A New Radioactive Decay Mode, True Ternary Fission, the Decay of Heavy Nuclei Into Three Comparable Fragments
EEPJ manuscript No. (will be inserted by the editor)
A New Radioactive Decay Mode, True Ternary Fission, theDecay of Heavy Nuclei Into Three Comparable Fragments.
W. von Oertzen A. K. Nasirov Helmholtz-Zentrum Berlin, Hahn-Meitner Platz 1, 14109 Berlin, Germany, and Fachbereich Physik Freie Universitaet Berlin, Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980 Russia, Institute of Nuclear Physics, Ulugbek,Tashkent,100214, UzbekistanReceived: date / Revised version: date
Abstract.
The ternary cluster decay of heavy nuclei (e.g. in spontaneous fission of
Cf(sf,fff)), has beenobserved in several experiments with binary coincidences between two fragments using detector telescopes(the FOBOS-detectors) with very large solid angles and placed on the opposite sides from the source offissioning nuclei. The binary coincidences at a relative angle of 180 deg. correspond to binary fission or tothe decay into three cluster fragments by registration of two of them in coincidences with missing nuclei ofdifferent masses (e.g. Sn, − Ca, − Ni). This marks a new step in the physics of fission-phenomenaof heavy nuclei. The new decay mode has been observed more then 10 years ago by the FOBOS groupin JINR (Dubna) Russia. These experimental results for the collinear cluster tripartition (CCT), refer tothe decay into three clusters of comparable masses. In the present work we discuss the various aspectsof this ternary fission (FFF) mode, with different mass partitions.The question of collinearity is analysedon the basis of recent publications. Further insight into the possible decay modes is obtained by thediscussion of the path towards larger deformation, towards hyper-deformation and by inspecting details ofthe potential energy surfaces (PES). The PES is determined as the total sum of the masses, including theshell effects, which enter with Q ggg , the three-body Q-value for the separation into three fragments, andthe total interaction (nuclear and Coulomb) between all three nuclei. In the path towards the extremelydeformed states leading to ternary fission, the concept of deformed shells is most important. At the scissionconfiguration the phase space determined by the PES’s leads to the final mass distributions. The possibilityof formation of fragments of almost equal size ( Z i = 32, 34, 32, for Z =98) and the observation of severalother fission modes in the same system can be predicted by the PES. The PES’s show pronounced minimaand valleys, where the phase space for the decay reaches maximum values, namely for several mass/chargecombinations of ternary fragments, which correspond to a variety of collinear ternary fission (multi-modal)decays. The case of the decay of Cf(sf,fff) turns out to be unique due to the presence of deformed shellsin the total system and of closed shells in all three nuclei in the decay.
PACS.
Binary fission of nuclei after irradiation of Uranium withslow neutrons has been observed by Otto Hahn and FritzStrassman in 1938, [1]. The fragments of Barium (charge Z = 56) have been identified by methods of radio-chemistry.This group from Berlin, together with Lise Meitner, whohad to escape from Nazi-Germany in 1938, had previ-ously studied a variety of nuclear processes induced byneutrons. Actually L. Meitner urged Hahn repeatedly tostudy in more detail the reaction products observed fromthe irradiation of Uranium-foils with neutrons. Differentgroups, in particular Fermi (see Ref. [2]) had published re-sults, where new radio-activities have been observed andattributed to new elements heavier then Uranium. Verycontroversial results of other groups followed in the 1936- 38’s. Fermi actually received the Nobel-price for this workof finding new elements (incorrect!) in 1938.Lise Meitner had worked together with O. Hahn inBerlin for many years. L. Meitner as an Austrian citizen ofJewish origin, had to escape from Nazi-Germany, she wentto Stockholm, Sweden in the fall of 1938, when Hitler in-vaded Austria (“Anschluss”). Due to these circumstancesshe could not participate in the final phases of the workof her colleagues in Berlin by O. Hahn and F. Strass-man. They published their results without Lise Meitner, in Naturwissentschaften , p11, (1939). However, O. Hahnwas eager to tell L.Meitner about his final result, alreadyearlier and with an exchange of letters in the fall of 1938and in Jan. 1939 between Stockholm, Copenhagen andBerlin. Lise Meitner was able to produce a paper withher nephew Otto Frisch (at that time in Copenhagen) a r X i v : . [ nu c l - t h ] J u l W. von Oertzen: Tue ternary Fission which gave the first interpretation of the results of fis-sion: L. Meitner und O. Frisch, Ref. [3]: “
Disintegration ofUranium by Neutrons, a new type of Nuclear Reaction ”.Frisch actually introduced here the concept/expression of“fission” for the splitting of Uranium-nuclei. Meitner andFrisch were the first to realize the very large energy-releaseinvolved in fission: >
200 MeV, a very large (enormous!)value for the energy-release in a nuclear decay, at thattime such a high value has never been observed before.With this result an avalanche of publications started in1939. For their first publication in the Nature-journal, thedraft-paper had been presented by Frisch to Niels Bohr(one of the giants of atomic/nuclear physics at that time)in January 1939 in Copenhagen, before his departure tothe USA. Reading their draft, Bohr exclaimed “what foolswe have been”! In fact the result could have been obtainedmany years before, using the by that time well known for-mula for the mass(energy) of nuclei with the liquid-dropmodel.The liquid drop model for the calculation of the bind-ing energy of nuclei has been developed in the years 1933 −−
35. The total binding energy of nuclei is illustrated inFig. 1, as function of the binding energy per nucleon.From this figure we deduce that the values reach theirmaximum for nuclear masses of A = (60 −
50 years with nuclear fission , National Academy of Sci-ence, Washington D.C., USA, and National Institute ofStandards and Technology, Gaithersburg, Maryland USARef. [6]. Another conference has been held in Leningrad(Soviet Union) October 16-20, 1989: Proceedings ed. L.W.Drabchinski,
Fission of Nuclei, 50 Years , Chlopin RadiumInstitute, 1992, Vol. 1 and 2 [7], (dominantly in Russian). The study of binary fission over the last decades hasproduced a large body of experimental results with em-phasis on a detailed description of several parameters suchas the mass and charge distribution of fission fragmentsand their kinetic energies. For the heavier nuclei the fissiondecay may become the dominant decay channel. An im-portant step for the understanding of the fission processis the appearance of shell structures in heavy deformednuclei, as introduced by Strutinsky [8,9]. The shell effectsplay decisive role in the yield of fragments, which is vary-ing as function of charge and neutron number. The vari-ation of the mass distribution of fission fragments withthe given proton and neutron numbers has been foundto depend mainly on the spherical and deformed shells.These phenomena are described with a large variety oftheoretical approaches, many of these have been put for-ward during the last decades. In the recent book by H.Krappe and K. Pomorski [10] many of these establishedapproaches are described.For an earlier overview on scientific aspects of (binary)fission we suggest the book edited by C. Wagemans 1991:“The Nuclear Fission Process” [11], covering all impor-tant aspects of this process. The fact which dominates themass-distributions of the fission fragments are the shell ef-fects for the charges Z = 20, 28, 50, and for the neutronswith N = 20, 28, 50 and 82. Thus the mass distributionof binary fission is asymmetric as shown in Fig.2.In addition this figure (from Ref. [12]) shows the yieldfor light fragments with masses from A = 4 up to A =30, observed in coincidence with binary fission channels.These are typically emitted perpendicular to the fissionaxis, determined by the vectors of the two heavy frag-ments. These events usually have been defined as “ternaryfission” [12]. The “true ternary fission” of heavy nucleidiscussed in the present survey, has been predicted manytimes in theoretical works since the early 1950-60’ties, adecay with an increasing probability for increasing totalcharge of the nuclei. Fission involves a rearrangement of nucleons in a collec-tive macroscopic motion (evolution in shapes) towards anelongated, deformed structure with energies dominated bythe liquid drop aspects and the quantal properties of nu-clei, the shell effects. These aspects have been describedearlier, as an example we cite the work of Swiatecki [13]and of Diehl and Greiner [14]. In the first it has beenindicated that apart of binary decays, ternary (and mul-tiple) decays are energetically possible in heavy nuclei.The probability for the decay depends on the barriersfor the individual combinations of fragments and on thephase space in these decays. Quite important is the varia-tion of fission properties in dependence on the total massand charge of the fissioning nucleus and the appearanceof shells, particularly for protons. For the heavier nuclei . von Oertzen: Tue ternary Fission 3
Fig. 1.
The binding energies per nucleon ( E B /nucleon ) in nuclei as function of mass A. The maximum is reached for masseswith A = 60-110. Ternary fission of heavy nuclei into three fragments with these masses is possible. The liquid drop modeldescribes the variation of the total binding energy as shown in the figure. Fig. 2.
