Structure properties of 226 Th and 256,258,260 Fm fission fragments: mean field analysis with the Gogny force
aa r X i v : . [ nu c l - t h ] D ec APS/123-QED
Structure properties of
Th and , , Fm fission fragments:mean field analysis with the Gogny force
N. Dubray, ∗ H. Goutte, and J.-P. Delaroche
CEA/DAM ˆIle-de-France DPTA/Service de Physique Nucl´eaire, Bruy`eres-le-Chˆatel91297 Arpajon cedex, France. (Dated: November 11, 2018)The constrained Hartree-Fock-Bogoliubov method is used with the Gogny interaction D1S tocalculate potential energy surfaces of fissioning nuclei
Th and , , Fm up to very largedeformations. The constraints employed are the mass quadrupole and octupole moments. In thissubspace of collective coordinates, many scission configurations are identified ranging from symme-tric to highly asymmetric fragmentations. Corresponding fragment properties at scission are derivedyielding fragment deformations, deformation energies, energy partitioning, neutron binding energiesat scission, neutron multiplicities, charge polarization and total fragment kinetic energies.
PACS numbers: 21.60.Jz, 24.75.+i, 27.90.+b
I. INTRODUCTION
Our knowledge of the fission process has made hugeprogress in recent years with the measurement of massand charge distributions of fission fragments for 70 fis-sioning systems [1], performed at the secondary beamfacility at GSI. The measured fragment yield distribu-tions have revealed new kinds of systematics on shellstructure in nuclear fission, such as transitions fromsingle- and double-humped mass distributions to a triple-humped structure in the vicinity of
Th. From a theo-retical point of view, microscopic self-consistent meth-ods appear to be well suited to study structure ef-fects in fissioning systems, where the sole input is thenucleon-nucleon force. Many studies based on meanfield approaches using Gogny or Skyrme forces have re-cently been devoted to the different fission modes, asfor example in − Fm isotopes [2, 3, 4, 5, 6, 7],where bimodal fission has been experimentally identi-fied [8, 9, 10, 11, 12, 13, 14, 15, 16, 17] and an-alyzed [18, 19]. Furthermore, two-dimensional time-dependent calculations have also been performed for the
U isotope in the elongation-asymmetry plane, whereit appears that fragment mass and total kinetic energydistributions are well reproduced. These calculationshave employed the Time-Dependent Generator Coordi-nate Method treated at the Gaussian Overlap Approxi-mation and used Hartree-Fock-Bogoliubov states [20].The present work, based on the constrained Hartree-Fock-Bogoliubov (HFB) method and the D1S force, is fo-cused on the calculation of structure properties of nascentfission fragments of light and heavy actinides, namely
Th and , , Fm. Fragment deformations, de-formation energies, energy partitioning, neutron bindingenergies, neutron multiplicities, charge polarization, andtotal fragment kinetic energies are calculated for a widerange of fragmentations. This large scale study has been ∗ Electronic address: [email protected] made possible thanks to the new generation of fast com-puters made available to our laboratory. By the mean-time, it is hoped that the calculated structure informa-tion here collected for a wide variety of fission fragmentswill serve as guideline for updating inputs (excitation en-ergy, energy partitioning, neutron binding energy, etc. . . )to phenomenological evaporation models aimed at calcu-lating prompt neutron emission from, and γ -ray decay offission fragments [21, 22, 23].The paper is organized as follows. In Sec. II is outlinedthe constrained HFB method in which, like in Ref. [20],quadrupole and octupole mass operators are adopted forexternal fields. This section also presents the mean fieldmethods used to describe: i) the scission mechanism aswell as ii) nascent fission fragments in low energy fis-sion. In Sec. III results are discussed, among which po-tential energy landscapes, scission configurations, and fis-sion fragment properties. Fission fragment yields are notconsidered in this work as they require a dynamical treat-ment [20]. Comparisons are made between present pre-dictions and experimental data for total fragment kineticenergy ( Th,
Fm) and prompt neutron multiplicity(
Fm) of fission fragments.
