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The role of neck degree of freedom in nuclear fission
Nuclear Physics A 808 (2008) 1–16 www.elsevier.com/locate/nuclphysa
The role of neck degree of freedom in nuclear fission Santanu Pal ∗ , Gargi Chaudhuri 1 , Jhilam Sadhukhan Physics Group, Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700 064, India Received 16 January 2008; received in revised form 22 March 2008; accepted 2 May 2008 Available online 9 May 2008
Abstract The dynamics of nuclear fission is investigated using Langevin equations in order to study the role of the neck degree of freedom in fission. The time-dependent fission widths are calculated in both one (with only elongation coordinate) and two (with elongation plus neck) dimensions. A quantitative estimate is also made of the distribution of the fission trajectories in two dimensions. Comparing the results from one and two-dimensional calculations, we find that the stationary fission widths are nearly the same in both the calculations. However, the most probable fission path beyond the saddle ridge in two dimensions deviates considerably from that in one dimension. © 2008 Elsevier B.V. All rights reserved. PACS: 05.40.-a; 24.60.-k; 24.60.Lz; 25.70.Jj Keywords: Nuclear fission; Neck parameter; Langevin equation; Fission trajectories
1. Introduction It is now an established fact that a dissipative dynamical model is essential for fission of highly excited heavy nuclei in order to reproduce a wide variety of experimental data. In particular, the statistical theory of nuclear fission due to Bohr and Wheeler [1] grossly underestimates the prescission multiplicities of light particles and GDR γ ’s [2–4]. The statistical theory also underestimates the evaporation residue cross sections in heavy ion induced fusion reactions [4,5]. Following the work of Kramers [6], a dynamical model for nuclear fission was initially developed employing the Fokker–Planck equation [7,8]. Subsequently, the Langevin equation was * Corresponding author. Tel.: +91 33 2359 8431; fax: +91 33 2334 6871.
E-mail address: [email protected] (S. Pal). 1 Present address: Physics Department, McGill University, Montreal, Canada.
0375-9474/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2008.05.001
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found to be more useful for practical applications [9]. Considering the elongation of a compound nucleus along the symmetry axis as the most relevant dynamical coordinate to describe nuclear fission, one-dimensional Langevin equation was found to account for the experimental values for prescission neutron multiplicity and evaporation residue cross sections for a number of heavy compound nuclei formed in heavy ion collisions [2,3,5,10]. The Langevin equation was subsequently solved in higher dimensions in order to explain the energy and mass distributions of the fission fragments [11–15]. In the present work, we would study the fission dynamics in elongation and neck coordinates using the Langevin equation. It is already known that addition of the neck degree of freedom in fission dynamics increases the stationary fission rate marginally compared to that obtained by considering only the elongation coordinate [12]. Therefore it would be of interest to find how closely the fission path in two dimensions (elongation plus neck coordinates) follows that in one (only elongation) dimension. One would also like to know the extent of excursions made by the Langevin trajectories along the neck degree of freedom and its effect on the scission point shape of the compound nucleus. Consequently the question arises regarding the role of neck dynamics in giving rise to the dispersion in fission fragment kinetic energy since this is mainly governed by the configuration of the compound nucleus at scission. We would address such concerns in this paper. We shall begin with a brief description of the basic equations in the next section. Various input quantities to the Langevin equation will also be described in this section. In addition to the timedependent fission rates, we shall also obtain numerical results for a number of suitably defined quantities depicting the distribution of the Langevin trajectories in two dimensions for the 224 Th nucleus. Section 3 will contain these results. We shall present a summary of the main results in the last section. 2. The Langevin equations In the Langevin description of nuclear fission, one usually assumes that the motion associated with the collective degrees of freedom relevant for fission can be modeled as that of a Brownian particle in a viscous heat bath [6,9]. The remaining nuclear degrees of freedom are considered to constitute the heat bath represented by a temperature while the interaction between the Brownian particle and the intrinsic motion of the heat bath is assumed to be random in nature. In order to specify the collective coordinates for a dynamical description of nuclear fission, we will use the “funny hills” shape parameters (c, h, α) as suggested by Brack et al. [17]. The elongation and neck degrees of freedom are denoted by c and h respectively while α corresponds to the asymmetry parameter. Since we are going to concentrate upon the neck dynamics in the present study, we will consider only symmetric fission (α = 0) in order to make unambiguous interpretation of the results. The surface of a nucleus of mass number A with elongation and neck coordinates as c and h, respectively, is defined as z2 2 2 ρ (z) = 1 − 2 a0 c0 + b0 z2 , b0 0, a0 0, c0 z2 b0 c0 z 2 2 , b0 < 0, a0 0, = 1 − 2 a0 c0 exp (2.1) R3 c0 where
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c0 = cR, 1
R = 1.16A 3 and c−1 , 2 in cylindrical coordinates. a0 and b0 are related by [17] b0 = 2h +
1 b0 − , b0 0, 3 5 c b0 4 , =− √ √ 1 p 3 e + (1 + 2p ) −πp erf( −p)
a0 =
b0 < 0,
(2.2)
where p = b0 c3 and erf(x) is the error function. The parameter h describes the variation of the thickness of the neck without changing the length 2c (in units of R) of the nucleus along the symmetry axis. Positive values of h implies that the neck formation starts at a lower value of c as compared to the case of h = 0 and hence scission of the nucleus into two fragments also takes place for a lower value of c. The neck region at scission is also smaller in such cases. On the other hand, a negative value for h shifts the scission point to a higher value of c. Consequently the neck region at scission is also longer. The two-dimensional Langevin equation in (c, h) coordinates has the following form [13], pj pk ∂ −1 ∂F dpi =− m jk − − ηij m−1 j k pk + gij Γj (t), dt 2 ∂qi ∂qi dqi −1 = m ij pj (2.3) dt where q1 and q2 stands for c and h respectively and pi represents the respective momentum. F is the free energy of the system and mij and ηij are the shape-dependent collective inertia and dissipation tensors respectively. The time-correlation property of the random force is assumed to follow the relation Γk (t)Γl (t ) = 2δkl δ(t − t ) and the strength of the random force is related to the dissipation coefficients through the fluctuation–dissipation theorem and is given as gik gj k = ηij T , where the temperature T of the compound nucleus at any instant of its evolution is given as T = Eint /a(q). The intrinsic excitation energy Eint is calculated from the total excitation energy E ∗ of the compound nucleus using the energy conservation, E ∗ = Eint + Ecoll + V (q), where Ecoll is the collective kinetic energy including the rotational energy and V (q) is the potential energy of the system. The level density parameter a(q) depends on the collective coordinates and is taken from the works of Ignatyuk et al. [18]. We will make the Werner–Wheeler approximation for incompressible irrotational flow to first calculate the collective inertia tensor and hence the components of its inverse (m−1 )ij [19]. The
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Fig. 1. Potential energy contours (in MeV) for 224 Th. The minimum in the potential energy valley is marked by the dotted line. The dashed line corresponds to scission configuration.
