Collinear true ternary fission as the consequence of the collective nuclear model
aa r X i v : . [ nu c l - t h ] M a y Collinear true ternary fission as the consequence of thecollective nuclear model
F. F. KarpeshinMendeleev All-Russian Research Institute of Metrology190005 Saint-Petersburg, RussiaMay 7, 2019
Abstract
The concept of collinear spontaneous true ternary fission of
Cf is subject to critical anal-ysis. The conclusion is that the collinear flight of the fragments turns out to be a natural andmost probable mode. The collinearity arises in the model on the prescission stage as a resultof the account of the principles of the collective Bohr?s model. It is partly destroyed at thepost-scission stage of spreading of the fragments due to their Coulomb interaction, with theallowance for the spin effects arising at the moment of scission. The final angular distribution ofthe fragments is calculated by means of the trajectory simulations. The calculated relative angleof the heavy and light fragments is kept 180 ◦ with an uncertainty within 0.4 ◦ , which justifiessearch for a collinear tri-partition at the modern stage of experiment. Introduction
The question of fission into three comparable fragments has a long, chal-lenging and fascinating story. As distinct from traditional ternary fission,where emission of two massive fragments is accompanied with a ternarylight particle, like an α particle, sometimes it is called true ternary fission(TTF). Strutinsky et al. were the first who proposed search for this process[1]. Attempts of creating a theory of TTF were undertaken by many theo-rists. Proceeding from typical initial conditions on the top of usual fissionbarrier, within the framework of the liquid drop model, Nix [2] demon-strated appearance of a third very light fragment which arose between twoother massive fragments in the case of very heavy fissioning systems with A & n fission of actinide nuclei by thermal neutrons [8] and α particles [9].The measurements were also conducted in heavy-ion collisions [10], andspontaneous fission of Cf [11]. However, only upper limits of the prob-ability of the processes were established at the level of 10 − – 10 − . It isworthy of noting that there was a tacit contradiction between theory andexperimental search. From the theoretical point of view, the linear form offissile nuclei is more favorable than a clover-leaf shape ( e. g. [5] and refs.cited therein). However, experimental efforts were mainly aimed at de-tecting fragments at approximately similar angles, i.e., ∼ ◦ . Based ongeneral considerations, the experimenters likely believed that the mutualelectrostatic repulsion could align the spreading angles.Solyakin et al. proposed the collinear mode of tripartition [12], whensearching for TTF of U by 1-GeV protons. This concept was mostsuccessfully realized in JINR experiments on FOBOS and mini-FOBOSsetups [13, 14, 15]. All the results presented below are obtained within theframework of the “missing-mass” approach. In fact, only two fragmentswere detected in each decay event at a relative angle of 180 ◦ . The mass andvelocity of the “missed” fragment were calculated based on the mass andmomentum conservation. The second principal feature of the spectrometerwas the presence of the blocking grid, which prevented cases when light andternary fragments could strike the same detector. Use of the missing-massmethod in combination with the supporting mesh on one of the detectors ed to conclusion that the collinear mode of TTF of U and spontaneousfission of
Cf may be at the level of up to 10 − – 10 − . At first sight, thismode is in contrast with ordinary ternary fission [16], where α particlesor protons are emitted approximately perpendicularly to the fission axis.It is no coincidence that in Ref. [17] the authors doubted a possibility ofa “perfectly” collinear flight in the case of three massive fragments. Atthe same time, it is known that there is a small fraction ∼
10 percent ofpolar alpha particles, emitted along the fission axis (e.g. [18] and refs.cited therein). Furthermore, direct estimations of possible scenarios ofthe fragment spread made in Refs. [19, 20, 21] show that the collineartrajectory only can be expected if all the three fragments are born in oneline. Minor displacement of the middle fragment from the axis ∼ α particlesare formed and then spread. In the case of TTF, all the three nascent ragments form an axially symmetric nucleus. This means that they movecoaxially until separation.The next question, however, arises whether this pre-scission collinearitymay survive in the course of post-scission propagation of the fragments.They make a complex three-body motion under the action of the mutualCoulomb