II. SELF-CONSISTENT APPROACH TOSCISSIONA. Constrained Hartree-Fock-Bogoliubov method
The deformed states of the nuclei under study havebeen determined using the constrained Hartree-Fock-Bogoliubov (HFB) [24] theory based on the minimizationprinciple of the energy functional, namely δ < Φ( { q l } ) | ˆ H − λ N ˆ N − λ Z ˆ Z − X l λ l ˆ Q l | Φ( { q l } ) > = 0 , (1)where ˆ H is the nuclear microscopic Hamiltonian, ˆ Q l amultipole operator, and λ N , λ Z , and λ l the Lagrangeparameters associated to constraints on nucleon numbersN, Z and average deformations q l , respectively, < Φ( { q l } ) | ˆ N | Φ( { q l } ) > = N,< Φ( { q l } ) | ˆ Z | Φ( { q l } ) > = Z,< Φ( { q l } ) | ˆ Q l | Φ( { q l } ) > = q l , (2)and where ˆ Q l is defined asˆ Q l = (1 + δ l, ) r π l + 1 A X i =1 r li Y l ( θ i , φ i ) . (3)In the present study, the Hamiltonian ˆ H is built using thefinite range and density-dependent nucleon-nucleon D1Sforce [25, 26]. One-body and two-body corrections forcenter of mass motion are taken into account in ˆ H . Manycalculations have shown that the energy functional de-rived from this Hamiltonian provides a very satisfactoryreproduction of nuclear properties over the whole masstable [27] and especially in the actinide region [28]. InEq. (2) the set of constraints { ˆ Q l } includes the isoscalaraxial dipole, quadrupole, and octupole mass momentsˆ Q , ˆ Q and ˆ Q , respectively. The dipole moment hasbeen constrained to zero so that the mean position of thenucleus center of mass is located at the origin of the coor-dinate system. The HFB energy of the deformed systemis defined as E HFB ( q , q ) = < Φ( q , q ) | ˆ H | Φ( q , q ) > . (4)In the present study, the Bogoliubov space has been re-stricted by enforcing axial symmetry along the z − axisand the self-consistent ˆ T ˆΠ symmetry, where ˆ T is thetime-reversal operator and ˆΠ the reflection with respectto the xOz plane. The system of Eqs. (1) and (2) hasbeen solved numerically by iterations for each set of de-formations by expanding the single particle states ontoaxially symmetric harmonic oscillator (HO) bases. Forsmall elongations ( q <
200 b) a one-center HO basiswith N = 14 major shells has been used while for largeelongations ( q ≥
200 b) a two-centers HO basis withtwice N = 11 major shells has been preferred [29]. Theparameters of the one- and two-centers HO bases havebeen optimized for each set of deformations, and we havechecked that the basis sizes are large enough. The poten-tial energies discussed below are the HFB energies definedin Eq. (4). B. Scission mechanism in the ( q , q ) plane At large quadrupole moment, it becomes energeticallymore favorable for a fissioning system to split into twoseparated fragments, rather than to take on a very elon-gated shape with a neck. In deformation space, this tran-sition corresponds to an evolution from the so-called fis-sion valley to the so-called fusion valley [30]. If a point Ain the fission valley leads to a point B in the fusion valley [ E - E ( p s ) ] ( M e V ) Th Fm024 Q ( b ) -10 0 10[ q - q (ps)20 ] (b)00.1 ρ N (f m - ) ρ N = 0.06 fm -3 post-scission point scission point FIG. 1: Comparison between symmetric fission of the
Thand