conservative driving force in the compound nucleus, which is treated here as a thermodynamic system, is derived from its free energy F for which we will use the following expression, F (c, h) = V (c, h) − a(c, h)T 2 , where the nucleus is considered as a noninteracting Fermi gas [20]. The potential energy V (c, h) is obtained from the finite-range liquid drop model by a double folding procedure [21]. The calculated potential along with locus of the minimum in the valley for 224 Th is shown in Fig. 1. We shall use the one-body model for nuclear dissipation in our calculations. The original wall friction (WF) due to Blocki et al. [22] and a weaker version of it, the chaos-weighted wall friction (CWWF) [23] will be considered here. A window friction will also be included in our calculation [24]. We choose the above two forms of dissipation essentially to investigate the dependence of the fission dynamics on the strength of dissipation. Further, the chaos-weighted wall friction has already been shown to give a better account of dissipation in fission [5,10]. The wall friction coefficients read as follows [25], zN −1 2 2 ∂D ∂ρ 2 2 ∂D 2 2 2 ∂ρ ∂ρ 1 ∂ρ ∂ρ 1 1 1 WF ρ2 + = πρm v¯ + + dz ηij 2 ∂qi ∂z ∂qi ∂qj ∂z ∂qj 2 ∂z zmin
zmax
+ zN
∂ρ 2 ∂ρ 2 ∂D2 + ∂qi ∂z ∂qi
∂ρ 2 ∂ρ 2 ∂D2 + ∂qj ∂z ∂qj
−1 2 2 2 ∂ρ 1 ρ2 + dz , 2 ∂z
(2.4)
where ρm is the mass density of the nucleus, v¯ is the average nucleon speed inside the nucleus and D1 , D2 are the positions of the centers of mass of the two parts of the fissioning system relative to the center of mass of the whole system. zmin and zmax are the two extreme ends of the nuclear shape along the z axis and zN is the position of the neck plane that divides the nucleus into two parts. Based on the argument that the degree of randomization of the single-particle motion within a nucleus depends upon its shape, it was shown earlier that the CWWF dissipation coefficient is given as [23] CWWF WF (c, h) = μ(c, h)ηij (c, h), ηij
(2.5)
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Fig. 2. Components of wall friction tensor. Contour lines are in h¯ unit. WF/CWWF
where ηij are the WF and CWWF dissipation coefficients respectively and μ is a measure of chaos (chaoticity) in the single-particle motion and depends upon the shape of the nucleus. In a classical picture, this will be given as the average fraction of the nucleon trajectories which are chaotic and is evaluated by sampling over a large number of classical trajectories for a given shape of the nucleus. Each of such trajectories is identified as a regular or as a chaotic one by considering the magnitude of its Lyapunov exponent over a long time interval [26]. The value of chaoticity μ changes from 0 to 1 as the nucleus evolves from a regular (e.g. spherical) to a non-integrable (usually deformed) shape. It would be of interest at this point to examine the two-dimensional landscape of the various input quantities in our calculation which we have defined in the above. We first show the contour plots for the three components of WF dissipation tensor in Fig. 2. We note here that while ηcc has a modest dependence on c and a strong dependence on h, ηhh increases steeply with c and has a weak dependence on h. On the other hand, variations of ηch with c and h are of similar magnitudes. We also make similar observations in Fig. 3 which shows the contour plots of the elements of inverse mass tensor. The non-diagonal components are found to be of similar magnitudes to the respective diagonal components for both the dissipation and inverse inertia tensors. This feature arises because the (c, h) coordinates are not orthogonal and was also noticed earlier [15]. The contour plots for chaoticity or the chaos factor μ are next shown in Fig. 4. We note here that, starting from a spherical shape (c = 1, h = 0) μ increases with c while h is held constant (h = 0). Similarly, μ also increases with increase in the magnitude of h starting from the spherical configuration. It further turns out that a valley exists in the landscape which extends
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Fig. 3. Components of mass inverse tensor. Contour lines are in MeV/h¯ 2 unit.
Fig. 4. Contour plot of the chaos factor (μ).
to large values of c and the value of μ along the bottom of the valley is small (μ < 0.1). This corresponds to the fact that the present shape parameters admit highly deformed yet integrable shapes. Subsequently the CWWF dissipation coefficients are obtained from Eq. (2.5) and are plotted in Fig. 5. As expected, introduction of the chaos factor makes the CWWF dissipation
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Fig. 5. Components of chaos-weighted wall friction tensor. Contour lines are in h¯ unit.
much smaller than the WF dissipation over a considerable region of the collective coordinate space. In the present work, we shall also study the Langevin dynamics of fission in one dimension. Here we will assume that fission would proceed along the minimum of the potential landscape (valley) in (c, h) coordinates, though we shall consider the Langevin equation in elongation coordinate (c) alone. Consequently, the one-dimensional potential in the Langevin equation will be defined as V (c) = V (c, h) at valley. Other quantities such as inertia and friction will be scalars in the one-dimensional case and will also be similarly defined. We shall determine the scission configuration of a compound nucleus from the following considerations [15,16]. The condition of zero neck radius is clearly not suitable for the present calculation since the liquid drop model description of the nucleus loses its validity when the neck radius becomes comparable with the separation between two nucleons inside the nucleus. Further, the Werner–Wheeler approximation for inertia tensor breaks down for zero neck radius [15]. Consequently, a Langevin trajectory calculated with the above inertia near zero neck radius would be in error. Thus the Langevin equations with the present set of input quantities are not suitable to model fission dynamics close to zero neck radius. On the other hand, Davies et al. [16] have proposed that the neck can undergo a sudden rupture when the Coulomb repulsion becomes larger than the nuclear attraction between the two future fragments. This can happen at a nonzero neck radius and it was shown in Ref. [16] that the neck radius at which rupture occurs is about 2 fm for actinide nuclei. This value equals approximately to 0.3R where R is the
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radius of the initial spherical shape of the nucleus. We shall consider scission to take place at this value of the neck radius. The locus of the scission configurations (the scission line) is shown in the potential landscape in Fig. 1. It may however be noted that a detailed understanding of the scission configuration in nuclear fission requires further investigations. The Langevin equations are numerically integrated in second order using a small time step of 0.0001h¯ /MeV. All the input quantities are first calculated on a uniform two-dimensional grid with 341 × 451 grid points covering the ranges of c ∈ (0.6, 4.0) and h ∈ (−1.5, 3.0). Calculations are performed for a compound nucleus at specified values of its spin and temperature. The initial collective coordinate is chosen as that of a spherical nucleus and the initial momentum distribution is assumed to follow that of a equilibrated thermal system. A Langevin trajectory is considered to have undergone fission whenever it crosses the scission line. 3. Results and discussions We first calculated the time dependent fission widths of the 224 Th nucleus at different values of its spin and temperature. The fission widths were obtained from the time rate at which the Langevin trajectories cross the scission line using an ensemble of 105 trajectories for each value of spin and temperature. The time-dependent fission widths calculated from both one- and two-dimensional Langevin equations are displayed