repulsion, at the same time keeping the memory of the rotationof the fission axis and that impulse they received at the moment of rupture.As it is shown in the next section, devoted to a qualitative consideration,combination of these factors destroys a collinear picture, but to a someextent. In section 3, formulas used for calculation are derived. The resultsof calculation for representative fragments of TTF are reported in section4. They are discussed in the concluding section. As is known, projection K of the total angular momentum of a nucleuson the nuclear axis, related to the intrinsic nuclear coordinate system,is a good quantum number. In quantum mechanics, the body cannotrotate around the axis of symmetry, and thus the rotational momentum isperpendicular to the axis of symmetry, and its projection K onto the axisof symmetry is zero. At the same time, a nonzero value of K can also beobserved in an axially symmetric system due to quasiparticle excitations nrelated to rotation. If the system is a bit non-axially symmetric, thenbasically it will also rotate around an axis perpendicular to the axis ofsymmetry, and it will only twist around the axis of symmetry slightly. Inquantum mechanics, this is reflected in the fact that the wave function ofthe fissile nucleus has the following form [22]:Ψ IM ( r i ) = X K a K D IM K ( θ, φ, ϑ ) χ K ( r ′ i ) . (1) I in Eq. (1) is the nuclear spin, M — its projection in the laboratoryframe. Wigner’s D functions from the Euler angles θ, φ, ϑ define the orien-tation of the nucleus in space and determine the angular distribution of thefragments. r i and r ′ i are the nuclear variables ( e. g. , nucleon coordinates)in the laboratory system and intrinsic coordinate system, respectively.The z ′ axis thus coincides with the fission axis. And let us direct the x ′ axis in the plane of symmetry of the fissile nucleus in the case of K > r to the intrinsic system r ′ can be performed in three steps. First two ofthem, the rotations by θ about the z axis and by φ about the new axis x ′ ,respectively, impose the z and z ′ axes with each other [22]. After which, itremains the third rotation by ϑ about the new z ′ axis, in order to imposethe x axis at the x ′ one. In the case of a axially-symmetric nucleus, thisthird rotation evidently might not be needed, as all the azimuthal angles re equivalent. The only way to combine this picture with Eq. (1) is to put K = 0. And vice versa : in the case of axially-asymmetric — triangularconfiguration of the fissile nucleus, rotation by the angle ϑ is essential.Correspondingly, this excludes values of K = 0 in Eq. (1), leaving onlyvalues K >
U. As I ≥ K , non-zero K values are only possible if fissilenucleus has a non-zero spin. This is not the case if spontaneous fission of Cf is considered. Therefore, it is only the symmetric configuration “threein line” which survives fission.Furthermore, even such a “co-axial” initial configuration is not enoughyet for the final collinearity. It can be destroyed during spreading of thefragments, as a result of interplay of the accelerating Coulomb force be-tween them and initial velocity conditions. Most essential is perpendicularto the fission axis component of the initial velocity, which arises at themoment of scission due to big spins and large relative angular momentumof the fragments. In the case of binary fission of actinide nuclei, the meanvalue of the fragment spin is about 7 – 8 [24, 25]. In papers [24, 26],appearance of the spins in the fragments at scission is explained by exci-tation of the collective modes of wriggling and bending vibrations. In thefirst case, the fragments are formed with spins parallel to each other, and erpendicular to the fission axis. Arising total spin is compensated by theorbital angular momentum L of the relative motion of the fragments. Thelatter is thus in the opposite direction to the total spin and also perpen-dicular to the fission axis, as shown in Fig. 1. The observed value of themean spin of the fragments can be explained in this way (e.g. [27]).One concludes from the above consideration that in the case of TTF,appearance of wriggling vibrations in all three fragments can lead to thetotal spin of the fragments, and, respectively, to their total orbital momen-tum as much as L ∼
20. And the larger the L value, the greater the finalangle of divergence between the fragments. Baring this in mind, we variedpossible L values within 0 ≤ L ≤