Mass distribution of binary fission fragments from the spontaneous decay of
Cf(sf), as function of fragment mass.The shell model explains the two hump distribution with enhanced yields for masses around A = 80 – 110 and A = 120–140.The left side of the figure shows the yields of lighter third fragments in coincidence with two binary fission fragments, for twosystems as indicated, from Ref. [12]. the fission decay may become the dominant decay chan-nel. As already mentioned an important step for the un-derstanding of the fission process is the appearance ofshell structures in heavy deformed nuclei , as introducedby Strutinsky [8,9]. This very unique approach has beenessential for nuclear structure studies by describing the super - and hyper-deformed shapes. Further super- andhyper-deformation played an important role in the morerecent studies in γ -spectroscopy of high spin states [15].The hyper-deformed configurations at low spin show im- portant manifestations with the occurrence of fission iso-mers [16,17,18], which have been studied extensively 40years ago [19,20,21]. Thus it appears that the ternary de-cay passes through hyper-deformed shapes of the fission-ing nucleus (see Fig. 3).In the following we list the factors governing the prob-abilities (the phase space ) for these decays:i) the details of the potential energy surface, the PES,ii) its valleys and hills,iii) the internal barriers at the two necks, W. von Oertzen: Tue ternary Fission
Fig. 3.
The potential energy as function of deformationfor a hyper-deformed nuclear shape, which shows the dou-ble humped fission barrier suggesting a path to the collinearternary fission decay in
U, adapted from Ref.[21].
Fig. 4.
Maximum values of the kinetic energies of fragments A and A in the model with sequential decay (as shown inFig. 19) leading to ternary decays, (from Ref. [22]). A similarresult has been obtained in Ref. [23]. iv) the Q ggg -values, the latter determining the kinetic en-ergies and the number of possible fragment (isotope) com-binations,v) the excitation energy range in the individual fragments,vi) the momentum range for these,vii) the number of excited states (or the density of states)in each of the fragments, the combinations consisting of 3isotopes, and byviii) the spin ( J )-multiplicity (phase space factor (2 J +1))in these excited states, with spins expected up to valuesof (6-8) + .More general considerations for a three-body decayhave been analysed in Ref. [25]. From this work we canstate that the radioactive ternary fission decay, being al-lowed from Q -value considerations, will proceed sequen- Fig. 5.
Comparison of the total potential energies, dominatedby the Coulomb interaction, of three fragments in an oblateconfiguration compared to a prolate (collinear) shape for threecluster decay, courtesy of Royer [24]. tially, because the two barriers and the phase space favourthe individual steps, the first steps are determined by thelower barriers and their phase space. The first step usuallywill start with the higher probability and proceed with thelighter mass combination of fragments with a lower barrierand the higher energy balance, which has the larger pre-ferred phase space. For this case of sequential decay we cancalculate the kinetic energies of the ternary fission decayof
Cf(sf,fff) with the decay mode into the three frag-ments: Ni + Ca +
Sn, which are most likely. Theresult is shown in Fig.4, and discussed further in sect.2.3,and illustrated in Fig. 19. Actually in this simple modelthe kinetic energies of the outer fragments are high, thefigure shows obtained maximum values, they are too high(as indicated in Ref.[26]) because no excitation energy ofthe fragments is considered.An important approach is to consider the connectionbetween ternary fission and extreme deformations, namelyhyper-deformation, illustrated in Fig. 3. In this contextthe work of Brosa et al.[27] is of importance, in this workdifferent fission paths are obtained, there extremely long(“super long”) shapes with very large deformations havebeen predicted. Fission must be considered as a macro-scopic rearrangement of all nucleons into a hyper-deformedshape with two necks. There will be two neck ruptures ina very short time sequence.The term “ternary fission” has been used for binary fis-sion accompanied by light particle (like α -particles) emis-sion. For such decays with a third light particle emittedperpendicular to the binary fission axis, compilations areavailable in Ref. [12]. These must be considered to be emit-ted from an prolate-binary configuration with a neck, thecorresponding yields decrease strongly (see Fig. 2) as func-tion of increasing mass(charge) of the third particle, fromRef. [12]. For the general survey of the possible ternarydecays (FFF), and the discussion of the relative probabil-ities of oblate to prolate ternary fission, we must look intothe Coulomb energy of the total system. In Fig. 5 we show . von Oertzen: Tue ternary Fission 5 Fig. 6.
The total potential energy of three fragments in acollinear configuration, it is smallest with the smaller fragmentat the center. The collinear configuration dominates the threecluster decay.
24 26 28 30 32 34 36 38 40 42242628303234363840 Z Z -275.0-273.0-271.0-269.0-267.0-265.0-263.0-261.0-259.0-257.0-255.0-253.0-251.0-249.0-247.0-245.0-243.0-241.0 A G e A S e Q ggg
64 68 72 76 80 84 88 92 96 100 104 646872768084889296100104 A Fig. 7.
The Q ggg -values (-2n) for Cf, the ternary fragmentswith Z i = 32 −
34, correspond to the highest Q-values. the mostsymmetric decays, however, with the highest internal barriers. the comparison, from Ref. [24], of the PES of an oblateversus a prolate arrangement of three equal (comparable)sized fragments, which can be related to the ternary decayof
Cf(fff).Clearly the linear arrangement will be preferred. Fur-ther, the potential energies, actually the PES as a functionof mass asymmetry, contain the most relevant quantitiesdetermining the phase space of the decays (discussed indetail later). The dominance of the Coulomb interactionpoints to distinct geometrical arrangements of the threeclusters as shown in Fig. 5 and Fig. 6. The latter impliesthat the collinear arrangements with the smallest frag-ment in the center will have the largest phase space andtherefore the highest probability.The almost symmetric combination of three fragmentsgives maximal Q ggg values, as indicated in Fig. 7, however,as shown later, this mode has a very low yield, because thecorresponding potential energy barriers are high, and theprobability of its population is small, because the highestinternal barriers occur, making the decay difficult fromthese configurations. Still the symmetric decays into threecomparable fragments with the largest Q -value, can beextracted from the data (see sect. 4).We have already introduced the concept of hyper -deformation, which must be considered in the dynamicalpath towards a three fragment channel. In this the com- Fig. 8.
The macroscopic geometry and coordinates chosen forthe ternary fission study of Karpov in the three center-shellmodel. The coordinates are defined appropriate to the ternarydecay, from Ref. [28]. The formation of shapes reminiscent ofsuper- and hyper-deformation in nuclei is observed. The radiusof the spherical nucleus (CN) is denoted as R . bination of macro- and micros-copic aspects of the po-tential energies as function of smoothly varying nuclearshapes become essential for the understanding of super-and hyper-deformation. Related to this quite importantand convincing results concerning the physical circum-stances for ternary fission have been obtained in the workof Karpov [28]. In this work the asymmetric three-centershell model is used, which is related to the two-center shellmodel, originally developed in 1972 by Greiner et.al. bythe Frankfurt group [29]. We will see that the structureof this PES is dominated by the shell effects in all threefragments like in Sn, Ca and Ni see Figs. 9, 10 andFig. 11. In this work appropriate shapes and coordinatesare introduced in order to describe the ternary decay, theyare shown in Fig.8.Macroscopic potential energy (dashed line), shell cor-rection (dotted line), and total macro-microscopical po-tential energy (solid line) of the
Cf nucleus correspond-ing to the
Sn+ Ca+ Ni ternary splitting.In these calculations the favored fission path (dashedline in Fig. 11 passes through two barriers. The first bar-rier is higher the second barrier appears at rather largeelongations(deformations). In the discussion of the results,the author analyses the macroscopic and microscopic fea-tures of the PES. The barriers are shown in Fig. 9. Thesecond neck appears in the macroscopic calculation, witha very high second barrier (see Fig. 9), ternary fissionis not possible with the liquid drop shapes. The inclu-sion of the shell effects in all three fragments producesthe important effects: the fission barrier becomes double-humped. The hyper-deformation appears (see Fig. 3), afeature giving rise to the fission-isomers in the actinides.The ternary fission barrier is reduced dramatically, thebarriers exist in the elongation parameter (deformation)and in the mass asymmetry. Actually the binary fission issuppressed and the probability for the ternary decay in-creases. Apart from the case discussed, other channels will
W. von Oertzen: Tue ternary Fission
Fig. 9.
The macroscopic (dashed line) potential energy of thethree-center system as function of the reduced elongation. Theshell correction ( δE , red dotted line), and the full macro-micro-scopic potential energy of Cf with a shape corresponding tothe splitting into (
Sn, Ca and Ni), from Ref. [28]. Thethird barrier (in the mac-mic approach) is strongly reduced.
Fig. 10.
The potential energy surface for
Cf as function ofthe elongation of the system, it shows the binary fission path,this figure is without the shell effects, see the next figure. Re-sults obtained with the three-center shell model. The dominantdecay is always with
Sn. The approach also secribes (dashedline) the binary fission path, from Ref.[28]. be favored like (“tin-sulfur-germanium”) and the combi-nation (“nickel-oxygen-samarium”). The ternary decay of
Cf observed in the experiments is due to the uniqueconfigurations with deformed shells and the shells in thefragments.
With the evolution of the nuclear shape towards the ternarymass split through a hyper-deformed shape with the for-
Fig. 11.
