Fission fragment mass yields of Th to Rf even-even nuclei
Krzysztof Pomorski, Jose M. Blanco, Pavel V. Kostryukov, Artur Dobrowolski, Bozena Nerlo-Pomorska, Michal Warda, Zhigang Xiao, Yongjing Chen, Lile Liu, Jun-Long Tian, Xinyue Diao, Qianghua Wu
FFission fragment mass yields of Th to Rf even-even nuclei ∗ Krzysztof Pomorski † , Jos´e M. Blanco , Pavel V. Kostryukov , Artur Dobrowolski ,Bo˙zena Nerlo-Pomorska , Micha(cid:32)l Warda , Zhigang Xiao ‡ , Yongjing Chen , Lile Liu ,Jun-Long Tian , Xinyue Diao , Qianghua Wu Institute of Physics, Maria Curie Sk(cid:32)lodowska University, 20-031 Lublin, Poland Department of Physics, Tsinghua University, Beijing 100084, China China Institute of Atomic Energy, Beijing 102413, China School of Physics and Electrical Engineering, Anyang Normal University, Anyang 455000, China
January 12, 2021
Abstract:
Fission properties of the actinide nuclei are deduced from theoretical analysis. We investigate potentialenergy surfaces and fission barriers and predict the fission fragment mass-yields of actinide isotopes.The results are compared with experimental data where available. The calculations were performed inthe macroscopic-microscopic approximation with the Lublin-Strasbourg Drop (LSD) for the macroscopicpart and the microscopic energy corrections were evaluated in the Yukawa-folded potential. The Fouriernuclear shape parametrization is used to describe the nuclear shape, including the non-axial degree offreedom. The fission fragment mass-yields of considered nuclei are evaluated within a 3D collective modelusing the Born-Oppenheimer approximation.
Keywords: nuclear fission, mac-mic model, fission barrier heights, fragment mass-yields
PACS: ∗ Supported by the Polish National Science Center (Grant No. 2018/30/Q/ST2/00185) and by the National Natural Science Foundationof China (Grant No. 11961131010 and 11790325). † Email [email protected] ‡ Email: [email protected] a r X i v : . [ nu c l - t h ] J a n Introduction
Good reproduction of fission barrier heights and fissionfragments mass-yields is a test of the theoretical mod-els describing the nuclear fission process. An interest-ing review of the existing fission models can be found inRefs. [1, 2, 3]. Extended calculations of the fission bar-rier heights can be found in Refs. [4, 5]. Readers who areinterested in the theory of nuclear fission can find moredetails in the textbook [6].In the present paper, the fission fragment mass yields(FMY) are obtained by an approximate solution of theeigenproblem of a three-dimensional collective Hamilto-nian, of which the coordinates correspond to the fission,neck, and mass-asymmetry modes. Here presented modelis described in details in Refs. [7, 8, 9]. The potential en-ergy surfaces (PES) of fissioning nuclei obtained by themacroscopic-microscopic (mac-mic) method in which theLublin-Strasbourg Drop (LSD) model [10] has been usedfor the macroscopic part of the energy, while the micro-scopic shell and pairing corrections are evaluated usingsingle-particle levels of the Yukawa-folded (YF) mean-field potential [11, 12]. The Fourier parametrization isused to describe shapes of fissioning nuclei [13, 14]. Itis shown in Ref. [15] that this parametrization describesvery well the shapes of the nuclei even close to the scissionconfiguration.The paper is organized in the following way. In Section2, we present first the details of the shape parametriza-tion and the theoretical model. Then we show the collec-tive potential energy surface evaluated within the mac-mic model for the selected isotopes and our estimates ofthe fission barrier heights. The calculated FMY are com-pared with the existing experimental data in Section 3.The estimates of FMY for Th isotopes and their depen-dence on two adjustable parameters are further discussedin details in Section 4. Conclusions and perspectives offurther investigations are presented in Section 5.
The evolution of a nucleus from the equilibrium statetowards fission is described here by a simple dynamicalapproach based on the PES. We assume that at large de-formations, the shape of the nucleus should depend onthree collective degrees of freedom describing its elonga-tion, left-right asymmetry, and the neck-size. At smallerdeformations, up to the second saddle, the non-axialshapes are also considered. In the following subsection,we present shortly a Fourier type parametrization of thenuclear shape which is used in the paper. zyz sh z neck -z +z sh z +z sh z l z r R = z r - z l x ρ ϕ a bc Fig. 1: Shape of a very elongated fissioning nucleus.