20. In the case of bending vibrations, thefragments are formed with spins antiparallel to one another. Therefore,the relative orbital momentum, together with the related destruction ofthe collinearity, is expected to be even smaller.Allowance for the initial transverse velocity results in the final diver-gence of the spreading fragments. If the fragments could move completelyfreely after scission, then with reasonable initial conditions, all the frag-ments would stay collinear on a rotating fission axis. The necessary condi-tion for the fragments to remain on the axis is that both the transverse andlongitudinal velocity components remain proportional to the distance fromc. m. When the Coulomb force is switched on, it changes only the longitu-dinal component. This violates the proportionality: the middle (ternary) ragment is pushed to the c. m. by the both outer fragments. This reflects,specifically, in its small final kinetic energy [19, 20, 21]. In turn, the middlefragment itself pushes both the side fragments out, which also works as toviolate the proportionality. As a result, as soon as the middle fragmentdescends from the axis, this immediately triggers the transverse compo-nent in the Coulomb repulsion between the fragments, which enhances thefurther destruction of collinearity.Note that the mechanism described has much in common with the ROTeffect, which arises in fission of nuclei with spins, different from zero, bypolarized neutrons [28, 29]. The ROT effect is the triple angular correlationbetween neutron spin and the momenta of fragments and ternary particles.The ROT effect can be explained as due to rotation of the fission axisbefore scission, which is transferred to the fragments at scission as thetransverse initial velocity. Naturally, this mechanism would also give acontribution to destruction of collinearity in TTF, if the fissile nucleushad a spin. There is, however, a big difference in the mechanism of ROTeffect and that discussed above. First, the mechanism, destroying thecollinearity in TTF, appears to be much stronger, as it is related withmuch higher momenta L .
20. Second, it does not contribute to the ROTeffect because of angular averaging: there would be no correlation of the L direction with the spin of the fissile nucleus, even if the spin were differentfrom zero. Contrary, in the case of TTF, where each event is detected ndependently of the others, relative momentum L could easily manifestitself through violation of the collinearity.To sum it up, we will assume that the total relative momentum ofthe fragments may reach as high as L ≈
20 in the case of wrigglingvibrations. This value is compensated by the total spin of the fragments.The latter may be smaller or even zero in the case of bending vibrations.This will reply to smaller angular momenta L . Respectively, still morecollinear trajectories of the fragments will be expected. Let us turn to thenumerical estimations. Numerical simulation of trajectories of representative fragments is a clas-sical method. Its applicability follows a known fact that the wavelengthrelated with the fragment translation is small as compared to its size. Suchcalculations were found to work well in description of the spectra and an-gular distributions of α particles, emitted in ternary fission (e. g.[30]),specifically, of the ROT effect [31].Let the fission axis coincide with the quantization axis z at the mo-ment of scission. In view of the axial symmetry of the problem, let x e the transverse direction axis. After scission, further trajectories of thefragments are determined by the repulsive Coulomb forces between them.Denote the side fragments with indices 1 and 2, and the middle frag-ment as No. 3 (Fig. 1). Representative trajectories are simulated in thenext section by solving the Newton equations of motion for each of thefragments: d r i dt = F i /M i , i = 1 , , , (2)where M i is mass of the i -th fragment, and F i is the resulting force actingon it from the two other fragments. For simplicity, the latter is calculatedunder natural assumption of spherical fragments. System of coupled differ-ential equations of the second order (2) has to be solved numerically withthe proper initial conditions concerning the positions of the fragments andtheir velocities at scission. I consider the generic mechanism of the TTF, when the both scissions occurnearly simultaneously. The choice of the initial conditions is illustrated inFig. 