Influence of angular momentum on fission fragment mass distribution: Interpretation within Langevin dynamics

Transcription

1 Nuclear Physics A 765 (2006) Influence of angular momentum on fission fragment mass distribution: Interpretation within Langevin dynamics E.G. Ryabov a,, A.V. Karpov b,g.d.adeev a a Department of Theoretical Physics, Omsk State University, Prospect Mira 55-A, Omsk, Russia b Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, Russia Received 26 August 2005; received in revised form 11 October 2005; accepted 13 October 2005 Available online 10 November 2005 Abstract Dependence of fission fragments mass distribution on the angular momentum within Langevin dynamics is studied. The calculations are performed in the framework of the rotating temperature-dependent finiterange liquid drop model. The calculations are done for the five nuclei, representing heavy fissioning nuclei, medium fissioning nuclei and light fissioning one with the angular momentum varied in the wide range from l = 0tol = 70 h. The dependence coefficients dσm 2 /dl2 for the investigated nuclei are extracted. The comparison of the extracted values with the experimental data reveals a good agreement for all the cases (the heavy, medium, and light fissioning nuclei). It is found out that the obtained dependence of σm 2 on l can be explained with the help of temperature at scission as a function of l. The latter dependence is determined by dependence of the mean prescission neutron multiplicity on l. The analysis of this dependence is done as a competition between fission process and neutron evaporation. Remembering of the former large fluctuations of mass asymmetry coordinate during descent from the saddle to scission is considered. It is shown that the remembering effect takes place, but does not play a crucial role for the investigated dependence of σm 2 on l Elsevier B.V. All rights reserved. PACS: i; Gg; Ma; Gv Keywords: Langevin fission dynamics; Angular momentum dependence; Mass distribution; Mean neutron multiplicity * Corresponding author. address: ryabov_e@rambler.ru (E.G. Ryabov) /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.nuclphysa

2 40 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Introduction The influence of the projectile angular momentum (l) on the fission fragment mass-energy distribution (MED) has been under discussion for almost twenty years. The various theoretical aspects of this influence were considered in many works (see work [1] and references therein) with contradictory results. We should mention here works of Gregoire and Scheuter [3], Faber [4] and Glagola et al. [5]. Nevertheless, there are still many deficiencies in the theoretical investigations of this influence. Estimations and dependencies based on the experimental values were published almost ten years ago [1] and later summarized in the review [2] for a wide range of fissility parameters. However, theoretical works of Adeev et al. [6,7] have been the only full-scale investigations of l-dependence of the MED until the present work. It has been shown [1] that results on l-dependence of the mass distribution (MD) from analysis based on experimental data can be summarized as follows: two groups of nuclei can be defined. For the first group (Z 2 /A 31) the variance of the mass distribution (σm 2 ) increases as the angular momentum of the compound nucleus becomes higher, i.e., dσm 2 /dl2 > 0. For the second group (light nuclei with Z 2 /A < 30) the mass distribution variance decreases as the angular momentum increases, i.e., dσm 2 /dl2 < 0. For the nuclei in the range of fissility between the two cases mentioned above, for example for Pt and Os, one type of dependence turns into another, i.e., dσm 2 /dl2 0. The results of research done by Adeev et al. [6,7] report qualitative agreement with conclusions of the above summarized analysis of Itkis et al. [1]. Although values of the ratio dσm 2 /dl2 are underestimated in comparison with the experimental results [1], the main features of dσm 2 /dl2 dependence on fissility Z 2 /A are well reproduced. Even the transition region, where dσm 2 /dl2 turns into 0, is nearly the same. Many recent results have demonstrated the successful application of the multidimensional Langevin equations to the fission of excited compound nuclei formed in reactions induced by heavy ions [8 11]. From the physical point of view, the Langevin equations are equivalent to the Fokker Planck one, which was widely used for modeling the fission of excited nuclei in the framework of the diffusion model. Particularly, the calculations of MED characteristics by Adeev et al. [6,7] were made by means of the Fokker Planck equation. The multidimensional Langevin equations are more suitable for computer modeling and do not require extra assumptions and approximations during the integration procedure in contrast to the Fokker Planck equation. A very important feature of our models based on the Langevin equations [8,9] is the capability to take into account particles evaporation preceding the fission via combining the dynamical model with the statistical one, as it was proposed by Mavlitov, Fröbrich, and Gontchar [12,13]. Thus, our combined dynamical-statistical model allows us to describe MED parameters along with the mean prescission neutron multiplicity, charged particle multiplicities, and gamma-quanta. So the aim of our study is to theoretically investigate the dependence of the mass distribution variance σm 2 on the angular momentum of the compound nuclei l. The used dynamical model along with the constituting ingredients is described in Section 2. The results and discussion can be found in Section 3. In Section 4 we summarize the obtained results and draw several conclusions. 2. Model The dynamical model has been described in our previous paper [10]. But in the present work we use a generalized temperature dependent finite-range model, proposed by Krappe [14], to calculate the driving force in the Langevin equations instead of the finite-range model with Sierk