in Fig. 6. Fission widths obtained with both CWWF and WF dissipations are shown in this figure. In all the plots in this figure, we first note that the value of the fission width starts from zero and subsequently reaches its stationary value after a certain time interval (transient time). It is also observed that the transient time becomes smaller at higher values of temperature for both the one- and two-dimensional cases. The transient time τf obtained from the over damped solution of Fokker–Planck equations in one dimension is given as [27], 10bf β , (3.1) τf = ln T 2ω2 where β represents the dissipation strength, ω defines the potential inside the barrier, bf is the fission barrier height, and T is the nuclear temperature. The transient time decreases at higher temperatures according to the above expression. The present results show that this trend persists for two-dimensional motion. It is further observed that the transient time is larger for two-dimensional fission trajectories than that in one dimension for all values of spin and temperatures. This observation reflects the fact that two-dimensional trajectories explore a larger phase space compared to those in one-dimension before they reach the scission configuration resulting in an additional delay for the former. We shall next compare the transient time associated with the fission widths obtained with the WF and CWWF dissipations. It is observed that the transient times for WF are substantially larger than CWWF for both one- and two-dimensional trajectories for all the cases considered here. It is interesting to note that the increase of transient time with the strength of dissipation as seen in Eq. (3.1) and which was obtained for a constant dissipation is also seen in the two-dimensional motion with a strongly shape dependent dissipative force. We shall next consider the stationary values of the fission widths which we have obtained at different spins and temperatures. We first note that the stationary widths from one- and two-dimensional calculations are quite close to each other. The stationary widths from twodimensional calculations however tend to become larger by about 10–15% than those from one-dimensional results at higher values of spin and temperatures. The above observations are true for fission widths calculated using both WF and CWWF dissipations. An enhancement of
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Fig. 6. Time-dependent fission widths of 224 Th at different spins and temperatures calculated with chaos-weighted wall friction (CWWF) as well as with wall friction (WF). The solid and dashed lines are obtained from calculations in two dimensions and one dimension respectively.
two-dimensional fission rate by about 15% over one-dimensional case was reported earlier [2, 11]. The present results extend the comparison to a wider range of dissipation strength, nuclear spin and temperature. 3.1. Trajectory density distribution in two dimension In the above, we find that the stationary fission rates in one and two dimensions are nearly equal though the c and h motions are strongly coupled through the non-diagonal elements of the inertia and dissipation tensors. It makes one curious to know how the trajectories in two dimensions are distributed and how much they deviate from the trajectories in one dimension. In order to study the distribution of trajectories in two dimensions, we have introduced a quantity to be referred as “trajectory density” and denoted by d(c, h) in the subsequent discussions. We define the trajectory density in the following manner.
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Fig. 7. Contours of trajectory density (see text) calculated at different spins and temperatures of the 224 Th nucleus using wall friction (WF). The minimum in the potential energy valley is marked by the solid line. The dashed line corresponds to scission configuration.
A counter is set for each cell in the (c, h) grid such that the count in a cell is increased by 1 if a Langevin trajectory is found within that cell at an instant during its time evolution. The counters in each successive cell tick as the trajectory moves in the two-dimensional space. The cumulative count in each cell considering all the fission trajectories divided by the total number of fission events would now provide us with an average, time-integrated distribution of the density of fission trajectories in the (c, h) coordinates. Figs. 7 and 8 display the contour plots of trajectory density distributions obtained with WF and CWWF dissipations respectively. The line which corresponds to the locus of the local minimum of potential energy and along which the one-dimensional Langevin trajectories are confined is also shown in these figures. The scission line is also marked in all the figures. A common feature which we note immediately in all the plots in both the figures is a welldefined ridge-like structure. The ridge corresponds to regions in the (c, h) plane which are most frequently visited by the two-dimensional Langevin trajectories. The ridge therefore defines the most probable coordinates traversed by the fission trajectories. Beyond the saddle region, the height of the ridge starts decreasing which indicates a reduction in the to-and-fro motion and thus signifies the onset of a fast saddle to scission transition. Our next observation in the contour plots concerns the spread of the trajectories along h, the neck degree of freedom. A significant oscillatory motion in this coordinate is clearly observed. As one might expect, the profile of the contours closely follow the potential energy landscape.
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Fig. 8. Same as in Fig. 7 but obtained with the chaos-weighted wall friction (CWWF).
The amplitude of the h-oscillation is largest at the ground state configuration (c ≈ 1, h ≈ 0) and it decreases gradually at larger elongations due to narrowing down of the potential energy profile in the saddle region. Further, a peak can be noticed in most of the contour plots in the saddle region. It is possibly a consequence of the constriction in the potential energy profile in the saddle region. A narrow potential pocket would make the oscillatory motion in the h-direction faster which, in turn, would cause a more frequent crossing of the (c, h) cells in this region. We shall now compare the fission paths followed in one- and two-dimensional motions. We note here that the ridge deviates significantly from the one-dimensional fission path. In particular, the two paths follow opposite trends beyond the saddle region. This feature is a consequence of the large magnitude of non-diagonal terms in both inertia and dissipation tensors. Therefore, a nucleus has a shorter neck at scission in one-dimensional fission compared to that in two dimensions. A longer neck at larger elongations is clearly the preferred configuration at scission in two-dimensional motion. This difference in the shapes at scission would be reflected in the fission fragment kinetic energy distributions as we would see later. So far we have discussed features which are common to both Figs. 7 and 8 obtained with WF and CWWF dissipations respectively. One distinguishing feature, however, is the observation that a peak appears at the ground state configuration (c ≈ 1, h ≈ 0) in all the plots in Fig. 8 which are absent in the plots of Fig. 7. This peak corresponds to an enhancement of oscillatory motion in this region for dynamics with CWWF dissipation compared to those with WF dissipation. We have already noted earlier in Figs. 2 and 5 that ηcc is much smaller for CWWF compared to WF while ηhh is comparatively small for both WF and CWWF in the above region. The oscillation
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Fig. 9. Kinetic energy distribution at scission calculated with wall friction (WF) at different spins and temperatures of 224 Th. The solid and dashed lines are obtained from calculations in two dimensions and one dimension respectively.
