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Determination of shell correction energies at saddle point using pre-scission neutron multiplicities
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Nuclear Physics A 913 (2013) 157–169 www.elsevier.com/locate/nuclphysa
Determination of shell correction energies at saddle point using pre-scission neutron multiplicities Golda K.S. a,b,∗ , A. Saxena c , V.K. Mittal b , K. Mahata c , P. Sugathan a , A. Jhingan a , V. Singh d , R. Sandal d , S. Goyal e , J. Gehlot a , A. Dhal a , B.R. Behera d , R.K. Bhowmik a , S. Kailas c a Inter University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi 110067, India b Physics Department, Punjabi University, Patiala, Punjab 147002, India c Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India d Department of Physics, Panjab University, Chandigarh 160014, India e Department of Physics and Astrophysics, University of Delhi, New Delhi 110007, India
Received 30 August 2012; received in revised form 8 March 2013; accepted 23 May 2013 Available online 14 June 2013
Abstract Pre-scission neutron multiplicities have been measured for 12 C + 194, 198 Pt systems at matching excitation energies at near Coulomb barrier region. Statistical model analysis with a modified fission barrier and level density prescription have been carried out to fit the measured pre-scission neutron multiplicities and the available evaporation residue and fission cross sections simultaneously to constrain statistical model parameters. Simultaneous fitting of the pre-scission neutron multiplicities and cross section data requires shell correction at the saddle point. © 2013 Elsevier B.V. All rights reserved. Keywords: N UCLEAR REACTIONS 194,198 Pt(12 C, f ), E ∗ = 49.3, 54.3, 59.4, 62.2 MeV; measured fission fragments, En , In (θ) using ToF; deduced neutron multiplicity vs En and θ , pre-scission neutron multiplicity vs E ∗ , fusion–fission σ ; calculated fission barrier, shell effects, statistical model parameters, pre-scission neutron multiplicity vs E ∗ , fusion–fission σ , fission σ vs E ∗ , evaporation residue σ vs E ∗ . Compared with published 208 Pb + α system
* Corresponding author at: Inter University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi 110067, India.
E-mail address: [email protected] (K.S. Golda). 0375-9474/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysa.2013.05.016
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1. Introduction The significance of nuclear shell structure in the formation of a new element in heavy ion induced nuclear reaction is a problem of current interest [1,2]. Few attempts have been made recently to understand the role of shell correction energies on the fission threshold as a function of the nuclear deformation [3–5]. Such studies bring out the interplay of macroscopic properties of nuclear matter to the microscopic effects such as deformation and shell structure of the nuclei [6]. The quantitative determination of fission barrier height is the key to understand the dynamics of heavy ion fusion–fission process and the prediction of the super heavy elements. Many researchers [3,7,8] have carried out theoretical calculations for the fusion–fission probability in the super heavy mass region in the frame work of a potential energy landscape in multidimensional deformation space. The existence of the island of stability has also been proposed using the nuclear shell model [9]. However, very few attempts have been carried out [5] to extract the shell correction to the fission barrier from the experimental fission data in heavy ion induced fusion–fission reactions. It is well established that the microscopic (shell) correction in the liquid drop nuclear energy surface [10,11] results in the multiple humps, valleys and saddle points along the nuclear trajectory in the potential energy landscape. Nevertheless the quantification of the exact shell correction values required to explain all the characteristics of the fissioning nucleus is extremely difficult. Moreover there are limited experimental observations [4] indicating the possible effects of shell structure in the fusion–fission reaction mechanism. Shrivastava et al. [4] have reported that the measured fission fragment angular anisotropies for 12 C + 198 Pt system have been found to be larger compared to the Statistical Saddle Point Model (SSPM) calculations, where as the observed anisotropies for 12 C + 194 Pt system have been found to be in good agreement with the SSPM calculations. The disagreement between the SSPM predictions and the experimental data for 12 C + 198 Pt system, which populates 210 Po as compound nucleus (CN), has been found to decrease with increasing excitation energy. This was interpreted as a signature of shell effects as the CN has closed shell neutron number and which could provide an extra stability against fission. Later it has been shown that [5] the shell corrections at saddle point is necessary to explain the measured fission and evaporation residue cross sections and the pre-scission neutron multiplicity data simultaneously in mass ∼200 region. It was also shown that more than one set of statistical model parameters could be used to explain the experimental fission as well as evaporation residue cross sections satisfactorily. However, a unique parameter set could be achieved if the pre-fission neutron multiplicity data are included in the analysis. The pre-scission neutron multiplicity data was not available so it was taken from the Baba’s systematics [12] in this analysis. In this regard experimental data for pre-scission neutron multiplicity is necessary to constrain the statistical model calculations to quantify the shell correction energies. With this motivation, we have carried out measurements of neutron spectra for 12 C + 194,198 Pt systems. The specific selection of these systems enabled us to do a meaningful statistical model analysis as the partial evaporation residue cross sections as well as the fission cross sections are already measured [4,13]. In the present study, care has been taken to select the systems which are asymmetric on the entrance channel and well above the Businaro–Gallone [14] critical mass asymmetry to avoid the contamination of non-compound fission events. 2. Experimental details The experiment was carried out using the heavy ion facility at Inter University Accelerator Centre (IUAC), New Delhi. Pulsed beam of 12 C delivered by the 15UD Pelletron, was
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Fig. 1. Schematic of the experimental setup.
bombarded on isotopically enriched self supporting rolled foils of 194 Pt and 198 Pt of thickness’s ∼1.75 mg/cm2 and ∼1.45 mg/cm2 , respectively. The beam energies had been adjusted to have same excitation energies for both the systems. Fission fragments were detected by a pair of 5 × 3 position sensitive multi-wire proportional counters kept at ±40◦ degree with respect to beam direction at a distance of 17 cm from the target position. The fast timing signals from the fission counters were used to get the flight time information of the fission fragments which enabled us to separate fission events from other reaction products. Neutron time of flights (TOF) were measured using 24 liquid scintillators (BC501) kept at different angles around the target chamber. Out of these 24 detectors, 16 detectors (5 diameter × 5 deep) were kept in reaction plane and the remaining 8 detectors (5 × 3 ) were kept at 15◦ up and down with respect to the reaction plane. Four of the reaction plane detectors were placed at 1 meter away from the target. All the remaining detectors were placed at 2 m away from the target in a cylindrical fashion as shown in Fig. 1. In order to reduce the gamma background seen by the detectors, the beam was dumped 4 m downstream from the target and was well shielded with paraffin and lead bricks. The time spread (∼1 ns)in the beam pulse was monitored using a BaF2 detector placed near to the beam dump. Discrimination between neutrons and γ rays was made by using pulse shape discrimination based on the zero-crossing technique and the TOF. The TOF of neutrons were converted into neutron energy by considering the prompt gamma peak in the TOF spectrum as the reference time. The obtained energy distribution was corrected for energy dependent detection efficiency of the neutron detectors. The intrinsic efficiency of neutron detectors were determined by measuring the neutron energy spectra from a 252 Cf spontaneous fission source of known strength kept at the target position. 3. Data analysis and results The pre- and post-scission components of neutron multiplicities and temperatures were obtained from the measured neutron energy spectra by using a multiple source least-square fitting procedure, using Watt expression [15,16]. Three moving sources of neutrons (compound nucleus and two fully accelerated fission fragments) were considered while determining the multiplicities from the fits. The neutrons emitted from these moving sources were assumed to be isotropic in their respective rest frames. Thus the measured neutron multiplicities are given as
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Fig. 2. (Colour online.) Double differential neutron multiplicity spectra (filled circles) for 12 C + 194 Pt system at 59.4 MeV excitation energy along with the fits for the pre-scission component (dashed curve) and post-scission components from fragment 1 (dotted curve) and fragment 2 (dot dashed curve). The solid curve represents the total contribution.