Fm nuclei through total energy, hexadecapole momentand minimal density along z − axis in the neck ρ N . q (ps)20 and E (ps) represent elongation and HFB energy of the first post-scission (ps) point for each fissioning system, respectively. through a small increment in either one of the deforma-tion parameters, then point A is here defined as a scissionpoint and point B as a post-scission point. Unfortunately,there is no universal way to distinguish a point in the fis-sion valley from a point in the fusion valley, and severalcriteria have been used in previous studies to achieve thisclassification. For instance, Bonneau et al. [31, 32] con-sider that scission occurs when the nuclear interactionbetween fragments is less than 1 % of the Coulomb repul-sion energy, whereas in refs. [20, 30] it was noted that thescission mecanism in U and
Pu is associated to thefollowing three properties: i) the neck between the frag-ments suddenly vanishes, ii) the hexadecapole moment ofthe system decreases, and iii) there is a drop in the poten-tial energy of the fissioning system. Whereas these threecriteria appear to be equivalent in the U-Pu region, thisis no longer the case for some of the nuclides studied here.As an example, in Fig. 1 we show the evolution of HFBenergy, mass hexadecapole moment and minimal densityin the neck as functions of the quadrupole moment forthe symmetric fission of
Th and
Fm. In the toppanel, the evolution of the HFB energy shows that scis-sion can either correspond to a sudden loss (
Th curve)or to a smooth decrease (
Fm curve) of the binding en-ergy. The same difference in behaviors can be observedfor the evolution of the mean values of the hexadecapolemoment h ˆ Q i . These examples clearly illustrate thatscission points can be determined neither by an energy-nor an hexadecapole moment-based criterion for the scis-sioning system in Fm. The lower panel of Fig. 1 showsthat in the symmetric fission of both
Th and
Fm,the density in the neck displays two different values be-fore and after scission, with an abrupt drop at scission.
FIG. 2: Symmetric scission configurations of
Th in the( z, r ) space coordinates, before and after scission (upper andlower panels, respectively). The isolines are separated by0 .
01 fm − . The dashed isoline corresponds to ρ = 0 .
16 fm − . In this situation, we define a post-scission configurationas one for which in the matter density along the sym-metry axis there is a local minimum that is lower than ρ = 0 .
06 fm − . Using this criterion, we define for eachnucleus a set of scission points in the ( q , q ) plane,that is called the scission line.Depending on the nucleus and fragmentation, the scis-sion transition is either smooth (e.g. symmetric frag-mentation of Fermium isotopes) or abrupt, in the presentsubspace of collective coordinates. In the first case, theenergy of the fissioning nucleus evolves smoothly withoutdiscontinuity from outer saddle to scission, becoming theCoulomb repulsion between nascent fragments at largeelongation. In the second case, there is an abrupt de-crease of energy and hexadecapole moment at scission.Figures 2 and 3 show the evolution of the nucleardensity at large elongation for the symmetric fragmen-tation of Th and
Fm, respectively. While
Thdisplays a very elongated shape at the scission point (up-per panel) and two prolate fragments at the post-scissionpoint (lower panel),
Fm symmetric fission leads totwo nearly spherical fragments separating smoothly. Inthe literature, these quite different ways of fissioningare called Elongated Fission (EF) and Compact Fis-sion (CF), respectively [33, 34]. In the
Fm sym-metric fission case, CF is currently explained by theproximity of double-magicity of the fragments ( Z = 50, N = 82) [35, 36]. FIG. 3: Same as Fig. 2 for the symmetric scission of
Fm.