As in figure 10 the potential energy surface for ternarymass split of
Cf as function of the elongation of the systemand the mass of the third (central) fragment, from Ref.[28].This figure compared to the previous figure shows the influ-ence of the shell effects, which distinctly change the landscape.Results obtained with the three-center shell model, with theshell effects included. The dominant decay is with
Sn asthe heaviest fragment with the smallest “Ca”-fragment at thecenter. The case with a closed-shell Ni-isotope is shown. Thedashed line shows the ternary fission path with two barriers. mation of fragments, the further decay (from the scissionpoint) is governed by the phase space. The fission pro-cess can proceed towards several mass partitions, theirindividual phase space will be different according to thepeculiarities.With the PES’s we obtain an overview of the differ-ent decay channels. The PES’s, which are discussed inRef. [30], are obtained by calculating the interaction be-tween all fragments: U ( R , R , Z , Z , A , A ) = Q ggg ++ V ( Coul )12 ( Z , Z , R + R )+ V ( R , Z , Z , A , A )+ V ( R , Z , Z , A , A ) , (1)where Q ggg = B + B + B − B CN is the balance ofthe fragments binding energies in the ternary fission. Thevalues of the binding energies are obtained from the masstables in Ref. [31]; V and V are the nucleus-nucleusinteraction of the middle cluster “3” ( A and Z ) withthe other two, their mass and charge numbers, with theleft “1” ( A and Z ) and right “2” ( A and Z ) fragmentsof the ternary system; V (Coul)12 is the Coulomb interactionbetween the two border fragments “1” and “2”, which areseparated by the distance R + R , where R and R are the distances between the middle cluster and two outerclusters placed on the left and right sides, respectively. Theinteraction potentials V and V consist of the Coulomband nuclear parts: V i ( R i , Z i , Z , A i , A ) = V (Coul)3 i ( Z i , Z , R i )+ V (Nucl)3 i ( Z i , A i , Z , A , R i ) , where i = 1 , . (2) . von Oertzen: Tue ternary Fission 7 A A Z Z -17.00-15.04-13.09-11.13-9.174-7.217-5.261-3.304-1.3480.60872.5654.5226.4788.43510.3912.3514.3016.2618.2220.1722.1324.0926.0427.00 Z = A T e CCT Z = A S n FFF
16 24 32 40 48 56 64 72 80 88 96 104 112 1201624324048566472808896
Fig. 12.
The contour plot of the potential energy surface(PES) it gives an overview of the different modes for thecollinear ternary decays characterized by two values of Z and Z of the ternary fragments for Cf. The pre-scission config-urations of the ternary system appear with the isotopes of tin, − Sn, Z =50, and Tellurium − Te, Z =52. The up-per rectangle shows the area of the expected mass (charge) dis-tribution of the symmetric ternary FFF-decay products withalmost equal fragment masses A ≈ A and A ≈ A . Thelower rectangle Z =16–20 corresponds to the expected regionfor ternary decays of the two products with the charge num-bers Z =28–34 and mass numbers A =72–88 of the ternaryconfigurations leading to the collinear cluster decays (CCT)observed in Refs.[32,33]. The nuclear interaction is calculated by the double fold-ing procedure with the effective nucleon-nucleon forces de-pending on the nucleon density distribution (see Ref. [30]).The Coulomb interaction is determined by the Wong for-mula [34], which also allows us to take into account thedeformed shape of fragments and the possibility to con-sider interactions under different angles of their axial sym-metry axis.Potential energy surfaceWe inspect the PES’s for the two cases of
Cf(sf) and
U(n,f) in two figures, Fig. 12 and Fig. 13, respectively.We note the “lower”(blue) regions connected to the com-binations with the Sn-fragments. Apart from the alreadyknown CCT and FFF - decays we observe a pronouncedregion with a minimum for A , with Z = 18, this mustbe combined as shown in the PES, with two fragmentswith Z = 40 (Zr), or the adjacent value of Z = 42, com-bined with Z = 38(Sr). The particular structure of theisotopes with Z = 18, have been observed in Refs. [35,36],they correspond to the new “shells” observed in neutron-rich isotopes, as in − Ar (but also in Ne). The samedepletion in the PES’s is observed in the PES for the caseof Uranium fission as shown in Fig. 13. These effects onthe masses, which are seen here, are already contained inthe recent mass table in Ref. [31]. The decays correspond-ing to these structures are actually contained in the data
CCT FFF Z = A S n U (0n) Z Z -25.00-22.00-19.00-16.00-13.00-10.00-7.000-4.000-1.0002.0005.0008.00011.0014.0017.0020.0023.0026.0029.0032.0033.00 Z Z Z A Fig. 13.
The potential energy surface for the ternary decaysof
U(n,f). The general features are similar to the Z − Z pattern for Cf, with a valley dominated by Sn-fragments, Z = 50. However, we observe the absence of pronounced effectsdue Z = 20, 28 (as in the case of Cf), because for thecase of U these fragments are absent. Similarly a region witha lower (green) valley for symmetric decays with comparablethree fragments with Z = 32+28+32 can be favoured (totalcharge of U = 92). Again the ternary decays for Z = 18 (asfor the case of Cf) can be favoured, where a proton shellin the fragment makes favorable Q -values. These decays arediscussed as multi-modal fissions later. and can be extracted as discussed in the chapter on mul-timodal fission (in sect. 4.3).With these figures we illustrate the role of the PES inthe formation of the ternary fragments, they point to thedifferences of the two cases. Clearly the charges have thestrongest influence on the phase space in the decays. Theeffect of the closed shells for protons ( Z , , = 20, 28 and50) is clearly visible with the low (deep blue) valleys inthe PES’s. The case of the almost symmetric decays withcharges of Z , , = 32, 34, 32 is seen as an depletion andpoints to the possible symmetric decays, these have beendiscussed in Refs. [37,38], and are later illustrated in thechapter on multimodal fission in the present survey.Quite remarkable is the structure of the PES shown inFig. 14 for the ternary decay of the super-heavy nucleus Fl, a neutron rich isotope of the recently (Ref. [39]) ob-served element with Z =114. However, with a much largerneutron-number, then observed. We observe the change ofthe favoured decays with the change of the total charge,and connected to this the influence of the variation of theclosed shells in the three fragments.It is interesting to analyse the situation for the ternaryfission for even heavier systems. In heavy systems shell ef-fects in the fragments are not dominant any more, thedecay is determined by the liquid drop energies. However,in the fragments the shell with the Z = 50 remain to bepresent. Further consequences concerning ternary decayscan be deduced from the study of the PES’s of very heavysystems like in Refs.[40,41]. In the first case the maxi-mal heavy system is studied, which will have no barrier W. von Oertzen: Tue ternary Fission
40 50 60 70 80 90 100 110 010203040506070010203040506070 (50,50,14) Z Z Z X a) Case - I (A +A +A ) V (MeV) (50,32,32) (78,18,18) (63,50,1) Fig. 14.
The potential energy surface for the spontaneousfission decay of a superheavy nucleus
Fl, which has beencalculated by Balasubramanian et al. Ref.[40]. With the in-crease of the total charge one region dominates, there are noclosed shells (except for Z = 50) in the fragments.
Fig. 15.
The potential energy surface for the three-body con-figurations formed in a collision of two U-nuclei:
U +
U,which has been calculated by Zagrebaev et al. Ref.[41]. Withthe increase of the tree-body clusterizations become more prob-able, the shell effect of Z =82 is still visible. for binary fission. In the reaction path of very heavy sys-tems, like for U +
U in Ref. [41], only a few exitchannels appear. A binary channel with No and aPb-isotope could be observed. Further decay channels areshown in Fig. 15, where the tree-body clusterizations withthe formation of neutron rich O-isotopes and some heavyclusters ( − Pu +
Pb + − O) are shown. Look-ing into these cases it becomes evident, that in the stud-ies of ternary fission the case of Cf-isotopes (and possiblyother heavy nuclei/isotopes in the vicinity), which we willdiscuss, is unique. Many aspects of nuclear structure con-nected to shells in the fragments and to deformed shells inthe total system are present in this case. More cases arediscussed by Balasubramanian et al. in Ref. [40], where theternary decay of heavy nuclei and for a variety of super-heavy elements have been studied. In this case shells inthe neutron number of the super-heavy elements can beimportant for the ternary decay. In very heavy systems, i.e. in reactions of very heavyions leading to binary and ternary reaction channels, colli-near decays have been reported. The dominance of theCoulomb interaction leads to aligned (collinear) ternaryfragmentations in higher energy collisions of heavy ions,in particular in deep inelastic collisions in the selection ofcentral impact parameters as described for
Au +
Aucollisions in Ref. [42].
As already mentioned a three-body decay can be con-sidered to proceed sequentially, each step being deter-mined by the phase space and Q -values of each step. Fromconsiderations of the PES’s the collinear decays are pre-dicted, actually the experimental results have been inter-preted as collinear cluster tripartion CCT). This claimfrom the experimental observations has been contestedrepeatedly. Most theoretical arguments are based on thestrong deep valleys in the potential energy surfaces, whichgive a favoured phase space for the collinear decays. Thecollinear decays are actually suggested by the hyper-defor-med shapes observed in heavy nuclei. This has been stud-ied experimentally in Ref. [21], for the case of U andshown in Fig. 3. These observations favor the collineargeometry relative to “oblate” shapes. In the collinear tri-partition of the hyper-deformed nuclear state one of thedeformed fragments in the first binary fission starts thesequential decay defining the axial symmetry axis, the fol-lowing fission process preserves this axis, thus both fissionaxes are parallel.These strongly deformed, hyper-deformed, configura-tions are favored because of the binding energy includingshell effects already mentioned, which depend non-linearlyon the degree of elongation (shape deformation). This phe-nomenon has been introduced by Strutinsky [8] and stud-ied experimentally extensively 40 years ago and describedin Refs. [18,19,20]. These deformed shells are quite pro-nounced at larger values of the quadrupole deformation.The hyper-deformed shapes (axis ratios of 3:1), appearin most of the heavier nuclei due to the deformed shelleffects in the total system, and give rise to the observa-tion of fission isomers [16,17,18]. For spontaneous fissionand fission induced by gamma-rays, neutrons or chargedparticles, the nucleus on its way to fission undergoes ex-treme deformations with a stretching due the Coulombforces and inevitably passes through super-deformed andhyper-deformed shapes before splitting. The shell effectsin nuclei will delay fission due to the structure with oneand two barriers in the collective potential.In the recent theoretical analysis of ternary fission byKarpov [28] the shell effect discussed previously by Struti-nsky [8], suggests that deformed shells are most importantfor the ternary decays, as well for the binary fissions. Inthis work [28], based on the three-center shell model, theshell structures in all fragments give rise to a complexdeformation of the total fissioning system, which is equiv-alent to hyper-deformation. The potential energies for the . von Oertzen: Tue ternary Fission 9 x (fm) y ( f m ) R =22 fm Fig. 16.
