Physics Letters B. Microscopic description of energy partition in fission fragments. M. Mirea. article info abstract

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1 Physics Letters B 717 (2012) Contents lists available at SciVerse ScienceDirect Physics Letters B Microscopic description of energy partition in fission fragments M. Mirea Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, Bucharest-Magurele, Romania article info abstract Article history: Received 19 July 2012 Received in revised form 12 September 2012 Accepted 12 September 2012 Available online 13 September 2012 Editor: J.-P. Blaizot Keywords: Excitation energy of fission fragments Time dependent pairing equations 234 U fission Fission dynamics The energy sorting at scission is treated fully microscopically in terms of the time dependent pairing equations. These equations for the intrinsic motion are mixed with a dynamical particle number projection condition on the primary fragments. The shape sequences during the disintegration resort from the minimal action principle. For this purpose, the potential energy is computed within the macroscopic microscopic approach and the inertia within semi-adiabatic craning approximation. The disentanglement of the wave function in two semi-spaces is realized within the Woods Saxon two-center shell model. That allows a separation of the pairing field onto two sub-spaces. It is shown that the excitation energies of the fission fragments depend on a delicate interplay between their structure at and their deformations at scission Elsevier B.V. All rights reserved. 1. Introduction In the macroscopic microscopic model, the basic properties of fission arise from a delicate competition between disruptive Coulomb forces, cohesive nuclear forces, fluctuating shell and pairing forces. It is believed [1] that these static forces are primarily responsible for experimental phenomena as ground states, fission into unequal size, sawtooth of neutron yields. It was experimentally evidenced that the neutron multiplicities, and hence the energy carried away from neutrons are larger for symmetric fission than for asymmetric one [2,3]. Consequently, conservation of energy requires that the inetic energy of the fragments must be less for symmetric fission. This behavior was considered as the evidence of the existence of two fission barriers [4,5]: one for the symmetric fission that leads to excited fragments, and another for the asymmetric fission with moderately excited fragments and high inetic energy. Later on, it was suggested that the neutron emission curves are mainly related to the structure of the deformed final fragments [6] and the excitation due to the dynamics [7]. However, the question of energy partition between fission fragments is not yet understood. Recently, this problem was intensively investigated within a wide range of methods, including Hartree Foc theories [8], the statistical mechanics [9,10], the phenomenological point by point model [11], em- address: mirea@ifin.nipne.ro. pirical evaluations [12], or the shell model in the sudden approximation [13]. In the following, a fully microscopic model is proposed to analyze systematically for the first time the energy sorting at scission on a wide range of partitions. This phenomenon is investigated within the Time Dependent Pairing Equations (TDPE) that resort from the variational principle. The model provides a measure for the average dissipated energy. A dynamical condition that fixes the number of particles on the two fission fragments [7] is introduced in the energy functional. Consequently, two fragments with good numbers of nucleons are recognized in the final stage of the fission process. The TDPE have a similar form as the time dependent Hartree Foc Bogoliubov equations. The quantities needed to solve these equations of motion are the single particle energies and the pairing interactions. The Woods Saxon two-center shell model is used for the disentanglement of the parent wave function in two semi-spaces. The neutron multiplicities of the fragments issued in fission are proportional with their excitation. The results of this wor underline two important phenomena that corroborate to realize the neutron multiplicity sawtooth behavior: the dynamics of the fission that fix the configuration at scission and the modality in which the structure of each formed fragment participates in the structure of the whole system. It is also suggested that microscopic effects at scission and the dissipative forces govern the neutron multiplicities. Hence, the neutron multiplicity is managed not only by the mixed fragment structures in the scission configuration, but also /$ see front matter 2012 Elsevier B.V. All rights reserved.

