aa r X i v : . [ nu c l - t h ] M a r Fission dynamics of Cf A. Zdeb, A. Dobrowolski, M. Warda
Department of Theoretical Physics, Maria Curie-Sk lodowska University, Lublin, Poland (Dated: November 5, 2018)The time-dependent generator coordinate method with the gaussian overlap approximation(TDGCM+GOA) formalism is applied to describe the fission of
Cf. We perform analysis offission from the initial states laying in the energetic range from the ground state to the state located4 MeV above the fission barrier. The fission fragment mass distributions, obtained for differentparity, energy of levels and types of mixed states, are calculated and compared with experimentaldata. The impact of the total time of wave packet propagation on the final results is studied as well.The weak dependence of obtained mass yields on the initial conditions is shown.
PACS numbers: 24.10.-i, 24.75.+i, 27.90.+b
I. INTRODUCTION
The properties of fission and its residues play a signif-icant role for the nuclear power industry and the relatedfields (e.g. nuclear waste management). One of the mainobservables of fission - fragment mass distribution - isan important input in the r-process calculations whichallow to explain the abundances of isotopes in the Uni-verse. An extensive experimental and theoretical studiesof this phenomenon (see e.g. reviews [1–4]) are carried onsince the first evidence of fission has been observed [5].Proper description of fission fragment mass distributionis still a challenging task for nuclear theory. Several mod-els have been developed so far to describe experimentalobservations. The first theoretical explanation of fissionwas given on the ground of the liquid drop model. Thefission fragment mass distributions calculated within thismethod are symmetric, as a consequence of ignoring mi-croscopic effects [6].A more sophisticated approach - statistical model [7] -allows to determine the probability of division of nucle-ons between nascent fragments at the scission point. Thecalculated mass distributions are overestimated in com-parison with the experimental ones.The distribution of fission fragments and the mean valueof total kinetic energy (TKE) can be obtained using thethe improved scission point model [8]. This is an exten-sion of the previous approach [7] and it allows to calculateinteraction energy between fragments and deformationenergy of the scission point configuration.The microscopic scission point method [9] is based onthe analysis of deformation and mass asymmetry of frag-ments that may be created after scission. The energiesare computed within microscopic self-consistent model.The assumed deformations and masses allow to calculatethe total kinetic energy of fragments and the probabilitythat the certain mass asymmetry would be observed.Fission fragment mass distributions may be also obtainedusing Langevin formalism [10–12]. This approach allowsto include plenty important dissipation and pairing ef-fects in description of fission process. Additionally, itis also possible to estimate the fission time scale in thismodel. Such calculations are performed under the as- sumption of the same deformation of both fragments.Interesting results were obtained within the similar ap-proach that treats the nuclear shape evolution as Brow-nian motions of nucleons. The possible directions in themulti-dimensional deformation space and their statisti-cal weights are found using Metropolis method [13, 14].Although the model reproduces experimental data withhigh accuracy, it contains phenomenological parameter - critical radius constant which value results much on theaccuracy in data reproduction.The authors of the general description of fission observ-ables (GEF method) [15, 16] obtained very good agree-ment with observed mass yields of most of measured fis-sioning isotopes. In this model the macroscopic potentialenergy surface (PES) is corrected by adding shell effectswhich are simulated by parabolic potential.There were also several attempts to describe fission frag-ment mass distributions in a fully microscopic way, i.e.using the Hartree-Fock-Bogolubov (HFB) method withthe Skyrme energy functional [17] or the time-dependentHartree-Fock (TDHF) method [18, 19], explained moredetailed in the next section. The pure self-consistentmodels (HFB) produce only the most probable fragmentmass asymmetry [21]. Dynamic effects, added on the topof the static results, are essential to obtain the full frag-ment mass distribution.Recently the time-dependent density functional theory(TDDFT) was applied to calculate TKE and mass yieldsof
Fm [20]. The authors analyzed fluctuations in scis-sion time, pre- and post-scission emissions of neutronsand protons. Also the correlations between TKE andcollective deformation of daughter nuclei were studied.The obtained mass yield is too narrow in comparison tothe measured one.The aim of the present work is to examine how the initialconditions affect the obtained mass yield, especially howthis quantity depends on the excitation energy and theparity of the initial state. We studied fission of
Cf us-ing the TDGCM+GOA approximation explained in de-tail in Refs. [18, 19, 22, 23] and briefly described in thenext Section. In Section II we present the theoreticalframework. Section III contains results of our investiga-tions. Conclusions are presented in the last section.