The potential energy surface V ( R , x , y ) as a func-tion of the position x of the middle fragment “3” (Ca) alongaxis R connecting the centres of the outer fragments “1” and“2” of the tri-nuclear system and y , which is perpendicularto R at the value of R = 22 fm. The minimum value ofthe potential well for Ca is V min ( R =22 fm, x =9.4 fm, y =0fm)=239.75 MeV. fission path have been shown in Figs. 10 and 11. Withthis work the most complete description for the dynamicsof the decay, particularly related to the barriers has beenobtained. There still remains the final dynamics of theternary decay, which will be governed by the phase space,the latter being determined by the potential energy sur-faces PES’S (generally obtained with an approach withthree deformed clusters).The ground state shape of the actinide nuclei is prolate[43], the shape is the result of the balance between the re-pulsive Coulomb and attractive surface tension forces. Theprolate shape in the ground state is favorable energeticallyfor the three massive fragments of the collinear ternary fis-sion. Authors of Ref. [44] found that the potential energyof the collinear shape is significantly lower than the onefor the oblate shape and the relative yield of the ternaryfission products is much larger for the prolate case thanfor the latter shape.Actually with the formation of the necks due to de-formed shells, quantum mechanical fluctuations are ex-pected, an almost simultaneous (a sequential within a verysmall time delay) break up of the two necks is expectedfor the ternary scission as described in Refs. [45,46]). Inthese calculations also the more asymmetric binary fissionis predicted and finally a three cluster configuration dueto the shell structures for protons and neutrons appears inthe exit channel. Actually the shell effects give for both,binary and ternary decays a dominance for the decay withfission fragments with mass A = 132 ( Z = 50, N = 82).In the work of Tashkhodjaev presented in Refs. [45,46]an explicit quantum mechanical calculation of the motionof the fragments towards scission and separation has beendone. According to results of this calculations in Ref. [47],the nuclear interaction between fragments plays a decisiverole in the pre-scission geometry of the tri-nuclear sys-tem. The collinear configuration of the tri-nuclear system R =23 fm y ( f m ) x (fm) Fig. 17.
The same as in Fig. 18 but for the larger distance R = 23 fm. The minimum value of the potential well for Ca isnow V min ( R = 23 fm, x =9.0 fm, y =2.2 fm)= 236.83 MeV. -100-80-60-40-200204060 0 5 10 15 20 25 30 35 40 45 50 55 60-2,0-1,6-1,2-0,8-0,40,00,40,81,2 R i x ( f m ) R R R i x ( c m / n s ) t (10 -22 s) Fig. 18.
The x component of the coordinates (upper part) andthe velocities (lower part) in the decay into three fragments asfunctions of time, from Ref. [47]. (TNS) is preferable for the values of the distance betweenouter nuclei “1” and “2” being in region of R = 21 − R >
22 fm the over-laps of the nucleon densities of the TNS nuclei decreaseand, as a result of the decrease in the nuclear attractionand the Coulomb repulsion the depth of the potential welldecreases. In that way the condition for the separation ofthe middle fragment Ca from Ni can arise. As a resultthe potential well for Ca-fragments moves away (perpen-dicular) from the x -axis at values of R =23 fm and itsminimum value decreases: V min ( R = 23 fm, x =9.0 fm, y =2.2 fm)=236.83 MeV (see Fig. 17). Fig. 19.
Illustration of the model of a sequential decay forthe collinear ternary fission proceeding in two steps. In the firststep the combination A + fragment A are formed, followed(within a very short time sequence, in 10 − seconds) by thefission of A into A and A . The local minimum at point R x = 9 . R y = 0fm (see Fig. 17) moves to the left (to the side of the Ninucleus), and starting from R = 23 fm this minimumpoint is transferred to a saddle point. In Ref. [46] the au-thors have considered the case when initially all nuclei areplaced in one line, which means that R y ( t = 0) = R y ( t =0) = R y ( t = 0) = 0, since the energy of the collinearconfiguration in the pre-scission state is the smallest, andthe values of the x -coordinates of these nuclei (with thesmaller relative distance between nuclei) correspond to thelocal minimum in the Fig. 17. Both components ( x and y ) of the initial velocities of the three nuclei are zero. Inother words the formation of fragments of the TNS is soslow that the fragments have zero (or very small) veloci-ties. Nevertheless, the assumption that all initial velocitiesare zero, means that there is no net force, which acts onthe nuclei in the equilibrium state. Results of the calcu-lations of the equations of motion with the above men-tioned initial conditions are shown in the Fig. 18. Thereit is shown that from the beginning, the Sn nucleus isgoing to breakup into the Ni+Ca system, and then at atime of t ≈ . × − s the Ni+Ca system has de-cayed. Moreover, an important result has been obtained,namely that the central third nucleus (Ca) has almost notchanged it’s coordinate, because it’s velocity is about zero.This means, that the detection of the middle nucleus (Ca)is almost impossible in an experiment. All together theseconditions lead to collinear fission of the tri-nuclear sys-tem. In fact in a similar recent study by Vijayaraghavan et al . [48] the trajectories of fragments in a ternary decayhave been calculated explicitly for equatorial and collineartrajectories. For lighter fragments an emission at relativeangles of 90 ◦ is obtained. For third fragments of interme-diate mass for an initial configuration with the fragmenton the axis results in a collinear decay.As a further consideration we look into the kinetic en-ergies of the three fragments with a model of a sequentialbreak up of the fissioning nucleus (illustrated in Fig. 19).The collinear decay mode is quite important in theanalysis of features of the fission fragments, in particulartheir kinetic energies. Several authors have addressed thispoint [22,49]. In Ref. [49] an overall study of the proba-bilities of ternary decays is given, comparing oblate andprolate fission. Ternary decays become dominant for cases where the central mass becomes larger than A = 30–40.Considering the potential energy as function of the sizeof the central (smaller) fragment, A , it has been shownthat for larger central fragments, the ternary mass splitsdominate the decays of heavy nuclei by orders of magni-tude. Further the dependence of the potential energy forternary mass splits on the geometry of the ternary sys-tem has been analysed by Vijayaraghavan et al [22] fordifferent orientation angles between the three fragmentsas shown in Fig. 20.In order to put the collinear geometry into the per-spective of the potential energy of the system, we canconsider the potential between three clusters in depen-dence on an angle between the axial symmetry axis of atwo-fragment subsystem and the third fragment, definingdifferent shapes of the decaying system (Fig. 20.) In thiswork of Vijyaraghavan [22] the potential energies of threefragments in different configurations have been calculated.In Fig.21 we show the potential energies of the fragmentsfor the three-cluster system, where the collinear and oblatearrangements are defined by the angles shown in Fig. 20,by varying the orientation angle for the three clusters. Wenote the lowest potential energy for the arrangement withCa as the central fragment.The potential energies are clearly lowest for the collineargeometry. Around the lower region of the potential en-ergy quantum mechanical fluctuations around the mini-mum must be expected. This is an effect of the uncer-tainty principle discussed in Ref. [52]. This has also beenconsidered in Ref. [23], where the question of the angulardistribution of the ternary fragments is addressed. Theconsiderations of the macroscopic features using clustersin the ternary nuclear system can, however, only give ageneral view, which describes mainly the properties of theasymptotic phase space of three fragments reached by theternary decay.For the discussion of the experimental results the ki-netic energies of the fragments are of importance, theyhave been calculated in Refs. [23,50]. For this study thecollinear cluster tripartition is defined to proceed in twosteps, in the first step an intermediate fragment definedas A is formed with some excitation energy, as shown inFig. 19, in the second step the fragments A and A areformed in a binary decay. The fission barrier of the firststep is small in comparison with the barrier of the sec-ond step, which is considerably higher then the first one.Note the fission barriers defined here are obtained withthe depth of the potential well in the landscape of thepotential energy surface, i.e. the value of the barrier rela-tive the bottom of the minimum. The absolute value of thebarrier is the Coulomb barrier, which determines the totalkinetic energy of the fission products. These kinetic ener-gies have been shown in Fig. 4, as function of the mass ofthe central fragment, A and for different excitation en-ergies of the intermediate fragment A . A considerablevalue for the excitation energy of A appears for a largefission probability in the second step. Actually a value ofat least 30 MeV must be assumed, in this case the kineticenergy of the central fragment A is close to zero, similar . von Oertzen: Tue ternary Fission 11 Fig. 20.