√ √ 3 d 2M Mi E n En − 2 En Ei /Ai cos θi + Ei /Ai = × exp − . dEn dΩn Ti 2(πTi )3/2
(1)
i=1
Where En is the laboratory energy of neutron and Ai , Ei , Ti and Mi are the mass, energy, temperature and multiplicity of each neutron emitting source i. θi is the direction of neutron with respect to the neutron emitting source. The kinetic energy of the compound nucleus ECN was determined by assuming full momentum transfer. Fission fragment energies and folding angles were obtained from Viola’s [17] systematics assuming fission to be symmetric. Simultaneous fitting was done for all the available fission coincident neutron data set at the highest excitation energy. Though the array consists of 24 detectors we have used only 5 × 5 detectors for the moving source fits due to their higher efficiency and better statistics. For the lower excitation energies we have considered the data exclusively from the detectors placed at 1 m distance due to the limitations in sufficient statistics. Fig. 2 represents the typical fits to the double differential neutron multiplicity spectra for 12 C + 194 Pt system at 59.4 MeV excitation energy. The angles of neutron with respect to CN and each of the fission fragments are indicated in the plots. The temperature of the fissioning nucleus, Tpre was calculated as T = (E ∗ /a) ˜ where E ∗ is the compound nucleus excitation energy and a˜ is the level density, by assuming a˜ to be ACN /9 MeV−1 . Tpre was scaled down to (11/12)T to account for the cascade of sequential particle evaporation [15] and its value was fixed in the moving source fits. All the remaining parameters were obtained by fitting the data with Eq. (1). Fitting was also done with Tpre as a free parameter. Thus obtained neutron multiplicities are within the errors of the values obtained with fixed Tpre . The post-scission multiplicity and the temperature were taken to be the same for both the fission fragments assuming symmetric fragmentation and were determined from the
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Table 1 Experimentally obtained temperatures and multiplicities for 12 C + 194 Pt system as determined from moving source fits. E∗
Tpre
Mpre
Tpost
Mpost
49.3 54.3 59.4
1.36 1.42 1.48
1.30 ± 0.13 1.59 ± 0.16 2.61 ± 0.24
1.09 ± 0.13 0.78 ± 0.14 1.05 ± 0.16
0.72 ± 0.11 0.71 ± 0.13 0.70 ± 0.10
Table 2 Same as Table 1 for 12 C + 198 Pt system. E∗
Tpre
Mpre
Tpost
Mpost
49.3 54.3 62.2
1.11 1.41 1.49
1.11 ± 0.1 1.31 ± 0.13 2.10 ± 0.19
0.88 ± 0.10 1.09 ± 0.18 1.11 ± 0.17
0.66 ± 0.11 0.69 ± 0.13 0.71 ± 0.12
Fig. 3. Experimental total neutron multiplicities as a function of total available excitation energy.
fits. The angular acceptances of neutron and fission detectors were incorporated in the moving source fits. The parameters determined from the moving source fits are listed in Tables 1 and 2. The Mpre values increase with excitation energy for both systems while the Mpost values show negligible variation. In order to further confirm the values of experimentally obtained pre-scission neutron multiplicities, an exclusive moving source fitting was carried out by considering the data only from the neutron detectors perpendicular to the fission direction, where contribution from post-scission component is expected to be small, for 54.3 MeV excitation energy for both the systems. The pre-scission multiplicity value thus obtained match with value determined after including all detectors within the uncertainties of fits. The total neutron emission (Mtotal = Mpre + 2Mpost ) in a fusion–fission reaction corresponds to the available excitation energy of the system which includes the excitation energy brought by the projectile, excitation energy of fragments due to dissipation of potential energy during saddle to scission and deformation energy. In Fig. 3 the total evaporative neutron multiplicities for both 12 C + 194, 198 Pt systems are plotted against the available excitation energy (E ∗ +Qfiss -TKE). The fission Q-value (Qfiss ) has been calculated by taking the respective experimental mass values
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from the 2003 Atomic Mass Evaluation by Audi et al. [18] and assuming fission to be symmetric. The total kinetic energy TKE released has been calculated using Viola’s [17] empirical relationship. As it is expected the total neutron multiplicity increases linearly with excitation energy [19]. The slope of the linear fit is 0.11 ± 0.02 which agrees with the value (0.11 neutrons/MeV) extracted by Hilscher et al. [19] for spontaneous and thermal, proton and alpha induced fission at low excitation energies. This observation suggests cost of total neutron evaporation is consistent with the systematics in this mass region of compound nucleus. On the average the measured pre-scission neutron multiplicity values (Tables 1 and 2) for 12 C + 194 Pt system are larger than that for 12 C + 198 Pt system at all energies. This observation is against the general argument [20] of dependence of pre-saddle particle emission on neutron to proton ratio (N/Z) of the compound nucleus at low excitation energies. Experimental results [21] have led us to consider angular momentum to be a possible cause for this anomaly. The initial J distribution of the decaying compound nuclei are obtained from the fits to the experimental fusion excitation functions [13] using the coupled channel code CCDEF [22]. The difference in the average angular momenta brought by the two systems at matching excitation energies is small (19.3 and 18.2 at 49.3 MeV excitation energy and 29.9 and 29.3 at 65 MeV in units of h¯ for 12 C + 194 Pt and 12 C + 198 Pt systems respectively) the angular momentum can be ruled out for the difference in the measured pre-scission multiplicities. In order to confirm this argument further, it has been verified by interchanging angular momentum distribution for 12 C + 194 Pt system with that of 12 C + 198 Pt systems at 54.3 MeV excitation energy in the statistical model code PACE2 [23]. The pre-scission multiplicities thus obtained match within 2% and the difference in angular momentum distribution for the two systems cannot explain the observed differences in the pre-scission neutron multiplicities of the two systems. To understand this feature, detailed statistical model calculations have been performed with the incorporation of shell effects. 4. Statistical model analysis The statistical model code PACE2 with modified level density and fission barrier prescription [5] was used to fit the pre-scission neutron multiplicities along with fission and evaporation residue excitation functions. These calculations consider emission of light particles also along with fission as the possible decay channels of the compound nucleus. The compound nucleus excitation energy and the particle separation energy are calculated by using the experimental masses [24]. The relative density of levels available for these processes determines the probability of decay through each of the channels. The neutron evaporation is governed by the available excitation energy (Un ) and the density of single particles states at the Fermi energy in the ground state of the residual nuclei and the fission decay is governed by the level density of the fissioning nuclei at the saddle point configuration. The available excitation energy (Uf ) for the fission channel is determined with respect to fission barrier height (Bf (J )). The fission barrier is calculated as given below Bf (J ) = BLD (J ) − n + f .
