Fission fragment angular distribution - Analysis and results

Transcription

1 Chapter 6 Fission fragment angular distribution - Analysis and results Angular distribution of the fission fragments in heavy ion induced fission is an important probe to understand the dynamics of heavy ion collisions. Fission fragment angular distributions were generally explained by the SSPM. In SSPM, fragment angular distribution is related to the angular momentum (J) distribution and the width (K 0 ) of the K (projection of J along the nuclear symmetry axis) distribution of the fissioning nuclei. K 0 in turn is related to the nuclear temperature and effective moment of inertia at the saddle point. This makes the angular distribution measurements a powerful tool to understand the shape of the fissioning nuclei at saddle point and also the angular momentum involved in the fusion process. It also answers to the diffusion process of viscous nuclear medium during the fission decay. Fission fragment angular distribution has been widely used to probe [, ] the dynamics of fusion-fission process in actinide region. Experimental measurements over years have shown that fragment angular distribution is sensitive to entrance channel parameters as well as the statistical aspects of intermediate system as it evolves in time. In this chapter we describe the data analysis, statistical model calculations and results obtained from the fragment angular distribution measurements for 6 O+ 94 Pt reaction. 6. The analysis 6.. Fission cross section and angular distributions The fission yields for each angle were taken and were normalized by the monitor detector kept at 40 o degree. The measured fission fragment angular distributions were transformed from laboratory to center-of-mass frame using Viola systematics for sym- 46

2 metric fission [3]. Energy loss corrections of the beam in the half target thickness were applied before the conversion to centre-of-mass. In heavy ion induced fission, the fragments are preferentially emitted in forward and backward directions with respect to the beam and the angular anistropy A is defined as W(80o ) (or W(0o ) ) where W(90 o ) W(90 o ) W(80o ) is the differential fission cross section at 80 o and W(90 o ) is the same at 90 o (perpendicular to the beam direction). The differential fission cross section was calculated using the expression Y fis W(θ cm ) = dσ fis dω = dσ R Y mon dω (θ lab) Ω mon G (6.) Ω fis where G is the Jacobian of laboratory frame to centre-of-mass frame transformation, Y fis and Y mon are the yields recorded by the fission detector and monitor (Rutherford) detector, respectively. Ω fis and Ω mon are the solid angle subtended by the fission dσ detector and monitor detector, respectively. R (θ dω lab) is the differential Rutherford cross section. It was assumed that events recorded in the monitor detectors were due to pure Rutherford scattering. Total fission cross section was obtained by integrating the differential cross section dσ fis. This was done by fitting the differential cross section dω data using the equations dσ fis dω (θ) = a + b cos θ + c cos 4 θ (6.) and, ( σ(e) = 4π a + b 3 + c ) 5 (6.3) Table 6. shows the experimental fission cross section along with the statistical errors at different beam (c.m.) energies. Table 6.: Fission cross section for 6 O + 94 Pt reaction at different beam energies. E c.m. (MeV) σ fiss (mb) σ fiss (error) (mb)

3 Fragment angular distributions (normalized differential cross section plotted against centre-of-mass angle) at different beam energies for the present system are shown in Fig. 6.. W(θ) / W(90) E lab =89.8 MeV 3 E lab = 87.8 MeV E lab =85.8 MeV 3 E lab =8.8 MeV E lab = 80.8 MeV 3 E lab = 78.8 MeV θ (Degrees) cm Figure 6.: Fission fragment angular distributions for 6 O + 94 Pt reaction. Solid lines are the best fits to the angular distribution using the standard expressions [4, 5]. The angular distributions were fitted using the standard expressions [4, 5]. In the standard theory of fission fragment angular distributions, it is assumed that final direction of fragments is given by the orientation of the nuclear symmetry axis as the nucleus passes over the saddle point. It is assumed that nucleus is formed initially in a configuration inside the fission barrier. It is also assumed that the Coriolis force and other effects violating the conservation of K quantum number are of insufficient strength to alter its value during the rapid descent from saddle to scission. Both these arguements are not valid for reactions involving heavier projectiles. Under the standard assumptions, angular distributions can be represented by the symmetric top 48