C. From scission to fragments
The main purpose of identifying the scission configu-rations of the nuclear shape using the method describedin the preceding section is to obtain information on frag-ment properties and distributions. The underlying as-sumption is that, once a scission configuration is reached,splitting of the nucleus will occur irremediably yieldingtwo separated fragments moving away from each otherunder the action of their mutual Coulomb repulsion. Ob-servable fragment properties such as kinetic energy orexcitation energy can then be inferred from the charac-teristics of the nascent fragments - as distance betweencenters of mass, deformations, . . . - at scission. It is im-portant to stress that the fragment properties derived insuch an analysis will not necessarily all correspond tothose observed in experiments, since some of the config-urations found at scission may not occur with significantprobability in the fission process.For each scission point, a sharp cut is made at the neckposition z N on the z − axis, which serves to define the light(L) and heavy (H) fragments. Some fragment properties,namely quadrupole and octupole deformations, massesand charges, distances between centers of charge andmass, are next calculated as mean values h ˆ O i L ≡ π Z z N −∞ dz Z ∞ r.dr ˆ Oρ ( z, r ) , (5) h ˆ O i H ≡ π Z ∞ z N dz Z ∞ r.dr ˆ Oρ ( z, r ) , (6)where ρ is the nuclear density and ˆ O a one-body operator.We have checked that all first multipole moments from Q to Q of the fissioning system are continuous alongscission lines, which ensures that the scission configura-tions analyzed form a continuous set from which fragmentproperties can be consistently derived. At this stage afew remarks are in order, namely: i) the adopted sharpcut assumption inevitably leads to non-integer values forcalculated fragment charges and masses; ii) as our modelis restricted to two collective coordinates, only one scis-sion configuration is predicted for any fixed ( q s , q s )value. As a consequence, the set of fragment pairs herededuced is only a fraction of all possible pairs whichwould be formed if the constrained HFB calculationswere extended to include other collective coordinates.Nascent fragments associated with different scission con-figurations may be found having nearly the same protonand neutron numbers. As charge and mass fission frag-ment yields are outside the scope of the present staticmodel, such fragmentations are considered having thesame weight in figures shown below where they will dis-play multiple-values, for example when plotted as a func-tion of fragment mass. III. RESULTSA. Potential energy landscapes
The potential energies have been calculated on a ( q , q ) mesh from ( q = 0, q = 0) to ( q = q s , q = q s ), where ( q s , q s ) belong to the scission lines. Withthe chosen mesh dimensions ∆ q = 10 b and ∆ q =4 b / , each of the potential energy landscapes shown inFig. 4 are generated with approximately 600 calculatedvalues. For convenience, the range of potential energiesshown is limited to 20 MeV for Th (see Fig. 4(a)) and to 50 MeV for the three Fm isotopes (Figs. 4(b), 4(c),4(d)). Isolines are separated by 1 MeV.The topological properties displayed by the four land-scapes are quite contrasted. We first notice that the low-est potential minima of
Th and − Fm are all softagainst quadrupole and octupole deformations, whichshould favor coupled quadrupole and octupole vibrationsat low excitation energies. These are the common fea-tures expected for these nuclides at normal deformations.As axial deformation increases beyond the inner barrier,a well defined superdeformed (SD) potential minimumis taking place only for
Th. The SD potential mini-mum is vanishing for − Fm as discussed previouslyfor actinides with neutron number
N >
156 [28].Beyond the SD potential minimum, a valley a few MeVdeep is showing up in
Th for asymmetric deformationall the way to a scission point with large left/right asym-metric fragmentation. An isomeric minimum appears for q = 140 b, q = 20 b / . At elongation q >
150 b, asymmetric valley is also observed until the scission point q s = 500 b is reached. As scission energies are similarin both valleys, symmetric and asymmetric fission modesare expected to compete in this nucleus. For − Fm,the potential landscapes display similar and smooth pat-terns beyond the first axial barrier. In contrast to
Th,we observe that: (i) a shallow asymmetric valley is iden-tified for q >
30 b / , (ii) the fall-off of the poten-tial landscapes versus elongation for the latter nuclidesis smooth for asymmetries q >
50 b / , (iii) the scissionlines display approximately smooth and linear trajecto-ries over the ( q , q ) plane, and (iv) a symmetric valleyis gradually developing beyond q = 100 b as N growsfrom 156 to 160. This last feature is not inconsistentwith the observation of a transition from asymmetric tosymmetric mass division in fission, in going from Fmto
Fm [11, 33, 37, 38]. Whether or not this transitioncan be further analyzed with the present static mean fieldapproach will be discussed below.