1. And let us specify the atomic and mass numbers of the fragmentsas Z i and A i , respectively, with the distances r , r and r betweenthe fragments. The positions of the fragments must be defined, baring inmind their asymptotic total kinetic energy (TKE), which must not exceedreaction heat Q . For the parameterization purposes, the total Coulomb Initial conditions for the trajectory simulations. V , V and V are the transverse velocities to the fissionaxis of the fragments 1 – 3, which comprise the total relative angular momentum of the collective rotation of thefragments (directed towards us). D is the distance between extreme fragments. energy of the fragments is minimized, based on the position of the ternaryfragment at fixed distance D = r between the side fragments: r = D √ Z √ Z + √ Z . (3)In this way, the initial positions of all three fragments are fixed by thesingle parameter D , which in turn is defined by the TKE value T , T ≤ Q : T = ( Z Z r + Z Z r + Z Z r ) e . (4) In the laboratory system, the most general motion of the fragments can berepresented as a superposition of a linear translation and rotation around heir center of mass. The former gives linear velocity of the fragment,the latter is nothing more than the spin of the fragment. A small initialvelocity of the fragments in the direction of fission is not important forthe present purposes. In order to calculate the initial transverse velocityof the fragments, let us designate the masses of the fragments and theirpositions on the axis of fission as M , z , M , z and M , z , respectively.The center of gravity of the fragments, determined during fission, is set as ζ = ( M z + M z + M z ) /M , (5)where the total mass M = M + M + M . The total angular momentumof the fragments L is defined as follows: ω [ M ( z − ζ ) + M ( z − ζ ) + M ( z − ζ ) ] = L ~ , (6)and the initial transverse velocity of fragment i is V i = ω ( z i − ζ ) . (7) A landscape of the potential deformation energy was calculated in Ref.[33] for the case of TTF of
Cf. It suggests the following mode as a likelycandidate: Cf → Sn + Ca + Ni , Q = 251 MeV , (8) ith the light and heavy side fragments of Ni and Sn, and the ternaryfragment of Ca in the middle. The Q value in fission (8) was calculated,using AME2012 atomic mass evaluation [32]. The presence of two magicor semi magic fragments in the final state provide a great released en-ergy Q . The situation is like in three-partition of the atomic clusters of N a +++ → N a + into three magic clusters of N a + [5]. The final TKEvalues depend on the scission configuration: position of the fragments,thickness of the necks. Deformation of the fragments takes a part of en-ergy from the Q value, diminishing TKE of the fragments. For this reason,I consider various representative TKE values and total angular momenta L . In the landscape of the potential energy [33], pronounced valleys favor-able for ternary fission were found. One of them, which may be relatedwith channel (8), lies after a saddle point at r ≈ R = 22 fm, where R is the radius of the mother nucleus. At this distance, formation of the fu-ture fragments starts. The valley presents a good opportunity for scissionand separation of all three fragments somewhere at r &
30 fm. Indeed,the TKE value T = Q would be achieved if scission occurred at r = 25.56fm. In practice, part of the released energy is stored in the deformationenergy of the fragments, while scission occurs at a larger distance. Baringthis in mind, we varied the parameter D = r in the range up to D = 40fm. Experimental results [15] confirm such an expectation. quations of motion (2) with initial conditions (3) and (7) were solvedby means of the Runge—Kutta—Nystr¨om method. The results of thetrajectory simulations are presented in Fig. 2 and Table 1. The calculatedkinetic energies of each of the fragments, together with their TKE, arepresented in Fig. 2 versus the distance between the side fragments D at scission. All the energies smoothly decrease with increasing D , whilethe TKE changes from T = Q = 251 MeV for D = 25.6 fm down to T = 160 MeV for D = 40 fm. As well as in ordinary binary fission,the heavy fragments are produced with lower kinetic energies. We note acharacteristic feature of TTF, which follows Fig. 2: the ternary fragments,which are formed between the heavy and light ones, turn out to be veryslow, with the kinetic energies of approximately 5 MeV. This is 15 – 20times as small as the energies of the main fragments. Such low energiesare in accordance with refs. [19, 21, 34]. Qualitative reason is that in acollinear flight, the motion of the ternary fragment is confined by the twoouter fragments.Results concerning the angular distribution of the fragments are pre-sented in Table 1. As a consequence of the rotation of the fission axis,none of the fragment trajectories remains on the z axis after scission, if L = 0. For the configuration presented in Fig. 1, where the momentum L is aimed at the reader, the light Ni fragment goes down from the z axis.In turn, the heavy Te fragment comes up. As a result, the two main frag- T K E ( M e V ) (cid:13) D (fm)(cid:13)
Figure 2:
Kinetic energies of the fragments and their TKE values against the scission point D : main fragments of Ni and