(2)
Here, n and f are shell corrections at the ground state and at the saddle point, respectively and the spin dependent liquid drop component of fission barrier BLD (J ) is taken from the rotating finite range model (RFRM) [25]. The value of shell correction at the saddle point f is assumed to be kf × n where kf is determined by fitting the experimental data. We have used the difference of experimental masses [24] and liquid drop masses according to Lysekil (1966)
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Table 3 Relevant statistical model parameters used in the SM calculations for 12 C + 194, 198 Pt systems. kf =0.0
System
BLD (0) (MeV)
n (MeV)
kf
a˜ f /a˜ n
Bf (0) (MeV)
Bf
12 C + 194 Pt
10.3 10.8
−8.09 −10.8
0.69 ± 0.04 0.70 ± 0.02
1.016 ± 0.006 1.010 ± 0.003
12.8 14.0
18.4 21.6
12 C + 198 Pt
(0) (MeV)
[26] mass formula with no shell correction to calculate shell correction at equilibrium deformation and the values were found to be consistent with the microscopic calculations of Moller and Nix [24] for shell correction energies. The calculations show that the estimation of ground state shell correction energies with above two cases agrees well. Though pairing energy has been included in the estimation of shell correction energies, its value is rather small compared to ground state shell correction energy. Fermi gas model with energy dependent shell corrections as prescribed by Ignatyuk et al. [27] as given below were used to calculate level density parameters at equilibrium and saddle point deformations. (3) ax = a˜ x 1 + (x /Ux ) 1 − e−0.054Ux where x = n or f corresponding to equilibrium or saddle point deformation. The asymptotic level density parameter for the equilibrium configuration (a˜ n )was taken to be A/9 MeV−1 , which gave the best fits to the measured fragment-neutron angular correlations in moving source fits. Due to the difference in the nuclear shape in the ground state and saddle point configurations the level density at these two levels may differ and the asymptotic level density parameter at saddle point a˜ f is taken as a˜ f /a˜ n × a˜ n . The available excitation energy of the compound nucleus at the equilibrium configuration is taken as Un = E ∗ − Erot (J ) − δP , where E ∗ , Erot (J ), and δP are the total excitation energy, the rotational energy, and the pairing energy, respectively. The available excitation energy at the saddle point deformation is taken as Uf = Un − Bf . The initial J distribution of the decaying nuclei are taken from the fits to the measured fusion excitation functions using coupled channel code CCDEF. A simultaneous fitting of all the experimental data available; viz fission and evaporation residue cross sections and pre-scission neutron multiplicities was carried out by varying (kf , a˜ f /a˜ n ) by using χ 2 minimization method. The values of kf and a˜ f /a˜ n thus obtained are given in Table 3. 5. Discussions In Fig. 4 the experimental pre-scission neutron multiplicities are compared with the statistical model predictions for the decay of CN with the modified fission barrier and level density prescription by fitting the pre-scission neutron multiplicities along with fission and partial ER cross sections as described earlier. The bottom panels of Fig. 4 shows the SM calculations (continuous line) using the best fitted (kf , a˜ f /a˜ n ) parameters along with the experimental fusion, fission and evaporation residue excitation functions [13]. Initially the χ 2 minimization was performed by varying a˜ f /a˜ n for fixed values of kf to fit the fission and partial ER excitation functions. As can be seen from Fig. 4 the calculated pre-scission neutron multiplicities without any shell correction at the saddle point (kf = 0.0) as shown in dotted–dashed lines are much less than the experimental values of Mpre though the calculations describe the fission and ER excitation functions well. k =0.0
The fission barriers (Bff (0)) thus obtained are given in Table 3. Whereas, the calculated pre-scission neutron multiplicities with full shell corrections at the saddle point (kf = 1.0) as
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Table 4 Sensitivity of kf and a˜ f /a˜ n on a˜ n . 12 C + 194 Pt
a˜ n A/8.0 A/9.0 A/11.0
12 C + 198 Pt
kf
a˜ f /a˜ n
kf
a˜ f /a˜ n
0.70 ± 0.05 0.69 ± 0.04 0.70 ± 0.02
1.010 ± 0.005 1.016 ± 0.006 1.017 ± 0.007
0.71 ± 0.02 0.70 ± 0.02 0.72 ± 0.04
1.008 ± 0.004 1.010 ± 0.003 1.011 ± 0.006
shown in dotted lines are much larger than the measured data. As the excitation energy of the fissioning system increases, the dynamical effect comes into picture which eventually increases the number of neutrons emitted before scission. Hence the simultaneous fitting of fusion and fission cross sections and pre-scission neutron multiplicity to obtain the best fitted (kf , a˜ f /a˜ n ) values was done at the lower two excitation energies. Thus obtained kf was used to estimate the saddle point shell correction and thereby to determine the fission barrier height (Table 3). The uncertainties in the fitted parameters are obtained by minimizing the χ 2 . A sensitivity analysis has been carried out to study the variation of the fitted parameters with different values of a˜ n . The values of a˜ n was changed from A/8 to A/11 and the variation in the fitted parameters were found to be negligible and well within the fitting uncertainty (Table 4). It is interesting to note that in case of 12 C + 198 Pt system the best fitted value of (kf , a˜ f /a˜ n ) thus obtained could satisfactorily explain the pre-scission multiplicity at the highest excitation energy also. However it under predicts the data at the highest energy for 12 C + 194 Pt system. A fission delay of 30 × 10−21 s [19] was incorporated to estimate the dynamical correction in the pre-scission neutron multiplicity. The dynamical emission of neutron was calculated by following the method discussed in Ref. [16]. The data shown in open circle in Fig. 4 is obtained by considering the dynamical effect. As the dynamical effects are more prominent at higher excitation energies (an excess of 0.8 neutrons at 59.4 MeV excitation energy and reduces to 0.04 at 49.3 MeV), inclusion of fission delay is essential to satisfactorily explain the measured prescission multiplicities at the highest excitation energy for the 12 C + 194 Pt system. However, the dissipation effects are not significant even at the highest excitation energy for the 12 C + 198 Pt system as the experimental value is observed to be only marginally higher than the statistical model prediction. Hence it could be conjectured that dissipation effects are observed for shell closed nuclei only at higher excitation energy compared to nuclei away from closed shell. Dissipative dynamics slows down the fission process and consequently a larger number of neutrons can be evaporated than that allowed by the standard statistical model. The role of dissipation in fusion–fission dynamics has been observed in different experimental observables like prescission particle and gamma emission, evaporation residue, etc. Back et al. [28] have shown that strength of the dissipation required to reproduce the experimental data not only depends on the excitation energy but also on the shell structure of the compound nucleus. For closed shell nuclei only moderate dissipation strength is required up to high excitation energy where as it was shown that dissipative strength dramatically increases for mid shell systems for excitation energy above ∼50 MeV. They have also shown that threshold for dissipation effects is higher for closed shell compound nucleus with N = 126. The higher dynamical threshold for the pre-scission multiplicity for 12 C + 198 Pt system compared to 12 C + 194 Pt system is consistent with the conclusions of Back et al. Further investigations are required to confirm this conclusion by extending the experimental measurements at higher excitation energies for 12 C + 198 Pt system.
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Fig. 4. (Colour online.) Experimental pre-scission neutron multiplicities and fusion–fission and evaporation residue excitation functions along with the statistical model calculation results. Solid lines shows the SM calculations with the best fitted (kf , a˜ f /a˜ n ) and the dotted and dashed–dotted lines show the SM calculations with kf = 1.0 and kf = 0.0 respectively.