B. Scission lines over the ( q , q ) plane The mesh sizes ∆ q and ∆ q so far adopted are wellsuited for performing a survey of potential energy land-scape properties. In the vicinity of scission points thestep sizes have been dramatically reduced to ∆ q = 2 band ∆ q = 1 b / , in order to define scission points withhigh precision. For each nucleus, approximately two tothree hundred scission points are used to define a scis-sion line. This is illustrated for Th in the upper panelof Fig. 5, where each point in the ( q , q ) plane corre-sponds to a single HFB calculation. Only configurationsbefore scission are shown. The curve in red color is thescission line, which is made of all exit points ( q s , q s ). Toease forthcoming discussions, a few scission points havebeen labeled with letters a, b, c,. . . , j.Most of the constrained HFB calculations at given ( q , q ) values are performed using as a starting point the FIG. 4: (Color online) Potential energies (MeV) as functions of the q (b) and q (b / ) mass moments for Th (a),
Fm(b),
Fm (c), and
Fm (d). Post-scission points are not plotted. generalized density matrix R [39] obtained at ( q − ∆ q , q ). However, in a few cases, it has been necessary tostart from the density matrix calculated at either ( q , q − ∆ q ) or ( q + ∆ q , q ), in order to reach all pos-sible fragmentations. For example, the segments definedbetween the labels b and c and between the labels d ande were determined increasing asymmetry and decreasingelongation, respectively.The scission lines determined for Th and for
Fm,
Fm and
Fm are shown in the upper and bottompanels of Fig. 5, respectively. The lines for the Fm iso-topes display similar features. The symmetric scissionconfigurations are found at q s = 270 b. Beyond thispoint, q s and q s increase gradually until q s reaches amaximum for q s ≃
500 b where asymmetry takes on val-ues in the range q s = 80 −
100 b / . For higher asymme-tries, the scission lines display wiggling patterns and arequite similar. The trajectory followed by the Th scis- sion line over the ( q , q ) plane is quite different. First,the nucleus stretches and gets an elongation nearly twiceas large as the one for Fm nuclides before symmetric scis-sion takes place. Next, elongation decreases as asymme-try increases until q s reaches a minimum for q s = 250 b,(label e in Fig. 5). Except for the point on the scissiontrajectory marked with the label f, q s and q s increasesmoothly until the point labeled i is reached. Beyondthis point located at ( q s = 444 b, q s = 142 b / ), both q s and q s decrease until the scission point labeled j isreached. Although the scission line is defined beyond thepoint labeled j, this segment lies in a ( q , q ) regionwhere potential energy is sharply raising. Therefore, thecorresponding scission configurations will not be reachedin low-energy fission and they will not be considered inthe rest of this work. For the same reason, the Fm scis-sion points beyond ( q s = 410 b, q s = 111 b / ) will alsobe discarded. q ( b / ) abcde fg ihj Th200 250 300 350 400 450 500 550q (b)020406080100 q ( b / ) Fm Fm Fm FIG. 5: (Color online) Upper panel: the scission line for
This shown over the ( q , q ) plane as a continuous curve inred color along which are marked symbols a, b, c,. . . j. Theblack dots, representing single HFB calculations, are shown toillustrate the densening of the mesh used close to the scissionline. Lower panel: scission lines for − Fm.
C. Energy along scission line
The potential energies E HFB along scission lines areshown for
Th and − Fm as functions of the frag-ment mass A frag in Figs. 6 and 7, respectively. Theseenergies take on identical values on both sides of thesymmetric fragmentation where A frag = A/
2. On thesefigures, each solid dot is for a single HFB calculation.In
Th one principal and two secondary minima areobserved which are likely to represent the most probablefragmentations in low energy fission. Hence, both sym-metric ( A frag ≃ A frag ≃
132 and A frag ≃ Z frag ≃
45) andthe two asymmetric modes ( Z frag ≃
52 and Z frag ≃ Z frag = 45),standard I ( Z frag = 54) and standard II ( Z frag = 56)fission channels [40].
80 100 120 140 160A frag (u)-1720-1715-1710-1705 E H F B ( M e V ) a bc d e fg hi j Th FIG. 6:
Th. Potential energy along the scission line as afunction of fragment mass. The symbols a, b, c,. . . j have thesame meaning as in Fig. 5. See text for more details.