3 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) coefficients [15]. Therefore, in order to make our article self-enclosed we depict all the main features of the used model Multidimensional Langevin equations In our dynamical calculations we use the well-known {c,h,α} parametrization [16]. Here c is the elongation parameter, h describes neck thickness evolution and α determines the mass ratio of future fragments. In [10] we introduced a new mass-asymmetry parameter scaled with elongation α = αc 3, which retains the same meaning as α. So, we used (c,h,α ) as collective coordinates in our modeling. The multidimensional Langevin equations will be given [17] in discretized form convenient for numerical simulation as ( µjk (q) p (n+1) i q (n+1) i = p (n) i + θ (n) ij ( 1 2 p(n) j p (n) k τ, ξ (n) j = q (n) i µ(n) ij (q)( p (n) j q i ) (n) K (n) i ) (q) + γ (n) ij (q)µ (n) jk (q)p(n) k τ + p (n+1) ) j τ, (1) where q = (c,h,α ) are the collective coordinates, p = (p c,p h,p α ) are the conjugate momenta, K i (q) is a conservative driving force, m ij ( µ ij = m ij 1 ) is the tensor of inertia, γ ij is the friction tensor, θ ij ξ j is a random force, τ is the integration time step, and ξ j is a random variable satisfying the relations (n) ξ = 0, i ξ (n 1 ) i ξ (n 2) j = 2δij δ n1 n 2. (2) The upper index n in Eqs. (1) and (2) denotes that the related quantity is calculated on the nth time step. In these equations, and further in this paper, we use the convention that repeated indices are to be summed over from 1 to 3, and that angular brackets denote averaging over an ensemble. The strengths of the random force are related to the diffusion tensor D ij by the equation D ij = θ ik θ kj, which, in turn, satisfies the Einstein relation D ij = Tγ ij.heret is the temperature of the heat bath constituted by internal degrees of freedom. The temperature of the heat bath is connected with the internal excitation energy E int of the nucleus through the level-density parameter a(q) by the relation of the Fermi-gas model T = (E int /a(q)) 1/2.The internal excitation energy is determined by using the energy conservation law E = E int + E coll (q, p) + V(q) + E evap (t), (3) where E is the total excitation energy of the nucleus, E coll (q, p) = 1 2 µ ij (q)p i p j is the kinetic energy of the collective degrees of freedom, V(q) is the potential energy, and E evap (t) is the energy carried away by evaporated particles by the time t. The question of the coordinatedependent level-density parameter a(q), the potential energy V(q), and the driving force K(q), entering Langevin equations (1) is considered particularly in Section 2.2. A modified one-body mechanism of nuclear dissipation [18,19] has been used to determine the dissipative part of the driving forces. The expression applied to calculate the friction tensor for the so-called surface-plus-window dissipation can be found in our previous works [10,11]. We

4 42 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) used the reduction coefficient value k s = This value is very close to the one extracted from experimental analysis [18] on the widths of giant resonances (k s = 0.27). Also, our previous calculations [10,11] have shown that the experimental data on the mean prescission neutron multiplicities and mass energy distribution variances can be reproduced with the k s coefficient in the range k s = The inertia tensor is calculated by means of the Werner Wheeler approximation for incompressible irrotational flow. A description of the method is given, for example, in Ref. [20]. It was shown in Ref. [21] that the Werner Wheeler method allows, with surprisingly high accuracy, to perform a calculation of the inertia tensor for almost all shapes of the fissioning nucleus, with the exception of the zero neck radius configurations. The Langevin trajectories are simulated starting from the ground state with the excitation energy E of the compound nucleus. The initial conditions were chosen by the Neumann method with the generating function { P(q 0, p 0,l 0,t = 0) exp V(q 0) + E coll (q 0, p 0 ) T } δ(q q 0 )δ(l l 0 ). (4) In the approach used it is necessary to choose the initial state of the fissioning system at its ground state for the each angular momentum value l. But for large values of l a potential energy pocket disappears, and the choice of initial collective coordinates becomes undefined. Therefore, we start modeling fission dynamics from a spherical nucleus, i.e., q 0 = (c 0 = 1.0,h 0 = 0.0,α 0 = 0.0). The initial state is assumed to be characterized by the thermal equilibrium momentum distribution. A spin distribution is very simple the angular momentum for compound nuclei in the ground state is constant, i.e., we simulate fissioning of the product of the heavy-ion complete fusion with the given l 0 value. It has been supposed that the scission occurred when the neck radius of the fissioning nucleus was equal to 0.3R 0 [16,22] (R 0 is the radius of the initial spherical nucleus). This scission condition determines the scission surface in the space of the collective coordinates. This value for the neck thickness for fission configuration was defined on the basis of the criterion of instability of a nucleus against variations in the thickness of its neck [16]. It has been shown [23], that the Strutinsky criterion for fission (0.3R 0 ) is a rather good approximation to the probabilistic criterion for scission of a fissile nucleus into fragments in the framework of the Langevin model employing the one-body nuclear viscosity mechanism. All the more, the question of the scission criterion is more crucial in the case of studying the energy distribution characteristics, not the ones of mass distribution, which are under consideration in the present work. Evaporation of the prescission light particles (j = n, p, α, γ ) along Langevin fission trajectories was taken into account using a Monte Carlo simulation technique [12,17]. All dimensional factors were recalculated when a light prescission particle was evaporated, only dimensionless functionals of the rotational, Coulomb, and nuclear energies were not recalculated. This procedure provides a good accuracy in calculating the potential energy. The loss of the angular momentum was taken into account by assuming that the light particles carry away l j = 1, 1, 2, 1 ( h) [17]. In other words, we recalculate the potential energy, the level-density parameter (see our previous work [24] for details), and the factors for the inertia and viscosity tensors after each evaporation event. These computations are obligatory, because the evaporation of light particles results in the change of the nucleonic composition of the initial nucleus. In the present work we have studied few compound nuclei with a rather low fissility (especially for the 162 Yb nucleus). This has been possible only by switching over to a statistical model description with a Kramers-type fission decay width after delay time, when stationary flux over

