fit is 10(±1)x10-21 sec. This rv value is consistent with the conclusion reached from the universal scaling in fission probabilities [18,25], and also consistent with the recent rv values reported in (13)(14)(15)(16).
The a 1 I an value given by the fit is 1.062. It is found that fits of comparable quality can be achieved for other a 1 1an values in a small range centered at 1.062 (see the x 2 values in the upper panel of Fig. 3). Higher estimates for fission probabilities resulting from a larger a 1/ an value seem to be (to a substantial extent) compensated by a larger value of rv (bottom panel of Fig. 3), and vice versa. This correlation between a 1 I an and rv values makes it difficult to obtain a unique value for rv. A good fit can be obtained with a TD value as small as zero, but not with a rv value larger than 25x10-21 sec, above which the fit not only requires an even larger a/ /an value (>1.075), but also the x 2 of the fit become greater than twice the minimum value.
The fusion cross, sections u 0 (see Fig. 1), which were calculated with the Bass Model interest to this work (>70 MeV), that could be used to judge the correctness of the Bass Model calculations. If the actual fusion cross sections are lower than the Bass predictions, which is likely [19,26], the resulting value for the transient time rv will be smaller.
In summary, we have found a new and straightforward way to estimate the transient time of fissioning systems, by utilizing the cumulative fission prob~bilities of neighboring isotopes. For five Os isotopes, the fission transient time TD is estimated to be smaller than 25x 10-21 sec, and the most likely value of rv is about lO(±l)x 10-21 sec. The quality of the fit for TD=O is such that no modification of the standard theory is demanded. This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Nuclear Physics Division of the U.S. Department of Energy, under Contract No. DE-AC03-76SF00098.