Figs. 5 and 6 show that there is a correlation betweenthe structures in the potential energy along the scissionline and the behavior of the scission line in the ( q , q )plane. In order to better visualize this correspondence,the potential energy of characteristic scission configura-tions labeled as a, b, c,. . . in Fig. 5 is displayed in Fig. 6.One observes that, as A frag increases, i) the scission lineshifts from symmetric to asymmetric mass division fol-lowing an irregular trajectory over the ( q , q ) plane,and ii) to each labeled scission configuration is associateda break in the E HFB energy values. It thus seems that thecompetition between symmetric and asymmetric fissionof
Th is tied with the static structure properties of thefissioning system along the scission line. The asymmet-ric scission configurations calculated for A frag ≃
132 and A frag ≃
145 coincide with the points marked with thesymbols f and i, respectively, in Fig. 5.The absolute minima in E HFB for − Fm alongthe scission line take place for asymmetric fragmentationwith A frag ≃ A frag ≃
142 in the
Fm mass-yieldmeasurements[11]. Symmetric fission is not energeticallyfavored as E HFB displays a maximum for A frag = A/
2, incontrast to the above results for
Th. However we ob-serve that the difference in energy between the maximumand minimum values taken by E HFB for A frag = A/ A frag ≃
145 decreases from 22 MeV to 16 MeVas the mass of Fm isotopes increases from A = 156 to A = 160. Although this feature would favor a transitionfrom asymmetric to symmetric fission, making a moredefinite conclusion on this transition requires a full dy-namical calculation in which both potential energy andtensor of inertia from ground state deformation to scis-sion configurations play a role.
80 100 120 140 160 180A frag (u)-1940-1930-1920-1910-1900 E H F B ( M e V ) Fm Fm Fm FIG. 7: (Color online) Same as Fig. 6 for − Fm.
80 100 120 140 160 180A frag (u)010203040 < Q ^ > ( b ) Th Fm Fm Fm FIG. 8: (Color online) Axial mass quadrupole moments h ˆ Q i of the nascent fission fragments for Th and − Fm.
D. Fragment deformations
The axial mass quadrupole moment of the nascent fis-sion fragments along scission lines is plotted on Fig. 8for the four studied fissioning systems. The most strik-ing feature is that the fragment deformations do notsignificantly depend on the fissioning system. The fourcurves are almost superimposed and have the expectedsaw-tooth structure: minima are found for A ≃
86 and A ≃ A ≃
112 and A ≃ N = 80 and Z = 50 stabilize spherical fragments ofTin isotopes at scission. In the case of Th a mini-mum with h ˆ Q i ≃ A ≃
86. This effect is drivenby the neutron magic number N = 50. On the otherhand, well-deformed Ruthenium isotopes ( Z = 44 and N ≃
68) are here predicted with h ˆ Q i ≃
22 b. This de- -30 -20 -10 0 10 20 30 40 50 60 70 (b)-940-935-930-925-920-915-910-905-900 E H F B ( M e V ) Ce Ru FIG. 9:
Ru and
Ce potential energy curves from con-strained HFB calculations restricted to axially-symmetric andleft-right-symmetric shapes as a function of axial quadrupoledeformation [41]. The potential energy of
Ce has been ar-bitrarily increased by 295 MeV to ease comparison betweencurves. formation corresponds to a shallow secondary minimumof the potential energy curve of the Ruthenium isotopesas a function of quadrupole deformation, as illustrated inFig. 9 for
Ru. Very heavy fragments around A ≃ Z = 58 and N = 92 at h ˆ Q i ≃
15 b. The potential energy curve for
Ce is also plotted in Fig. 9 as a function of the axialquadrupole moment, and it appears that h ˆ Q i ≃
15 bcorresponds to the ground-state deformation.Axial mass octupole moments of fission fragments areplotted in Fig. 10 as functions of the fragment mass. Theoctupole moments display almost the same behavior ver-sus A frag as the one for the quadrupole moments: minimaare observed for A ≃
86 and A ≃
130 and maxima for A ≃
112 and A ≃ E. Fragment deformation energy