2 M. Mirea / Physics Letters B 717 (2012) by its history. This suggestions are based on microscopic and dynamical calculations as detailed in the following. 2. Model First of all, the main steps in deriving the microscopic equations of motion constrained by the projections of number of particles are presented. The starting point of the model is the energy functional L = ϕ H i h t λ(n 1 ˆN 2 N 2 ˆN1 ) ϕ (1) where H(t) = ( ɛ a + ) a + a + a G >0,i>0 a + a+ a i aī (2) is the Hamiltonian with pairing residual interactions, ɛ being singleparticleenergies, ϕ = ( u + v a + ) a+ 0 (3) is the Bogoliubov wave function with BCS occupation v and vacancy u amplitudes. λ represents a Lagrange multiplier and Nˆ 1 = ( a1 a + ) 1 + a a 1 + 1, 1 Nˆ 2 = ( a2 a a + 2) a 2 2 are operators for the number of particles in the pairing active level space pertaining to the nuclei A 1 and A 2, respectively. The summation indices, 1 and 2 run over all states considered in the pairing active space, the states of the fragment one, and the those of the fragments two in the same space, respectively. N 1 and N 2 are the correspondent number of particles belonging to the two fission fragments. Obviously, the following identity ϕ (N 1 ˆN2 N 2 ˆN1 ) ϕ =0 (5) can be used to fix the number of particle N 1 and N 2 in the two fission fragments. The condition (5) is added to the energy functional (1) by mean of the Lagrange multiplier as an implicit condition. Therefore, Eq. (1) is subject to a supplementary condition that project the number of particles onto the two fragments. The next equations are obtained for the TDPE: i h ρ 1 = κ 1 1 κ 1 1, i h ρ 2 = κ 2 2 κ 2 2, i h κ 1 = (2ρ 1 1) 1 2κ 1 (ɛ 1 + λn 2 ), i h κ 2 = (2ρ 2 1) 2 2κ 2 (ɛ 2 λn 1 ) (6) where ρ = v 2 are occupation probabilities, κ = u v are pairing moment components, and = G κ is the pairing gap. 1 = G 1 1 κ 1 + G 12 2 κ 2 and 2 = G 12 1 κ 1 + G 2 2 κ 2 are the gaps for the two fragments. It must be noticed that if G 12 = 0, then we obtain two sets of non-coupled equations, one set for each fission fragment [7]. A particular form of these equations was deduced for the first time in Refs. [14,15]. Some remars must be made about the system (6). In the monopole approximation, the pairing interaction G is considered as a constant and it is renormalized by solving the BCS equations. Nevertheless, the matrix element of the pairing interaction G is in principle dependent of the overlap of the wave functions of the pairs [16,17]. In the simplest description, (4) as long as a single nucleus is involved, the monopole approximation is considered to perform well [18]. After the scission, the matrix elements of the pairing interaction between wave functions belonging to states in different fragments G 12 must be zero. In general, the pairing interaction matrix elements of two different fragments are not the same. This observation allows us to use different values for the pairing interactions G relative to the nucleus involved, that is G 1 G 2. It is possible to evaluate 1 and 2 if the states that correspond to the heavy fragment and to the light one could be disentangled. These TDPE are solutions of an optimization problem and can offer a measure of the average dissipated energy at a given deformation during the disintegration process provided that the velocity of the deformations is nown. The difference between the total energy value E obtained within the TDPE and E 0 given by the static BCS-equations represents an approximate measure for the dissipation E : E = E E 0. (7) E is expressed simply in terms of ρ and κ E = 2 ɛ ρ G 2 κ G ρ 2, (8) and E 0 corresponds to the values ρ 0 and κ 0 associated to the lower energy state, that is, obtained from BCS equations. This definition was introduced in Ref. [14] where the nuclear viscosity coefficient is determined by comparing microscopic results with hydrodynamic ones. So, the TDPE provide the values of ρ and κ in each fragment. By calculating the BCS solutions of the same fragment in the average deformation, in the corresponding pairing active space, the intrinsic excitation is obtained with formula (7). 