4 wave functions WMK J J + (θ) = 4π DJ MK (φ, θ, ψ) (6.4) where D J MK (φ, θ, ψ) are the symmetric top wave functions. For spin zero nuclei, M, the spin projection, onto the beam axis becomes zero and the above expression reduces to W0K J J + (θ) = 4π dj 0K (θ) (6.5) For comparing with the experimental data, it is necessary to average over J and K. Halpern and Strutinsky [6] derived the expression for K distribution using a constant temperature level density assuming the statistical arguements. exp( K K ) 0 P ρ(k) = J for K J K K= J exp( K 0 ) 0 for K > J where K 0 = I eff T, is the variance of the K distribution and I eff is the effective moment of inertia, discussed in section.4.. Using these, the angular distribution of the fragments can be represented as, J K= J W(θ) = (J + )T J J=0 (J + )dj 0K (θ) exp[ K J K= J K 0 (J)] exp[ K K 0 (J)] (6.6) where T J is the transmission coefficient for fusion of the J th partial wave and K 0 (J) is the standard deviation of the K distribution for the J spin states. d-functions were calculated following method given in the ref. [5]. Fission fragment angular anisotropies (A = W(80o ) ) were obtained from the above W(90 o ) fit. Table 6. gives the experimental fragment anisotropy along with errors at different beam (c.m.) energies. 49

5 Table 6.: Experimental fission fragment angular anisotropy in 6 O + 94 Pt reaction at different beam energies. E c.m. (MeV) A expt A (error) Statistical model analysis SSPM relates the fragment angular anisotropy to the angular momentum J at the saddle point and the projection of this total angular momentum on the nuclear symmetry axis K. In the simplified form the fragment angular anisotropy is given by the approximate expression, A = + < l > 4K 0 (6.7) where K 0 is the variance of K distribution, that is, the statistical spread in K distribution, explained earlier in this chapter. Larger the value of K 0, more isotropic will be the angular distribution. If the re-separation occurs before the equilibration of K degrees of freedom, the system will have memory of its entrance channel. In such cases, narrower will be the K distribution, which leads to larger anisotropy values. Such large anisotropy values were experimentally observed for many systems. The effective moment of inertia, I eff, depends on the shape and hence the deformation of the nucleus at the saddle point. I eff is given by, = I eff I I (6.8) The saddle point temperature T is calculated using Eq B f (l) and E rot were calculated using the Sierk prescription [7] based on the rotating finite range model (RFRM). The mean square angular momentum (< l >) values of the fissioning nuclei were calculated using statistical model code PACE in trace back mode. As the CN 0 Rn decay via fission and particle evaporation, one cannot use the fusion < l >-values directly in SSPM calculations. The fusion l-distributions were obtained using the coupled channel code CCFULL by fitting the experimental fusion cross sections (sum 50

6 of ER cross sections and fission cross sections). This fusion l-distribution was used as the input to PACE in trace back mode to get the fission < l >-values. The CCFULL parameters used in the calculations for 6 O + 94 Pt reaction were discussed in chapter 4 In heavy ion induced fusion reactions the CN are formed at high excitation energy and angular momentum. These hot, rapidly rotating nuclei decay to their ground state via particle emission, gamma emission and fission. The competition between various decay channels is governed by the transmission coefficients and level densities of the final states. The particle emission and fission can occur from the CN itself or from its decay products until the excitation energy becomes less than particle separation energies and fission barrier, respectively. When the excitation energy becomes less than separation energy and fission barrier energy, gamma decay takes over the decay process. The remaining excitation energy and angular momentum will be removed in this way. Fission can take palce from the CN itself or after the emission of one or two neutrons. In mass 00 region, the neutron separation energy and fission barrier are comparable which assist the multi-chance fission to become a dominant decay channel. During the chance fission, when a neuton evaporates before the fission process it reduces the CN excitation energy by 8-0 MeV. However, the angular momentum carried away by the neutron evaporation is much less compared to the CN angular momentum. The experimentally observed angular distribution contains contribution from various steps of this decay process and anisotropy calculations assuming average angular momenta and excitation energies may give ambiguous results. Shell effects also play a very improtant role in fission dynamics. It is well known that nuclei with magic (, 8, 0, 8, 50, 8 and 6) proton or neutron numbers are more stable comapared to non-magic nuclei. Single-particle effects are also responsible for many other familiar nuclear phenomena. These include the occurance of deformed rather than spherical ground state shapes for mid-shell nuclei, the occurance of secondary minimum in the fission barriers of many actinide nuclei etc. These secondary minima are responsible for the occurance of fission isomers and intermediate structures in fission cross sections. The division of heavy nuclei at low excitation energies into fragments of unequal mass is also believed to be caused by these single particle effects. The sole reason for the existence of super-heavy nuclei (with vanishingly small liquid drop fission barriers) are also due to the shell effects in the potential energy surface. In mass 00 region, shell correction may not lead to any significant secondary minima in the nuclear deformation potential energy. This is due to the rapid variation of LDM potential energy with deformation. However, significant shell corrections to LDM 5