II. METHOD
We perform our analysis in two steps: (I) static calcula-tions of the PES and mass parameters, and (II) dynamicpart of the evolution of the wave packets. (I) The PES iscalculated using the Hartree-Fock-Bogolubov model withthe D1S Gogny-type interactions. The HFB equationsare solved with constraints on quadrupole Q and oc-tupole Q moments of the total nuclear density. Detailsof the calculation can be found in Refs. [24–26]. Themass parameters are computed within the adiabatic time-dependent Hartree-Fock (ATDHF) formalism [27, 28].We analyze the evolution of the wave packet from theinitial state up to the rupture of a nucleus into two frag-ments. Therefore we need to determine possible scis-sion configurations with two touching daughter nuclei. Inpractice, self-consistent calculations allow to find the pre-scisson shape of a nucleus as the last point in the defor-mation space before the rupture of the neck [29]. Most ofthe fission fragments’ properties are determined betweensaddle configuration and this point. The set of thesepoints referring to various octupole deformations createspre-scission line (p-sl) which is presented in Fig. 1a withthe white solid line. (II) The dynamic calculations aredone within the TDGCM+GOA formalism. The theo-retical framework of this method is explained in detail inRefs. [18, 19, 22] and references therein.The main constituent of the model is the collectiveHamiltonian which is taken in the form: b H coll = − ~ √ γ P i,j =2 ∂∂ Qi √ γB ij ( Q , Q ) ∂∂ Qj + V ( Q , Q ) , (1)where B ij = M − are the mass parameters of the collec-tive inertia tensor B with M − defined as: M i,j = P i,j =2 , = ( M ( − ik ) − ( M ( − kl ) − ( M ( − lj ) − . (2)The moments of order − m are given by: M ( − m ) i,j = P µν h Φ( Q ,Q ) | b Q i | µν ih µν | b Q j | Φ( Q ,Q ) i ( E µ − E ν ) m , (3)where Φ( Q , Q ) are solutions of the constrained HFBvariational principle and | µν i are the quasi-particle stateswith energies E µ and E ν . The quantity γ is the determi-nant of a metric tensor in the two-dimensional space ofcollective variables ( Q , Q ).Recently the improved approach to the moments of in-ertia calculations has been used in the fission barrier pen-etration analysis [30]. The non-perturbative mass param-eters were obtained in the Q − Q deformation space.It has been shown that minimization of action integralwith the non-perturbative mass parameters modifies thefission trajectory in the barrier region. The penetrationprobability is higher in comparison to that resulting from dynamic calculations within perturbative inertias. Thisis an important constituent of the fission half-lives stud-ies. However, the distribution of the probability currentalong the p-sl depends essentially on the evolution direc-tions beyond the barrier, rather than the trajectory ofthe system through the saddle [31]. The values of per-turbative and non-perturbative masses differ around theground state minimum but at large elongations both ap-proaches produce fairly similar collective parameters [32].Thus fission mass yields should not be strongly modifiedwhen the perturbative ”cranking” inertias were replacedby the non-perturbative ones.To find the initial collective wave function of the n − thstate g π n ( Q , Q , t = 0) with parity π , the eigenproblem b H coll g π n = E n g π n is solved in the two-dimensional groundstate well (0 ≤ Q ≤
55 b, − ≤ Q ≤
40 b / ). Sinceeigenstates of the Hamiltonian of the mother nucleus arestationary, the initial collective wave functions (for t = 0)are generated in the ground state well V ′ ( Q , Q ) thatis slightly modified HFB potential V ( Q , Q ). The bot-tom of the potential well stays unchanged while the re-gion beyond the barrier of V ′ ( Q , Q ) is generated bylinear extrapolation to large values of energy. The wavepackets obtained in this way may be treated as the eigen-states of fissioning nucleus and the procedure of time evo-lution through the realistic nuclear potential may be ef-ficiently applied.The dynamic part of calculations is based on the numer-ical code originally developed by Goutte et al. [18] andlater on enhanced by the implementation of determinantof the metric tensor γ . The probability flux ~J ( Q , Q , t )flowing through each point along the p-sl coordinates( Q sc20 , Q sc30 ) (see