3. Results The low energy fission of 234 U will be investigated. In the macroscopic microscopic method (MMM), the whole system is characterized by some collective coordinates that vary in time leading to a split of the nucleus into two fragments of masses A 1 and A 2. The macroscopic microscopic model provides a single particle potential in which the nucleons move independently. As mentioned in Ref. [14], such a description is within the spirit of the more rigorous Hartree Foc method which defines the potential in terms of the instantaneous positions of the nucleons. The basic ingredient of the MMM is the nuclear shape parametrization. An axial symmetric nuclear shape is obtained by smoothly joining two spheroids of semi-axis a i and b i (i = 1, 2) with a nec surface generated by the rotation of an arc of circle. By imposing the condition of volume conservation we are left with five independent generalized coordinates that are associated to five degrees of freedom: the elongation R denoting the distance between the centers of the fragments; the necing parameter C = S/R 3 related to the curvature of the nec; the eccentricities ɛ i =[1 (b i /a i ) 2 ] 1/2 if a i b i (or ɛ i = [1 (a i /b i ) 2 ] 1/2 if a i < b i ) associated to deformations of the two fragments (a i and b i denoting the semi-axis) and the mass asymmetry parameter η = a 1 /a 2 given by the ratio of the major semi-axis. This parametrization was widely used previously in the calculation concerning cluster decay [19,20], the dissipation during the fission process [21], the evidence of a new pair breaing effect [22] or the generalization of the microscopic time dependent equations [23] or the craning inertia [24]. As specified in Ref. [25], first of all a calculation of the trajectory in our five-dimensional space, beginning with the ground state of

3 254 M. Mirea / Physics Letters B 717 (2012) the system up to the exit point of the barrier must be performed. The minimal action principle was used to determine the best path in the configuration space and hence the potential barrier as realized in Ref. [27]. The dependencies of ɛ i (i = 1, 2), C and η as function of the elongation R were obtained. It is not yet understood how the compound nucleus is transformed in a variety of different fragmentations [26]. It is also believed that the models for mass distributions have limited predictive power. To overcome these difficulties, some simple assumptions are made in our wor. We rely on the results of Ref. [28] where the mass distribution of the fragments was relatively well reproduced by considering that the variation of the mass asymmetry is linear from the saddle configuration of the outer barrier up to the exit point. It is considered that under the rapid descent from the top of the second barrier different mass partitions are obtained. To obtain these different mass partitions, the simplest way is to vary the mass asymmetry parameter η. We made a simplifying assumptions by considering that the generalized coordinate associated to the nec follows the variation obtained along the minimal action path in all chan- Fig. 1. The calculated ground state eccentricities ɛ 1,2 of the primary fragments are given with a full line as function of the fragment masses A 1,2. The averaged eccentricities used in this wor are displayed with a dashed line. nels. In order to select the channels at different asymmetries, we choose partitions that give considerable values of the independent yields. From the maximal values of even even independent fission yields of Ref. [29] the binary partitions corresponding to the next isotopes were selected: 66,68,70,72 Ni, 74,76 Zn, 78,80 Ge, 82,84,86 Se, 88,90,92 Kr, 94,96 Sr, 98,100,102 Zr, 104,106,108 Mo, 110,112 Ru, 114,116,118,120 Pd, 122,124 Cd, 126,128,130 Sn, 132,134,136 Te, 138,140 Xe, 142,144,146 Ba, 148,150,152 Ce, 154,156 Nd, 158,160 Sm, 162,164,166,168 Gd. The ground state deformation of each nucleus was calculated. The deformations are represented in Fig. 1. To avoid a rapid staggering of the deformations that could not be realistic, we preferred to wor with smoothed interpolations of the ground state fragment eccentricities. The averaged eccentricities used in this wor are plotted in Fig. 1. Only prolate deformations were used because the oblate ones produces unfavorable population of the fission channels. The potential energies and the inertia were computed for each partition as represented in Fig. 2. The inertia is computed within the semi-adiabatic craning model described in Ref. [24] by taing into account the blocing effect [30]. That allows the calculation of the Gamow factor of each channel, and hence the penetrabilities. The action integral gives the probabilities to obtain these partitions and therefore the relative yields can be evaluated. TherelativeyieldsareplottedinFig. 