7 potential at saddle point were predicted in this mass region [8, 9]. The nuclear level density parameter, an important parameter in statistical model calculations, is also found to be very much sensitive to shell corrections. The level density parameter shows a dramatic reduction in its values near the shell closure. Shell corrections at the saddle point were not considered seriously in many previously reported works. This correction is particularly important for the nuclei having significant ground state shell corrections around A CN = 0. Aradhana et al [0] reported anomalous angular anisotropies in C + 98 Pt system (forming the CN 0 Po with neutron number = 6) and normal anisotropies in C + 94 Pt (forming the CN 06 Po with neutron number = 4). It was observed that the measured anisotropies were significantly larger than SSPM calculations assuming average excitation energy and angular momentum and the deviations from the calculations increased with decrease in energy. The authors speculated that shell effects in potential energy could be the possible reason for this anomalous behavior. Mahata et al [] further extended this study, measuring fission fragment angular distributions in 9 F + 94,98 Pt systems. 9 F + 94 Pt has a neutron shell closure (N = 6) which was expected to deviate from SSPM predictions if the earlier speculations were correct. Deviation from SSPM calculations were observed at few energies for this system. Statistical model calculations incoporating the multi-chance nature of fission as well as shell corrections at the saddle point could explain the experimental anisotropies satisfactorily. It was also observed that the re-analysis of C + 94,98 Pt data, with multi-chance fission taken into account in the calculations, reduced the difference between the experimental data from SSPM calculations considerably in C + 98 Pt reaction. In the present work, statistical calculations were performed using PACE, with fusion l-distribution as the input. The broadening of angular momentum at near barrier energies [] were taken into account by including the rotational couplings of the target nuclei. Fermi gas model was used to calculate the level denisty parameters for ground state deformations and saddle point deformations. As the level density parameter and fission barrier heights are sensitive to the shell correction, these parameters could be varied to fit the excitation functions. However, it was found that different sets of B f (l) and the ratio of level density parameter at saddle point and equilibrium deformation, a f a n, give good fit to the experimental data. Hence, in addition to the measured fission cross section and ER cross section, prefission multiplicities, ν pre, were also used to constrain the statistical parameters. To the best of our knowledge, ν pre values of 6 O + 94 Pt system are not reported in literature. Hence ν pre values were obtained from the systematics of Saxena et al [3]. In the calculations, the energy dependent shell 5

8 Fission probability (%) Experiment Statistical model E * (MeV) Figure 6.: Fission probabilty for 6 O + 94 Pt reaction plotted against CN excitation energy. Solid line is the statistical model fit to the experimental data. correction form (Ignatyuk level density parameter [4]) with asymptotic value A CN 9 was used for a n. Shell corrections were incorporated in fission barrier as well as in level density parameter in a systematic manner in the present study. The form of the fission barrier and level density parameters used in the calculations are given below. B f (l) = B RFRM f (l) n + k n (6.9) a n = a [ + n U ( e ηu ) ] (6.0) a f = ( a f a n ) a [ + k n U ( e ηu ) ] (6.) where a, n and η are the liquid drop level density parameter, shell correction at ground state and damping factor, respectively. k is the scaling factor and k n is the shell correction at saddle point deformation. k = correspond to equal shell corrections 53

9 3 Systematics Statistical model.5 ν pre E * (MeV) Figure 6.3: Pre-fission neutron multiplicities (obtained from the systematics) for 6 O + 94 Pt reaction plotted against CN excitation energy. Solid line is the statistical model fit. at the saddle and equilibrium deformation. On the contrary k = 0 correspond to zero shell correction at saddle point. Using the above prescriptions ER cross section, fission probability and ν pre values were simultaneously fitted. The fission barrier B f (l) and a f a n were varied during this procedure. The best fit to the experimental values were obtained for k = 0.76 and a f a n = Fig. 6. and Fig. 6.3 show the experimental fission probability and neutron multiplicities plotted against CN excitation energy. The solid line is the statistical model fit. Fig. 6.4 shows fission cross sections compared with the statistical model predictions. SSPM calculations were performed using the average excitation energy and < l > values obtained from the PACE output. Table 6.3 gives different parameters used to obtain the fragment anisotropies at different beam energies. The SSPM calculations assuming average excitation energy and angular momentum values overpredicted the angular anisotropies at higher energies. At lower excitation energies, however, the calculations matched with experimental results. It may be noted that in the energy range studied in the present experiment, the contribution from the fast fission process was negligible as the maximum angular momentum populated in the 54