It is to be noted that the heights of fission barrier that explains the obtained Mpre values within the present formalism are lower than the earlier reported values [29–31]. If the SM calculations are carried out without any shell correction at the saddle point (kf = 0.0), the resultant k =0.0
fission barrier heights (Bff (0)) as shown in Table 3 are 18.4 MeV and 21.6 MeV for 206 Po and 210 Po respectively. These values match very well with those (19.0 MeV and 22.1 MeV respectively) obtained by Moller et al. [29] and Capote et al. [32]. However as discussed earlier calculations with these barriers under-predict the pre-scission multiplicities even though it explains the fission and ER excitation functions (refer Fig. 4). The analysis was further extended for 18 O + 192 Os system [30] which populates 210 Po, one of the CN of the present measurements. It should be noted that the same data was analyzed by Sagaidak et al. [30] without including prescission neutron multiplicities in the statistical model analysis and extracted fission barriers are found to be higher than the present values for 210 Po compound system. The fission and ER excitation functions for 18 O + 192 Os system are simultaneously fitted along with pre-scission neutron multiplicities from the present measurement at the lower two excitation energies. Interestingly the best fitted (kf , a˜ f /a˜ n ) parameters (0.705 ± 0.040, 1.023 ± 0.004) show that an introduction of considerable saddle point shell correction is required to explain the experimental observables and it also demonstrates the importance of the measurement of average pre-scission multiplicity in extracting fission barrier heights. It is generally observed that the fission barriers from rotating finite range model (RFRM) after suitable scaling describes the heavy ion induced fission and evaporation residue (ER) data for nuclei of mass A 179. In particular, the fission barrier extracted by Charity et al. [33] by fitting
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Fig. 5. (Colour online.) Comparison of experimental fission excitation function [31,34] for α + 206 Pb system with statistical model results for different values of (kf , a˜ f /a˜ n ).
the fission and ER excitation function for 210 Po compound nuclei agrees with those given by RFRM model. The present analysis gives the fission barriers which are about 30% higher than RFRM fission barrier. The present formalism was also applied for the measured fission excitation function of α + 206 Pb system [31,34] which populates the compound nucleus 210 Po in the excitation energy range 25–50 MeV as shown in Fig. 5. However, pre-scission neutron multiplicity data for this entrance channel is not available. It is observed that at low excitation energies, the slope of the excitation function is better reproduced by a smaller shell correction at saddle point with larger value of a˜ f /a˜ n (1.20) but at higher excitation energies the slope of the excitation function could be better reproduced by a larger shell correction at the saddle point with a˜ f /a˜ n = 1.00. The same approach was also used to analyze the experimental data from literature of 213 At populated through 4 He + 209 Bi reaction [35]. A considerable shell correction was required to explain the ER and fission excitation functions here as well [36]. Measurements of pre-scission neutron multiplicity for these systems will be helpful to fix the statistical model parameters. The small shell effects at the saddle point are quite often neglected, while the ground state shell corrections are considered for the estimation of the fission barrier height along with the scaled down liquid drop component of fission barrier. It is a general practice to adjust the scaling factor to reproduce the evaporation residue (ER) and fission excitation functions. Andreyev et al. [37] have carried out a systematic study of the evaporation residue cross sections for neutron deficient nuclei of Bi and Po which showed that a significant reduction of the theoretical fission barrier (up to 35%) was required for the satisfactory description of the experimental data. It was suggested that the reduction of the LD fission barriers is a more general phenomenon and its physical interpretation is not very clear, and one has to search for a new approach to explain the extended data set. The phenomenological statistical analysis carried out by Iljinov et al. [38] by
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Fig. 6. (Colour online.) Shell correction energies at saddle point (top panel), ground state (middle panel) and their ratio (bottom panel) for 12 C + 194, 198 Pt systems along with the data from Ref. [41] for Z = 88–90. The dotted horizontal line drawn at ratio = 0.7 is to guide the eye.
analyzing the data of neutron emission and fission partial width ratio showed that a saddle point shell correction of −7.0 MeV with a reasonable value of the parameter a˜ f /a˜ n was required to obtain the best agreement between calculation and experiment. The statistical model code used in the present calculation does not include the collective enhancement of the level densities explicitly. It has been argued by Mahata et al. [5] that the combined effect of collective enhancement and Kramer’s factor [39] is to maintain kf nearly unchanged. The effects of above two factors in fission width are in opposite directions and mutually cancel each other. As shown in Ref. [5], the collective enhancement and Kramer’s factor (of 0.1) for 19 F + 192 Os system change the values of (kf , a˜ f /a˜ n ) from (0.78, 1.01) to (0.75, 0.93). It is to be noted that the saddle point shell correction energies are not sensitive to the choice of Kramer’s factor and collective enhancement. However, inclusion of collective enhancement leads to unphysical values of a˜ f /a˜ n in this mass region. Hence it is not included in our calculations of shell correction energies at saddle point. Moreover collective enhancement and its fade-out is an open problem which is yet to be established experimentally [40,30]. The shell correction energies at the saddle point and their ratios to the ground state shell correction energies for the 12 C + 194 Pt and 12 C + 198 Pt extracted from the present analy´ et al. [41], for nearby sis are compared with those values reported by Siwek-Wilczynska systems (Z = 88–90) in Fig. 6. It is observed that similar ratio of saddle point to ground state shell correction energies were found for Z = 88–90 also though the absolute values of shell correction energies are higher in the case for both ground state as well as saddle point.
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6. Summary and conclusion The average pre-scission neutron multiplicities for 12 C + 194 Pt (forming 206 Po) and for 12 C + (forming the shell closed compound nucleus 210 Po) systems were measured. The statistical model parameters were varied in a global manner to fit the neutron multiplicity data along with the ER and fission cross section data simultaneously. It is observed that pre-scission multiplicities play a major role in fixing the statistical model parameters. The importance of the present work is on the fact that experimental values of Mpre were included while extracting the fission barrier. It has been observed that a considerable amount of shell correction at the saddle point is required to fit the measured experimental data within the present formalism. However, if the reaction mechanism included a considerable component from the dynamical processes at these excitation energies, the required value of shell correction at the saddle point will be much reduced. Further the dynamical time required to explain the pre-scission neutron multiplicity would be very large (dynamical delay of 75 × 10−20 s at 49.3 MeV as compared to the available systematics [12,19,16]) if we consider the excess neutron multiplicity observed is entirely due to dynamical process and there is no shell correction at the saddle point. The other possible contribution to excess pre-scission neutron multiplicities could be given by Fuller mechanism [42] of sudden neck rupture giving rise to emission during scission process but the component of scission emission is expected to be very small [21]. The absolute value of fission barrier extracted using the present analysis are higher by about 30% with that reported earlier [33] for 210 Po. To summarize, pre-scission neutron multiplicity should be included for deducing reliable fission barrier in statistical model analysis. The SM analysis discussed here suggests an approach of determining the fission barrier height by including saddle point shell correction. The present formalism suggests the presence of significant shell corrections at the saddle points for the present systems. Measurement of pre-scission neutron multiplicities in the α induced fission for the same compound nucleus may help in determining fission barrier more accurately. 198 Pt
Acknowledgements We are thankful to the accelerator crew of IUAC for providing good quality pulsed beams as per the requirement. We gratefully acknowledge Dr. S.S. Kapoor for the stimulating discussions and critical suggestions. We are thankful to Dr. A. Roy, Dr. V.S. Ramamurthy and Prof. M.B. Chatterjee for their support at different stages of this work and Mr. Abhilash S.R. for the target fabrication. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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Nuclear Physics A 913 (2013) 157–169 www.elsevier.com/locate/nuclphysa
Determination of shell correction energies at saddle point using pre-scission neutron multiplicities Golda K.S. a,b,∗ , A. Saxena c , V.K. Mittal b , K. Mahata c , P. Sugathan a , A. Jhingan a , V. Singh d , R. Sandal d , S. Goyal e , J. Gehlot a , A. Dhal a , B.R. Behera d , R.K. Bhowmik a , S. Kailas c a Inter University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi 110067, India b Physics Department, Punjabi University, Patiala, Punjab 147002, India c Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India d Department of Physics, Panjab University, Chandigarh 160014, India e Department of Physics and Astrophysics, University of Delhi, New Delhi 110007, India
Received 30 August 2012; received in revised form 8 March 2013; accepted 23 May 2013 Available online 14 June 2013
Abstract Pre-scission neutron multiplicities have been measured for 12 C + 194, 198 Pt systems at matching excitation energies at near Coulomb barrier region. Statistical model analysis with a modified fission barrier and level density prescription have been carried out to fit the measured pre-scission neutron multiplicities and the available evaporation residue and fission cross sections simultaneously to constrain statistical model parameters. Simultaneous fitting of the pre-scission neutron multiplicities and cross section data requires shell correction at the saddle point. © 2013 Elsevier B.V. All rights reserved. Keywords: N UCLEAR REACTIONS 194,198 Pt(12 C, f ), E ∗ = 49.3, 54.3, 59.4, 62.2 MeV; measured fission fragments, En , In (θ) using ToF; deduced neutron multiplicity vs En and θ , pre-scission neutron multiplicity vs E ∗ , fusion–fission σ ; calculated fission barrier, shell effects, statistical model parameters, pre-scission neutron multiplicity vs E ∗ , fusion–fission σ , fission σ vs E ∗ , evaporation residue σ vs E ∗ . Compared with published 208 Pb + α system
* Corresponding author at: Inter University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi 110067, India.