Energy partitioning in fission is a key input of modelsaiming at describing sequential neutron and γ -ray emis-sion from fission fragments [21, 22, 23]. In the presentstudy, the assumption will be made that the excitationenergy stored into fission fragment arises only from theirquadrupole and octupole deformations at the momentof scission. With this assumption possible intrinsic orthermal excitations prior to scission are neglected. Theestimates given below must therefore be considered aslower bound of fragment excitation energies.The fragment deformation energy is defined as [42] E def = E ff − E gs , (7)
80 100 120 140 160 180A frag (u)024681012 < Q ^ > ( b / ) Th Fm Fm Fm FIG. 10: (Color online) Axial mass octupole moments h ˆ Q i of nascent fission fragments for Th and − Fm. where E ff is the energy of the nascent fragment, and E gs the one of the fragment ground state. In this work, E gs has been deduced for all fragments from usual HFB cal-culations, whereas E ff is the HFB energy predicted in aconstrained HFB calculation where the axial quadrupoleand octupole moments are those obtained at scission con-figurations (see Figs. 8 and 10). Let us mention that inthese two sets of calculations, the neutron and protonnumbers of each fragment have been taken to be integervalues closest to the N and Z mean values calculated forthe nascent fragments. Such an approximation leads toan uncertainty in HFB energies which amounts to be lessthan 1 MeV.The FF deformation energies ( E def ) derived in this wayfor the four nuclei studied here are shown as functions of A frag and Z frag in Figs. 11(a) and 11(b), respectively.Strong variations are observed, with maxima reaching E def ∼ −
20 MeV near A frag ∼
120 and minima closeto zero near A frag ∼
145 and A frag ∼ Sn.When plotted as function of Z frag , regions with E def ∼ Z frag ∼
50 and Z frag ∼
56, that isto − Sn and
Ba, respectively. Furthermore, themaximum identified previously in the E def values at A frag ∼
120 gets split over two Z frag components, namely Z frag ∼
48 and Z frag ∼
52, that is for near symmetric andhighly asymmetric charge divisions in the Fm and Th nu-clides, respectively.Finally, Fig. 11(c) displays the difference ( E L − E H )between the deformation energies of light and heavyfragments. This difference takes on values rangingfrom 23 MeV to -15 MeV. Extrema are located at far-asymmetric mass divisions. More than 70% of the lightfragments display E L > E H values. FIG. 11: (Color online) Nascent fragment deformation ener-gies for
Th and − Fm as functions of: (a) fragmentmass, and (b) fragment charge. The differences between light(L) and heavy (H) fragment energies as functions of light frag-ment mass are shown in (c).
80 100 120 140 160 180A frag (u)2345678 B * n ( M e V ) Th Fm Fm Fm FIG. 12: (Color online) One-neutron binding energies ofnascent fission fragments as functions of fragment mass for
T h and − Fm.
F. Prompt fission neutrons
In the present section, we aim at calculating the mul-tiplicity ν frag of prompt neutrons emitted by each fissionfragment. For this purpose, we assume that the defor-mation energy of any fragment is converted into internalexcitation energy through collective vibrations and thatthe fragment will de-excite only through prompt neutronemission. As an estimate, the neutron emission multi-plicity of one fragment is taken as [43, 44] ν frag = E def h E k i + B ∗ n , (8)where B ∗ n is the one-neutron binding energy in nascentfragment, and h E k i the mean energy of the emitted neu-tron. The latter is assumed to be 2 MeV in Th [45]and 1.5 MeV in − Fm [46].
1. One-neutron binding energy
In the present work the one-neutron FF binding energy B ∗ n is taken equal to the neutron chemical potential ob-tained in the HFB calculations performed on the scissionline. The B ∗ n values are plotted in Fig. 12 as a functionof fragment mass. It is seen that they globally decreasefrom approximately 7 MeV to 3 - 4 MeV with increasingmass. The lowest B ∗ n values are obtained for A ≃ Z ≃ N ≃ B ∗ n andground state B n is plotted in Fig. 13. We find that thisdifference can be as large as 2 MeV in absolute value.Such differences are a consequence of the evolution ofsingle-particle neutron gaps as a function of deformation.
80 100 120 140 160 180A frag (u)-2-1012 [ B * n - B n ] ( M e V ) Th Fm Fm Fm FIG. 13: (Color online) Differences between one-neutronbinding energies of fragments as calculated for scission con-figurations and for ground states, plotted as functions of frag-ment mass.