10 Table 6.3: Different parameters used to obtain the fragment anisotropies at different beam energies. E c.m., < E >, B f (l), E rot and T are in MeV. I E c.m. < l > < E > B f (l) E eff rot K 0 < l > T reaction was much lower than the critical angular momentum at which the liquid drop fission barrier vanishes. The pre-equilibrium fission, another non-compound nucleus process that occurs when the temperature at the saddle point becomes comparable to the fission barrier, also did not contribute to the fragment angular distributions in the present system. The probability of occurance of pre-equilibrium fission is given by the approximate expression, P pef = e 0.5 B f T (6.) The maximum probability of pre-equilibrium fission in the energy range studied in 6 O + 94 Pt reaction is less than %. Fig. 6.5 shows the experimental anisotropies compared with SSPM calculations assuming average excitation energy and angular momentum. As mentioned earlier in this chapter, when multi-chance fission is a dominant decay process, calculations assuming average values of E and < l > give ambiguous results. The sensitivity of fragment angular distributions to fission occuring after particle emission from the CN has been pointed out long back in literature [4, 5, 6]. Hence fission fragment angular distributions were calculated using the expression W(θ) = J max m,e J=0 σ fission (m, E, J) J K= J (J + ) d J 0K (θ) exp[ K K0 (J)] J (6.3) K K= J exp[ K0 (J)] The distributions of the fissioning nuclei in different chance (m) in (E,J) space, σ fission (m, E, J), were calculated using PACE as mentioned earlier. However, fission fragment angular anisotropies calculated using the RFRM effective moment of inertia I eff, overpredicted the data. As the statistical parameters were already fixed 55

11 000 σ fiss (mb) 00 Experiment PACE E c.m. (MeV) Figure 6.4: Experimental fission cross sections at different energies (centre-of-mass) compared with PACE predictions. 3 Anisotropy (A).5 Experiment SSPM Calculations E (MeV) c.m. Figure 6.5: Fission fragment angular anisotropies in 6 O + 94 Pt reaction compared with SSPM calculations using average values of angular momentum and excitation energies. 56

12 in our calculations, the anisotropies were fitted by scaling up the RFRM effective moment of inertia. It was found that I eff scaled up by a factor of.0 ±0.04 reproduced the experimental anisotopies. The error in the scaling factor represents the standard deviation. Fig. 6.6 shows the experimental anisotropies compared with theoretical calculations assuming average values of < l > and E, RFRM effective moment of inertia I eff and I eff increased by 0%. Anisotropy (A) (MeV) E c.m. Experiment SSPM with Scaled I eff. + <l > / 4 K 0 SSPM Without I eff. scaling Figure 6.6: Fission fragment angular anisotropies in 6 O + 94 Pt reaction compared with SSPM calculations using average values of angular momentum and excitation energies, RFRM effective moment of inertia I eff and I eff increased by 0%. 6.3 Discussion The disagreement between experimental results and SSPM calculations assuming average values of angular momentum and excitation energies confirmed the role of shell corrections at saddle point in the calculations. It was also realised that when the fission barrier is comparable with neutron separation energies, multi-chance nature must be taken into account in the calculations, as the experimentally observed angular distribution contains contribution from all steps in the decay process. Detailed statistical analysis revealed that the ER cross sections and fission probabilty can be fitted satisfactorily by different sets of B f (l) and a f a n values, which may not yield proper angular distributions. It may be noted that the ν pre values were also influenced by B f (l) and 57

13 a f a n values. In order to have unambiguous values of J distribution and excitation energies, ν pre values must also be fitted simultaneously to constrain the statistical model parameters. The experimental results also showed that RFRM I eff -value has to be increased by 0% to fit the anisotropy values. This result is consistent with previous measurements mass 00 region. It may be noted that I eff values were multiplied by. ±0.7 in the case of 9 F + 94 Pt and.37 ±0.5 in the case of 9 F + 98 Pt to fit the experimental results []. Similarly a multiplication factor of 4 ±0. and 0.96 ±0.0 were required to fit anisotropy values in C + 94 Pt and C + 98 Pt reactions, respectively. Average scaling factor obtained for 9 F + 88(9) Pt system was.7 ±0.6 (.39 ±0.8) [7]. The larger I eff values observed in these systems could imply more compact shapes at the saddle point as compared to that predicted by RFRM. However more experimental results are required to have a systematic understanding of the role of multi-chance fission, shell closure and the shell corrections at saddle point, in fusion-fission process. 58

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