Sn — full and dashed lines, respectively, dash-dotted line — the energy of the ternary fragment of Ca,scaled by a factor of 10, and dotted line — the total kinetic energy. ments scatter in the opposite directions, with the angle Θ between themremaining close to 180 ◦ . The calculated Θ values are displayed in the Ta-ble for various momenta L . As one can see from the Table, the differencefrom 180 ◦ does not exceed 0.4 ◦ . This deflection is a consequence of thetransverse force, discussed in Section 2. The calculated angles between thelight and heavy fragments satisfy the experimental conditions [13, 14, 15],where only collinear fission events with a relative angle of 180 ◦ ± ◦ wereselected. he ternary fragment always flies in the same direction as the lightone. The angle of divergence Φ between them is also presented in theTable against the total angular momentum L . The latter was varied ina wide range 0 ≤ L ≤
20, as explained previously. All three fragmentsremain in the same plane. Projections of the velocities of the light andternary fragments on the axis, perpendicular to the direction of the heavyfragment, have opposite signs. Scattering of the ternary fragment into theupper and lower half-planes in Fig. 1 is equally probable. The resultspresented clearly show that the Ni and Ca fragments, moving in the samedirection, diverge within one-two degrees at most, for all the considered L values. However, with the energies, presented in Fig. 2, the ternaryfragment arrives at the range of the FOBOS or mini FOBOS detectorwith delay of ∼ − s as compared to the light fragment. It follows from the considered model that the light and heavy fragments flyin the opposite directions with the relative angle 180 ◦ with the accuracyof 0.1 ◦ – 0.4 ◦ . This is in contrast with the first experiments [8] and others,aimed at detecting fragments at the angles of ∼ ◦ to one another.The above result thus justifies search for collinear mode of TTF as mostprobable. The result obtained bases on the two circumstances. The first Calculated angular distributions of the fragments of true ternary fission of
Cf (8) versus the scissiondistance D and the relative angular momentum L . Θ is the angle between the directions of the heavy and lightfragments, Φ — the divergence angle between the light Ni and ternary Ca fragments, moving in the same direction D , fm L Θ ◦ Φ ◦ circumstance is axial symmetry of the fissile nucleus on its path towardsscission. Because of the axial symmetry, there is a sole way of formation ofthe fragments, when they remain co-axial till scission. This is a remarkableillustration of the collective A. Bohr’s model. Furthermore, it is in spiritof the A. Bohr’s hypothesis about predominance of the channel with K = 0 in photofission of U. This hypothesis transfers the principles ofsymmetry from the collective model into fission. As is known, the A.Bohr’s hypothesis also works in the case of fission of
U by thermalneutrons, in which case the compound nucleus is characterized by strongmixing of the states with different K values, so as all the possible values rom 0 to 4 may become nearly equally probable due to the Coriolis mixing[35]. As a result, it was suggested that the channel J π K = 4 − J π = 4 − . Also in this case, mostprobable channels with a certain K reply to minimal energy over the fissionbarrier. As an example of application of the symmetry principles in fission,we also note paper [36] where, in addition to the J π K characteristics offission barriers, the signature quantum numbers s and r related to thesymmetry of the first and second fission barriers were introduced, aimedat further studying the properties of fission of nuclei with K = 0. One cansay that the barrier works as the filter, which selects the channels withminimum energy over the barrier. However, the angular distribution inthe case of fission of nuclei with spins becomes more complicated becauseof the strong mixing over K values on the stage of the compound nucleus[35]. A consideration of such cases can be performed elsewhere, in a similarapproach.The second circumstance is good conservation of the collinearity dur-ing the post-scission spreading of the fragments under the action of theCoulomb repulsive force. The repulsion leads to descent of the fragmentsfrom the axis, after which appearance of the transverse component of theCoulomb force forms the final value of the angles of divergence betweenthe fragments. The final angle between the light and ternary fragmentsremains at the level of two degrees. The kinetic energy of the ternary article turns out to be very low, around 4 – 7 MeV.In the trajectory simulations, I did not take into account the stronginteraction of the fragments, whereas in [19] the authors