E-mail address: [email protected] (K.S. Golda). 0375-9474/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysa.2013.05.016
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1. Introduction The significance of nuclear shell structure in the formation of a new element in heavy ion induced nuclear reaction is a problem of current interest [1,2]. Few attempts have been made recently to understand the role of shell correction energies on the fission threshold as a function of the nuclear deformation [3–5]. Such studies bring out the interplay of macroscopic properties of nuclear matter to the microscopic effects such as deformation and shell structure of the nuclei [6]. The quantitative determination of fission barrier height is the key to understand the dynamics of heavy ion fusion–fission process and the prediction of the super heavy elements. Many researchers [3,7,8] have carried out theoretical calculations for the fusion–fission probability in the super heavy mass region in the frame work of a potential energy landscape in multidimensional deformation space. The existence of the island of stability has also been proposed using the nuclear shell model [9]. However, very few attempts have been carried out [5] to extract the shell correction to the fission barrier from the experimental fission data in heavy ion induced fusion–fission reactions. It is well established that the microscopic (shell) correction in the liquid drop nuclear energy surface [10,11] results in the multiple humps, valleys and saddle points along the nuclear trajectory in the potential energy landscape. Nevertheless the quantification of the exact shell correction values required to explain all the characteristics of the fissioning nucleus is extremely difficult. Moreover there are limited experimental observations [4] indicating the possible effects of shell structure in the fusion–fission reaction mechanism. Shrivastava et al. [4] have reported that the measured fission fragment angular anisotropies for 12 C + 198 Pt system have been found to be larger compared to the Statistical Saddle Point Model (SSPM) calculations, where as the observed anisotropies for 12 C + 194 Pt system have been found to be in good agreement with the SSPM calculations. The disagreement between the SSPM predictions and the experimental data for 12 C + 198 Pt system, which populates 210 Po as compound nucleus (CN), has been found to decrease with increasing excitation energy. This was interpreted as a signature of shell effects as the CN has closed shell neutron number and which could provide an extra stability against fission. Later it has been shown that [5] the shell corrections at saddle point is necessary to explain the measured fission and evaporation residue cross sections and the pre-scission neutron multiplicity data simultaneously in mass ∼200 region. It was also shown that more than one set of statistical model parameters could be used to explain the experimental fission as well as evaporation residue cross sections satisfactorily. However, a unique parameter set could be achieved if the pre-fission neutron multiplicity data are included in the analysis. The pre-scission neutron multiplicity data was not available so it was taken from the Baba’s systematics [12] in this analysis. In this regard experimental data for pre-scission neutron multiplicity is necessary to constrain the statistical model calculations to quantify the shell correction energies. With this motivation, we have carried out measurements of neutron spectra for 12 C + 194,198 Pt systems. The specific selection of these systems enabled us to do a meaningful statistical model analysis as the partial evaporation residue cross sections as well as the fission cross sections are already measured [4,13]. In the present study, care has been taken to select the systems which are asymmetric on the entrance channel and well above the Businaro–Gallone [14] critical mass asymmetry to avoid the contamination of non-compound fission events. 2. Experimental details The experiment was carried out using the heavy ion facility at Inter University Accelerator Centre (IUAC), New Delhi. Pulsed beam of 12 C delivered by the 15UD Pelletron, was
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Fig. 1. Schematic of the experimental setup.
bombarded on isotopically enriched self supporting rolled foils of 194 Pt and 198 Pt of thickness’s ∼1.75 mg/cm2 and ∼1.45 mg/cm2 , respectively. The beam energies had been adjusted to have same excitation energies for both the systems. Fission fragments were detected by a pair of 5 × 3 position sensitive multi-wire proportional counters kept at ±40◦ degree with respect to beam direction at a distance of 17 cm from the target position. The fast timing signals from the fission counters were used to get the flight time information of the fission fragments which enabled us to separate fission events from other reaction products. Neutron time of flights (TOF) were measured using 24 liquid scintillators (BC501) kept at different angles around the target chamber. Out of these 24 detectors, 16 detectors (5 diameter × 5 deep) were kept in reaction plane and the remaining 8 detectors (5 × 3 ) were kept at 15◦ up and down with respect to the reaction plane. Four of the reaction plane detectors were placed at 1 meter away from the target. All the remaining detectors were placed at 2 m away from the target in a cylindrical fashion as shown in Fig. 1. In order to reduce the gamma background seen by the detectors, the beam was dumped 4 m downstream from the target and was well shielded with paraffin and lead bricks. The time spread (∼1 ns)in the beam pulse was monitored using a BaF2 detector placed near to the beam dump. Discrimination between neutrons and γ rays was made by using pulse shape discrimination based on the zero-crossing technique and the TOF. The TOF of neutrons were converted into neutron energy by considering the prompt gamma peak in the TOF spectrum as the reference time. The obtained energy distribution was corrected for energy dependent detection efficiency of the neutron detectors. The intrinsic efficiency of neutron detectors were determined by measuring the neutron energy spectra from a 252 Cf spontaneous fission source of known strength kept at the target position. 3. Data analysis and results The pre- and post-scission components of neutron multiplicities and temperatures were obtained from the measured neutron energy spectra by using a multiple source least-square fitting procedure, using Watt expression [15,16]. Three moving sources of neutrons (compound nucleus and two fully accelerated fission fragments) were considered while determining the multiplicities from the fits. The neutrons emitted from these moving sources were assumed to be isotropic in their respective rest frames. Thus the measured neutron multiplicities are given as
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Fig. 2. (Colour online.) Double differential neutron multiplicity spectra (filled circles) for 12 C + 194 Pt system at 59.4 MeV excitation energy along with the fits for the pre-scission component (dashed curve) and post-scission components from fragment 1 (dotted curve) and fragment 2 (dot dashed curve). The solid curve represents the total contribution.