Fig. 1) is defined as follows: ~J ( Q , Q , t ) = ~ ı √ γB ( Q , Q ) × [ g ∗ ( Q , Q , t ) ∇ g ( Q , Q , t ) − g ( Q , Q , t ) ∇ g ∗ ( Q , Q , t )] . (4)One can therefore obtain the probability that the fission-ing system reaches a certain point in a deformation spaceby the expression: P ( Q sc20 , Q sc30 ) = t = T propag R t =0 ~J ( Q sc20 , Q sc30 , t ) · ~n dt. (5)Above, ~n stands for the normal to the p-sl vector in thepoint ( Q sc20 , Q sc30 ) . In this formula T propag is a time of prop-agation which will be discussed in Section III A.Each point of the p-sl corresponds to a different molecu-lar shape of a nucleus. Few illustrative configurations areshown in Fig. 1b. As it was mentioned above, from thedynamic part of the calculations one can get the prob-ability P ( Q sc20 , Q sc30 ) that a nucleus takes a certain shapebefore splitting. It is usually assumed that the neck rup-ture takes place at z coordinate where the neck is thethinnest. Nevertheless, this is a rather simplified pictureignoring all possible fluctuations caused by collective nu-clear surface vibrations. These effects may be included E [ M e V ] (a) 0 50 100 150 200 250 Q [b] 0 20 40 60 80 100 Q [ b / ] -40-30-20-10 0 10 20 E [ M e V ] (a) 0 50 100 150 200 250 Q [b] 0 20 40 60 80 100 Q [ b / ] E [ M e V ] (a) 0 50 100 150 200 250 Q [b] 0 20 40 60 80 100 Q [ b / ] E [ M e V ] (a) 0 50 100 150 200 250 Q [b] 0 20 40 60 80 100 Q [ b / ] Q = 130 b, Q = 10 b A H / A L = 131 / 121(b) Q = 180 b, Q = 40 b A H / A L = 134 / 118Q = 220 b, Q = 70 b A H / A L = 142 / 110 FIG. 1. (a) The potential energy surface of
Cf. The p-sl isdepicted in white. (b) Density profiles corresponding to theconfigurations marked by dots are presented in panel. e.g. by the gaussian smoothing [22] or by applying therandom neck rupture (rnr) mechanism [33–36]. In thelatter method, for each pre-scission shape correspondingto the scission configuration ( Q sc20 , Q sc30 ) the probabilityof splitting of a nucleus at the certain position on thesymmetry axis OZ along the neck is evaluated. Thus themass distribution is given by: P ( Q sc20 , Q sc30 ) = exp[ − γσ ( z ) /T ] . (6)Here σ ( z ) = 2 π R ∞ r ⊥ ρ ( z, r ⊥ ) dr ⊥ is a linear density of anucleus along the symmetry axis z , T is a temperature of a nucleus in at pre-scission deformation and γ is thesurface tension coefficient with a standard parametriza-tion given in Ref. [37]. The temperature T is of Boltz-mann form and depends on excitation energy E ∗ : T = p E ∗ /A . The energy E ∗ is defined as a difference be-tween the eigenenergy E n of a propagated state g π n andthe potential energy of a nucleus at the pre-scission point: E ∗ = E n − E scHFB . This is a standard parametrization ofthe rnr model without any fitting procedures. However apossible modification of the γ/T ratio affects the broad-ness of the mass yields [33]. The final fragment massdistribution is obtained as a convolution of the densitycurrent probability distribution along the p-sl and thernr mechanism. III. RESULTS ζ ( t ) t [10 -21 s] E n < B f , π=+1 E n < B f , π=−1 E n > B f , π=+1 E n > B f , π=−1 FIG. 2. The tunneling probability ζ ( t ) (Eq. 7) of Cf eigen-states ( g + - red triangles, g − - green circles) as a function oftime. The open symbols refer to the states located under fis-sion barrier while the full ones correspond to the states layingabove the barrier. We have chosen the neutron-rich
Cf isotope for ourinvestigations which represents an asymmetric type offission. This nucleus has been extensively studied ex-perimentally and many important observables such asmass, TKE distribution and average prompt neutronsmultiplicities are measured [38–42] and available for com-parisons with theoretical predictions. The fission barrierof
Cf, calculated within the HFB model with Gognyforces D1S, is equal to B f =9.71 MeV. In the present workwe investigate the eigenstates of the collective Hamilto-nian (1) with the potential V ′ ( Q , Q ) located in theenergetic range from the ground state to B f +4 MeV. A. Propagation time
