Submitted to the Proceedings of the Third International Conference on Dynamical Aspects of Nuclear Fission

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1 Submitted to the Proceedings of the Third International Conference on Dynamical Aspects of Nuclear Fission August 30 - September 4, 1996, Casta-Papiernicka, Slovak Republic Dynamical Fission Timescales in Hot Rotating Nuclei M. Thoennessen National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA The eects of dissipation in ssion of hot nuclei is still not well understood and quite controversial. The extraction of dissipation parameters from statistical models depends on the ssion description within the model. The inuence of the curvatures of the potential energy surface will be discussed. In addition, recent measurements and data analyses which conrm the description of the onset of dissipation in terms of the ratio of the threshold temperature over the ssion barrier will be described. 1. INTRODUCTION Dissipation in ssion has been included in calculations of the saddle-to-scission motion for a long time [1]. The observation of an enhanced emission of pre-ssion neutrons, protons, and -rays in the ssion of hot nuclei opened up the opportunity to study the detailed inuence of dissipation on the total ssion process [2, 3]. These experiments were typically analysed within statistical models and discrepancies between the calculated and measured pre-ssion particle/-raymultiplicitieswere attributed to the inuence of dissipation. Subsequent modications of the codes incorporated this dissipation as ssion time delays and/or with the help of the nuclear friction coecient or the reduced dissipation constant. The relations between these constants and the ssion time delay is not straight forward and can depend critically on intrinsic parameters, for example, the curvature of the potential energy surface which will be discussed in the next section. Although the inuence of dissipation on the ssion process is well established, the temperature and/or deformation dependence is still controversial (see Ref. [4] and references therein). The importance of a phenomenological parameter, the ratio of the threshold temperature and the ssion barrier, and recent conrmation from new data analyses will be described in Section DISSIPATION PARAMETERS IN STATISTICAL MODELS Fission as a decay mode of heavy compound nuclei was rst described with the transition state model [5], where the ssion decay width, trans is given by:, trans = 1 2(E) Z E,EB 0 (E, E B, )d (1) where E B is the ssion barrier, E the excitation energy and the compound nuclear level density. is the \:::density of all levels which arise from excitation of all degrees of freedom 1

2 other than ssion itself." [5]. With the explicit distinction between and equation 1 can be approximated by, trans = ~! eq 2 exp(,e B=T ) (2) where! eq is the curvature at the equilibrium deformation (barrier assault frequency). However, statistical models do not depend on intrinsic quantities like the shape of the barrier and the ssion decay width diers from the transition state model. In the statistical model the two level densities of Equation 1 are set equal = and the ssion decay width used in statistical model codes can be expressed as:, stat = T 2 exp(,e B=T ) (3) Thus, the transition state value and the statistical model value are not identical and are related via:, trans = ~! eq T, stat (4) This dierence has actually been known for a long time [6] but it is usually not taken into account and the approximation ~! eq = T is typically made in the analyses. It is not obvious that this approximation should be valid, because the decay ofthe compound nucleus covers a wide range of temperatures and the curvature at the equilibrium deformation (assault frequency) certainly changes with increasing angular momentum until the barrier vanishes. The left side of Figure 1 shows the assault frequency as a function of compound nuclear spin for dierent temperatures of 200 Pb. The calculations were performed based on the nite range liquid drop model and the parameterization of Lestone [7, 8]. The average value of ~! eq is around 1 MeV, however, it varies considerably, especially towards larger spins close to the vanishing of the barrier. The right side of Figure 1 shows the ratio of the curvature over the temperature ~! eq =T. The approximation of the statistical model assumes this value to be constant equal to unity. As can be seen, the ratio is certainly not constant and varies in the present range from 0:5 to1:6, which could certainly have a signicant inuence on the calculation of pre-ssion particle emission probabilities and particle multiplicities. The eect of this dependence is illustrated for the case of 19 F+ 181 Ta at 120 MeV. Figure 2 shows the experimental evaporation residue cross section (bottom) and the neutron multiplicity (top) as shaded areas [9]. A standard statistical model calculation reproduces the cross section but underpredicts the pre-ssion neutron multiplicities. The incorporation of the ~! eq =T correction changes the results signicantly. Neither the cross section nor the multiplicity is reproduced. This discrepancy becomes even larger when the temperature dependence of the potential landscape is included in the calculation of the barrier curvature. The pre-ssion particle multiplicity aswell as the residue cross section are the important observables for the determination of dissipation parameters. Thus this eect should be included when statistical models are applied in order to calculate dissipation constants. Dissipation is included in these codes following the description by Kramers. He introduced ssion as a dynamical process and calculated the ssion decay width, Kra as a function 2