√ √ 3 d 2M Mi E n En − 2 En Ei /Ai cos θi + Ei /Ai = × exp − . dEn dΩn Ti 2(πTi )3/2
(1)
i=1
Where En is the laboratory energy of neutron and Ai , Ei , Ti and Mi are the mass, energy, temperature and multiplicity of each neutron emitting source i. θi is the direction of neutron with respect to the neutron emitting source. The kinetic energy of the compound nucleus ECN was determined by assuming full momentum transfer. Fission fragment energies and folding angles were obtained from Viola’s [17] systematics assuming fission to be symmetric. Simultaneous fitting was done for all the available fission coincident neutron data set at the highest excitation energy. Though the array consists of 24 detectors we have used only 5 × 5 detectors for the moving source fits due to their higher efficiency and better statistics. For the lower excitation energies we have considered the data exclusively from the detectors placed at 1 m distance due to the limitations in sufficient statistics. Fig. 2 represents the typical fits to the double differential neutron multiplicity spectra for 12 C + 194 Pt system at 59.4 MeV excitation energy. The angles of neutron with respect to CN and each of the fission fragments are indicated in the plots. The temperature of the fissioning nucleus, Tpre was calculated as T = (E ∗ /a) ˜ where E ∗ is the compound nucleus excitation energy and a˜ is the level density, by assuming a˜ to be ACN /9 MeV−1 . Tpre was scaled down to (11/12)T to account for the cascade of sequential particle evaporation [15] and its value was fixed in the moving source fits. All the remaining parameters were obtained by fitting the data with Eq. (1). Fitting was also done with Tpre as a free parameter. Thus obtained neutron multiplicities are within the errors of the values obtained with fixed Tpre . The post-scission multiplicity and the temperature were taken to be the same for both the fission fragments assuming symmetric fragmentation and were determined from the
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Table 1 Experimentally obtained temperatures and multiplicities for 12 C + 194 Pt system as determined from moving source fits. E∗
Tpre
Mpre
Tpost
Mpost
49.3 54.3 59.4
1.36 1.42 1.48
1.30 ± 0.13 1.59 ± 0.16 2.61 ± 0.24
1.09 ± 0.13 0.78 ± 0.14 1.05 ± 0.16
0.72 ± 0.11 0.71 ± 0.13 0.70 ± 0.10
Table 2 Same as Table 1 for 12 C + 198 Pt system. E∗
Tpre
Mpre
Tpost
Mpost
49.3 54.3 62.2
1.11 1.41 1.49
1.11 ± 0.1 1.31 ± 0.13 2.10 ± 0.19
0.88 ± 0.10 1.09 ± 0.18 1.11 ± 0.17
0.66 ± 0.11 0.69 ± 0.13 0.71 ± 0.12
Fig. 3. Experimental total neutron multiplicities as a function of total available excitation energy.
fits. The angular acceptances of neutron and fission detectors were incorporated in the moving source fits. The parameters determined from the moving source fits are listed in Tables 1 and 2. The Mpre values increase with excitation energy for both systems while the Mpost values show negligible variation. In order to further confirm the values of experimentally obtained pre-scission neutron multiplicities, an exclusive moving source fitting was carried out by considering the data only from the neutron detectors perpendicular to the fission direction, where contribution from post-scission component is expected to be small, for 54.3 MeV excitation energy for both the systems. The pre-scission multiplicity value thus obtained match with value determined after including all detectors within the uncertainties of fits. The total neutron emission (Mtotal = Mpre + 2Mpost ) in a fusion–fission reaction corresponds to the available excitation energy of the system which includes the excitation energy brought by the projectile, excitation energy of fragments due to dissipation of potential energy during saddle to scission and deformation energy. In Fig. 3 the total evaporative neutron multiplicities for both 12 C + 194, 198 Pt systems are plotted against the available excitation energy (E ∗ +Qfiss -TKE). The fission Q-value (Qfiss ) has been calculated by taking the respective experimental mass values
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from the 2003 Atomic Mass Evaluation by Audi et al. [18] and assuming fission to be symmetric. The total kinetic energy TKE released has been calculated using Viola’s [17] empirical relationship. As it is expected the total neutron multiplicity increases linearly with excitation energy [19]. The slope of the linear fit is 0.11 ± 0.02 which agrees with the value (0.11 neutrons/MeV) extracted by Hilscher et al. [19] for spontaneous and thermal, proton and alpha induced fission at low excitation energies. This observation suggests cost of total neutron evaporation is consistent with the systematics in this mass region of compound nucleus. On the average the measured pre-scission neutron multiplicity values (Tables 1 and 2) for 12 C + 194 Pt system are larger than that for 12 C + 198 Pt system at all energies. This observation is against the general argument [20] of dependence of pre-saddle particle emission on neutron to proton ratio (N/Z) of the compound nucleus at low excitation energies. Experimental results [21] have led us to consider angular momentum to be a possible cause for this anomaly. The initial J distribution of the decaying compound nuclei are obtained from the fits to the experimental fusion excitation functions [13] using the coupled channel code CCDEF [22]. The difference in the average angular momenta brought by the two systems at matching excitation energies is small (19.3 and 18.2 at 49.3 MeV excitation energy and 29.9 and 29.3 at 65 MeV in units of h¯ for 12 C + 194 Pt and 12 C + 198 Pt systems respectively) the angular momentum can be ruled out for the difference in the measured pre-scission multiplicities. In order to confirm this argument further, it has been verified by interchanging angular momentum distribution for 12 C + 194 Pt system with that of 12 C + 198 Pt systems at 54.3 MeV excitation energy in the statistical model code PACE2 [23]. The pre-scission multiplicities thus obtained match within 2% and the difference in angular momentum distribution for the two systems cannot explain the observed differences in the pre-scission neutron multiplicities of the two systems. To understand this feature, detailed statistical model calculations have been performed with the incorporation of shell effects. 4. Statistical model analysis The statistical model code PACE2 with modified level density and fission barrier prescription [5] was used to fit the pre-scission neutron multiplicities along with fission and evaporation residue excitation functions. These calculations consider emission of light particles also along with fission as the possible decay channels of the compound nucleus. The compound nucleus excitation energy and the particle separation energy are calculated by using the experimental masses [24]. The relative density of levels available for these processes determines the probability of decay through each of the channels. The neutron evaporation is governed by the available excitation energy (Un ) and the density of single particles states at the Fermi energy in the ground state of the residual nuclei and the fission decay is governed by the level density of the fissioning nuclei at the saddle point configuration. The available excitation energy (Uf ) for the fission channel is determined with respect to fission barrier height (Bf (J )). The fission barrier is calculated as given below Bf (J ) = BLD (J ) − n + f .