3. The most important behavior of the mass distribution is reproduced: the distribution exhibits two clearly separated symmetric mountains and the heavy pea is centered at A 1 134, while the experimental value is approximately 136 [2]. Another prominent feature of the mass distribution is an oscillation in yield. A such phenomenon was evidenced experimentally [31]. The excitation energy of the fragments is a sum between the intrinsic dissipation and the collective energy. In our case the shift of the deformations from the ground state values give a measure of the excitation accumulated into the collective modes. The difference between the potential energy of the emerging fragments and their ground state energy is considered as a collective excitation. The intrinsic excitation must be not involved in the calculation of the collective excitation, so that both deformation energies are determined adiabatically in the framewor of the MMM. Fig. 2. (a) Potential energy V as function of the elongation R and the masses of the light fragment A 2. (b) Contour plot of the potential energy in steps of 1 MeV. (c) Inertia B in the semi-adiabatic approximation as function of R and A 2. (d) Contour plot of the inertia in steps of 2 h 2 /MeV/fm 2 units.

4 M. Mirea / Physics Letters B 717 (2012) Fig. 3. Relative yields Y in a logarithmic scale as function of the masses of the fragments A 1 and A 2. Fig. 4. The total excitation energies E of the fission fragments are plotted versus their masses with a thic full line. A thin full line is used for the deformation energy, a thin dashed line gives the neutron dissipated energy while the thin dotdashed line gives the dissipated energy for the proton space. The sum of excitation energies of the fission partners is plotted as function of the heavy fragment mass with a thic dashed line. The collective excitations are represented in Fig. 4 as function of the mass of the primary fragments with a thin full line. In order to evaluate the intrinsic excitation, the microscopic equations of motion must be solved. To solve the TDPE, the velocity of the elongation R/ t is required. This velocity is considered 10 6 m/s, that gives a time for the descent between the saddle to scission of about s. This value is considered as a typical scission time. The initial conditions for ρ and κ are set by the ground state BCS solutions for the parent nucleus. Up to the outer barrier saddle point configuration, the value of λ is considered zero. That is, the system evolves for all possible partitions in the same manner. The classical TDPE are solved [14,15]. At the saddle point a non-zero value for λ is introduced such that in the descend of the outer barrier the particle number projection must be realized. This procedure is possible because the two-center shell model is able to disentangle the wave functions that belong to the two fragments even before the scission [7]. When the number of particles reaches the final value, the value of G 12 is set to zero. When G 12 is zero, two uncoupled systems of time dependent pairing equations are obtained characterizing the two fragments separately. It is not possible to extract directly the energy partition from experimental data. Nevertheless, the main de-excitation process is the neutron evaporation. Therefore, indirect information can be obtained from neutron multiplicities, for which accurate results are available [2]. The excitation energy of each fragment is computed within the relation (7). The results are plotted in Fig. 4. Several experimental features are reproduced by the theoretical data. The deeply minimum in the neutron multiplicity occurs close to the mass of the doubly magic nucleus 132 Sn. A maximal value of the neutron multiplicity is obtained for the mass 116, that is in the symmetric fission region. In general, the excitation energy of the light fragment is larger than that of the heavy one. We can compare also the behavior of the average neutron multiplicity with the sum of total excitation energies