To avoid the reflections at the edges of the ( Q , Q )grid we apply the absorbing complex potential [43] whichis active in the region beyond the scission line. Since thewave packet is absorbed after crossing the p-sl, it is pos-sible to calculate the reduction of the density probability ζ ( t ) in the considered collective space ( Q , Q ) at eachtime step: ζ ( t ) = 1 − Z | g π ( q , q , t ) | dq dq . (7)This quantity gives the information about the survival ζ ( t = T p r opag ) g + g - FIG. 3. The energy dependence of the tunneling probability ζ ( t = T propag ) (Eq. 7) of Cf eigenstates g π . The fissionbarrier height B f =9.41 MeV is marked by the vertical blackline. rate against fission after time t . It may be also inter-preted as a tunneling probability of a particular statethrough the fission barrier.In Fig. 2 changes of ζ ( t ) as a function of time for eachof the considered eigenstates of a mother nucleus with E g . s . ≤ E ≤ B f +4 MeV are displayed while Fig. 3 showsthe dependence of the value of ζ ( t = T propag ) on theenergy of an initial state. There is a visible tendencythat the states with higher energy propagate faster -rapid increase of ζ ( t ) may be observed. The lowest P [ A U ] A t t t t t t exp ζ ( t ) t [10 -21 s]t t t t t t FIG. 4. The tunneling probability ζ ( t ) (Eq. 7) of eigenstate n = 36 , π = − t , ..., t ). The experimental data weretaken from [38] and corrected by an average number of emit-ted prompt neutrons [42]. states propagate very slowly and after T propag fissionprobability is negligibly small - below 1%. In the sametime levels with E n > B f + 2 MeV propagate rapidlyand at t = 50 · − s and ζ ( t ) saturate at value close to1. It means that the wave function completely run awayfrom the ground state well. Several levels laying around E n ≈ B f ± T propag only about 1% of the wave packet crossedp-sl. The other levels from this range propagate muchfaster and after T propag leave the vicinity of the groundstate. There are 2 eigenstates ( g +34 , E = 11 .
82 MeV; g +35 , E = 11 .
86 MeV) which behavior diverges fromthis main tendency - after fast reduction of the densityprobability at the very beginning of the time evolutionthey stabilize at some value of ζ ( t ). The fission proba-bility ζ ( t ) changes its value just by around 0 . − . g − (inset of Fig. 4). The shape ofmass distribution does not depend significantly on theduration of the time evolution - the curves resulting fromall time steps overlap. Similar behavior is typical for alltested levels. This allows us to stop the time evolutionafter arbitrary chosen T propag = 2 . · − s. Furthercalculations show that even if time of propagation isextended twice, the tunneling probability stays almostconstant and the final distribution of the probabilitycurrent density along the p-sl does not significantlychange.One can also observe that ζ ( t = T propag ) is smallerfor states with negative parity than for positive oneswith similar energies. This tendency may be under-stood when one looks at the shape of the PES inthe ground state well and fission barrier region. Thesaddle is located at Q =0 and energy grows withincreasing octupole deformation. Since the states withnegative parity prefer Q = 0 channel their propaga-tion is hindered by the potential around the saddle point. B. Parity dependence P [ A U ] A π =+1 π =- 1exp FIG. 5. The fission fragment mass distributions as a result ofpropagation of positive (blue solid line) and negative (greendashed lines) states g π in the considered range of eigenvalues. In Fig. 5 we show the mass yields, as a result of timeevolution of positive and negative parity states in consid-ered energy regime, are displayed. The states of the sameparity lead to very similar shapes of yields - heavy andlight fragments peaks keep almost the same position inde-pendently on the initial energy. Furthermore, mass dis-tributions have very similar broadness. The most prob-able masses of fragments obtained from negative parityare shifted by 2 mass units to the center of distributionin comparison to the ones resulted from the positive par-ity functions. Some of considered states (with positiveparity and
E < B f ) lead to a slightly different mass dis-tribution around A = 126 than the others. The obtainedmass yields show that in these cases the symmetric fissionchannel has a small contribution. C. Energy dependence
To investigate the impact of the energy of the initialstate on the final fragment mass distribution we havecompared the results obtained after time evolution of thelowest state E = 2 .