A typical shape of the nucleus on the way from the saddleto the scission configuration is shown in Fig. 1, where ρ ( z ), the distance from the z-axis to the surface of thenucleus as a function of z , is plotted. Here by the nuclearsurface is treated the surface of the nuclear liquid drop,or the half-density surface when the microscopic densitydistribution is considered.The function ρ ( z ) corresponding to the nuclear surfacecan be expanded in the Fourier series in the followingway [13]: ρ s ( z ) = R ∞ (cid:80) n =1 (cid:104) a n cos (cid:16) (2 n − π z − z sh z (cid:17) + a n +1 sin (cid:16) nπ z − z sh z (cid:17)(cid:105) . (1)Here z is the half-length of the total elongation of thenucleus and z sh locates the center of mass of the nucleusat the origin of the coordinate system. The expansion pa-rameters a i can serve as parameters describing the shapeof nucleus. The length parameter c = z /R is fixed bythe volume conservation condition, where R is the ra-dius of spherical nucleus having the same volume as thedeformed one.Contrary to frequently used spherical harmonics ex-pansion (conf. e.g., Refs. [4, 5]), the Fourier series con-verges much earlier for the realistic shape of nuclei [13, 14]and only a few first terms are sufficient in practical use.Although one can work directly with these Fourier expan-sion coefficients treating them as free deformation param-eters, it is more suitable to use their combinations { q n } ,2alled optimal coordinates [14], as following q = a (0)2 /a − a /a (0)2 q = a q = a + (cid:113) ( q / + ( a (0)4 ) q = a − ( q − a / q = a − (cid:113) ( q / + ( a (0)6 ) . (2)The functions q n ( { a i } ) were chosen in such a way thatthe liquid-drop energy as a function of the elongation q becomes minimal along a trajectory that defines theliquid-drop path to fission. The a (0)2 n in Eq. (2) arethe expansion coefficients of a spherical shape given by a (0)2 n = ( − n − π (2 n − . The above relations proposedin Ref. [14] transform the original deformation parame-ters a i to the more natural parameters q i , which ensurethat only minor variations of the liquid-drop fission pathsoccur around q = 0. In addition, more and more elon-gated prolate shapes correspond to decreasing values of a , while oblate ones are described by a >
1, which con-tradicts the traditional definition of the elongation pa-rameter. The parametrization (2) is rapidly convergent.It was shown in Ref. [15] that the effect of q and q onthe macroscopic potential energy of nuclei is negligiblefor small elongations of nuclei up to the saddle pointsand contributes within 0.5 MeV around the scission con-figurations.Non-axial shapes can easily be obtained assuming that,for a given value of the z -coordinate, the surface cross-section (blue dashed oval in Fig. 1) has the form of anellipse with half-axes a ( z ) and b ( z ) [14]: (cid:37) s ( z, ϕ ) = ρ s ( z ) 1 − η η + 2 η cos(2 ϕ ) with η = b − aa + b , (3)where the parameter η describes the non-axial deforma-tion of the nuclear shapes. The volume conservation con-dition requires that ρ s ( z ) = a ( z ) b ( z ). The nuclear potential energies of actinide nuclei areevaluated in the following equidistant grid-points inthe 4D collective space built on the q , q , q , and η deformation parameters: q = − .
60 (0 .
05) 2 . ,q = 0 .
00 (0 .
03) 0 . ,q = − .
21 (0 .
03) 0 . ,η = 0 .
00 (0 .