5 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) the saddle point is reached. This procedure has been first proposed in Ref. [12]. An appropriate expression for the fission width that is the generalization of the Kramers formula to the multidimensional case [25,26] reads as follows ( detω 2 ) ij (q gs ) 1/2 Γ f = ω K det Ωij 2 (q exp ( ( F(q sd) F(q ) gs) /T ), (5) sd) where the frequency tensors at the ground state and saddle point ( Ωij 2 2 ) (q F(q) gs) = µ ij (q gs ), q i q j q=q gs ( Ωij 2 2 ) (q F(q) sd) = µ ij (q sd ) q i q j q=q sd are determined by the inverse mass and by the curvature tensors. The Kramers frequency ω K is determined from the algebraic equation det ( E(2πω K / h) + (2πω K / h)µ ij (q sd )γ ij (q sd ) + Ω 2 ij (q sd) ) = 0, (6) where E is the unit matrix, and the coordinates q sd and q gs determine the saddle point and the ground state, respectively. The fission probability P f is calculated as P f = N f /(N f + N ER ), where N f is the number of fission events and N ER is the number of evaporation residue events. A dynamical trajectory will either reach the scission surface, in which case it is counted as a fission event; or if the excitation energy for a trajectory which is still inside the saddle reaches the value E int + E coll (q, p) + V(q)<min(B j,b f ) the event is counted as evaporation residue (B j is the binding energy of the particle j). Within statistical description the mode of decay (the specific light particle evaporation or fission event) was chosen by the Monte Carlo sampling. If the fission event had occurred, then we switched back to the dynamical description. We choose the initial condition according to generating function P(q 0, p 0 ) exp{ V(q 0)+E coll (q 0,p 0 ) T }, but for the start point on the ridge line instead of the ground state. V(q 0 ) is the potential energy of the nucleus at the start point on the ridge line. Initial momenta ph 0,p0 α can be negative or positive with the equal probability, whereas initial momentum pc 0 is directed to the scission. Thus the nucleus is not allowed to move back over the barrier after exit from the statistical description due to fission event. The key quantity in any statistical model of excited nuclei decay is the level density ρ(e,l,q). The functional form used in our model for the level density at a fixed excitation energy E and angular momentum l is given by [27] ( (2l + 1)a(q)1/2 h 2 ) 3/2 exp [2(a(q)E) 1/2 ] ρ(e,l,q) = 12 2J (q) E 2 K rot (E)K vib (E), (7) where J (q) is the rigid-body moment of inertia of the nucleus with respect to the axis perpendicular to the nuclear symmetry axis and K rot(vib) (E) is the rotational (vibrational) collective enhancement of the level density K rot (E) = [ Krot 0 1] f(e)+ 1, K vib = 1, [ ( )] E 1 Ecr f(e)= 1 + exp. (8) E

6 44 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) We use the constant values of E cr = 40 MeV and E = 10 MeV for the rotational enhancement of the level density [28]. In expression (8) Krot 0 looks as follows [27] K 0 rot = J T. The importance of the collective effects in the nuclear level density and vanishing of these effects with increasing nuclear temperature have been pointed out in many papers (e.g., [28 30]). Nevertheless, these effects are seldom incorporated in Langevin models. Our calculations [24] show that the experimental data on the mean prescission neutron multiplicity and fission probability are described better if the collective enhancements and their damping are considered. It is well known that the rotational enhancement for the spherical nucleus equals one due to symmetry effects. At the same time, the spherical deformation {c = 1,h= 0,α = 0} is never realized in the calculations except for the first step. Therefore, the rotational enhancement of the level density has been taken into account for each integration step of the Langevin equations. In contrast with our work [24] we use K vib = 1, i.e., we do not account for the vibrational collective enhancement of the level density. As noted in work of Junghans and coauthors [28], there is a problem how to estimate the collective enhancements for weakly deformed shapes (the parameter of quadrupole deformation β 2 < 0.15). In addition to that, our estimations showed that the coefficient under consideration dσ 2 M /dl2 is weakly dependent on whether vibrational enhancement is taken into account or not. Therefore, we supposed K vib = 1 in our present calculations Generalized temperature-dependent finite-range liquid-drop model Detailed description and study of the various properties of the model was done in our previous works [24,31]. Here we recall all main features of used model and our results, which are important for the multidimensional Langevin modeling. Free Helmholtz energy in the finite-range LDM based on Yukawa-plus-exponential mass formula as a function of the mass number A = N + Z, relative neutron excess I = (N Z)/A,the temperature, the angular momentum, and a set of collective coordinates q has been suggested [14] in the following form: F(A,Z,q,T,l) ( = a v 1 kv I 2) ( A + a s 1 ks I 2) B n (q)a 2/3 + c 0 A 0 Z 2 + a c A 1/3 B c(q) ( ) 5 3 2/3 Z 4/3 a c 4 2π A 1/3 + h 2 l(l + 1), (9) 2J(q) where a v, a s, and a c are the usual volume, surface, and Coulomb energy parameters of the finite-range LDM at zero temperature and k v and k s are the corresponding volume and surface asymmetry parameters. The deformation dependence is taken into account through the shape functions B n (q), B c (q), and J(q) 1 B n (q) = 8π 2 a 4 r0 2A2/3 15 B c (q) = 32π 2 r0 5A5/3 V V V V ( 2 r ) r exp( r r /a) a r r dr dr, (10) /a [ ( r ) r exp ( r ] r ) dr dr 2a d a d r r. (11)