of partners. The average neutron multiplicity is defined as the sum of neutrons from both fragments [2]. The minimal value of the average neutron multiplicity appears around 130u. This fact is reproduced by theoretical data. Our results plotted with a thic dashed line in Fig. 4 give a minimum of the excitation. An enhanced neutron emission is obtained in the symmetric fission and the very asymmetric region. Between these extreme, the total neutron emission shows a structure with a maximal value around A 1 = 140. These behavior are also exhibited by the total excitation energy [2]. From another respect, it must be mentioned that the excitation energy exhibits a structure with strong fluctuations, especially for A 2 < 82. A possible explanation for this peculiar aspect could be related to the existence of another fission path associated to large mass asymmetries. Therefore, the deformations and the dynamics assumed in this region could be not realistic. It is important to discuss about the validity of the model when the initial excitation varies, for instance in induced fission for different inetic energies of the bombarding particles. A variation of the initial excitation energy could be translated in deviations from the optimal fission path or in changes of the collective internuclear velocity. In our treatment, the cold fission path was considered as a consistent ingredient. There are some arguments in the favor of our treatment. For example, emphasizing the role of the small values of shell effects in the outer barrier region, the topographic theorem claims [32] that the fission trajectory does not vary very much with the initial excitation energy. This assumption is supported by older affirmations that the trajectories for spontaneous and induced fission are identical or at least similar [33]. In our model, up to the outer saddle point the paths of all partitions are the same. Due to its simplicity, the integral formalism of the minimal action principle is used in our calculations. The differential formalism in terms of Euler Lagrange equations must give an equivalent description of the fission path. So the path must be the same in the classical region (characterized by an excitation energy larger than the deformation energy) and in the forbidden region (where the excitation energy is lower than the deformation one) if no large variations of the barriers are possible. On another hand, the modification of the collective internuclear velocity produce a larger effect on the intrinsic excitations. As showed previously in Refs. [7,34], the magnitude of the dissipated energy is proportional to the internuclear velocity no matter the channel. Therefore, the structure of the dissipated energy distribution as function of the mass will be preserved, and only magnitude of the dissipated energies change. When the collective energy is large, a part of it is lost into intrinsic dissipation. If the collective energy is small, the internuclear velocity must decrease to very small values. But for small values of the internuclear velocity, the dissipation becomes negligible leading to an attenuation of the structure. In our formalism, the deformations of the complementary fragments play a very important role. The excitations are obtained by calculating only the occupation probabilities of the single particle states in the common active pairing space, that is for a finite number of levels above and below the Fermi energy. When the scission is reached, the levels of the two separated fragments are superimposed. The dynamical occupation probabilities of the levels depend on the way in which the levels of complementary fragments are interlaced and bounced in shells. On another hand, the ground state BCS probabilities in each nucleus depend