The coordinates of three clusters defined with anorientation angle of fragment A relative to the axis betweenthe two other fragments, from Ref. [22]. The correspondingpotential energies are shown in Fig.21. to the other approaches discussed in Ref. [23,47] and inRef. [28]. The kinetic energy E kin A of A is maximumand reaches values of 150 MeV. This is actually an arte-fact of the model as pointed out in Ref.[26], because nointrinsic excitation of the three fragments are taken intoaccount. In this reference the energy correlations betweenthe two outer fragments ( A and A , obtained with theexperimental setup COMETA) show that the kinetic ener-gies show several groups with much lower kinetic energies.Most important for the experimental circumstances is thefact that the central fragment usually is lost due to it’slow kinetic energy by being absorbed in the target and/orin the target-backing.The same result for the kinetic energies has been ob-tained in the work of Holmwall et al. [23] in explicit calcu-lations of the trajectories. Their approach to calculate thekinetic energies is based on the “Almost sequential decay”,as used in the approach of Ref. [47]. In this work also thedependence on various parameters of the decay are shown,namely the dependence on the possible neutron emission(neutron-multiplicities), the effect of lateral momenta andthe variations in initial geometrical positions of the cen-tral clusters. For deviations from the central position, itis concluded that they destroy the collinearity in the finalchannel. Actually a detailed analysis of the kinetic ener-gies of the fragments and the dynamical constraints forthe decay as done in Ref. [23] within a model with exist-ing clusters (as in many other approaches) gives results,which prevent the ternary decay, in contradiction to theexperimental observations. The authors of Ref. [23] didnot include the nuclear interaction in the calculation ofthe nucleus-nucleus potential. As a result the kinetic en-ergy of the outgoing fragments had been overestimated. Asa conclusion we can state that these approaches based onpreformed clusters without deformation of the fragments,fail to describe the experimental results.More relevant are the concepts with a continuous de-formation path like on hyper-deformation, discussed be-fore and in the approach of Karpov[28], where the totalsystem with three nuclear centers is used all along the fis-sion path. In the latter approach also two barriers exist,which are strongly influenced by the shells, in particularthe second barrier (which is higher) is very strongly re-duced by the shell effects in the fragments.Initially ternary fission has been defined by the obser-vation of a third light particle emitted perpendicular to . .
28 40 .
132 50
Sn+
116 46
Pd+ He .
132 50
Sn+ He+
116 46 Pd V ( M e V )
132 50
Sn+
106 42
Mo+
14 6 C . .
132 50
Sn+
14 6 C+
106 42 Mo
132 50
Sn+
Ni+ Ca Angle (degree)
132 50
Sn+
Ca+ Ni Fig. 21.
The potential energies of three clusters defined fordifferent orientation angles for the three clusters for the com-bination A + A as defined in Fig.20. Three cases for thecentral particle are considered, with He, C and Ca. the fission axis in coincidence with the binary fission frag-ments. The most important feature, which appeared withthe study of true ternary fission is the fact that the decayinto three comparable mass fragments becomes collinear.In the experiment by Schall et al. [51], the question ofthe decay from oblate deformed shapes is addressed, typ-ically a decay with three similar vectors was expected.The experiment has been designed to detect ternary fis-sion events with the emission of three heavier fragmentsat relative angles of 120 o . The experiment covered a largesolid angle using several ionisation chambers (detectors)each covering a large angular range of (90 o ) degrees, de-signed to observe over a large solid angle the t riangularshape of the decay-vectors. This experiment gave a nega-tive result, with an upper limit of the probability for thisdecay of 1 . · − /(binary fission). The failure in observa-tion of true ternary fission in this experiment is explainedby the very small probability of the population of the con-figuration which is presented in Fig. 20(c). The potentialenergy barrier of the last configuration is very high and itspopulation needs higher excitation energy for the fissionto occur. This argument is illustrated in Fig. 21 wherethe potential energies of three clusters are presented fordifferent orientation angles between lines connecting thecentres of mass of the three clusters. Fig. 22.
Very mass asymmetric fission yields of the
Cm(n,f)reaction. (adopted from Ref. [55]). The fission yield aroundmass A = 70 ( Z = 28) is enhanced. In fission the interplay of the macroscopic (liquid drop)and microscopic (nucleon orbits) effects are most impor-tant [8,9,53,54]. A triple humped barrier is observed inseveral cases, giving rise to important phenomena, like fis-sion isomers, multi-modal binary fission and creating theway to ternary fission as illustrated in Fig.3. Binary fis-sion has an asymmetric distribution of the masses with theisotope
Sn appearing in almost all fission decays [11].In the very asymmetric binary fission lighter fragmentswith masses A = 60–80 can be observed. In these cases“cold” fragments may appear and and deformed shell ef-fects are important. In several works [55] experimentalresults on very asymmetric binary fission have been ob-tained, these are single arm measurements. An exampleof such experimental results is shown in Fig. 22. In thesecases the yields of masses around Ni are enhanced [55],in particular for the higher kinetic energies of the frag-ments. These yields in the region of the lighter fragmentsof A = 70, can be due to special shell effects, in particularthe deformed shells around Ni ( Z = 28, N = 52), or dueto ternary fission.. U(n,fff) and
Cf(sf)
Inspecting the recent experimental observations and thenumerous theoretical predictions [44,56,57] we can statethat in the heavy systems and for ternary fragments withlarger charge, ternary collinear decay from a prolate con-figuration is preferred. We refer to the ternary decays inthe present work as to “true ternary fission”, see alsoe.g. Zagrebaev et al. in Refs. [41,57]. A more recent sur-vey of clustering effects in binary and ternary fission canbe found in the articles by G. Adamian, N. Antonenkoand W. Scheid [58]. In the work by D. Poenaru and W.Greiner [59] it is shown, that in heavy nuclei collinear ternary decays are observed with increasing probabilitiesfor increasing charge of the total system. For even heaviersystems quarterny fission must be considered.Concerning the experimental evidence for true ternaryfission several experiments have been performed at theFLNR in Dubna, to list most of them:a) with two complete detector telescopes (called FOBOS[32,60]) with 180 degrees relative angle for binary coin-cidences in spontaneous fission of
Cf(sf) and with thesame experimental deviceb) neutron induced fission
U(n,fff), performed at thereactor in Dubna.c) Experiments of binary fission for
Cf(sf) in coinci-dence with neutrons with a system called “MiniFobos”,containing smaller detectors of similar structure as withFOBOS, Ref. [33].d) Binary coincidences for
Cf(sf), using PIN-diodes (forenergy and timing signals) called COMETA.The most complete data set has been obtained withthe system described in Refs. [32,33,60]. Quite impor-tantly, supporting the claims made in these works, themulti-modal ternary fission decays predicted from the cal-culations of the PES’s, have been extracted from the com-plete data in a later stage, see Refs. [37,38].
For the observation of the ternary decays with the missingmass method the decay vectors of two fragments, whichare identified with all parameters (mass, energy and an-gle) in a measurement of the two fragments in coincidenceis needed. In the rather common single arm experimentsone fragment is identified with high precision (mass, en-ergy and angle) with the partner being known under theassumption of a binary decay. However, in the recent ex-periments, coincidences between two fully identified frag-ments have been measured with two FOBOS-detector-telescopes [32,60] placed at 180 o (see the illustration inFig.23). The FOBOS detectors allowed the measurementof time of flight, energy, energy-loss and angle with highprecision with a very large solid angle .The details of the detector structure are shown in ascheme in Fig.24. In Ref. [32] the experimental evidencefor the missing mass effect had been explained. Due to theenergy loss and momentum dispersion in the support foilsfor the low pressure parallel plate counters in the frontof the Bragg-Ionisation chambers a particular scheme hasbeen considered. In this scheme the third fragment afterdispersion in the foils is blocked by the support structurefor the foils of the ionisation chambers. Actually in therecent calculations, cited in this work, it appears that thecentral (third) fragment, will have very low kinetic ener-gies, thus it most probably gets stuck in the material ofthe support of the target or already in the target itself.Thus the missing mass effect has also been observed withdetector systems consisting of solid state detectors (PIN-Diodes, COMETA-set-up), with a measurement of massesby the time of flight (TOF) and energy parameters. . von Oertzen: Tue ternary Fission 13 Fig. 23.
The two FOBOS detectors placed at a relative angle of 180 o degrees for the measurement of coincidences between twofragments. Two of the leading experimentalists, D. Kamanin and Y. Pyatkov (in the center) are depicted. Fig. 24.