03) 0 . . (4)Here, the numbers in the parentheses are the step size,while the numbers on the left (right) side are the lower (upper) boundaries od the grid, respectively. The energyof a nucleus is obtained in the mac-mic model, where thesmooth energy part is given by the LSD model [10], andthe microscopic effects have been evaluated using the YFsingle-particle potential [11, 12]. The Strutinsky shell-correction method [16, 17, 18] with a 6 th order correc-tional polynomial and a smoothing width γ S = 1 . (cid:126) ω is used to determine the shell energy correction, where (cid:126) ω = 41 /A / MeV is the distance between the spher-ical harmonic-oscillator major shells. The BCS theory[19] with the approximate GCM+GOA particle numberprojection method [20] is used for the pairing correla-tions. An universal pairing strengths written as G N / =0 . (cid:126) ω , with N = Z, N for protons or neutrons, was ad-justed in Ref. [21] to the experimentally measured massdifferences of nuclei from different mass regions. It wasassumed in Ref. [21] that the “pairing window” contains2 √ N single-particle energy levels closest to the Fermilevel. All the above parameters were fixed in the past,and none of them was specially fitted to the propertiesof actinide nuclei.A typical PES for actinides is shown in Fig. 2, wheretwo cross-sections ( q , η ) and ( q , q ) of the 4D poten-tial energy surface of Pu are shown. As one can see,
Fig. 2: Potential energy surface of
Pu minimized with re-spect to q at the ( q , η ) (top map) and ( q , q ) (bottom)planes. the inclusion of the non-axial deformation is importantup to elongations corresponding to the second saddle3 q ≤ . q (cid:54) = 0 in the ground-state, the left-rightasymmetry begins to play an important role at large elon-gations of nuclei, from the second saddle ( q ≈
1) up tothe scission configuration ( q (cid:38)
0 2 4 6 8 10 Th E [ M e V ] Cf
0 2 4 6 8 10 U E [ M e V ] Fm
0 2 4 6 8 10 Pu E [ M e V ] No
0 2 4 6 8 10 220 230 240 250 Cm E [ M e V ] A E A E B exp E A exp E B
240 250 260 270 Rf A Fig. 3: Fission barrier heights of even-even actinide nuclei inour 4D mac-mic model.
The first ( E A ) and the second ( E B ) fission barrierheights obtained in our model for nuclei from Th to Rfare compared in Fig. 3 with the experimental data takenfrom Ref. [22, 3]. The agreement of our estimates withthe data is rather satisfactory, and is comparable withinan accuracy obtained in other theoretical models. The M s add m a c - M g . s . e x p [ M e V ] AexpLSD
Fig. 4: Fission barrier heights of even-even actinide nu-clei evaluated using the topographical theorem and the LSDmodel compared with the experimental barrier heights as afunction of mass number A [24]. largest deviation between our estimates and the exper-imental values are observed in thorium isotopes, wherethey are underestimated. The main origin of these dis-crepencies is mostly from the inaccuracy of determiningthe ground-state masses in our model. To prove it, wehave estimated the fission barrier heights using the socalled topographical theorem of Myers and ´Swi¸atecki [23],where the barrier height (the largest one) is defined as E barr = M saddmac − M expg . s . , (5)where M saddmac is the first barrier saddle point mass evalu-ated in the macroscopic model (i.e., without microscopicenergy correction) and M expg . s . is the experimental ground-state mass of the nucleus.Using the LSD model [10] to evaluate the macroscopic(read LD) mass one obtains the ’´Swi¸atecki’ estimates ofbarrier heights, which deviate from the experimental dataonly by 310 keV on the average, as shown in Fig. 4. Itmeans that additional work to improve the estimates ofthe ground-state masses has to be done. In particular, tomake a better fit of the pairing strength. Our ”universal”pairing force [21], used in the present work, reproduceson average the pairing gaps of nuclei from different massregions, but it might be that it does not reproduce per-fectly the pairing properties in actinides. The present research is a continuation and extensionof our previous works [7, 8, 9], where more detaileddescription of the collective fission model was given.The fundamental idea of this approach is the use of theBorn-Oppenheimer approximation (BOA) to separatethe relatively slow motion towards fission, mainly in q q and q collective coordinates. The BOA allows usto treat these both types of motion as decoupled, whatleads, in consequence, to the wave