did. At firstglance, this may seem to be a shortcoming. However, at deeper insight,my present model is more realistic, than used in [19]. Paper [19] containsmany physical errors, among which I note the following.1) Unrealistic initial configuration of three touching spheres, withoutattention to the dynamics of fission. The distance between the side frag-ments R = 20 fm is rather typical for binary fission of a smaller actinidenucleus, such as U. In that case, one has to take into account that be-fore scission, a long neck is formed. After its rupture, the remnants of theneck snap back [37]. Thus, in the landscape of the potential energy [33],pronounced valleys favorable for ternary fission were found. One of them,which may be related with channel (8), lies after a saddle point at r =22 fm. The TKE value T = Q would be achieved if scission occurred onlyat r = 25.56 fm. In practice, part of the released energy is stored inthe deformation energy of the fragments, while scission occurs at a largerdistance.2) And only after this, the fragments can be considered as approximatelyspherical, being at a considerable distance from one another, where short-range strong interaction really does not play a noticeable role. Choiceof the initial configuration in the present work: R = 27 – 35 fm is in greement with the above picture and microscopic dynamical calculation[33].3) Moreover, one expects that the configuration [19] will rather lead tothe merging of the three spheres into one mother nucleus, than to theirfurther separation.4) Not to mention, that there are tens versions of the effective strongforces in the nuclei in the literature, and making use of each of them willyield in different results, including absurd. In the case of any realisticscenario of fission, this does not make any problem. But it bares problemsin the case of extreme scenario, like in [19]. For example, if more advancedpotential from [38] is used, the total energy of the configuration will exceedthe Q-value by 15 MeV.5) Even with the interaction described in Ref. [19], the total energyof the considered configuration exhausts the total Q value for the givenchannel, so that the total kinetic energy (known as TKE) of the fragmentswill correspond to the case of so-called cold fission, never seen in doubleor triple fission.6) Another absurdity arises when considering the initial velocity in [19].The authors in [19] do not write, how to combine availability of the trans-verse initial velocity v y on the middle fragment with the momentum con-servation, if the other velocities are zero: v x = v x = v x = v y = v y = 0.My model, presented in Section 3.3, harmonically satisfies the momentum onservation.7) The attention is drawn to a long meaningless chain of mathematicalformulas, the result of which is... the derivation of the Newton’s equations,starting from (3), (4) to (6), (7) in [19], in the framework of the Lagrangianformalism. I am forced to remind you that the Newton’s laws, being such,do not need to be justified.The first error seems to be the heaviest and fatal one. Even mistakeNo. 6, though related with violation of a fundamental law, can be under-stood as a consequence of incomplete thinking. But so much was writtene.g. about non-reversibility of fission with fusion channel, which arises be-cause of absence of the formations like remnants of the neck in the fusionchannel. The authors write about dynamics in their calculation. But thedynamics in fission occurs at the prescission stage. Cf. e. g. classicalworks by Swiatecki, Nix, Koonin, Sierk et al. [2, 3, 4] and others. Therein,calculation of the dynamics starts on the saddle, not at scission. The au-thors [19] also forgot about scission dynamics, model of which is proposedherein for the first time. Regretably, error No. 1 makes evidence of thegeneration gap.Dwelling on these and other shortcomings of [19] is not among mypresent purposes, although. The main difference is that the principal ques-tion: collinear or not collinear case will be realized, actually remains unan-swered in [19]. I want to show the deeper physical reason, which makes ollinear TTF most probable. It is remarkable that the final collinearityof the fragments gives a strong evidence of the ideal axial symmetry ofthe fissile system in its evolution up to scission. This collinearity would bealready broken by the very minimum shift of the middle fragment from thefission axis by . e tc., in none of these examples does the description achieve suchan accuracy, may be, ∼
10 percent at most. Therefore, TTF turns outto be a process where the merits and the underlying symmetries of thecollective model manifest themselves in full shine.
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