along the c-direction in the potential energy pocket at ground state is therefore weakly damped for CWWF and strongly damped for WF. The faster c-oscillation with CWWF therefore makes additional contribution to the build up of trajectory density at the equilibrium shape and hence the peak appears. 3.2. Kinetic energy distribution We have calculated the collective kinetic energy (Esc ) at scission for both one- and twodimensional dynamics of fission and the results are displayed in Figs. 9 and 10. We first note that the kinetic energy distributions are almost identical for both WF (Fig. 9) and CWWF (Fig. 10) dissipations. All the distributions resemble Boltzmann distribution indicating that the systems are not far from equilibrium. We further note that the peak position of a distribution obtained from two-dimensional calculation is at a higher energy than that from one dimension. The peak position is also found to shift towards higher values at higher compound nucleus temperatures. These observations also suggest that the energy sharing between the collective motion and the heat bath approximately follows the equipartition of energy. Using Esc , the collective kinetic energy at scission, we have next calculated the total kinetic energy of the fission fragments (Eff ) from the following mass-energy conservation equation, Eff = + Vsc + Esc − Vn
(3.2)
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Fig. 10. Same as in Fig. 9 but obtained with the chaos-weighted wall friction (CWWF).
where represents the mass difference between the compound nucleus and the fission fragments, Vsc is the collective potential energy at scission while Vn denotes the energy that has to be spent from kinetic energy in order to overcome the nuclear attraction between the nascent fragments. The last quantity is calculated as the work done in creating two additional surfaces, each of radius 0.3R as defined by the scission condition [15], and using the surface energy coefficient as given by Sierk [21]. We have assumed in Eq. (3.2) that Esc is fully converted into relative kinetic energy of the fission fragments. The distribution of the fission fragment kinetic energies thus obtained are shown in Figs. 11 and 12 for WF and CWWF dissipations respectively. We first note here that the distributions obtained with the two dissipations are quite similar. However, two-dimensional fission trajectories give rise to fission fragment kinetic energy distributions which are distinct from those obtained in one-dimensional motion. Since the potential energy at scission is same for all the trajectories in one dimension, Eff and Esc differ by a constant term and a Boltzmann-like distribution is retained for fission fragment kinetic energy distribution. On the other hand, the potential energy at scission can be different for different two-dimensional trajectories which, in turn, results in a fission fragment kinetic energy distribution which is more symmetric than the one-dimensional distribution. The shape of the fission fragment kinetic energy distribution thus becomes indicative of the number of collective degrees of freedom. One can also expect that the dispersion of potential energy at scission would increase with addition of more degrees of freedom resulting
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Fig. 11. Fission fragment kinetic energy distribution calculated with wall friction (WF) at different spins and temperatures of 224 Th. The solid and dashed lines are obtained from calculations in two dimensions and one dimension respectively.
in a broader and more symmetric distribution. Experimental fission fragment kinetic energies displaying Gaussian distributions therefore suggest a multi-dimensional dynamics of fission. Our next observation in Figs. 11 and 12 concerns the peak positions in the energy distributions. We note that in each plot, the peak appears at a lower energy for two-dimensional motion compared to that in one dimension. This aspect essentially reflects the fact that the scission configurations are quite different in one- and two-dimensional motions, as we have already observed in Figs. 7 and 8 while discussing the distribution of trajectories near the scission line. A nucleus with (c, h) degrees of freedom is more elongated at scission than the one with only c degree of freedom. This results in a lower Coulomb barrier and hence a smaller kinetic energy of the fission fragments for the former case compared to the later. Subsequently, we calculated the variance (σ 2 = E 2 − E2 ) of fission fragment kinetic energies and also that of kinetic energy distributions at scission. The values are shown in Fig. 13 for two nuclear temperatures. We first note that σ 2 increases with temperature which again indicates that the systems under consideration are not far from equilibrium. We next observe that the variance of Esc increases with the number of degrees of freedom as one would expect. The dispersion of potential energy among the fission trajectories would make the variance of Eff to be larger than that of Esc which we also find in this figure. We shall note at this point that the experimental variance [28] for 224 Th at an excitation energy of 53.8 MeV (corresponding to 1.5 MeV of nuclear temperature) is 137 MeV2 . Evidently, the present calculation accounts for only a small
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Fig. 12. Same as in Fig. 11 but obtained with the chaos-weighted wall friction (CWWF).
Fig. 13. Kinetic energy variances at different temperatures.
fraction (≈ 10%) of the experimental variance. This however is not surprising since we have not included the asymmetry degree of freedom in our calculation. Karpov et al. [15] have shown that a three-dimensional Langevin dynamics calculation can reproduce the above experimental variance. The importance of the asymmetry degree of freedom in producing a dispersion of the fission fragment kinetic energy has also been discussed earlier in detail by Abe et al. [2].
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4. Summary We have investigated the dynamics of fission in one and two dimensions considering the elongation and the neck degree of freedom for collective motion. We have calculated the timedependent fission width, the distribution of fission trajectories and the kinetic energy distribution of the fission fragments. Comparing the results from one- and two-dimensional calculations, we have found that though the stationary fission widths are nearly the same in both the calculations, the most probable fission path beyond the saddle ridge in two dimensions deviates considerably from that in one dimension. Further, the two-dimensional fission trajectories are found to make extensive excursions along the neck degree of freedom. The scission configurations are also different in the two cases. Two-dimensional trajectories prefer to undergo fission with a longer neck at scission compared to those in one dimension. Consequently, the kinetic energies of fission fragments in two dimensions are lower than those in one dimension. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
N. Bohr, J.A. Wheeler, Phys. Rev. 56 (1939) 426. Y. Abe, S. Ayik, P.-G. Reinhard, E. Suraud, Phys. Rep. 275 (1996) 49. P. Fröbrich, I.I. Gontchar, Phys. Rep. 292 (1998) 131. I. Dioszegi, N.P. Shaw, I. Mazumdar, A. Hatzikoutelis, P. Paul, Phys. Rev. C 61 (2000) 024513. G. Chaudhuri, S. Pal, Eur. Phys. J. A 18 (2003) 9. H.A. Kramers, Physica (Amsterdam) 4 (1940) 284. P. Grange, H.A. Weidenmüller, Phys. Lett. 96B (1980) 26. P. Grange, Q. Li-Jang, H.A. Weidenmüller, Phys. Rev. C 27 (1983) 2063. Y. Abe, C. Gregoire, H. Delagrange, J. Phys. C 4 (1986) C4-329. G. Chaudhuri, S. Pal, Phys. Rev. C 65 (2002) 054612. T. Wada, N. Carjan, Y. Abe, Nucl. Phys. A 538 (1992) 283c. G.-R. Tillack, Phys. Lett. B 278 (1992) 403. T. Wada, Y. Abe, N. Carjan, Phys. Rev. Lett. 70 (1993) 3538. D.V. Vanin, G.I. Kosenko, G.D. Adeev, Phys. Rev. C 59 (1999) 2114. A.V. Karpov, P.N. Nadtochy, D.V. Vanin, G.D. Adeev, Phys. Rev. C 63 (2001) 054610. K.T.R. Davies, R.A. Managan, J.R. Nix, A.J. Sierk, Phys. Rev. C 16 (1977) 1890. M. Brack, J. Damgard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky, C.Y. Wong, Rev. Mod. Phys. 44 (1972) 320. A.V. Ignatyuk, M.G. Itkis, V.N. Okolovich, G.N. Smirenkin, A. Tishin, Sov. J. Nucl. Phys. 21 (1975) 255. K.T.R. Davies, A.J. Sierk, J.R. Nix, Phys. Rev. C 13 (1976) 2385. P. Fröbrich, I.I. Gontchar, N.D. Mavlitov, Nucl. Phys. A 556 (1993) 281. A.J. Sierk, Phys. Rev. C 33 (1986) 2039. J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A.J. Sierk, W.J. Swiatecki, Ann. Phys. (N.Y.) 113 (1978) 330. S. Pal, T. Mukhopadhyay, Phys. Rev. C 54 (1996) 1333. G. Chaudhuri, S. Pal, Phys. Rev. C 63 (2001) 064603. A.J. Sierk, J.R. Nix, Phys. Rev. C 21 (1980) 982. J. Blocki, F. Brut, T. Srokowski, W.J. Swiatecki, Nucl. Phys. A 545 (1992) 511c. K.H. Bhatt, P. Grange, B. Hiller, Phys. Rev. C 33 (1986) 954. M.G. Itkis, Yu.Ts. Oganessian, G.G. Chubarian, V.S. Salamatin, A.Ya. Rusanov, V.N. Okolovich, in: Yu.Ts. Oganessian, R. Kalpakchieva, W. von Oertzen (Eds.), Proceedings of the XV EPS Conference on Low Energy Nuclear Dynamics (LEND-95), St. Petersburg, 1995, World Scientific, 1995, p. 177.