7 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Table 1 The temperature-dependent finite-range LDM coefficients. Values at zero temperature (first row) and temperature coefficients x i (second row) r 0 (fm) a (fm) a d (fm) a v (MeV) k v a s (MeV) k s a i (0) x i (MeV 2 ) The last term in Eq. (9) represents the rotational energy with the shape-dependent rigid-body moment of inertia and takes into account the diffuseness of a realistic nuclear density [32]. The functionals of the generalized nuclear energy and Coulomb energy given by Eqs. (10) and (11), respectively, model effects of finite range of nuclear forces and the realistic distributions of charge and nuclear densities. These functionals in the finite-range LDM are no longer independent of the size of the nuclear drop and must be calculated for any specific nucleus in contrast to the LDM with a sharp surface. For spherical nuclei they are also no longer equal to 1 but they can be evaluated in closed form [14]. The temperature dependence of the 7 coefficients entering Eq. (9) a v, a s, k v, k s, r 0, a, and a d is parameterized in the form a i (T ) = a i (T = 0) ( 1 x i T 2), (12) that can be expected [33] valid for T 4 MeV. The most complete information about the thermal coefficients x i has been obtained through self-consistent extended Thomas Fermi calculations with SkM interaction [33,34]. A liquid drop expansion of these results for the Gibbs free energy has been converted to the structure of Eq. (9) in Ref. [14]. The values of 14 parameters recommended in [14] and used in the present work are listed in Table 1. If the Helmholtz free energy is available, the entropy and the level-density parameter can be obtained from thermodynamic relation and well-known formula of the Fermi-gas model ( ) F(q,T) S(q,T)= T V, a(q,t)= S(q,T). (13) 2T A valuable conclusion was made in our previous work [24]: a(q,t)is weakly dependent on the nuclear temperature T. Thus, we suppose a(q,t) in the finite-range LDM being independent of the nuclear temperature and calculate it at T = 1.5 MeV. It appreciably simplifies application of the finite-range LDM, especially,to the multidimensional Langevin simulations. We note here that we calculate only the level-density parameter with T = 1.5 MeV. In our Langevin modeling of the fission of the compound nucleus we dealt with the nuclei at different temperatures, which depend on their excitation energies (see the beginning of Section 3). From the Fermi-gas model one has E int (q,t)= E(q,T) E(q,T = 0) = a(q)t 2, (14) where E int (q,t) and E(q,T) are the internal and total excitation energies of the nucleus, respectively. It follows from Eqs. (13) and (14) that temperature dependence of the Helmholtz free energy can be approximated by the expansion F(q,T)= E(q,T) TS(q,T)= V(q) a(q)t 2, (15) where V(q) is the zero-temperature potential energy surface. The microscopic self-consistent Thomas Fermi calculations with SkM interaction [34] have shown that Eq. (15) can be a reasonable approximation for T 4MeV.

8 46 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) So, in the framework of the temperature-dependent finite-range model we can calculate consistently the two main input parameters for the Langevin simulations combined with the statistical model: the driving force and the level-density parameter. It can be easily seen that the zero-temperature potential energy surface V(q), which enters the equation (15) is just the potential energy surface in the finite-range model with the Sierk coefficients [15]. Thus, the usage of the term generalized in the name of the model used in our present work must be clear. Deformation dependence of the level density parameter is often approximated by a(q) = a v A + a s A 2/3 B s (q), (16) where B s (q) is the dimensionless functional of the surface energy in the LDM with a sharp surface [16], a v and a s are the volume and surface coefficients of the level density parameter. In dynamical modeling the two sets of the coefficients a v and a s are frequently used [35,36]. Ignatyuk and coworkers proposed that a v = and a s = 0.095, while the coefficients of Töke and Swiatecki read as a v = and a s = It was shown in our works [24,31], that the level-density parameter from the finite-range LDM [14] is close to the values of a(q) in the Ignatyuk parametrization, regardless of the dependence one examines: deformation dependence of a(q) for a given nucleus, or a fissility parameter dependence of a(q) for the spherical nuclei. This fact can be explained by the results of our approximation, which was done in our recent work [24]. We have carried out an approximation procedure of the deformation dependence of the level-density parameter by expression (16) in order to extract the values of the coefficients a v and a s. Extracted values are a v = and a s = The coefficient a v has close values in all approximations of a(q), while the value of the a s extracted in [24] is quite close to the Ignatyuk coefficient, and differs more than twice from the Töke and Swiatecki value. To make it clear, we should mention that we use the exactly calculated level density parameter in the Langevin calculations. 3. Results and discussions In the present work we have carried out calculations for five reactions which result in the following compound nuclei: 244 Cm (E = 77 MeV,T = 1.9MeV), 224 Th (E = 184 MeV,T = 3.1MeV), 195 Hg (E = 75.7MeV,T = 2MeV), 184 Pt (E = 117.3MeV,T = 2.65 MeV), and 162 Yb E = 117.5MeV,T = 2.67 MeV. The initial excitation energies and the temperatures for the respective nuclei are presented in the brackets above. We calculated temperatures of the nuclei using the following relation T = (E /a(q)). All our results of dynamical calculations presented in the figures were obtained with those conditions. The first two compound systems represent the region of heavy fissioning nuclei, the third and fourth systems are the medium nuclei and the last one is an example of the light fissioning nucleus. The obtained results for these three regions of the mass number A are presented in this section. It is worth mentioning here that results for compound nucleus 224 Th are quite similar to those obtained for the 244 Cm nucleus. The picture is the same for two medium fissioning nuclei: results for the 195 Hg are qualitatively similar to the results of the 184 Pt modeling. Therefore, we present here the results for three compound systems only: 244 Cm, 184 Pt, and 162 Yb. They are representative enough to shed light on the question of the angular momentum dependence of the fission fragment mass distribution Results and comparison with the experimental analysis The results of our calculations for the mass distribution variances are presented in Fig. 1. There are three graphs showing the dependence of the variance of the mass distribution on l 2