5 256 M. Mirea / Physics Letters B 717 (2012) on the succession and the energies of the levels. These considerations emphasize the role of the deformations. If the deformation of one of the nucleus is changed, the levels change their energies leading to a modification of the intermission with the single particle states of the complementary fragment, and even the number of levels in the common pairing space of the two fragments could not be the same. Due to these properties, for some deformations, it is even impossible to find solutions of the equations or the dissipated energy found in one fragment is very large. 4. Conclusion Excepting the strong fluctuations related to the large mass asymmetries, the results agree qualitatively well with the data that correspond to neutron multiplicities. It is a first microscopic description of the intrinsic energy partition in a wide range of fission channels that succeed to reproduce the main behavior of the neutron multiplicities. Concluding, it is important to stress that the internal energy at scission is not a free energy which may be distributed in any way between the two fragments. The energy flow depends on the shapes at scission. The shapes at scission are directed by a dynamical trajectory. This trajectory depends on the internal structure of the system through the shell effects. Once the scission is obtained, the nucleons must be distributed onto the microscopic levels of the two fragments. The internal excitation is produced by this rearrangement of nucleons. But in the same time, the single particle energies distribution depends on the deformations of the fragments. Therefore, a subtle interplay exists in permanence between the intrinsic energy and the excitation energy due to deformations. A shift of the final deformations from the best configuration not only produces collective excitations, but also a variation of the dissipated energy. Other microscopic descriptions are available in the literature. The most fundamental one is based on the Hartree Foc formalism [8]. Possible excitations states are constructed by mixing different sets of quasiparticles. Some of these combinations lead to well defined combinations of binary systems. This method is able to deduce the probability of populating different scission configurations and provides some criteria to define the scission configuration. On a more phenomenological way, in the independent single particle model of Ref. [13], the conservation of the number of particle requires the introduction of a quantity that reflect the prescission emission/absorption of neutrons. Acnowledgement This wor was supported by CNCS-UEFISCDI, project number PN-II-ID-PCE References [1] J.R. Nix, Nucl. Phys. A 502 (1989) 609c. [2] K. Nishio, M. Naashima, I. Kimura, Y. Naagome, J. Nucl. Sci. Tech. 35 (1998) 631. [3] V.F. Apalin, Yu.N. Gritsyu, I.E. Kutinov, V.I. Lebedev, L.A. Miaelian, Nucl. Phys. 71 (1965) 553. [4] J.C.D. Milton, J.S. Fraser, Phys. Rev. Lett. 7 (1961) 67. [5] J. Terrel, Phys. Rev. 127 (1962) 880. [6] R. Vandenbosch, Nucl. Phys. 46 (1963) 129. [7] M. Mirea, Phys. Rev. C 83 (2011) [8] W. Younes, D. Gogny, Phys. Rev. Lett. 107 (2011) [9] K.-H. Schmidt, B. Jurado, Phys. Rev. Lett. 104 (2010) [10] K.-H. Schmidt, B. Jurado, Phys. Rev. C 83 (2011) [11] C. Morariu, A. Tudora, F.J. Hambsch, S. Oberstedt, C. Manailescu, J. Phys. G 39 (2012) [12] Yong-Jing Chen, Jing Qian, Ting-Jin Liu, Zhu-Xia Li, Xi-Zhen Wu, Neng-Chuan Shu, Int. J. Mod. Phys. E 21 (2012) [13] N. Carjan, F.-J. Hambsch, M. Rizea, O. Serot, Phys. Rev. C 85 (2012) [14] S.E. Koonin, J.R. Nix, Phys. Rev. C 13 (1976) 209. [15] J. Bloci, H. Flocard, Nucl. Phys. A 273 (1976) 45. [16] D.S. Delion, M. Baldo, U. Lombardo, Nucl. Phys. A 593 (1995) 151. [17] R.C. Kennedy, Phys. Rev. 144 (1966) 804. [18] J. Dobaczewsi, W. Nazarewicz, T.R. Werner, J.F. Berger, C.R. Chinn, J. Decharge, Phys. Rev. C 53 (1966) [19] M. Mirea, A. Sandulescu, D.S. Delion, Nucl. Phys. A (2011) 23. [20] M. Mirea, A. Sandulescu, D.S. Delion, Eur. Phys. J. B 48 (2012) 86. [21] M. Mirea, Int. J. Mod. Phys. E 21 (2012) [22] M. Mirea, Phys. Lett. B 680 (2009) 316. [23] M. Mirea, Phys. Rev. C 78 (2008) [24] M. Mirea, R.C. Bobulescu, J. Phys. G 37 (2010) [25] M. Brac, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsy, C.Y. Wong, Rev. Mod. Phys. 44 (1972) 320. [26] J. Randrup, P. Moller, Phys. Rev. Lett. 106 (2011) [27] M. Mirea, L. Tassan-Got, Cent. Eur. J. Phys. 9 (2011) 116. [28] M. Mirea, D.S. Delion, A. Sandulescu, Phys. Rev. C 81 (2010) [29] A.C. Wahl, At. Data Nucl. Data Tables 39 (1988) 1. [30] M. Mirea, Rom. Rep. Phys. 63 (2011) 676. [31] I.S. Grant, Rep. Progr. Phys. 39 (1976) 955. [32] W.D. Myers, W.J. Swiateci, Nucl. Phys. A 601 (1996) 141. [33] T. Ledergerber, H.C. Pauli, Nucl. Phys. A 207 (1973) 1. [34] M. Mirea, L. Tassan-Got, C. Stephan, C.O. Bacri, Nucl. Phys. A 735 (2004) 21.