function in form ofthe following product:Ψ nE ( q , q , q ) = u nE ( q ) φ n ( q , q ; q ) . (6)The function u nE ( q ) is the eigenfunction correspondingto the motion towards fission, while the φ n ( q , q ; q ) sim-ulates the n − phonon “fast” collective vibrations in the“perpendicular” to the fission mode { q , q } plane.To determine the u nE ( q ) function for a single q modeone can use the WKB approximation as it has been donein Ref. [7]. To obtain the function φ n ( q , q ; q ), one hasto solve numerically for each value of q the eigenprob-lem of the underlying Hamiltonian in the perpendicular { q , q } space. However, for the low energy fission, it issufficient to take only the lowest wave function in the per-pendicular mode and evaluate the density of probability W ( q , q ; q ) of finding the system for a given elongation q within the area of ( q ± dq , q ± dq ) as W ( q , q ; q ) = | Ψ( q , q , q ) | = | φ ( q , q ; q ) | . (7)Further simplification we have made is to approximatethe modulus square of the total wave function in Eq. (7)by the Wigner function in the following form W ( q , q ; q ) ∝ exp V ( q , q ; q ) − V min ( q ) T ∗ , (8)where V min ( q ) is the minimum of the potential for agiven elongation q and T ∗ is a generalized temperature[25] which takes into account both thermal excitation offissioning nucleus and the collective zero-point energy E T ∗ = E / tanh( E /T ) . (9)The temperature ( T ) of nucleus with mass-number A isevaluated from its thermal excitation energy ( E ∗ ) usingthe phenomenological relation E ∗ = aT , with a = A /(10MeV). The generalized temperature T ∗ is approximatelyequal to the zero-point energy when T is small while forsufficiently high temperatures ( T (cid:29) E ) it approaches to T . In the following E is treated as one of two adjustableparameters of our model. Of course, one expects E ofthe order of 1 to 2 MeV as implied by the energy levelpositions of typical collective vibrational states.To obtain the FMY for a given elongation q one hasto integrate the probabilities (8) over the full range ofthe neck parameter q w ( q ; q ) = (cid:90) W ( q , q ; q ) dq . (10)It is rather obvious that the fission probability maystrongly depend on the neck radius R neck . Following Ref. [7] one assumes the neck rupture probability P tobe equal to P ( q , q , q ) = k k P neck ( R neck ) , (11)where P neck is a geometrical factor indicating the neckbreaking probability proportional to the neck thickness,while k /k describes the fact that the larger collectivevelocity towards fission, v ( q ) = ˙ q , implies that the neckrupture between two neighboring q configurations is get-ting less probable. The constant parameter k plays therole of scaling parameter which is finally eliminated inthe calculation of the resulting FMY. The expression forthe geometrical probability factor P neck ( R neck ) is chosenhere in a form of Gauss function [8]: P neck ( R neck ) = exp [ − log(2)( R neck /d ) ] , (12)where d , our second adjustable parameter, is the “half-width” of the neck-breaking probability. The momentum k in Eq. (11) simulates the dynamics of the fission pro-cess, which, as usual, depends both on the local collectivekinetic energy ( E kin ) and the inertia ( M ) towards the fis-sion mode (cid:126) k M ( q ) = E kin = E − E ∗ − V ( q ) , (13)with ¯ M ( q ) standing for the (averaged over q and q de-grees of freedom) inertia parameter at a given elongation q , and V ( q ) is the potential corresponding to the bot-tom of the fission valley. In the further calculations weassume that the part of the total energy converted intoheat E ∗ is negligibly small due to rather small frictionforces in the low energy fission. A good approximationof the inertia ¯ M ( q ), proposed in Ref. [26], is to use theirrotational flow mass parameter B irr , which is derivedinitially as a function of the distance between fragments R and the reduced mass µ of both fragments¯ M ( q ) = µ [1 + 11 . B irr /µ − (cid:18) ∂R ∂q (cid:19) . (14)In order to make use of the neck rupture probability P ( q , q ; q ) of Eq. (11), one has to rewrite the integralover q in probability distribution (10) in the followingform: w ( q ; q ) = (cid:90) W ( q , q ; q ) P ( q , q , q ) dq , (15)in which the neck rupture probability is now taken intoaccount. The above approximation describes a very im-portant fact that, for a fixed q value, the fission mayoccur within a certain range of q deformations with dif-ferent probabilities. Therefore, to obtain the true fission5robability distribution w (cid:48) ( q ; q ) at a strictly given q ,one has to exclude the fission events occurred in the “pre-vious” q (cid:48) < q configurations, i.e., w (cid:48) ( q ; q ) = w ( q ; q ) 1 − (cid:82) q (cid:48) 55 and q = 0,while the second one is at q = 1 . 