The role of neck degree of freedom in nuclear fission Santanu Pal ∗ , Gargi Chaudhuri 1 , Jhilam Sadhukhan Physics Group, Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700 064, India Received 16 January 2008; received in revised form 22 March 2008; accepted 2 May 2008 Available online 9 May 2008
Abstract The dynamics of nuclear fission is investigated using Langevin equations in order to study the role of the neck degree of freedom in fission. The time-dependent fission widths are calculated in both one (with only elongation coordinate) and two (with elongation plus neck) dimensions. A quantitative estimate is also made of the distribution of the fission trajectories in two dimensions. Comparing the results from one and two-dimensional calculations, we find that the stationary fission widths are nearly the same in both the calculations. However, the most probable fission path beyond the saddle ridge in two dimensions deviates considerably from that in one dimension. © 2008 Elsevier B.V. All rights reserved. PACS: 05.40.-a; 24.60.-k; 24.60.Lz; 25.70.Jj Keywords: Nuclear fission; Neck parameter; Langevin equation; Fission trajectories
1. Introduction It is now an established fact that a dissipative dynamical model is essential for fission of highly excited heavy nuclei in order to reproduce a wide variety of experimental data. In particular, the statistical theory of nuclear fission due to Bohr and Wheeler [1] grossly underestimates the prescission multiplicities of light particles and GDR γ ’s [2–4]. The statistical theory also underestimates the evaporation residue cross sections in heavy ion induced fusion reactions [4,5]. Following the work of Kramers [6], a dynamical model for nuclear fission was initially developed employing the Fokker–Planck equation [7,8]. Subsequently, the Langevin equation was * Corresponding author. Tel.: +91 33 2359 8431; fax: +91 33 2334 6871.
E-mail address: [email protected] (S. Pal). 1 Present address: Physics Department, McGill University, Montreal, Canada.
0375-9474/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2008.05.001
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found to be more useful for practical applications [9]. Considering the elongation of a compound nucleus along the symmetry axis as the most relevant dynamical coordinate to describe nuclear fission, one-dimensional Langevin equation was found to account for the experimental values for prescission neutron multiplicity and evaporation residue cross sections for a number of heavy compound nuclei formed in heavy ion collisions [2,3,5,10]. The Langevin equation was subsequently solved in higher dimensions in order to explain the energy and mass distributions of the fission fragments [11–15]. In the present work, we would study the fission dynamics in elongation and neck coordinates using the Langevin equation. It is already known that addition of the neck degree of freedom in fission dynamics increases the stationary fission rate marginally compared to that obtained by considering only the elongation coordinate [12]. Therefore it would be of interest to find how closely the fission path in two dimensions (elongation plus neck coordinates) follows that in one (only elongation) dimension. One would also like to know the extent of excursions made by the Langevin trajectories along the neck degree of freedom and its effect on the scission point shape of the compound nucleus. Consequently the question arises regarding the role of neck dynamics in giving rise to the dispersion in fission fragment kinetic energy since this is mainly governed by the configuration of the compound nucleus at scission. We would address such concerns in this paper. We shall begin with a brief description of the basic equations in the next section. Various input quantities to the Langevin equation will also be described in this section. In addition to the timedependent fission rates, we shall also obtain numerical results for a number of suitably defined quantities depicting the distribution of the Langevin trajectories in two dimensions for the 224 Th nucleus. Section 3 will contain these results. We shall present a summary of the main results in the last section. 2. The Langevin equations In the Langevin description of nuclear fission, one usually assumes that the motion associated with the collective degrees of freedom relevant for fission can be modeled as that of a Brownian particle in a viscous heat bath [6,9]. The remaining nuclear degrees of freedom are considered to constitute the heat bath represented by a temperature while the interaction between the Brownian particle and the intrinsic motion of the heat bath is assumed to be random in nature. In order to specify the collective coordinates for a dynamical description of nuclear fission, we will use the “funny hills” shape parameters (c, h, α) as suggested by Brack et al. [17]. The elongation and neck degrees of freedom are denoted by c and h respectively while α corresponds to the asymmetry parameter. Since we are going to concentrate upon the neck dynamics in the present study, we will consider only symmetric fission (α = 0) in order to make unambiguous interpretation of the results. The surface of a nucleus of mass number A with elongation and neck coordinates as c and h, respectively, is defined as z2 2 2 ρ (z) = 1 − 2 a0 c0 + b0 z2 , b0 0, a0 0, c0 z2 b0 c0 z 2 2 , b0 < 0, a0 0, = 1 − 2 a0 c0 exp (2.1) R3 c0 where
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c0 = cR, 1
R = 1.16A 3 and c−1 , 2 in cylindrical coordinates. a0 and b0 are related by [17] b0 = 2h +
1 b0 − , b0 0, 3 5 c b0 4 , =− √ √ 1 p 3 e + (1 + 2p ) −πp erf( −p)
a0 =
b0 < 0,
(2.2)
where p = b0 c3 and erf(x) is the error function. The parameter h describes the variation of the thickness of the neck without changing the length 2c (in units of R) of the nucleus along the symmetry axis. Positive values of h implies that the neck formation starts at a lower value of c as compared to the case of h = 0 and hence scission of the nucleus into two fragments also takes place for a lower value of c. The neck region at scission is also smaller in such cases. On the other hand, a negative value for h shifts the scission point to a higher value of c. Consequently the neck region at scission is also longer. The two-dimensional Langevin equation in (c, h) coordinates has the following form [13], pj pk ∂ −1 ∂F dpi =− m jk − − ηij m−1 j k pk + gij Γj (t), dt 2 ∂qi ∂qi dqi −1 = m ij pj (2.3) dt where q1 and q2 stands for c and h respectively and pi represents the respective momentum. F is the free energy of the system and mij and ηij are the shape-dependent collective inertia and dissipation tensors respectively. The time-correlation property of the random force is assumed to follow the relation Γk (t)Γl (t ) = 2δkl δ(t − t ) and the strength of the random force is related to the dissipation coefficients through the fluctuation–dissipation theorem and is given as gik gj k = ηij T , where the temperature T of the compound nucleus at any instant of its evolution is given as T = Eint /a(q). The intrinsic excitation energy Eint is calculated from the total excitation energy E ∗ of the compound nucleus using the energy conservation, E ∗ = Eint + Ecoll + V (q), where Ecoll is the collective kinetic energy including the rotational energy and V (q) is the potential energy of the system. The level density parameter a(q) depends on the collective coordinates and is taken from the works of Ignatyuk et al. [18]. We will make the Werner–Wheeler approximation for incompressible irrotational flow to first calculate the collective inertia tensor and hence the components of its inverse (m−1 )ij [19]. The
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Fig. 1. Potential energy contours (in MeV) for 224 Th. The minimum in the potential energy valley is marked by the dotted line. The dashed line corresponds to scission configuration.