9 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 1. Dependence of the variance of the mass distribution on l 2 for 244 Cm, 184 Pt and 162 Yb (solid line) and linear approximation (dash line) with linear approximation coefficient for these nuclei. Two values for 184 Pt and three values for 162 Yb nuclei of dσm 2 /dl2 coefficient represent the change in dependence of σm 2 on l. The summarized results are in the right down panel, where the dσm 2 /dl2 coefficient is plotted against Z 2 /A. The connected black squares indicated by the arrows on this graph are our present results (for the 162 Yb, 195 Hg and 184 Pt nuclei several black squares connected by the vertical lines represent the different values of the dσm 2 /dl2 coefficient for these nuclei). The solid curve and the dash-dotted curves are the results and error boundaries for the dσm 2 /dl2 coefficient from the analysis based on the experimental data [1]. The dash curve are the results of the theoretical calculations by means of the Fokker Planck equations, made by Adeev and colleagues [6]. for 244 Cm, 184 Pt and 162 Yb with excitation energies as stated above. In the right down panel in Fig. 1 there is a resulting graph, where the approximation coefficients dσm 2 /dl2 are shown for these three nuclei along with the results of the experimental analysis of Itkis, Rusanov and others [1,37] and previous theoretical results within the Fokker Planck dynamics. For the 184 Pt and 162 Yb nuclei the variance dependence coefficient dσm 2 /dl2 takes several values and even changes the sign as l increases. We estimated the magnitude of the dσm 2 /dl2 coefficient for the ranges where σm 2 monotonously depend on l for each nuclei. That is why there are three points for 162 Yb and two points for 184 Pt and 195 Hg on the right down panel in Fig. 1. It can be clearly seen from the very beginning of Section 3 that we have carried out our calculations with different excitation energies and initial temperatures of the compound nuclei under consideration. We have carried out calculations for three nuclei ( 244 Cm, 184 Pt, 162 Yb) with equal initial temperatures in order to pick out the influence of the variation in angular momentum on the fission fragment mass distribution only. We have chosen two values of the initial temperatures T = 2 MeV and T = 3.1 MeV for the nonrotating nucleus. Results of these calculations

10 48 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) have shown that we revealed just the same dependencies σm 2 on l as presented in Fig. 1. The coefficients dσm 2 /dl2 differ slightly (in the range of one hundredth) from the values presented in Fig. 1, which we obtained using different initial temperatures. Thus, all the further results and discussions are concerned with our calculations with different initial temperatures as stated in the very beginning of Section 3. The analyses of the experimental data on the fission fragment mass distribution made by Itkis, Rusanov et al. [1,37] are well known. The peculiarities of the MD variance dependence on the angular momentum are treated in those analyses in the terms of the stiffness (d 2 V/dη 2 )or, in other words, stability of the nucleus against the mass-asymmetric deformation in the saddle configuration. That is, the statical characteristics of the finite-range LDM [15] are only included into analysis: the configurations of the saddle and ground states and the angular momentum dependence of the stiffness of the nucleus. In the above mentioned analyses, it is stated that, for the medium and light nuclei observable values of σm 2 are determined by the saddle value of the stiffness, because there is practically no descent from the barrier for those nuclei. In the case of the light nuclei this stiffness increases with augmentation of l. This fact results in narrowing of the mass distribution. For medium nuclei the increase of the angular momentum leads to about the same shifts of the saddle and ground deformations along the elongation collective coordinate (c in our parametrization) towards each other. The stiffness for the medium nuclei d 2 V/dη 2 remains nearly the same regardless of the angular momentum value, so dσ 2 /dl 2 0 [37]. As stated in work [1], for the heavy nuclei the descent from the barrier plays an important role. The stiffness of the nucleus near the saddle point decreases with increasing l. Therefore, the mass distribution for heavy nuclei becomes broader in case of the remembering of the system about descent from the barrier. This assumption of the remembering of the former large fluctuations of mass asymmetry coordinate during descent from the saddle to scission holds true for the diffusion model used in works of Adeev et al. [6,7]. In other words the value of dσm 2 /dl2 > 0is explained with the help of effective stiffness, or stiffness averaged along the descent from the barrier. This assumption about remembering effect of the nuclear system about its descent is quite reasonable in the framework of the diffusion model and the two-body mechanism of viscosity, used in [6,7]. The mean fission time was estimated in the work [6]. The extracted values (2 5) s are comparable with the experimental estimations of the relaxation time (τ rel ) of the mass-asymmetric mode [38] (5.3 ± 1) s, which governs the forming of the mass distribution of fission fragments. It should be stressed that the mean fission times extracted with the help of the Langevin modeling and the one-body type of viscosity in the present work are much greater than estimations [6] (see Fig. 2). The difference can reach two or three orders of magnitude. Even in the case of heavy nuclei we get the mean fission times close enough to results of work [6] only for the large angular momentum (l >60 h). It is obvious from Fig. 2 that in this case the fission barriers turn practically into zero and the compound nucleus splits into fission fragments rapidly. One can guess from the comparison of the mean fission times that the evolution of the nuclear system proceeds much slower in the case of the Langevin dynamics with the one-body type mechanism of viscosity. Our calculations show that the stiffness of the nuclei along the mean fission trajectory are insensibly different for the different values of the angular momentum (see Fig. 3). Two curves in Fig. 3 do practically coincide. The angular momentum increase leads to the shift of the ground state of the nucleus along the elongation coordinate c to the lager values, i.e., ground state moves towards the position of the saddle point on the c axis. In other words this

11 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 2. The mean fission time t f and the fission barriers B f as functions of the angular momentum for the 244 Cm, 184 Pt, and 162 Yb compound nuclei.