3 eq eq Figure 1: Curvature of the potential (left) and the ratio of the curvature over the temperature (right) as a function of spin for 200 Pb at temperatures of 0.8 MeV (solid), 1.2 MeV (shortdashed), 1.6 MeV (dot-dashed) and 2.0 MeV (long-dashed). of dissipation [10] which can be written as:, Kra = ~! eq 2 (p 1+ 2, )exp(,e B =T ) (5) with the temperature T and the nuclear friction coecient. Thus, the Kramers ssion width corresponds to the transition state width, modied by the factor p 1+ 2,. It is now possible to include the curvature of the barrier explicitly into the statistical codes by combining equations 3 and 5:, Kra =( p 1+ 2, ) ~! eq T, stat (6) instead of earlier calculations where, stat =, trans was assumed. Such a dynamical calculation with = 10, which had been applied in previous calculations for this system [11] reproduces the multiplicity, however, the cross section is overestimated (see Figure 2). This overprediction was also present in the calculation of Ref. [11], however, the data were reproduced by reducing the overall ssion barrier. The last column of Figure 2 which reproduces both evaporation and pre-ssion neutron multiplicity is a calculation with a temperature dependent dissipation coecient in a description similar to the one applied in Ref. [12]. This temperature dependence of the dissipation coecient as a function of the excitation energy (top) and the resulting reduction of the ssion decay width compared to the transition state calculation is shown in Figure 3. At high excitation energies the inuence of the dissipation is large whereas at low excitation energies the full statistical rate is restored. 3

4 Figure 2: Pre-ssion neutron multiplicities (top) and evaporation residue cross sections (bottom) for the reaction 19 F+ 181 TaatE Beam = 120 MeV calculated for conditions explained in the text. The experimental values (with uncertainties) are indicated by the shaded areas. Figure 3: Dissipation coecient (top) and the resulting reduction of the ssion width (, Kra =, trans ) (bottom) as a function of excitation energy for the reaction 19 F+ 181 Taat 120 MeV. Figure 4: The curvature of the saddle point as a function of spin for 200 Pb at temperatures of 0.8 MeV (solid), 1.2 MeV (short-dashed), 1.6 MeV (dot-dashed) and 2.0 MeV (long-dashed). 4

5 The analysis presented in Figure 2 is not meant to extract details of the temperature dependent dissipation coecient, but rather to demonstrate the importance of the detailed shape of the ssion barrier. Another approximation which has been applied relates to the conversion between the friction parameter and the dissipation coecient : = =2! sp : (7) Typically a constant! sp = s,1 is used which yields = =2 10,21 s. However, Figure 4 shows the curvature as a function of the compound nucleus spin for dierent temperatures for 200 Pb. Again, these calculations were performed with the nite range liquid drop model using the parameterization of Lestone [7, 8]. Although the average value is ~! sp 0:75MeV corresponding to! sp s,1 there is a rather large spread which is strongly temperature as well as spin dependent. Thus no direct conversion between and is possible, and the comparison of results with analyses based on the two dierent dissipation constants should be treated with caution. 3. THRESHOLD OF DISSIPATIVE FISSION One of the main open questions is the temperature dependence of dissipation. Whereas one-body dissipation has at most a p T dependence, the predictions for the temperature dependence of two-body dissipation varies from stronger than T 2 to 1=T 2 (see Ref. [2, 3]). Experimentally, the temperature dependence can be explored by excitation function measurements of several dierent observables. The rst excitation function of pre-ssion neutron multiplicities showed that only at low energies the statistical model reproduces the data well [9]. However, at higher beam energies the model underpredicted the pre-ssion neutron multiplicities indicating a longer ssion time scale. A rather rapid onset of the dissipation was observed as a function of excitation energy in GDR -ray experiments [12, 13]. Recently, the analysis of prescission neutron multiplicity measurements in strongly damped reactions was consistent with the GDR results [14]. Although a variety of experiments have been performed measuring long ssion times at higher excitation energies, the transition from low energies to high energies has not been studied in detail. A compilation of a large data set found an empirical relation for the onset of the observation for this dynamical eect [15]. The ratio of the temperature above which the statistical model fails to reproduce the data (dened as the threshold temperature T thresh )over the (temperature dependent) ssion barrier is 0.26 independent of the system and analysis. The analysis included heavy-ion fusion, proton induced fusion, and peripheral reactions with the detection of charged particle, neutron and GDR -ray multiplicities [15]. Only two measurements did not seem to t the systematic. These two reactions 32 S+ 184 W and 32 S+ 208 Pb are the only reactions that have considerable contributions from quasi-ssion reactions and it is thus not surprising that these reactions do not follow a systematic trend that depends on the barrier. The analysis of excitation functions of ssion cross section measurements which were 5