Scheme of (Ex1) for coincidence measurements oftwo fragments of the collinear decay of
Cf. Here: 1 – Cfsource, 2 – source backing, 3 – micro-channel plate (MCP)based timing ”start” detector, 4 – position sensitive avalanchecounter (PSAC) as ”stop” detector, 5 – ionization chamber(BIC) with the supporting mesh, 6 – mesh of the entrancewindow. The side view of the mesh is shown in the insert (a),a mesh section is presented in the insert (b). After passageof the two fragments through the source backing, two lightfragments ( L and L , originally fragment L ), are obtainedwith a small angle divergence due to multiple scattering. Oneof the fragments ( L ) can be lost hitting the metal structure ofthe mesh, while the fragment L reaches the detectors of thearm1, insert (b). The source backing (2) exists only on one sideand causes an angular dispersion in the direction towards theright side (arm1). In these experiments the binary decay appears as twostrong regions of the registered binary fragments in coin-cidence, as shown in Fig. 26. The determination of all pa-rameters in coincidence with two vectors (angles, mass and energy) gives the complete kinematics for the ternary de-cay, in these experiments the determination of the missingmass indicated by “7” is well determined. This approachhas been established, and the phenomenon of collinearcluster tripartition, the CCT-decay, has been described,Refs. [32,33]. The CCT-decay has been observed indepen-dently in three different experiments with a missing masspeak: for the spontaneous fission of
Cf(sf,fff), and inanother experiment (at the neutron beam of the nuclearreactor at the JINR) with the same experimental set-upfor neutron induced fission in
U(n th ,fff), see Refs. [32,33]. In the latter case with U-nuclei the missing massturns out to be smaller (as expected), namely it is ob-served as isotopes of Si ( Z =14), corresponding to smallercharges and masses due to the smaller total mass of thefissioning nucleus, with A = 236. Further observations,which confirmed these results have been obtained withvery different experimental arrangements with the systemcalled COMETA. In the fission mode in the two cases Cf(sf,fff) and in
U(n th ,fff), the decay is dominatedby fragments which are strongly bound isotopes (clusters,nuclei with closed shells) of Sn, Ni, and Ca. The latter,Ca (or Si), as the smallest third particles, if positionedalong the line connecting the fragments Sn and Ni, theycorrespond to a minimum value of the potential energy.The PES calculated for these cases were compared andillustrated in Fig.12 and in Fig. 13.In the experiments of Refs. [32,33], an important ex-perimental feature appears, which leads to unsymmetricexperimental observations in the two detector arms. This Fig. 25.
The energy loss signal derived from the the ionisa-tion chambers of the FOBOS-telescopes. The signals have beenextracted from the drift time of the fission fragments tracks inthe ionization chambers. Two sides relative to the target arereferred to as “Modul 1” and “Modul 2”, only one shows themissing fragment yield, according to Fig.26. effect occurs due to the target backing (and the materialin front of the detector telescopes) pointing only to one ofthe detector arms (arm1). Two of the three collinear frag-ments (2 and 3), quasi-bound in the fragment A movetowards one of the detectors (called arm1). Fragments A and A are dispersed in angle while passing through thematerial of the source/target backing and the entrancefoils of the detector telescopes. This originally introducedinterpretation of the observed effect, can be almost com-pletely relaxed in view of the results (obtained later) onthe kinetic energies. These calculations have shown thatthe central (smaller) fragments attain very small kineticenergy, and they most probably already get lost by ab-sorption in the target and in the target support pointingonly into one direction (towards amr1).With the knowledge on the kinetic energies derivedfrom the more recent calculations (see the kinetic ener-gies in Fig. 4) we can expect that the central fragment A due to its very low kinetic energy gets lost in all citedexperiments. For the alternative case with a small an-gular dispersion due to the target backing material, itgets dispersed in angle and it is stopped in arm1 on asupport-structure (label 6) of the thin foils in front of theBragg-detectors, see Ref. [32]. This fact gives the char-acteristic side peak in the mass-mass-correlations with asmall maximum (label 7) corresponding to a missing mass( A = 40 −− U(n,fff) as shownin Fig. 27 the same features are observed as before, forFig. 26, however, with a smaller (charge Z =14) missingmass of 28 - 36 due to the smaller total mass, as describedin Refs. [32,33]. The side peak in this case correspondsto some isotopes of Silicon. Therefore the missing masspeak appears closer to the sum of the masses of the bi-nary fragments (sum of the binary masses is now 236).Due to the support (metallic) grid in front of the thinfoils in front of the Bragg-ionisation chambers, scatteredfragments with high intensity appear in both cases, theyform the background shown in Figs. 26 and Fig. 27. Thisscattered background can be taken from the other detec-tor arm, where this missing mass effect is absent, it definesthe background with high counting rates. The final signalis shown in the figures by subtraction of the counting ratesin the two arms. Due to the high counting rate in these ex-periments, the background subtraction can be done withvery good precision. We repeat the fact that the two de-tector arms give different yields, due to the orientationof the target-backing to one of the detector arms, whichcauses the important difference in the spectra observed,with respect to the detector which “sees the pure targetside”. The yield in the bump is now 5.1x10 − /(binary fis-sion) and it corresponds to the sum all isotopes in the A fragments occuring in the decay. Further studies of fission decays with coincidences of heavyfragments at relative angles of 180 o and with neutrons de-tected in coincidence have been performed as describedin Ref. [33]. The experimental arrangement is shown inFig. 28. The neutrons are registered in neutron detec-tors with an assembly, which surrounds the center, wherethe source for Cf(ff,fff) is placed. For this purpose amodified smaller FOBOS system has been used, shownin Fig. 28, which allows the observation of binary coinci-dences in coincidence with neutrons.Fig. 29 shows the mass-mass correlation for
Cf fis-sions in coincidence with neutrons. We notice the absenceof the “tail” of scattered binary fragments seen in theother cases (Figs.26 and 27). We compare in the lower . von Oertzen: Tue ternary Fission 15
Fig. 26.
Upper part: the projection of the mass spectra forthe detectors in arm1(a) and arm2(b) for spontaneous fissionin
Cf. The missing mass spectrum (c) is obtained by thedifference. Lower part: shows the original raw data, the coin-cidences in the two FOBOS-detectors, the contour map of themass-mass distribution (in logarithmic scale the steps betweenthe lines are a factor 2.5) of the fragments of
Cf(sf), de-tected in coincidence in the two opposite arms of the FOBOSspectrometer. The specific bump (missing mass) in the yieldsin arm1 is indicated by an arrow, 7. The sum line with M s =225 amu is shown as an illustration. part the projections on the mass scales denoted as ( M and M ), the first showing the bumps observed in theNi-region in the three experiments. For Cf(sf), Ex2,denotes the result with neutron coincidences, where notail of scattered binary fragments appear, for the case of
Cf(sf) they are visible. The spectra show the sum ofmasses without the mass of the missing fragment.Quite remarkable is the observation of the yields inthe experimental correlation plot with a definit multiplic-ity ( n =2), shown in Fig. 30 related to decays with missinglighter fragments as central nuclei with the neutron richisotopes of Ne, O, and C. In fact, with these lighter frag- Fig. 27.
Mass-mass correlation of the two registered massesmeasured with the FOBOS detector system for the n-inducedfission
U(n,fff) at the reactor in DUBNA. The extra missingmass peak indicated by an arrow (compare previous figure) ofthe ternary decay (due to a smaller missing mass now isotopesof Silicon (Si)) has now a smaller distance to the binary fissionfragments compared to the case of Fig.26, due to the smallertotal mass of 236. Projections of the sum of the masses areshown in Fig.29
Fig. 28.
Experimental Mini-FOBOS set-up for the measure-ment of correlations of the two fission fragments in coincidencewith neutrons. Labels: 1-Minifobos-detectors (Bragg-ionisationchambers), 2-neutron detectors, 3 - source(target). ments we anticipate an increased neutron emission fromthe two neck-ruptures, with two heavier fragments at theoutside borders of the chains. The projection onto the axiswith M is also shown on the left side. We see an enhancedyield for masses with A = M = 30 and A = M = 70-80,which indicate the emission of lighter central masses with Z =8,10 as well as for Z =20-28.In the work in Ref. [33], the neutron detectors havebeen arranged perpendicular to the fission axis of the bi- nary coincidences, defined by the two arrays of MiniFO-BOS detectors. This orientation has been chosen to havean increased efficiency for the spontaneous neutrons emit-ted during fission, potentially from the two necks with thelighter fragments, this explains the observations in Fig.30.a) Contour map of the mass-mass distribution (loga-rithmic scale, with lines approximately a step factor of 1.5)from a coincidence in the two opposite arms of Ex2. Thebump in the spectrometer arm (arm1) facing the backingof the Cf source is marked by the arrow. b) Projectionsonto the M1-axis for comparison with the experiment Ex1,and with the results of the 235U(nth, f) reaction [1]. Posi-tions of the magic isotopes of Ni are marked by the arrows(see text of sect. 4.2 for details). c) Projections onto thedirection Ms = M2 +M1. The result for Ex1 is presentedby two curves marked by the arrows 1 and 2 (dotted) forthe arm1 and arm2, respectively. For Ex2 the yield of arm2is subtractedWith the knowledge from the calculations of the ki-netic energies described before (shown in figure 4), wehave concluded that Ni-isotopes can be observed as fis-sion fragments with the missing mass of Ca-isotopes inthe ternary decay of Cf. As before in Ex1 and Ex2the “Ni”-yield can be observed in the COMETA-stup, be-cause the central fragment is absorbed in the source andthe source-backing. The experimental result is shown inFig. 31 and with the projection on the mass scale for theobserved Ni-isotopes in Fig. 32. Their overall (summedover the isotopes) yield is equal to 2.5 x 10 − per binaryfission. A further discussion of the COMETA-experimentwith their results is given in Ref. [26], where energy spec-tra in correlations between the two heavier fragments (FF and FF ) are discussed. In this work three different energygroups appear, which are interpreted by three different de-cay scenarios, with FF and FF in different geometricalarrangements. A big variety of geometrical arrangementsof the intermediate system (FF +FF ) (in the form of pic-tograms) are proposed, which can produce the differentcorrelated energy groups observed. Most importantly thenumber of different groups in the energy correlations canbe explained, the intermediate system consisting of thefragments FF and FF gives different energies, in one casethe rotation of the intermediate system of (FF +FF ) isassumed, which gives particular low values of the energiesof FF and FF , as observed. Further a critical assessmentof other theoretical works is given. Ternary fission into fragments with comparable masses isa process, which occurs in heavy nuclei under conditionsof large values of the fissibility parameter: X , for the ratios Z /A >
31. The decay into three heavier fragments (trueternary fission) is found to be collinear, as observed in therecent experiments and discussed in the previous sections.