12 50 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 3. The stiffness of the nucleus 184 Pt along the mean fission trajectory for the different values of the angular momentum. The solid line is for the zero angular momentum, circles are for the l = 50 h case. The arrows indicate the c coordinates of the saddle points for the two different values of the angular momentum. The right arrow is for the saddle at l = 0 h and the left one for l = 50 h. shift results in the abridgment of the corresponding curve of the stiffness along the mean fission trajectory (see Fig. 3 and its caption for details). In particular, this coincidence of the curves means the independence of the calculated stiffness for the scission shape on the angular momentum (see Fig. 4). So, one can see that the stiffness along the mean fission trajectory is weakly dependent on the angular momentum in the case of our dynamical modeling, therefore the value of the averaged or effective stiffness on the descent is also independent on l. That is why we should include additional factors into our analysis in order to understand the angular momentum dependence of the MD variance. The variance of the mass distribution having the Gaussian like form can be estimated in the framework of the statical approach with the following well-known equation [37]: σ 2 M = A2 T/16 ( d 2 V/dη 2). We define the mass-asymmetry coordinate η following Strutinsky s definition (see our previous work [31] for detail). As long as we consider the heated nucleus as the thermodynamic system we should use some thermodynamic potential instead of the potential energy V,forexample, the Helmholtz free energy F. In other words, we should calculate the stiffness values as d 2 F/dη 2, so Eq. (17) should be rewritten as following: σ 2 M = A2 T/16 ( d 2 F/dη 2). It is evident from this simple relation that, along with the stiffness, the temperature can also be a very important factor. As we supposed, only the short time interval (τ rel ) of the evolution of the nucleus along the mean trajectory is responsible for the process of the MED formation in our calculations. Therefore we use the temperature at scission (T sc ) in our analysis. Another reason for it is the fact that the most part of the prescission neutrons is evaporated before saddle [11], so the temperature of the system changes noticeably before the nucleus reaches the saddle. (17) (18)

13 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 4. The stiffness of the nuclei in the scission as a function of the angular momentum for the 244 Cm (squares), 184 Pt (triangles), and 162 Yb (circles) nuclei. We define the temperature at scission as the value of temperature in the mean scission point. The latter can be determined as intersection of a mean dynamical trajectory with a scission surface. To make it clear, we note here that if one calculates variance of the mass distribution using equation (18) with the stiffness values and the nucleus temperature at the scission for the different values of l, then one obtains the same angular momentum dependence as in the case of the dynamical modeling. In other words, if we calculate σm 2 st in the statistical limit (using Eq. (18)), we get just the same dependence on the angular momentum as for the dynamically calculated variance of MD σm 2 dyn, i.e., the fraction σ M 2 dyn /σ M 2 st is slightly depend on l. The difference in those two values of the mass distribution variances is illustrated by Fig. 5. One can clearly see that in case of heavy nuclei this ratio is very big, it is about 2.8 forthe 244 Cm nucleus. This fact manifests that the nucleus effectively remembers the former large fluctuations of mass asymmetry coordinate during descent from the saddle to scission, i.e., it remembers the descent along the fission trajectory with lower values for stiffness. But, as it was stated above those values are independent on l, sotheσm 2 dyn /σ M 2 st value is insensibly dependent on the angular momentum in the case of heavy and medium fissioning nuclei. One should not be confused with the evident decrease of the σm 2 dyn /σ M 2 st ratio for the 162 Yb nucleus with the increase of l: this is not caused by the angular momentum dependence of the stiffness value. On the contrary, it was shown above that the stiffness in the scission point is independent on l. Thus we see that remembering effect takes place especially for the heavy and the medium nuclei, but it does not cause the angular momentum dependence of the MD variance. Those effect is just the same, regardless of the angular momentum value of the compound nucleus. So, other factors are coming foreground. Appropriate computation of these factors can be done only in the dynamical modeling, which accounts for the evaporation of the light particles. This process leads to the change in the internal energy, which results in change of the temperature. Let us see now how this factor is influenced by the angular momentum of the nucleus. It is obvious from Fig. 6 that the obtained angular momentum dependence of T sc is in agreement with the dependence of the mass distribution variance of fission fragments on l. The temperature and the variance decrease in the same range of the angular momentum values. If the temperature at the

14 52 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 5. The angular momentum dependence of the ratio σm 2 dyn /σ M 2 stat. σ M 2 dyn are the results of our present dynamical calculations, while σm 2 stat are the statical limit values of the variances, calculated using Eq. (18) and the values of the stiffness and the temperature at the scission point. scission point is practically constant, then the variance is weakly dependent on l. The rise of T sc coincides with the rise of σm 2 with the variation of the angular momentum. The temperature at scission, in turn, is determined by the mean prescission neutron multiplicity ( n ). The more neutrons are evaporated, the colder the nucleus will be at scission. This inverse relationship between the mean prescission neutron multiplicity and the temperature at scission is clearly seen from Fig Detailed analysis The dependence of the evaporated neutrons on the angular momentum can be clarified using the fission barriers (B f ) and the neutron binding energies (B n ) for the compound nuclei with different l. The neutron binding energy is independent of the angular momentum value. In contrast, the value of the fission barrier is very sensitive to l. One can see from Fig. 8 that the whole investigated scope of l can be divided into the three ranges. One can define the range where B n >B f, the range where B n is very close to B f, and the range where B n <B f. These inequalities of B n and B f determine the inequalities for the neutron (Γ n ) and the fission (Γ f ) widths for the nuclei in the neutron evaporation cascade of the initial compound nucleus. For Γ n and Γ f we can also divide the scope of l into three ranges, with the same bounds as it was observed for the fission barriers and the neutron binding energies. For example, let us have a look at the 162 Yb nucleus (see Figs. 8 and 9). For l<45γ f Γ n and the decay time is mainly determined by the neutron width. The fission probability is determined by the ratio of the fission and the neutron widths at each step of the evaporation cascade and this ratio increases as the angular momentum increases. Due to that, the fission probability for the compound system which evaporates neutrons increases steadily with the comparison of the nonrotating nucleus. Therefore T sc decreases with the increase of l, because this increase of the angular momentum augments the fission probability for the system with low internal energy. In contrast, for l>55,b f <B n and decay time at each integration step can be determined