10 and q = 0 . 08. Asone can see in the upper panel, the non-axial deformation η does not influence the PES at lager q deformation. So,we do not take this degree of freedom into account in ouranalysis of the FMY’s. Let us notice that each of the 2Denergy maps shown in Fig. 2 is only a projection of thefull 4D PES, and one has to consider other cross-sectionsin order to analyze the fission process in details.The fission fragment mass yields obtained in our modelare presented in Figs. 5 to 8. Some experimental datafor the FMY were obtained for the fission of excited nu-clei. In such a case, we take this excitation into accountand reduce the microscopic energy correction accordingto the prescription found in Ref. [27]. Our estimates ofFMY correspond to the so-called pre-neutron yields, i.e.,the mass yields before neutron emission from fragmentsand with such data (red stars in Figs. 5 to 8) they haveto be compared. In the case of Th isotopes, we have usedthe fragment charge yields from Refs. [28, 29] and to ob-tain the mass-yields it is assumed that the Z/N ratio inthe fragment is the same as in the mother nucleus. Incases when the pre-neutron data were not available, wehave plotted the post-neutron data (blue crosses) just toget piece of information about the experimental situa-tion. It is shown that for the Th isotopes, although theagreement of the estimates with the experimental datais not very satisfactory, the general trend is reproduced,i.e., a transition from symmetric to asymmetric fissionis evidently reproduced with a growing mass number ofisotope. The best agreement was achieved for Th and − Th nuclei. The agreement with experimental datain the Uranium chain, presented in the bottom part ofFig. 5, is much better. Here the maxima and the widthsof the fragment mass distribution are well reproduced,and a similar transition between symmetric and asym-metric fission as in Th isotopes is evident.The prediction of the FMY’s for Pu and Cm isotopesare compared in Fig. 6 with the experimental data. Thepre-neutron experimental yields for Pu [34] and Cm[32] isotopes are obtained for the spontaneous fissioncase, while those for Cm and Cm are post-neutronyields taken from Ref. [3]. A nice agreement with the6 0 4 8 12 Th y i e l d Th Th ch Th ch 0 4 8 12 Th y i e l d ch Th ch Th ch Th ch 0 4 8 12 60 90 120 150 Th y i e l d A f ch 60 90 120 150 Th A f 60 90 120 150 Th A f 60 90 120 150 Th A f 0 4 8 12 U y i e l d U U U 0 4 8 12 U y i e l d U th U th U post 0 4 8 12 70 100 130 160 U y i e l d A f 70 100 130 160 U A f 70 100 130 160 U A f 70 100 130 160 U A f Fig. 5: Fission fragment mass-yields of Th (top part) and U(bottom part) isotopes. Experimental data (red stars) for Thisotopes are extracted from the charge-yields of Refs. [28, 29]while the mass-yields for U isotopes (botton part) are takenfrom Ref. [30, 31] for the thermal neutron induced fission (th).Just to guide the eye we have used for U the post-neutrondata (blue crosses) taken from Ref. [3]. data obtained for the two lightest Pu isotopes Pu and Pu is slightly spoiled when the number of neutronsincreases i.e. for Pu and Pu. It is mainly becausewe have used here the globally optimized values of E and d , which are not fitted to Pu data only as done inRef. [8]. In all investigated Pu and Cm isotopes here,the asymmetric fission is predicted with the mass of theheavy fragment A ≈ Cf [31]and Cf [33] have to do with pre-neutron yields. Therest of the experimental yields presented in Fig. 7 corre-sponds to the post-neutron data (blue crosses). One cansee that for lighter Cf and Fm isotopes, the asymmet-ric yields are predicted, while in the case of the heaviestCf and Fm nuclei, the symmetric fission is foreseen. Asone can deduce from the above results, our estimates arerather consistent with the experimental yields. A sim- 0 4 8 12 Pu y i e l d Pu Pu Pu 0 4 8 12 Pu y i e l d sf Pu sf Pu sf Pu sf 0 4 8 12 70 100 130 160 Pu y i e l d A f sf 70 100 130 160 Pu A f 70 100 130 160 Pu A f 70 100 130 160 Pu A f 0 4 8 12 Cm y i e l d Cm Cm Cm 0 4 8 12 Cm y i e l d Cm Cm Cm post 0 4 8 12 70 100 130 160 Cm y i e l d A f sf 70 