Fig. 29. a) Contour map of the mass-mass correlation of theMINI-Fobos experiment with coincident neutrons. Note the ab-sence of the scattering tails observed in the other cases shownin Fig.26 and Fig.27; Projections (onto the M (b) and of thesum, M s , of registered masses (c)) of the yields for compari-son with different experiments as indicated; For Ex2 yields ofmasses from the experiment with neutron coincidences, (Mini-Fobos) is substracted.. von Oertzen: Tue ternary Fission 17 Fig. 30.
The yields of masses from the experiment with a cho-sen multiplicity in neutron coincidences (n=2), (MiniFobos),shown in a correlation of (M1-M2) and projections on the axisof M for fragments with missing light fragments M (withcharges Z). Fig. 31.
Binary coincidences of fragments registered with theCOMETA set-up. The region of the heavy fragments aroundmasses A = 120 – 160 in the arm1 is shown for the dominantbinary mass. In addition we observe corresponding to the “Ni-bump”, now Ni isotopes well separated in mass (from Ref. [33]).The Ni-isotopes with mass 68 and 72 (indicated by labels 2 and3 in the 2D-plot of M - M ) appear in the projection on theM axis, as shown in Fig. 32. The sum-line M s of the fragmentsis shown. Fig. 32.
Projection of the masses observed for the binarycoincidences of fragments with masses in the Ni-region (seeFig. 31), registered in the arm1 (from Ref. [33]).
Fig. 33.
The potential energy surface for the ternary decayof
Cf(2n,sf), with the emission of two neutrons, fragments A − A A Z = 50. The com-bination of Z = 50 and Z = 20, or /and Z = 28 are clearlyseen to be favored. This important dynamical aspect of true ternary fissionhas in fact been often predicted in the last decades [14,44,56]. The ternary decay has Q -values which are largerthan for binary channels, for the favoured ternary chan-nels this amounts to additional 20 – 30 MeV see Fig. 7.This fact is observed by inspecting the potentials and thepotential energy surfaces for the decay which can be cal-culated as function of the masses of the fragments formedin the decay.Binary coincidences are obtained between fragments A , in arm Nr.2 and A in arm Nr.1 with the missing mass method. The masses and their vectors of two fragmentsare measured, thus the mass A is uniquely determined,it corresponds to Ca isotopes in the case of Cf. Thisnew exotic decay can be understood as the breakup ofvery prolate deformed elongated hyper-deformed shapes,as discussed for hyper-deformation in
U in Ref. [21],and shown in Fig.3. The decay is considered with twosequential neck ruptures [50,62], as illustrated in Fig.19.Actually the central fragment A has extremely low ki-netic energy (see Fig.4) and is mostly lost. The kineticenergies of the fragments have been calculated for the se-quential kinematics as shown in Fig.4. We find that thecentral fragment attains very low kinetic energies, if weassume that the nucleus A has some intermediate exci-tation energy of (10–30 MeV).The main effect in the missing mass FOBOS-experimentswith the binary coincidences is the difference in the count-ing rates (mass spectra) in the two arms of the coincidencearrangement. Two fragments of the ternary decay travelin one, arm1, (see Ref. [32]) through the dispersive me-dia, the source backing and the foils of the start detector.Thus the dispersive effect (angular dispersion of 1-2 o ) ofthe two fragments from the ternary decay is only presentin arm1(!) with the target/source backing pointing to thedetector of arm1. Thus the missing mass effect appearsin the counting rate difference and the difference in themass spectra of arm1 and arm2 (the difference N (arm1)- N (arm2)). In Fig.26 two mass spectra of arm1 and arm2and their difference are shown for the case of Cf, as wellas the raw data. The yield derived from the peak in thedifference is 4.7 x 10 − For the consideration of the absolute and relative prob-abilities, the phase space (see below) and the barriers haveto be discussed. With the PES for
Cf we observe sev-eral places favored for ternary decays. In order to judgethe importance of the various decay channels, the internaldecay barriers have been calculated. In the sequential de-cay mechanism of the ternary fission, the splitting systemgoes through two corresponding barriers at each step offission, these barriers are the same for a symmetric decay.These barriers are shown in Fig. 34, where the potentialenergy of the ternary system Ni+ Ca+
Sn is shownin a two dimensional plot for the collinear configuration.The motion along R ( R ) at the fixed value of R ( R ) is an example for the sequential fission. The case,when R and R are increased simultaneously, the si-multaneous decay of the ternary system into three partsat the same time will occur. In Fig. 7, the motion alongthe diagonal line corresponds to the last kind of decay. Itis seen from Fig. 34 that the barrier for the simultaneousfission is much higher than the barriers which appear atthe sequential fission. This figure allows to judge the rel-ative importance of the decay channels. The symmetricdecay with Ge + Se + Ge occurs if the symmetricconfiguration is populated in the corresponding valley ofthe potential surface. This figure further shows an impor-tant fact: the initially observed CCT-decay with the closedshell nuclei (Sn,Ca,Ni) has two different internal barriers,which are the lowest barriers. Variations of the neutron
10 11 12 13 14 15 Q ggg + V i n t ( R , R ) + V ( ) C ( R i + R ( m ) k ) ( M e V ) R i3 (fm) Ge+ Se+ Ge Zr+ Ar+ Zr Ni(B )+ Ca+ Sn Ni+ Ca+
Sn(B ) Ru+ Ne+
Fig. 34.
Upper part: Comparison of the barriers of symmet-ric ternary collinear cluster decays of
Cf and other ternarydecay modes. The most symmetric decays have the highestbarriers. Lower part: the barriers in a 3D plot for the nonsymmetric decay into (Ni+ Ca + Sn), in a correlation of twocharacteristic distances R and R . number adds further favored channels, so the total yieldbecomes quite large.The other symmetric channels with Ar and Nehave lower barriers, we expect that the probability in-creases for smaller values on the charge and the mass of thecentral fragment. This hierarchy corresponds to the exper-imental observations. The barriers of the most importantternary collinear mass splits are shown in the upper panelof Fig. 34. The probabilities for the observation of dif-ferent modes depend on the formation probability of frag-ments and on the barriers for the collinear decay channels.For mass-symmetric decays one barrier will be sufficientto characterize the channel, however, for the CCT-decayobserved in Ref. [32,33], with
Sn + Ca + Ni, frag-ments as the dominant channel, two barriers are relevant,see the bottom panel of Fig. 34. The experimental observa-tions clearly reflect with their probabilities the influenceof the barriers:, the most symmetric mass partition hasthe lowest yield, the CCT decay is of intermediate proba- . von Oertzen: Tue ternary Fission 19 bility, the CCT decay with a light fragment, like Ne, inthe middle, would have the highest yield [38,63].
The fission processes is a statistical decay of the total(compound) nucleus, CN, as already discussed in 1939 byN. Bohr and J.A. Wheeler [4]. Actually the ternary decayoccurs with two neck ruptures in a short time-sequence,as discussed in Ref. [30]. The probabilities of the rupturesare governed by the internal prescission barriers, by thephase space and by the PES in each configuration. ThePES are obtained as described in Refs. [30,64], and areshown in Figs.12 and 13. The total phase space of thesestatistical decays is determined by:i) the energy balance and thus the details of the poten-tial energy surface, PES, namely, its valleys and hills, ii)the internal barriers for the two necks, and iii) the Q ggg -values, the latter determining the kinetic energies and thenumber of possible fragment (isotope) combinations,iv) the excitation energy range in the individual frag-ments, v) their momentum range,vi) the number of excited states (or the density of states)in each of the fragments, the combinations consisting of2(or 3) isotopes, and byvii) the spin ( J ) multiplicity in these excited states withspins up to (6-8) + (phase space factor (2 J + 1)).We have seen that the PES for the case of neutron in-duced fission, U(n,fff), shown in Fig.13, shows differentdetails in structure compared to the
Cf(sf) case due tothe absence of shells in the favored fragments. Details ofthis decay have been discussed by Tashkhodjaev et al. inRef.[30], and we show in Fig.35 the yield of fragments ob-tained in this work. A much wider distribution in massesof fission fragments is observed as compared to the caseof the spontaneous fission in
Cf(sf).In this work attentions is paid to the fact of the se-quential nature of the ternary fission decay, which pro-duces in the first step two fragments with excitation en-ergies, Nd ∗ at 25.3 MeV and Ge ∗ , which leads to theevaporation of neutrons. In the second step the fission of Nd ∗ into Ni + Ge ∗ will occur. These predictions,with a yield of 1.5x10 − /(binary fission) are in very goodagreement with the experimental observations. In Fig. 30, we show the yield in an experiment Ref. [33]with coincident neutrons (multiplicity n=2). This experi-ment selects a region of lighter missing masses of ternaryfragments. These light neutron-rich fragments are typi-cally created at the center of the system with the emissionof neutrons. The area is seen in the lower part (blue area)of the PES’s (in Figs.8 and 9). The projection on the A1- axis shows that these are isotopes of Neon, Oxygen andCarbon. The examples indicate, that the multiple modesof the ternary fission decays, can successfully be predicted
Fig. 35.