15 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 6. The variance of the mass distributions (open squares) and temperatures (circles) at the scission for the 244 Cm, 184 Pt, and 162 Yb nuclei as functions of the angular momentum.

16 54 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 7. The temperatures at the scission T sc (circles) and the mean prescission neutron multiplicity n (open squares) for the 244 Cm, 184 Pt, and 162 Yb nuclei as functions of the angular momentum.

17 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 8. The fission barriers B f (the solid lines) and neutron binding energies B n (the dash lines) as functions of the angular momentum for the nuclei in the neutron cascade of initial 162 Yb compound nucleus: lines with the squares correspond to 162 Yb nucleus, lines with the circles to 159 Yb, with the triangles to 156 Yb. mainly by the fission width. The higher the angular momentum value, the lower the barrier. Hence the probability for the nucleus to evaporate many neutrons before scission becomes smaller and smaller. We see the opposite tendency in this case: the amount of the evaporated neutrons decreases as angular momentum increases. The nucleus undergoes fission more and more quickly, without evaporating a large number of neutrons. For the intermediate values where 45 <l<55 B n and B f get closer. The neutron and fission widths are also close. Dependencies of the neutron multiplicity, temperature, and mean fission time on the angular momentum change in this range of l. The temperature decreases, the neutron multiplicity and mean fission time increase with the angular momentum for small values of l. In contrast, for l>50 the mean prescission neutron multiplicity and mean fission time decrease and the temperature increases with the angular momentum of the compound nucleus. Figs. 7 and 2 (for 162 Yb) illustrate the preceding statements. We have a slightly different picture in the case of the 184 Pt nucleus. We get a very wide range of l, where B n is very close to B f (see Fig. 10) and, accordingly, Γ f is close to Γ n. Practically, for the 184 Pt nucleus the mean prescission neutron multiplicity and temperature are weakly dependent on the angular momentum in the range of l from 0 to 30 h. Thus, we observe the same behavior for the variance of the mass distribution σm 2. Further increase of the angular momentum leads to the increment of the fission probability and we get the same situation as for the 162 Yb nucleus at large l. In the recent work of Gontchar et al. [39] the analysis has been done for the mean prescission neutron multiplicity and the mean fission time for 190 Pt as the functions of the angular momentum for a given excitation energy in the framework of the Langevin dynamics. The details of the analysis in that work are quite similar to ours. The resonance-like behavior for mean fission time and the mean prescission neutron multiplicity against the angular momentum of the

18 56 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 9. The neutron Γ n (squares) and the fission Γ f (circles) widths as functions of the angular momentum in the neutron cascade of initial 162 Yb compound nuclei.

19 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 10. The fission barriers B f (the solid lines) and the neutron binding energies B n (the dash lines) as functions of the angular momentum for the nuclei in the neutron cascade of initial 184 Pt compound nucleus: lines with the squares correspond to 184 Pt nucleus, lines with the circles to 181 Pt, with the triangles to 178 Pt. compound system was found by Gontchar and coworkers. Our results on the mean fission time are in qualitative agreement, while our results on the mean prescission neutron multiplicity cannot likely be treated as the resonance-like. This discrepancy may be caused by the fact that the one-dimensional Langevin model was used in [39]. For the heavy nuclei the situation is very simple. The inequalities between the fission and the neutron widths and the dependencies of the fission barriers and the neutron binding energies for the 244 Cm compound nucleus are presented in Fig. 11. It is obvious that one has a situation, which is similar to the case of 162 Yb or 184 Pt nuclei at high angular momenta (60 h and 50 h, respectively). B n is greater than B f for the nuclei in the neutron evaporation cascade of 244 Cm with any l. This results in the following: the fission width becomes greater than the neutron one after some neutron evaporation events, thus the fission mode of decay becomes dominant over the neutron evaporation and the nucleus splits rapidly. For the 244 Cm nucleus the dependence of all the examined characteristics on l remains immutable. The mean fission time and mean prescission neutron multiplicity decrease, the temperature at scission increases correspondingly (see Fig. 7 for 244 Cm nucleus), which results in the obtained dependence of σ 2 M on l2, presented in Fig. 6. The increase of T sc causes the increase of the variance of MD with the angular momentum. The dependence of the mean fission time in our calculations for the 244 Cm nucleus is in a qualitative agreement with the results of the above mentioned work of Gontchar et al. [39] for 235 U: t f decrease as the angular momentum becomes higher, without any resonance-like behavior. In other words the resonance-like behavior of the t f disappears as the fissility parameter increases (see Fig. 2).