conservative driving force in the compound nucleus, which is treated here as a thermodynamic system, is derived from its free energy F for which we will use the following expression, F (c, h) = V (c, h) − a(c, h)T 2 , where the nucleus is considered as a noninteracting Fermi gas [20]. The potential energy V (c, h) is obtained from the finite-range liquid drop model by a double folding procedure [21]. The calculated potential along with locus of the minimum in the valley for 224 Th is shown in Fig. 1. We shall use the one-body model for nuclear dissipation in our calculations. The original wall friction (WF) due to Blocki et al. [22] and a weaker version of it, the chaos-weighted wall friction (CWWF) [23] will be considered here. A window friction will also be included in our calculation [24]. We choose the above two forms of dissipation essentially to investigate the dependence of the fission dynamics on the strength of dissipation. Further, the chaos-weighted wall friction has already been shown to give a better account of dissipation in fission [5,10]. The wall friction coefficients read as follows [25], zN −1 2 2 ∂D ∂ρ 2 2 ∂D 2 2 2 ∂ρ ∂ρ 1 ∂ρ ∂ρ 1 1 1 WF ρ2 + = πρm v¯ + + dz ηij 2 ∂qi ∂z ∂qi ∂qj ∂z ∂qj 2 ∂z zmin
zmax
+ zN
∂ρ 2 ∂ρ 2 ∂D2 + ∂qi ∂z ∂qi
∂ρ 2 ∂ρ 2 ∂D2 + ∂qj ∂z ∂qj
−1 2 2 2 ∂ρ 1 ρ2 + dz , 2 ∂z
(2.4)
where ρm is the mass density of the nucleus, v¯ is the average nucleon speed inside the nucleus and D1 , D2 are the positions of the centers of mass of the two parts of the fissioning system relative to the center of mass of the whole system. zmin and zmax are the two extreme ends of the nuclear shape along the z axis and zN is the position of the neck plane that divides the nucleus into two parts. Based on the argument that the degree of randomization of the single-particle motion within a nucleus depends upon its shape, it was shown earlier that the CWWF dissipation coefficient is given as [23] CWWF WF (c, h) = μ(c, h)ηij (c, h), ηij
(2.5)
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Fig. 2. Components of wall friction tensor. Contour lines are in h¯ unit. WF/CWWF
where ηij are the WF and CWWF dissipation coefficients respectively and μ is a measure of chaos (chaoticity) in the single-particle motion and depends upon the shape of the nucleus. In a classical picture, this will be given as the average fraction of the nucleon trajectories which are chaotic and is evaluated by sampling over a large number of classical trajectories for a given shape of the nucleus. Each of such trajectories is identified as a regular or as a chaotic one by considering the magnitude of its Lyapunov exponent over a long time interval [26]. The value of chaoticity μ changes from 0 to 1 as the nucleus evolves from a regular (e.g. spherical) to a non-integrable (usually deformed) shape. It would be of interest at this point to examine the two-dimensional landscape of the various input quantities in our calculation which we have defined in the above. We first show the contour plots for the three components of WF dissipation tensor in Fig. 2. We note here that while ηcc has a modest dependence on c and a strong dependence on h, ηhh increases steeply with c and has a weak dependence on h. On the other hand, variations of ηch with c and h are of similar magnitudes. We also make similar observations in Fig. 3 which shows the contour plots of the elements of inverse mass tensor. The non-diagonal components are found to be of similar magnitudes to the respective diagonal components for both the dissipation and inverse inertia tensors. This feature arises because the (c, h) coordinates are not orthogonal and was also noticed earlier [15]. The contour plots for chaoticity or the chaos factor μ are next shown in Fig. 4. We note here that, starting from a spherical shape (c = 1, h = 0) μ increases with c while h is held constant (h = 0). Similarly, μ also increases with increase in the magnitude of h starting from the spherical configuration. It further turns out that a valley exists in the landscape which extends
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Fig. 3. Components of mass inverse tensor. Contour lines are in MeV/h¯ 2 unit.
Fig. 4. Contour plot of the chaos factor (μ).
to large values of c and the value of μ along the bottom of the valley is small (μ < 0.1). This corresponds to the fact that the present shape parameters admit highly deformed yet integrable shapes. Subsequently the CWWF dissipation coefficients are obtained from Eq. (2.5) and are plotted in Fig. 5. As expected, introduction of the chaos factor makes the CWWF dissipation
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Fig. 5. Components of chaos-weighted wall friction tensor. Contour lines are in h¯ unit.