6 T thresh /E Bar (T) Ni Mo 16 O Sm A 260 Figure 5: Ratio of the threshold temperature T thresh over the temperature dependent ssion barrier E bar (T ) as a function of mass. The previously published data (open circles, [15]) are shown together with the new analysis of ssion cross section data (solid circles [17, 16]). The calculated values for the Temperature over the ssion barrier for the reactions 16 O+ 144 Sm (open square) and 60 Ni Mo (solid square) are also indicated. not included in the original analysis are shown in Figure 5 (solid circles) together with the previous data (open circles). The data were taken from Ref. [16, 17] and the extracted onset for discrepancies from statistical models agrees nicely with the systematics. In contrast, a recent analysis of -induced fusion-ssion reactions did not observe any deviations from the statistical model even for values above T thresh =E bar =0:26 [18]. However, the analysis of ssion cross section measurements only are not sucient because of the uncertainty of several parameters in the statistical models. It is crucial to measure both, cross sections and pre-ssion multiplicities, simultaneously. Another experiment which follows the systematics was performed by Fiore et al. [19]. They measured the reactions 16 O+ 144 Sm and 60 Ni Mo to form the compound nucleus 160 Yb at the same excitation energy of 75MeV. The measured pre-ssion neutron multiplicity for the oxygen induced reaction ( exp =1:3 +0:4,0:2) agreed with statistical model calculations ( SM =1:2) whereas the pre-ssion neutron multiplicity from the nickel induced reaction was signicantly higher ( exp =1:4 +0:2) than predicted (,0:7 SM =0:3). This eect can be explained with the dierent angular momenta populated in the two reactions which are shown in Figure 6. The ssion decay channel dominates for angular momenta larger than 55~. Thus, the average ssion cross section for 16 O+ 144 Sm is substantially smaller ( fis 73mb) compared to 60 Ni Mo ( fis 316mb). In addition, the average angular momentum for ssion is lower, 55~ compared to 65~. Thus, the ssion barrier is larger for the oxygen induced reaction yielding a smaller value for the ratio T thresh =E bar (the temperatures are only slightly dierent). The resulting values of T thresh =E bar for 16 O Sm (open square) and 60 Ni Mo (full square) are included in Figure 5. Based on the systematics it is thus expected that the pre-ssion neutron multiplicity for the reaction 6

7 σ (mb) O Sm L Max = 53 h 60 Ni Mo L Max = 78 h Angular Momentum (h) 80 Figure 6: The fusion cross section as a function of the angular momentum for the two reactions 16 O+ 144 Sm and 60 Ni Mo. The corresponding ssion cross sections are indicated by the vertical and horizontal lines, respectively. 16 O+ 144 Sm agrees with the statistical model, whereas the reaction 60 Ni Mo should be inuenced by dissipation and thus the statistical model underpredicts the pre-ssion neutron multiplicities. Another example is the recent analysis of large fragment emission which did not need any dynamical eects and could be reproduced with the statistical model in agreement with the present relation [20]. The large barriers for the emission of large fragments yield signicantly lower values for T=E Bar ( 0:1) than the expected onset of observation of dissipation. The validity and underlying physical meaning of this phenomenological parameter is still controversial and has to be studied by further experiments [21, 22, 23]. 4. CONCLUSIONS The curvatures of the potential energy surface can be important in determining dissipation parameters and should be included in statistical model calculations. The temperature dependence and/or the proposed deformation dependence which was not discussed in the present paper is still to be determined [24]. For future measurements it is absolutely essential to measure both, evaporation residue/ssion cross sections and pre-ssion particle/-ray multiplicities, simultaneously. Otherwise it is not possible to constrain statistical model parameters and extract the dissipation coecients. This work was supported in part by the National Science Foundation under grant no. PHY

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