05 MeV and the state laying at en-ergy E = 11 .
07 MeV (see Fig. 6). Obtained yields have P [ A U ] AE = 2.05 MeVE = 11.07 MeVexp FIG. 6. The comparison of mass yields obtained from timeevolution of the ground state at energy E = 2 .
05 MeV (blueline) and the state with eigenvalue E = 11 .
07 MeV (greenline). Both states have positive parity. very similar shape. Self-consistent calculations show thatthe PES depends weakly on the excitation energy [44],thus any qualitatively important changes in the fissionyields should not expected. Since both initial states havethe same (positive) parity, the positions of A H and A L peaks cover. In the case of E state the symmetric-fissionmode contribution is non-zero, what is not observed inthe experiment. Keeping in mind that for the first eigen-state ζ ( T propag ) <<
1% and the energy distance be-tween these eigenstates is rather large, the differences be-tween theoretical and experimental yields may be treatedas negligibly small. The most probable A H /A L ratio140/112 is slightly smaller than the measured - 142/110and the calculated mass yields are narrow in comparisonto the experimental one. Similar results are obtained forany initial state. We may conclude that the fragmentmass distribution of a nucleus excited to energy below oraround fission barrier height stays almost unchanged. D. Mixed states
According to the well established picture of inducedfission, the excited nucleus does not residue in its pureeigenstate but fissions from the state which is the super-position of its eigenstates. Measured mass yields of low-energy induced-fission does not differ significantly fromspontaneous-fission ones. Thus problem of state mixingin the theoretical description of the process should bealso considered. We examined various types of mixingstates, using the distributions of gaussian-shape: P ( E n ) = 1 σ √ π exp (cid:20) − ( E n − µ ) σ (cid:21) (8)and of the Fermi-shape: P ( E n ) = 1 B f (cid:2) exp (cid:0) E n − B f d (cid:1) + 1 (cid:3) (9) c on t r i bu t i on [ % ] ζ ( t = T p r opag ) E [MeV](a) π =+1 π = -1(i)(ii)(iii)(iv)(v) 0 0.01 0.02 0.03 0.04 0.05 80 100 120 140 160 180 P [ A U ] A(b) (i)(ii)(iii)(iv)(v)(vi)(vii)exp
FIG. 7. Upper panel (a): Types of statistical mixing of eigen-states and fission fragment mass distributions obtained as aresults of time evolution of these wave packets (lower panel(b)). The left axis of upper panel shows the percentage con-tribution of each state according to particular statistical dis-tribution used in the mixing procedure. The right axis refersto the tunneling probability ζ ( t ) of considered states. Mixing of eigenstates in several energy regimes areshown in Fig. 7(a). The initial wave packets are gen-erated as the linear combinations of all single stateslocated in this energy regime with statistical weightsgiven by the considered probability distributions (i)-(v). The Gaussian-type mixing are performed for (i) σ = 1 (MeV) , µ = B f + 1 MeV, (ii) σ = 2 (MeV) , µ = B f +2 , MeV , (iii) σ = 1 (MeV) , µ = B f +1 MeVand Fermi-type with the center of distribution locatedat B f and diffuseness parameters (iv) d = 0 . d = 0 . B f ≤ E n ≤ B f + 1 MeVand (vii) B f ≤ E n ≤ B f + 2 MeV. In Fig. 7(b) the fissionfragment mass distributions obtained as the result oftime evolution of the mixed wave packets are displayedand compared to the measured ones. The shapes of thesemass yields show that there are no important qualitativedifferences between final distributions resulted from theconsidered statistical methods of mixing. The resultedpeaks are shifted by 4 mass units in comparison tothe experimental ones. One can observe that the totalwidths of A H , A L peaks are now slightly broader thanthose resulting from the evolution of the single states.Moreover, the aforementioned symmetric mode of thelow laying states does not affect the yield in Fig. 7b. Theinfluence of this effect is washed out by the dominantcontribution coming from levels with higher energy andfaster barrier penetration. E. Discussion of the results
The present method allows to obtain the main charac-teristics of observed mass yield of