20 58 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) Fig. 11. The neutron Γ n (squares) and the fission Γ f (circles) widths and the fission barriers B f (the solid lines) and the neutron binding energies B n (the dash lines) as functions of the angular momentum for the nuclei in the neutron evaporation cascade of initial 244 Cm compound nucleus: lines with squares correspond to 244 Cm nucleus, lines with circles to 241 Cm, with triangles to 238 Cm. 4. Summary It should be stressed that present results are the first three-dimensional full scale work in the framework of the Langevin dynamics which aims at the question of the angular momentum dependence of the variance σm 2. In the present work we have carried out the calculations for the three groups of nuclei. The determined coefficients dσm 2 /dl2 for these three groups of nuclei are in good qualitative agreement with the estimations based on the experimental data. Our results for the dσm 2 /dl2 coefficients can be summarized as follows: (1) dσ 2 M /dl2 > 0 for the heavy fissioning nuclei ( 244 Cm and 224 Th in our calculations); (2) for the medium fissioning nuclei ( 184 Pt and 195 Hg) dσ 2 M /dl2 0 at the angular momentum l = 0 35 h and l = 0 40 h respectively, and dσ 2 M /dl2 > 0 for those nuclei at l>35 h and l>40 h respectively; (3) for the light fissioning nuclei ( 162 Yb nucleus in our calculations) dσ 2 M /dl2 < 0atl = 0 45 h, dσ 2 M /dl2 0atl = h and dσ 2 M /dl2 > 0forl>50 h. These inequalities illustrate the well-known fact that the rotated nucleus can be effectively considered as the heavier nonrotating nucleus. The coefficients under consideration have shown

21 E.G. Ryabov et al. / Nuclear Physics A 765 (2006) weak dependence on the initial temperature (or excitation energy) of the nucleus for T ranges between 2 and 3 MeV. The analysis of the above mentioned inequalities shows: other important factors should be included into analysis along with the stiffness of the nuclei. We revealed that remembering of the former large fluctuations of mass asymmetry coordinate during descent from the saddle to scission is present, but it does not cause the angular momentum dependence of the MD variance. We have found out that the temperature at scission point can be treated as the factor which defines the dependence of σm 2 on l. Our present Langevin dynamical calculations show that, as long as the stiffness is weakly dependent on l for all investigated cases, the variance of the mass distribution manifests similar dependence on l as the temperature at scission. In turn, the temperature is determined by the cooling of the nucleus during the fission dynamics by the prescission neutron evaporation. The analysis of the mean prescission neutron multiplicity was done in our work with the help of the dependence of the fission and the neutron widths on l, similar to the work of Gontchar et al. [39]. The inequalities between Γ f and Γ n determine the concurrence between the two decay modes the fission and the neutron evaporation. Thus, it becomes obvious that Langevin dynamics combined with the statistical model play a crucial role we just cannot understand obtained dependencies for σm 2 only in terms of the statical characteristics of the potential energy map (the stiffness particularly). In our investigations the stiffness along the mean Langevin trajectory is independent of l, therefore the temperature at scission which is determined by the evaporated neutrons and the whole fission dynamics becomes very important. Acknowledgements We are grateful to Dr. A.Ya. Rusanov for enlightening comments and suggestions, and to M. Stefan for carefully reading the manuscript. References [1] M.G. Itkis, Yu.A. Muzichka, Yu.Ts. Oganessian, et al., Yad. Fiz. 58 (1995) 2140, Phys. At. Nucl. 58 (1995) [2] M.G. Itkis, A.Ya. Rusanov, Phys. Part. Nucl. 29 (1998) 389. [3] C. Gregoire, F. Scheuter, Z. Phys. A 303 (1981) 337. [4] M.E. Faber, Z. Phys. A 297 (1980) 277; M.E. Faber, Phys. Rev. C 24 (1981) [5] B.G. Glagola, B.B. Back, R.R. Betts, Phys. Rev. C 29 (1984) 486. [6] G.D. Adeev, I.I. Gontchar, V.V. Pashkevich, N.I. Pischasov, O.I. Serdyuk, Phys. Part. Nucl. 19 (1988) [7] G.D. Adeev, I.I. Gontchar, L.A. Marchenko, N.I. Pischasov, Yad. Fiz. 43 (1986) 1137, Sov. J. Nucl. Phys. 43 (1986) 727. [8] Y. Abe, S. Ayik, P.G. Reinhard, E. Suraud, Phys. Rep. 275 (1996) 49. [9] G.D. Adeev, A.V. Karpov, P.N. Nadtochy, D.V. Vanin, Phys. Part. Nucl. 36 (2005) 732. [10] A.V. Karpov, P.N. Nadtochy, D.V. Vanin, G.D. Adeev, Phys. Rev. C 63 (2001) [11] P.N. Nadtochy, G.D. Adeev, A.V. Karpov, Phys. Rev. C 65 (2002) [12] N.D. Mavlitov, P. Fröbrich, I.I. Gontchar, Z. Phys. A 342 (1992) 195. [13] P. Fröbrich, I.I. Gontchar, Nucl. Phys. A 563 (1993) 326. [14] H.J. Krappe, Phys. Rev. C 59 (1999) [15] A.J. Sierk, Phys. Rev. C 33 (1986) [16] M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky, C.Y. Wong, Rev. Mod. Phys. 44 (1972) 320. [17] P. Fröbrich, I.I. Gontchar, Phys. Rep. 292 (1998) 131. [18] J.R. Nix, A.J. Sierk, in: M.I. Zarubina, E.V. Ivashkevich (Eds.), Proceedings of the International School-Seminar on Heavy Ion Physics, Dubna, USSR, 1986, JINR, Dubna, 1987, p. 453; J.R. Nix, Nucl. Phys. A 502 (1989) 609.