(2)
Here, n and f are shell corrections at the ground state and at the saddle point, respectively and the spin dependent liquid drop component of fission barrier BLD (J ) is taken from the rotating finite range model (RFRM) [25]. The value of shell correction at the saddle point f is assumed to be kf × n where kf is determined by fitting the experimental data. We have used the difference of experimental masses [24] and liquid drop masses according to Lysekil (1966)
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Table 3 Relevant statistical model parameters used in the SM calculations for 12 C + 194, 198 Pt systems. kf =0.0
System
BLD (0) (MeV)
n (MeV)
kf
a˜ f /a˜ n
Bf (0) (MeV)
Bf
12 C + 194 Pt
10.3 10.8
−8.09 −10.8
0.69 ± 0.04 0.70 ± 0.02
1.016 ± 0.006 1.010 ± 0.003
12.8 14.0
18.4 21.6
12 C + 198 Pt
(0) (MeV)
[26] mass formula with no shell correction to calculate shell correction at equilibrium deformation and the values were found to be consistent with the microscopic calculations of Moller and Nix [24] for shell correction energies. The calculations show that the estimation of ground state shell correction energies with above two cases agrees well. Though pairing energy has been included in the estimation of shell correction energies, its value is rather small compared to ground state shell correction energy. Fermi gas model with energy dependent shell corrections as prescribed by Ignatyuk et al. [27] as given below were used to calculate level density parameters at equilibrium and saddle point deformations. (3) ax = a˜ x 1 + (x /Ux ) 1 − e−0.054Ux where x = n or f corresponding to equilibrium or saddle point deformation. The asymptotic level density parameter for the equilibrium configuration (a˜ n )was taken to be A/9 MeV−1 , which gave the best fits to the measured fragment-neutron angular correlations in moving source fits. Due to the difference in the nuclear shape in the ground state and saddle point configurations the level density at these two levels may differ and the asymptotic level density parameter at saddle point a˜ f is taken as a˜ f /a˜ n × a˜ n . The available excitation energy of the compound nucleus at the equilibrium configuration is taken as Un = E ∗ − Erot (J ) − δP , where E ∗ , Erot (J ), and δP are the total excitation energy, the rotational energy, and the pairing energy, respectively. The available excitation energy at the saddle point deformation is taken as Uf = Un − Bf . The initial J distribution of the decaying nuclei are taken from the fits to the measured fusion excitation functions using coupled channel code CCDEF. A simultaneous fitting of all the experimental data available; viz fission and evaporation residue cross sections and pre-scission neutron multiplicities was carried out by varying (kf , a˜ f /a˜ n ) by using χ 2 minimization method. The values of kf and a˜ f /a˜ n thus obtained are given in Table 3. 5. Discussions In Fig. 4 the experimental pre-scission neutron multiplicities are compared with the statistical model predictions for the decay of CN with the modified fission barrier and level density prescription by fitting the pre-scission neutron multiplicities along with fission and partial ER cross sections as described earlier. The bottom panels of Fig. 4 shows the SM calculations (continuous line) using the best fitted (kf , a˜ f /a˜ n ) parameters along with the experimental fusion, fission and evaporation residue excitation functions [13]. Initially the χ 2 minimization was performed by varying a˜ f /a˜ n for fixed values of kf to fit the fission and partial ER excitation functions. As can be seen from Fig. 4 the calculated pre-scission neutron multiplicities without any shell correction at the saddle point (kf = 0.0) as shown in dotted–dashed lines are much less than the experimental values of Mpre though the calculations describe the fission and ER excitation functions well. k =0.0
The fission barriers (Bff (0)) thus obtained are given in Table 3. Whereas, the calculated pre-scission neutron multiplicities with full shell corrections at the saddle point (kf = 1.0) as
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Table 4 Sensitivity of kf and a˜ f /a˜ n on a˜ n . 12 C + 194 Pt
a˜ n A/8.0 A/9.0 A/11.0
12 C + 198 Pt
kf
a˜ f /a˜ n
kf
a˜ f /a˜ n
0.70 ± 0.05 0.69 ± 0.04 0.70 ± 0.02
1.010 ± 0.005 1.016 ± 0.006 1.017 ± 0.007
0.71 ± 0.02 0.70 ± 0.02 0.72 ± 0.04
1.008 ± 0.004 1.010 ± 0.003 1.011 ± 0.006
shown in dotted lines are much larger than the measured data. As the excitation energy of the fissioning system increases, the dynamical effect comes into picture which eventually increases the number of neutrons emitted before scission. Hence the simultaneous fitting of fusion and fission cross sections and pre-scission neutron multiplicity to obtain the best fitted (kf , a˜ f /a˜ n ) values was done at the lower two excitation energies. Thus obtained kf was used to estimate the saddle point shell correction and thereby to determine the fission barrier height (Table 3). The uncertainties in the fitted parameters are obtained by minimizing the χ 2 . A sensitivity analysis has been carried out to study the variation of the fitted parameters with different values of a˜ n . The values of a˜ n was changed from A/8 to A/11 and the variation in the fitted parameters were found to be negligible and well within the fitting uncertainty (Table 4). It is interesting to note that in case of 12 C + 198 Pt system the best fitted value of (kf , a˜ f /a˜ n ) thus obtained could satisfactorily explain the pre-scission multiplicity at the highest excitation energy also. However it under predicts the data at the highest energy for 12 C + 194 Pt system. A fission delay of 30 × 10−21 s [19] was incorporated to estimate the dynamical correction in the pre-scission neutron multiplicity. The dynamical emission of neutron was calculated by following the method discussed in Ref. [16]. The data shown in open circle in Fig. 4 is obtained by considering the dynamical effect. As the dynamical effects are more prominent at higher excitation energies (an excess of 0.8 neutrons at 59.4 MeV excitation energy and reduces to 0.04 at 49.3 MeV), inclusion of fission delay is essential to satisfactorily explain the measured prescission multiplicities at the highest excitation energy for the 12 C + 194 Pt system. However, the dissipation effects are not significant even at the highest excitation energy for the 12 C + 198 Pt system as the experimental value is observed to be only marginally higher than the statistical model prediction. Hence it could be conjectured that dissipation effects are observed for shell closed nuclei only at higher excitation energy compared to nuclei away from closed shell. Dissipative dynamics slows down the fission process and consequently a larger number of neutrons can be evaporated than that allowed by the standard statistical model. The role of dissipation in fusion–fission dynamics has been observed in different experimental observables like prescission particle and gamma emission, evaporation residue, etc. Back et al. [28] have shown that strength of the dissipation required to reproduce the experimental data not only depends on the excitation energy but also on the shell structure of the compound nucleus. For closed shell nuclei only moderate dissipation strength is required up to high excitation energy where as it was shown that dissipative strength dramatically increases for mid shell systems for excitation energy above ∼50 MeV. They have also shown that threshold for dissipation effects is higher for closed shell compound nucleus with N = 126. The higher dynamical threshold for the pre-scission multiplicity for 12 C + 198 Pt system compared to 12 C + 194 Pt system is consistent with the conclusions of Back et al. Further investigations are required to confirm this conclusion by extending the experimental measurements at higher excitation energies for 12 C + 198 Pt system.
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Fig. 4. (Colour online.) Experimental pre-scission neutron multiplicities and fusion–fission and evaporation residue excitation functions along with the statistical model calculation results. Solid lines shows the SM calculations with the best fitted (kf , a˜ f /a˜ n ) and the dotted and dashed–dotted lines show the SM calculations with kf = 1.0 and kf = 0.0 respectively.