much smaller than the WF dissipation over a considerable region of the collective coordinate space. In the present work, we shall also study the Langevin dynamics of fission in one dimension. Here we will assume that fission would proceed along the minimum of the potential landscape (valley) in (c, h) coordinates, though we shall consider the Langevin equation in elongation coordinate (c) alone. Consequently, the one-dimensional potential in the Langevin equation will be defined as V (c) = V (c, h) at valley. Other quantities such as inertia and friction will be scalars in the one-dimensional case and will also be similarly defined. We shall determine the scission configuration of a compound nucleus from the following considerations [15,16]. The condition of zero neck radius is clearly not suitable for the present calculation since the liquid drop model description of the nucleus loses its validity when the neck radius becomes comparable with the separation between two nucleons inside the nucleus. Further, the Werner–Wheeler approximation for inertia tensor breaks down for zero neck radius [15]. Consequently, a Langevin trajectory calculated with the above inertia near zero neck radius would be in error. Thus the Langevin equations with the present set of input quantities are not suitable to model fission dynamics close to zero neck radius. On the other hand, Davies et al. [16] have proposed that the neck can undergo a sudden rupture when the Coulomb repulsion becomes larger than the nuclear attraction between the two future fragments. This can happen at a nonzero neck radius and it was shown in Ref. [16] that the neck radius at which rupture occurs is about 2 fm for actinide nuclei. This value equals approximately to 0.3R where R is the
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radius of the initial spherical shape of the nucleus. We shall consider scission to take place at this value of the neck radius. The locus of the scission configurations (the scission line) is shown in the potential landscape in Fig. 1. It may however be noted that a detailed understanding of the scission configuration in nuclear fission requires further investigations. The Langevin equations are numerically integrated in second order using a small time step of 0.0001h¯ /MeV. All the input quantities are first calculated on a uniform two-dimensional grid with 341 × 451 grid points covering the ranges of c ∈ (0.6, 4.0) and h ∈ (−1.5, 3.0). Calculations are performed for a compound nucleus at specified values of its spin and temperature. The initial collective coordinate is chosen as that of a spherical nucleus and the initial momentum distribution is assumed to follow that of a equilibrated thermal system. A Langevin trajectory is considered to have undergone fission whenever it crosses the scission line. 3. Results and discussions We first calculated the time dependent fission widths of the 224 Th nucleus at different values of its spin and temperature. The fission widths were obtained from the time rate at which the Langevin trajectories cross the scission line using an ensemble of 105 trajectories for each value of spin and temperature. The time-dependent fission widths calculated from both one- and two-dimensional Langevin equations are displayed in Fig. 6. Fission widths obtained with both CWWF and WF dissipations are shown in this figure. In all the plots in this figure, we first note that the value of the fission width starts from zero and subsequently reaches its stationary value after a certain time interval (transient time). It is also observed that the transient time becomes smaller at higher values of temperature for both the one- and two-dimensional cases. The transient time τf obtained from the over damped solution of Fokker–Planck equations in one dimension is given as [27], 10bf β , (3.1) τf = ln T 2ω2 where β represents the dissipation strength, ω defines the potential inside the barrier, bf is the fission barrier height, and T is the nuclear temperature. The transient time decreases at higher temperatures according to the above expression. The present results show that this trend persists for two-dimensional motion. It is further observed that the transient time is larger for two-dimensional fission trajectories than that in one dimension for all values of spin and temperatures. This observation reflects the fact that two-dimensional trajectories explore a larger phase space compared to those in one-dimension before they reach the scission configuration resulting in an additional delay for the former. We shall next compare the transient time associated with the fission widths obtained with the WF and CWWF dissipations. It is observed that the transient times for WF are substantially larger than CWWF for both one- and two-dimensional trajectories for all the cases considered here. It is interesting to note that the increase of transient time with the strength of dissipation as seen in Eq. (3.1) and which was obtained for a constant dissipation is also seen in the two-dimensional motion with a strongly shape dependent dissipative force. We shall next consider the stationary values of the fission widths which we have obtained at different spins and temperatures. We first note that the stationary widths from one- and two-dimensional calculations are quite close to each other. The stationary widths from twodimensional calculations however tend to become larger by about 10–15% than those from one-dimensional results at higher values of spin and temperatures. The above observations are true for fission widths calculated using both WF and CWWF dissipations. An enhancement of
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Fig. 6. Time-dependent fission widths of 224 Th at different spins and temperatures calculated with chaos-weighted wall friction (CWWF) as well as with wall friction (WF). The solid and dashed lines are obtained from calculations in two dimensions and one dimension respectively.
two-dimensional fission rate by about 15% over one-dimensional case was reported earlier [2, 11]. The present results extend the comparison to a wider range of dissipation strength, nuclear spin and temperature. 3.1. Trajectory density distribution in two dimension In the above, we find that the stationary fission rates in one and two dimensions are nearly equal though the c and h motions are strongly coupled through the non-diagonal elements of the inertia and dissipation tensors. It makes one curious to know how the trajectories in two dimensions are distributed and how much they deviate from the trajectories in one dimension. In order to study the distribution of trajectories in two dimensions, we have introduced a quantity to be referred as “trajectory density” and denoted by d(c, h) in the subsequent discussions. We define the trajectory density in the following manner.
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Fig. 7. Contours of trajectory density (see text) calculated at different spins and temperatures of the 224 Th nucleus using wall friction (WF). The minimum in the potential energy valley is marked by the solid line. The dashed line corresponds to scission configuration.
A counter is set for each cell in the (c, h) grid such that the count in a cell is increased by 1 if a Langevin trajectory is found within that cell at an instant during its time evolution. The counters in each successive cell tick as the trajectory moves in the two-dimensional space. The cumulative count in each cell considering all the fission trajectories divided by the total number of fission events would now provide us with an average, time-integrated distribution of the density of fission trajectories in the (c, h) coordinates. Figs. 7 and 8 display the contour plots of trajectory density distributions obtained with WF and CWWF dissipations respectively. The line which corresponds to the locus of the local minimum of potential energy and along which the one-dimensional Langevin trajectories are confined is also shown in these figures. The scission line is also marked in all the figures. A common feature which we note immediately in all the plots in both the figures is a welldefined ridge-like structure. The ridge corresponds to regions in the (c, h) plane which are most frequently visited by the two-dimensional Langevin trajectories. The ridge therefore defines the most probable coordinates traversed by the fission trajectories. Beyond the saddle region, the height of the ridge starts decreasing which indicates a reduction in the to-and-fro motion and thus signifies the onset of a fast saddle to scission transition. Our next observation in the contour plots concerns the spread of the trajectories along h, the neck degree of freedom. A significant oscillatory motion in this coordinate is clearly observed. As one might expect, the profile of the contours closely follow the potential energy landscape.
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Fig. 8. Same as in Fig. 7 but obtained with the chaos-weighted wall friction (CWWF).