Cf. The calculatedmost probable fragment mass asymmetry fairly repro-duces experimental data. We predict also diminishingyield for symmetric mass split.The largest discrepancy is obtained in the most asym-metric part of the fragment mass distribution, where ourresults underestimate experimental evidence. This is be-cause nuclear configurations corresponding to such asym-metry appear on the p-sl only for large values of octupoledeformation: Q sc30 >
80 b / . Since they lay high in en-ergy (see Fig. 1a) the probability that the probability fluxflowing through this region is small.In order to obtain a better description of such asym-metric fragmentations, some extensions of the model arerequired. There are several options of possible modifica-tions. One of them is an extension of the deformationspace by another degree of freedom (hexadecapole mo-ment, strength of pairing correlations), including quantaleffects on a neck rupture mechanism or energy dissipa-tion effects between the intrinsic and collective degreesof freedom.Studies of the three-dimensional PES of Cf shownthat the hexadecapole moment plays an important role inthe description of fission mass asymmetry [45]. Namely,the scission configuration may be reached for lowerquadrupole moment within different mass asymmetrywhen the PES is spanned on Q − Q − Q space.Moreover, there were found several fission paths in suchthree-dimensional deformation space which lead to dif-ferent pre-scission shapes.As it was shown in the dynamical description of nu-clear fission, proton and neutron pairing correlations areimportant ingredients [46]. Pairing correlations have itsimpact on action integral minimization. The interplaybetween the potential energy and collective inertia af-fects the propagation direction chosen by the system inthe deformation space.The detailed, microscopic analysis of the pre-scissionconfiguration are also needed. One may investigate thesingle particle energies and density distributions just be-fore splitting [47]. It allows to study the formation ofthe nascent fragments and predefine the most possiblemass asymmetry. It is also possible to identify fissionfragments through the localization of the nucleons wavefunctions [48].The relevance of the fluctuations in the fission dynamicsalso has been recently demonstrated [49]. The Langevinequations were solved to find the time-dependent fissionpaths in the microscopically calculated multidimensionalspace. It was shown that the peaks positions of the yielddepend strongly on the topography of the PES in the pre-scission region, whereas the crucial role in reproductionof the broadness of the fragment mass distribution playthe dissipative collective dynamics and collective inertia. IV. SUMMARY
The following conclusions can be drawn from our in-vestigations: • The fission fragment mass distribution of
Cf de-pends weakly on the parity of the initial state.The peaks resulting from propagation of states withnegative parity are shifted by 2 mass units to thecenter of the mass yield in comparison to those ob-tained from the evolution of positive ones. • There is no strong correlation between fragment mass distribution and energy of propagated eigen-state up to E n = B f + 4 MeV. The peak positionand the broadness of the mass yield is practicallyindependent on the energy of the initial state. • The shape of mass yield does not depend on thetime of propagation. No difference is observed inthe fragment mass distribution at any stage of atime evolution of a particular state. • The mass distributions obtained from the initialstates taken as a various combinations of the indi-vidual states stay almost unchanged. The resultedpeaks are shifted by 4 mass units in comparison tothe experimental ones. • This approach is not sufficient to reproduce ex-perimental yields broadness, especially at the highmass asymmetry. To improve the broadness ofthe mass yield, the model needs several modifica-tions, e.g. proton and neutron pairing correlationsor hexadecapole moment should be taken as thecollective coordinates. It would be also worth tocheck whether the replacement of the perturbativeby non-perturbative ”cranking” inertia changes thedynamic landscape. It will be a subject of the fur-ther investigations.
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