It is to be noted that the heights of fission barrier that explains the obtained Mpre values within the present formalism are lower than the earlier reported values [29–31]. If the SM calculations are carried out without any shell correction at the saddle point (kf = 0.0), the resultant k =0.0
fission barrier heights (Bff (0)) as shown in Table 3 are 18.4 MeV and 21.6 MeV for 206 Po and 210 Po respectively. These values match very well with those (19.0 MeV and 22.1 MeV respectively) obtained by Moller et al. [29] and Capote et al. [32]. However as discussed earlier calculations with these barriers under-predict the pre-scission multiplicities even though it explains the fission and ER excitation functions (refer Fig. 4). The analysis was further extended for 18 O + 192 Os system [30] which populates 210 Po, one of the CN of the present measurements. It should be noted that the same data was analyzed by Sagaidak et al. [30] without including prescission neutron multiplicities in the statistical model analysis and extracted fission barriers are found to be higher than the present values for 210 Po compound system. The fission and ER excitation functions for 18 O + 192 Os system are simultaneously fitted along with pre-scission neutron multiplicities from the present measurement at the lower two excitation energies. Interestingly the best fitted (kf , a˜ f /a˜ n ) parameters (0.705 ± 0.040, 1.023 ± 0.004) show that an introduction of considerable saddle point shell correction is required to explain the experimental observables and it also demonstrates the importance of the measurement of average pre-scission multiplicity in extracting fission barrier heights. It is generally observed that the fission barriers from rotating finite range model (RFRM) after suitable scaling describes the heavy ion induced fission and evaporation residue (ER) data for nuclei of mass A 179. In particular, the fission barrier extracted by Charity et al. [33] by fitting
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Fig. 5. (Colour online.) Comparison of experimental fission excitation function [31,34] for α + 206 Pb system with statistical model results for different values of (kf , a˜ f /a˜ n ).
the fission and ER excitation function for 210 Po compound nuclei agrees with those given by RFRM model. The present analysis gives the fission barriers which are about 30% higher than RFRM fission barrier. The present formalism was also applied for the measured fission excitation function of α + 206 Pb system [31,34] which populates the compound nucleus 210 Po in the excitation energy range 25–50 MeV as shown in Fig. 5. However, pre-scission neutron multiplicity data for this entrance channel is not available. It is observed that at low excitation energies, the slope of the excitation function is better reproduced by a smaller shell correction at saddle point with larger value of a˜ f /a˜ n (1.20) but at higher excitation energies the slope of the excitation function could be better reproduced by a larger shell correction at the saddle point with a˜ f /a˜ n = 1.00. The same approach was also used to analyze the experimental data from literature of 213 At populated through 4 He + 209 Bi reaction [35]. A considerable shell correction was required to explain the ER and fission excitation functions here as well [36]. Measurements of pre-scission neutron multiplicity for these systems will be helpful to fix the statistical model parameters. The small shell effects at the saddle point are quite often neglected, while the ground state shell corrections are considered for the estimation of the fission barrier height along with the scaled down liquid drop component of fission barrier. It is a general practice to adjust the scaling factor to reproduce the evaporation residue (ER) and fission excitation functions. Andreyev et al. [37] have carried out a systematic study of the evaporation residue cross sections for neutron deficient nuclei of Bi and Po which showed that a significant reduction of the theoretical fission barrier (up to 35%) was required for the satisfactory description of the experimental data. It was suggested that the reduction of the LD fission barriers is a more general phenomenon and its physical interpretation is not very clear, and one has to search for a new approach to explain the extended data set. The phenomenological statistical analysis carried out by Iljinov et al. [38] by
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Fig. 6. (Colour online.) Shell correction energies at saddle point (top panel), ground state (middle panel) and their ratio (bottom panel) for 12 C + 194, 198 Pt systems along with the data from Ref. [41] for Z = 88–90. The dotted horizontal line drawn at ratio = 0.7 is to guide the eye.
analyzing the data of neutron emission and fission partial width ratio showed that a saddle point shell correction of −7.0 MeV with a reasonable value of the parameter a˜ f /a˜ n was required to obtain the best agreement between calculation and experiment. The statistical model code used in the present calculation does not include the collective enhancement of the level densities explicitly. It has been argued by Mahata et al. [5] that the combined effect of collective enhancement and Kramer’s factor [39] is to maintain kf nearly unchanged. The effects of above two factors in fission width are in opposite directions and mutually cancel each other. As shown in Ref. [5], the collective enhancement and Kramer’s factor (of 0.1) for 19 F + 192 Os system change the values of (kf , a˜ f /a˜ n ) from (0.78, 1.01) to (0.75, 0.93). It is to be noted that the saddle point shell correction energies are not sensitive to the choice of Kramer’s factor and collective enhancement. However, inclusion of collective enhancement leads to unphysical values of a˜ f /a˜ n in this mass region. Hence it is not included in our calculations of shell correction energies at saddle point. Moreover collective enhancement and its fade-out is an open problem which is yet to be established experimentally [40,30]. The shell correction energies at the saddle point and their ratios to the ground state shell correction energies for the 12 C + 194 Pt and 12 C + 198 Pt extracted from the present analy´ et al. [41], for nearby sis are compared with those values reported by Siwek-Wilczynska systems (Z = 88–90) in Fig. 6. It is observed that similar ratio of saddle point to ground state shell correction energies were found for Z = 88–90 also though the absolute values of shell correction energies are higher in the case for both ground state as well as saddle point.
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6. Summary and conclusion The average pre-scission neutron multiplicities for 12 C + 194 Pt (forming 206 Po) and for 12 C + (forming the shell closed compound nucleus 210 Po) systems were measured. The statistical model parameters were varied in a global manner to fit the neutron multiplicity data along with the ER and fission cross section data simultaneously. It is observed that pre-scission multiplicities play a major role in fixing the statistical model parameters. The importance of the present work is on the fact that experimental values of Mpre were included while extracting the fission barrier. It has been observed that a considerable amount of shell correction at the saddle point is required to fit the measured experimental data within the present formalism. However, if the reaction mechanism included a considerable component from the dynamical processes at these excitation energies, the required value of shell correction at the saddle point will be much reduced. Further the dynamical time required to explain the pre-scission neutron multiplicity would be very large (dynamical delay of 75 × 10−20 s at 49.3 MeV as compared to the available systematics [12,19,16]) if we consider the excess neutron multiplicity observed is entirely due to dynamical process and there is no shell correction at the saddle point. The other possible contribution to excess pre-scission neutron multiplicities could be given by Fuller mechanism [42] of sudden neck rupture giving rise to emission during scission process but the component of scission emission is expected to be very small [21]. The absolute value of fission barrier extracted using the present analysis are higher by about 30% with that reported earlier [33] for 210 Po. To summarize, pre-scission neutron multiplicity should be included for deducing reliable fission barrier in statistical model analysis. The SM analysis discussed here suggests an approach of determining the fission barrier height by including saddle point shell correction. The present formalism suggests the presence of significant shell corrections at the saddle points for the present systems. Measurement of pre-scission neutron multiplicities in the α induced fission for the same compound nucleus may help in determining fission barrier more accurately. 198 Pt
Acknowledgements We are thankful to the accelerator crew of IUAC for providing good quality pulsed beams as per the requirement. We gratefully acknowledge Dr. S.S. Kapoor for the stimulating discussions and critical suggestions. We are thankful to Dr. A. Roy, Dr. V.S. Ramamurthy and Prof. M.B. Chatterjee for their support at different stages of this work and Mr. Abhilash S.R. for the target fabrication. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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