100 130 160 Cm A f post 70 100 130 160 Cm A f 70 100 130 160 Cm A f Fig. 6: Fission fragment mass yields of Pu (top part) andCm (bottom part) isotopes. Experimental data (red stars)are taken from Ref. [34] for Pu chain and Refs. [3, 32] for Cmnuclei. ilar tendency as seen in the Cf and Fm chains can beobserved in Fig. 8 for the No and Rf isotopes, where theasymmetric fission is predicted in the lighter nuclei whilethe symmetric fission mode dominates for the isotopeswith N > No and Rf [35, 37] is evident.The overall good quality of our predictions in a broadmass region of the actinide elements is probably due tothe fact that in very heavy nuclei, the fission barrier isvery short, and the fission valley forms very early, i.e., ata relatively small elongation of the nucleus. An opositesituation occurs in the thorium nuclei, where the fissionbarriers are very broad. Fig. 9 presents the fission valleypotential as a function of the elongation parameter q .It is shown that the average slope of the curve from thelast saddle to scission in the thorium nuclei is almostthree times smaller than in nobelium. Obviously, such alarge difference in the slope towards fission influences thefission dynamics in these both types of nuclei. This is amain reason why one has to study in details the PES inTh nuclei to explain here observed change in the FMYsystematics.7 0 4 8 12 Cf y i e l d Cf Cf Cf 0 4 8 12 Cf y i e l d Cf Cf post Cf sf 0 4 8 12 70 100 130 160 Cf y i e l d A f 70 100 130 160 Cf A f sf 70 100 130 160 Cf A f 70 100 130 160 Cf A f 0 4 8 12 Fm y i e l d Fm Fm Fm sf 0 4 8 12 Fm y i e l d sf Fm Fm Fm post 0 4 8 12 70 100 130 160 Fm y i e l d A f post 70 100 130 160 Fm A f post 70 100 130 160 Fm A f 70 100 130 160 Fm A f Fig. 7: Fission fragment mass yields of Cf (top) and Fm (bot-tom) isotopes. Experimental data for pre-neutron yields (redstars) are taken from Refs. [31, 33] while the post-neutronyields (blue crosses) origin from Ref. [3, 36]. The agreement of our estimates of the FMY’s with theexperimental data in the Th chain of isotopes depictedin Fig. 5 is not quantitatively satisfactory. So, in thepresent section, we would like to look for the origin ofthese discrepancies. First, these yields for the Th nucleiare evaluated using E and d obtained by the fit to thedata for all nuclei. The PES’s for Th nuclei are verymuch different from those for heavier nuclei. This canbe seen by comparing the PES of Pu shown in Fig. 2(bottom) with the corresponding maps for − Th iso-topes presented in Fig. 10.In Pu, the fission path goes directly from the sad-dle point to the asymmetric fission valley in Pu whileit is not the case in Th where the system from the3rd minimum at q ≈ . < . ≈ Th nucleus prefers the sym- 0 4 8 12 No y i e l d No No No 0 4 8 12 No y i e l d No No No sf 0 4 8 12 70 100 130 160 No y i e l d A f post 70 100 130 160 No A f 70 100 130 160 No A f 70 100 130 160 No A f 0 4 8 12 Rf y i e l d Rf Rf Rf 0 4 8 12 Rf y i e l d post Rf sf Rf Rf 0 4 8 12 70 100 130 160 Rf y i e l d A f 70 100 130 160 Rf A f 70 100 130 160 Rf A f 70 100 130 160 Rf A f Fig. 8: Fission fragment mass yields of No (top) and Rf (bot-tom) isotopes. Experimental data (crosses) are taken fromRefs. [35, 36, 37]. -40-30-20-10 0 10 1.80 1.90 2.00 2.10 2.20 2.30 Th Pu Cf No V [ M e V ] q Fig. 9: Potential corresponding to the bottom of fission valleyas a function of the elongation parameter q metric fission, which is confirmed by the experimentalyield. In Th the situation is similar, while beyond Th the path leading to the asymmetric fission beginsto be preferred. To better understand this process, onehas to study the PES’s in the full 3D deformation space.8 ig. 10: Potential energy surface cross-sections of − Thisotopes minimized with respect the neck parameter q on theplane ( q , q ). In Fig. 11 the ( q , q ) cross-sections of the PES for Thcorresponding to different elongations ( q =1.8, 2.0, 2.2,and 2.3) are shown.Two minima, one corresponding to the symmetric ( q = 0) and the other to the asymmetric ( q ≈ . 12) con-figuration, are visible in each cross-section. At q = 1 . Fig. 11: Potential energy surface cross-sections of Th onthe plane ( q , q ). The panels from top to bottom correspondto elongations q = 1.8 to 2.3, respectively. The solid red linesdrawn in the bottom panels correspond to the neck radiusequaling to the nuclear radius constant. 