The amplitude of the h-oscillation is largest at the ground state configuration (c ≈ 1, h ≈ 0) and it decreases gradually at larger elongations due to narrowing down of the potential energy profile in the saddle region. Further, a peak can be noticed in most of the contour plots in the saddle region. It is possibly a consequence of the constriction in the potential energy profile in the saddle region. A narrow potential pocket would make the oscillatory motion in the h-direction faster which, in turn, would cause a more frequent crossing of the (c, h) cells in this region. We shall now compare the fission paths followed in one- and two-dimensional motions. We note here that the ridge deviates significantly from the one-dimensional fission path. In particular, the two paths follow opposite trends beyond the saddle region. This feature is a consequence of the large magnitude of non-diagonal terms in both inertia and dissipation tensors. Therefore, a nucleus has a shorter neck at scission in one-dimensional fission compared to that in two dimensions. A longer neck at larger elongations is clearly the preferred configuration at scission in two-dimensional motion. This difference in the shapes at scission would be reflected in the fission fragment kinetic energy distributions as we would see later. So far we have discussed features which are common to both Figs. 7 and 8 obtained with WF and CWWF dissipations respectively. One distinguishing feature, however, is the observation that a peak appears at the ground state configuration (c ≈ 1, h ≈ 0) in all the plots in Fig. 8 which are absent in the plots of Fig. 7. This peak corresponds to an enhancement of oscillatory motion in this region for dynamics with CWWF dissipation compared to those with WF dissipation. We have already noted earlier in Figs. 2 and 5 that ηcc is much smaller for CWWF compared to WF while ηhh is comparatively small for both WF and CWWF in the above region. The oscillation
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Fig. 9. Kinetic energy distribution at scission calculated with wall friction (WF) at different spins and temperatures of 224 Th. The solid and dashed lines are obtained from calculations in two dimensions and one dimension respectively.
along the c-direction in the potential energy pocket at ground state is therefore weakly damped for CWWF and strongly damped for WF. The faster c-oscillation with CWWF therefore makes additional contribution to the build up of trajectory density at the equilibrium shape and hence the peak appears. 3.2. Kinetic energy distribution We have calculated the collective kinetic energy (Esc ) at scission for both one- and twodimensional dynamics of fission and the results are displayed in Figs. 9 and 10. We first note that the kinetic energy distributions are almost identical for both WF (Fig. 9) and CWWF (Fig. 10) dissipations. All the distributions resemble Boltzmann distribution indicating that the systems are not far from equilibrium. We further note that the peak position of a distribution obtained from two-dimensional calculation is at a higher energy than that from one dimension. The peak position is also found to shift towards higher values at higher compound nucleus temperatures. These observations also suggest that the energy sharing between the collective motion and the heat bath approximately follows the equipartition of energy. Using Esc , the collective kinetic energy at scission, we have next calculated the total kinetic energy of the fission fragments (Eff ) from the following mass-energy conservation equation, Eff = + Vsc + Esc − Vn
(3.2)
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Fig. 10. Same as in Fig. 9 but obtained with the chaos-weighted wall friction (CWWF).
where represents the mass difference between the compound nucleus and the fission fragments, Vsc is the collective potential energy at scission while Vn denotes the energy that has to be spent from kinetic energy in order to overcome the nuclear attraction between the nascent fragments. The last quantity is calculated as the work done in creating two additional surfaces, each of radius 0.3R as defined by the scission condition [15], and using the surface energy coefficient as given by Sierk [21]. We have assumed in Eq. (3.2) that Esc is fully converted into relative kinetic energy of the fission fragments. The distribution of the fission fragment kinetic energies thus obtained are shown in Figs. 11 and 12 for WF and CWWF dissipations respectively. We first note here that the distributions obtained with the two dissipations are quite similar. However, two-dimensional fission trajectories give rise to fission fragment kinetic energy distributions which are distinct from those obtained in one-dimensional motion. Since the potential energy at scission is same for all the trajectories in one dimension, Eff and Esc differ by a constant term and a Boltzmann-like distribution is retained for fission fragment kinetic energy distribution. On the other hand, the potential energy at scission can be different for different two-dimensional trajectories which, in turn, results in a fission fragment kinetic energy distribution which is more symmetric than the one-dimensional distribution. The shape of the fission fragment kinetic energy distribution thus becomes indicative of the number of collective degrees of freedom. One can also expect that the dispersion of potential energy at scission would increase with addition of more degrees of freedom resulting
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Fig. 11. Fission fragment kinetic energy distribution calculated with wall friction (WF) at different spins and temperatures of 224 Th. The solid and dashed lines are obtained from calculations in two dimensions and one dimension respectively.
in a broader and more symmetric distribution. Experimental fission fragment kinetic energies displaying Gaussian distributions therefore suggest a multi-dimensional dynamics of fission. Our next observation in Figs. 11 and 12 concerns the peak positions in the energy distributions. We note that in each plot, the peak appears at a lower energy for two-dimensional motion compared to that in one dimension. This aspect essentially reflects the fact that the scission configurations are quite different in one- and two-dimensional motions, as we have already observed in Figs. 7 and 8 while discussing the distribution of trajectories near the scission line. A nucleus with (c, h) degrees of freedom is more elongated at scission than the one with only c degree of freedom. This results in a lower Coulomb barrier and hence a smaller kinetic energy of the fission fragments for the former case compared to the later. Subsequently, we calculated the variance (σ 2 = E 2 − E2 ) of fission fragment kinetic energies and also that of kinetic energy distributions at scission. The values are shown in Fig. 13 for two nuclear temperatures. We first note that σ 2 increases with temperature which again indicates that the systems under consideration are not far from equilibrium. We next observe that the variance of Esc increases with the number of degrees of freedom as one would expect. The dispersion of potential energy among the fission trajectories would make the variance of Eff to be larger than that of Esc which we also find in this figure. We shall note at this point that the experimental variance [28] for 224 Th at an excitation energy of 53.8 MeV (corresponding to 1.5 MeV of nuclear temperature) is 137 MeV2 . Evidently, the present calculation accounts for only a small
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Fig. 12. Same as in Fig. 11 but obtained with the chaos-weighted wall friction (CWWF).
Fig. 13. Kinetic energy variances at different temperatures.
fraction (≈ 10%) of the experimental variance. This however is not surprising since we have not included the asymmetry degree of freedom in our calculation. Karpov et al. [15] have shown that a three-dimensional Langevin dynamics calculation can reproduce the above experimental variance. The importance of the asymmetry degree of freedom in producing a dispersion of the fission fragment kinetic energy has also been discussed earlier in detail by Abe et al. [2].
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4. Summary We have investigated the dynamics of fission in one and two dimensions considering the elongation and the neck degree of freedom for collective motion. We have calculated the timedependent fission width, the distribution of fission trajectories and the kinetic energy distribution of the fission fragments. Comparing the results from one- and two-dimensional calculations, we have found that though the stationary fission widths are nearly the same in both the calculations, the most probable fission path beyond the saddle ridge in two dimensions deviates considerably from that in one dimension. Further, the two-dimensional fission trajectories are found to make extensive excursions along the neck degree of freedom. The scission configurations are also different in the two cases. Two-dimensional trajectories prefer to undergo fission with a longer neck at scission compared to those in one dimension. Consequently, the kinetic energies of fission fragments in two dimensions are lower than those in one dimension. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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