Results of the calculation by Tashkhodjaev etal., [30], for the fragments yields of sequential ternary fission in
U(n,fff). The wider (as compared to
Cf(sf)) overall spreadin masses of fragments is obtained, as predicted from the PESin Fig.13. Up triangles, results of the experimental data se-lected. Red diamonds and red squares the result of symmetric(sequential, with approximately equal momenta) decays U ∗ − > ( Nd ∗ + Ge ∗ ) − > Ge + Ni and ( Ce ∗ − > Zn+ Ni) + Ge ∗ . Fragments with comparable masses havebeen selected. with the PES’s, however, it can be very difficult to extractthe corresponding yield from the raw data.For the ternary decays of Cf with a collinear ar-rangement of the three fragments the PES’s, the contour-plot in Fig. 27 show distinct minima for various chargecombinations (multi-modal ternary decays) with Σ Z =98:i) for CCT Z = 20, and Z = 28, this is the main CCT-mode observed in Refs. [32] andii) less pronounced are charge combinations Z = 28, and Z = 20, as observed in Ref. [32,33]. The complementaryfragments with Z are isotopes of Sn ( Z = 50). The PESshows a pronounced valley with charge Z = 50 and neu-tron number N =126 , due to the closed shell for the num-ber of protons and neutrons. Because of the dominance ofthe Coulomb interaction, the proton shells are the mostimportant. The individual probabilities vary over severalorders of magnitude, see also Ref. [38].iii) The ternary fissions with Z = 18, are predicted.iv) We observe a pronounced region of minima for the sym-metric charge combinations with three comparable frag-ments, for the FFF-decays (this decay is marked as FFF):for ( Z = 32, 34, 32), and ( Z = 34, 32, 32), the fragment Z has an equivalent role as the other two Z -values Z = 32, 32, 34 (since Z + Z + Z = 98) they can be in-terchanged – we have an almost symmetric ternary decay.Because of this fact, permutations of the labels including Z in the figure of the PES (see also Fig.7) will producesimilar results, and a symmetric shape of the coincident Fig. 36.
Binary coincidences of fragments in the spontaneousternary decay (FFF) of
Cf(sf) selected for the symmetricdecay, from the inclusive data, measured in Ref. [32]. The outerfragments ( A , A , with masses M , M ) are selected with thecondition for the velocities and the momenta P i : ( V ≈ V , and P ≈ P ). Remnants of the binary fragments in coincidencesare seen. The region of the FFF-decays is marked. Missingfragments ( A ) here are again isotopes of nuclei with ( Z = 30- 36). Scattered points above this region are true coincidencesof other symmetric ternary decays, from Ref.[37]. Fig. 37.
Extraction of the yield from the data of Ref. [32,63]:the “Mo-Ar”-mode. The two different mass spectra, obtainedfrom the two arms (arm1 and arm2). The data are projectedto the A axis for arm1 and arm2 measured in Ref. [32] withbinary coincidences. events in the experiment. A quite equivalent decay pat-tern is observed in the ternary fission in the U(n,fff)reaction, where the dominant decays are connected to theproton-shells. We show in Fig.13 the PES for this case,the observed decays (Ref.[37]) are analogous to the spon-taneous fission in
Cf(sf). Several favored fragment com-binations are observed, including the symmetric decay dis-cussed in Refs. [37,38].
Fig. 38.
The derivative of the difference of the two spectrashown in Fig.37, events in the region of
Mo appear. Thecomplementary missing fragments, ( A ), here are isotopes ofAr with ( Z = 18). For
Cf(sf) we can select this symmetric decay modefrom the overall data of Ref. [32,33] by choosing conditionson the momenta (velocities) of Z and Z , with ( P = P ), see Fig. 36. This choice produces an rectangular two-dimensional field of the experimental binary coincidentevents in the favored region (similar result will be obtainedby the reflection of the labels), see also Fig. 7.An unexpected favored ternary decay mode is con-nected to the charge Z = 18 seen in the PES of both cases Cf(sf) and
U(n,fff) as a blue dip. A special procedurehad to be chosen to obtain the yield of this “Mo-mode”.The original spectra show only a small difference in thetwo detectors. Making a first derivative of this differencewe observe a strong peak for the mass region of A = 102– 110 (shown in Fig.38). Summary
In this survey we have discussed several theoretical ap-proaches to the ternary fission decay of heavy nuclei. Fromprincipal considerations we conclude that the decay willbe sequential. This conclusion comes from arguments withthe phase space, this is generally much smaller (threedecay vectors) for spontaneous three-body final channelsthan in a sequence of two two-body decays. In the lattertwo barriers appear in the respective two-body channels.For this case the kinetic energies have been calculated forthe three fragments in a collinear geometry. In this casethe central (smaller) fragment attains energies close tozero (0), giving a binary coincidence event with a miss-ing mass. Thus in the experiments discussed, the centralfragments are generally lost by absorption in the targetand/or in the target backing. Some detailed predictionsof the sequential ternary decay are consistently describedin the theoretical analysis in two of the studied cases.From the two experimental cases of the observed ternaryfission in heavy nuclei with U and Cf, which have large Q - . von Oertzen: Tue ternary Fission 21 values for a split into three nuclear fragments with com-parable masses studied in the present survey, we can con-clude that the appearance of deformed shells in the totalsystem and in the fragments (for certain proton and neu-tron numbers) turns out to be decisive for the occurrenceof true ternary fission at low excitation energy. In all casesthe fragments with Z = 50 (Tin isotopes) dominate theoutcome of the decay. The ternary decay in Cf(sf) isa unique case, because of deformed shells and because inall three fragments closed shells for proton numbers Z =20, 28 and 50 and for neutron numbers (20, 28, 50, 126)appear, these induce the decisive lowering of the secondbarrier. In the case of Cf(sf) the shells in the lighterfragments appear with, Z = 8,20,28. For the discussion ofthe decay modes the potential energy surfaces (PES’s) areshown.In the survey of the experimental data, we notice thatfor completely independent experimental setups with bi-nary coincidences for two fragments (for Cf(sf)) and forthe completely different physical case,
U(n,fff) with asmaller total mass and charge, we have the same physicalphenomenon, the observation of the missing mass in thecoincidence of two binary fragments, with a comparablemass for the third fragment. The latter (
U(n,fff)) caseshows the same feature with different and smaller (as ex-pected) ternary fragment masses. The decays are observedwith dominant fragments with Z = 50, which are consis-tent with the independent and different theoretical pre-dictions, mainly based on the PES’s. Based on the PES’sdifferent ternary decays, multi-modal decays, can be pre-dicted. These have been extracted by using correspondingconditions (gates in the experimental data) in the analy-sis of the experimental data. In the analysis of the variousdecay properties we find that the models based on pre-formed clusters show deficiencies, which are only partiallyremoved by introducing deformation in the fragments. Theapproaches with three preformed clusters allow the cal-culation of the PES‘s, which determine the phase spacefor the decay after scission, once the fragments have beenformed. In an detailed analysis of ternary fragmentations,based on an approach with three clusters by Holmval etal.[23] it is actually shown that in the case of Cf(sf)ternary fragmentation is not possible if nuclear interac-tion between fragment and deformed shell effects are notconsidered.The approach of Karpov [28] based on the three-centershell model, with strongly deformed shapes contains themaximum of necessary macroscopic and microscopic fea-tures and gives the best description for the dynamics withthe formation of two necks, and finally of the three frag-ments. This approach can be connected to the conceptof hyper-deformation with strongly deformed (elongated)shapes, which show the path towards ternary decays. Thepresent work shows the uniqueness of the case of
Cf(sf),namely that the ternary fragmentation occurs due theshell effects in the decaying nucleus and in all three frag-ments . Already in the dominant case of fragments with Z =50, the decay barriers are high in the liquid drop po-tential energies, fission can only be observed due the shell effects in the fragments with Z = 20, 28 and Z = 50.For the case of U(n,fff) the excitation energy broughtin by the neutron capture is decisive to induce ternaryfragmentation.From our analysis we conclude that the CCT-decayobserved in Ref. [32], the collinear cluster tripartition canbe considered as a manifestation of hyper-deformation inheavy deformed nuclei.
Acknowledgments
We thank A. Karpov, D. Kamanin and Y. Pyatkov formany useful discussions and preparation of figures.
Dedication
The authors dedicate this survey to the memory of two col-leges, who passed away in recent years, Walther Greiner(passed away in 2016) of Frankfurt (Germany) and Va-lerij Zagrebaev (passed away in 2015) of FLNR in Dubna,(Russian Federation) (e.g. Refs. [41,57]). They had con-tributed many ideas and important steps for the devel-opment of various aspects in the physics of true ternaryfission.
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