0 4 8 12 Th y i e l d Th Th ch Th ch 0 4 8 12 Th y i e l d ch Th ch Th ch Th ch 0 4 8 12 60 90 120 150 Th y i e l d A f ch 60 90 120 150 Th A f 60 90 120 150 Th A f 60 90 120 150 Th A f Fig. 12: Fission fragment mass yields of Th sotopes repro-duced by using constants E and d fitted to the experimentaldata (red stars) taken from Refs. [28, 29]. they are separated by a 4 MeV high barrier which be-comes smaller with growing elongation q , reaching fi-nally 0.5 MeV height at q = 2 . 1. At such elongations,the transition between the symmetric and asymmetricfission is possible. Both fission valleys are well separatedagain at the largest deformations close to the scission line(red line in the figure). So, the Th nuclei make the ”de-cision” where to go pretty early, i.e., at an early stage farbefore the scission configuration. It means that one hasto modify the adjustable parameters E and d in orderto better describe the transition between the symmetricand asymmetric fission modes observed in Th nuclei whenthe neutron number grows. The new fit performed to thedata for Th isotopes only gives E =1.5 MeV and d =2.5fm. The resulting mass yields are compared in Fig. 12with the experimental data. This time the agreement ismuch more satisfactory. The new value of the neck pa-rameter d is larger than that adjusted to all nuclei. Itsuggests that in Th nuclei, the choice of the preferablefission mode is made at a thicker neck, i.e., in a prettyearly stage. The smaller value of E used for Th iso-topes is probably related to the competition between thesymmetric and asymmetric minima. In order to briefly summarize our investigations, we canwrite: • The overall accordance of the theoretical FMY esti-mates with the experimental data indicates that themac-mic model with the LSD energy for the macro-scopic smooth part and the shell and pairing cor-rections evaluated on the basis of the Yukawa-foldedsingle-particle potential describes well the potentialenergy surfaces of actinide nuclei, • Three-dimensional set of the Fourier deformation pa-rameters used to describe the shape of fissioning nu-clei are fully capable to produce a wide variety ofthe shapes of nuclei on their way to fission, • The collective 3D model based on the Born-Oppenheimer approximation and comprising elonga-tion, mass asymmetry, and neck modes reproduceswell the mains features of the fission fragment massyields data, • The Wigner function used to approximate the prob-ability distribution related to the neck and massasymmetry degrees of freedom simmulates in aproper way this distribution for low-energy fission, • A neck-breaking probability depending on the size ofthe neck has to be introduced to improve the accor-dence of our FMY estimates with the experimentallymeasured values.Our mac-mic model and the collective 3D approach,which couples fission mode, neck, and mass asymme-try collective vibrations, can describe the main featuresof the fission process in actinide nuclei. The estimatedfission barrier heights deviate not much from their ex-perimental values. The measured fission fragment massyields are also reproduced in a satisfactory way. Onthe other hand, one has to treat the presented collec-tive model as a kind of rough tool which allows to obtainthe FMY by a relatively quick calculation. To get moreprecise results, one has to use more advanced models inwhich the whole fission dynamics and the energy dissi-pation will be taken into account. Such calculations mayuse the Langevin dynamics (conf. Ref. [6]) or the im-proved quantum molecular dynamics model (ImQMD).The latter method has been successfully applied to de-scribe the fission process in the heavy ion induced fissionreactions, where the excitation energy increases, leadingpossibly to a shorter fission time scale and even to theoccurrence of a ternary fission [38, 39].The Langevin type calculations, profiting of the PESgenerated in a mac-mic approach together with the 3DFourier shape parametrization as well with the use of theself-consistent method, are carried out in parallel by ourgroup. Acknowledgments The authors would like to thank Christelle Schmitt andKarl-Heinz Schmidt for supplying us with a part of theexperimental data. References [1] M. R. Mumpower, P. Jaffke, M. Verriere, J. Ran-drup, Phys. Rev, C , 054607 (2020).102] J. Randrup, P. M¨oller, Phys. Rev. C , 064606(2013).[3] K.-H.Schmidt, B.Jurado, C.Amouroux, C.Schmitt,Nucl.Data Sheets , 107 (2016).[4] A. Baran, M. Kowal, P.-G. Reinhard, L.M. Robledo,A. 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