2. Neutron multiplicity
The multiplicities calculated from Eq. (8) are shown inFigs. 14 - 16 as functions of fragment mass for the fourstudied nuclei. On figures 14 and 16, a solid line has beenadded to guide the eye. Typical saw-tooth structures areobserved, displaying maxima and minima. These struc-tures appear correlated with the quadrupole deformationof fragments at scission.In the case of
Th fission, the neutron multiplicitycurve displays pronounced structures separated by fivemass units from A frag = 110 to A frag = 150, that arelinked to: i) fragment deformations (Figs. 8 and 10), ii)the deformability of the fragments (the softness of po-tential energies with respect to axial quadrupole and oc-tupole deformations), and iii) the one-neutron bindingenergy (see Fig. 12).For Fm isotopes, the curves look more regular, andshow that neutron emission is almost vanishing around A frag = 130 and is maximum around A frag = 120. InFig. 15, comparison is made with the experimental datafor spontaneous fission of Fm [47]. The agreement be-tween theoretical values and measurements is rather sat-isfactory, as the global data pattern is well reproduced.However, calculations appear to underestimate the num-ber of emitted neutrons in the A frag = 90 −
130 regionand a second minimum is found around A frag = 144. As aconsequence, the calculated number of emitted neutronsis 30% smaller than experimental data.These discrepancies probably come from our model as-sumptions. Part of the overall underestimation of thenumber of emitted neutrons is presumably due to thefact that the deformation energy has been calculated withconstraints placed only on axial quadrupole and octupoledeformations of fragments. More realistic calculations0
70 80 90 100 110 120 130 140 150 160A frag (u)00.511.522.53 N e u t r on m u lti p li c it y Th FIG. 14: (Color online)
Th. Calculated neutron multiplic-ity as a function of fragment mass. The solid line is to guidethe eye.
80 100 120 140 160 180A frag (u)01234 N e u t r on m u lti p li c it y Fm (exp.)
Fm (th.)
FIG. 15:
Fm. Neutron multiplicity versus fragment mass.Comparison between predictions (solid symbols) and data [47](empty symbols). should include the effect of higher order multipole frag-ment deformations such as q and q . As for the secondminimum at A frag ∼
80 100 120 140 160 180A frag (u)01234 N e u t r on m u lti p li c it y Fm
80 100 120 140 160 180A frag (u)012345 N e u t r on m u lti p li c it y Fm FIG. 16: (Color online) Neutron multiplicities for
Fm (up-per panel) and
Fm (lower panel). Lines are to guide theeye.
G. Deviation from the unchanged chargedistribution
Introduced in 1962 by Wahl, the fragment unchangedcharge distribution Z UCD (i.e. charge polarization) is thecharge number of a fragment with a given mass A frag , ifits Z/A ratio were the same as the one of the fissioningnucleus [48]: Z UCD ≡ Z fs .A frag A fs . (9)The deviation ∆ Z frag = Z frag − Z UCD of the chargeof the fragment Z frag from the unchanged charge distri-bution Z UCD is plotted in Figs. 17 and 18 as a functionof the fragment charge for
Th and
Fm. Values of∆ Z frag for Fm and
Fm are very close to the ones of
Fm. We first observe that ∆ Z frag is globally positivefor light fragments and negative for heavy ones. This fea-1 E p a i r ( p ) ( M e V )
30 35 40 45 50 55 60Z frag (u)-1.5-1-0.500.51 ∆ Z fr a g ( c h a r g e un it s ) Th FIG. 17: Nascent fission fragment proton pairing energy (up-per panel) and deviation from unchanged charge distribution(bottom panel) as functions of fragment charge for
Th. ture stems from the fact that heavy systems may sustainstronger neutron excess than light ones, as observed anddiscussed for several fissioning systems [49, 50]. The pat-terns displayed by ∆ Z frag as functions of Z frag are quitecontrasted. While both sets of ∆ Z frag values show sharpstructures as Z frag increases, ∆ Z frag globally decreasesin Th from ∆ Z frag ≃ Z frag ≃ −
1. In con-trast, in
Fm the ∆ Z frag values reach a plateau with | ∆ Z frag | ≃ . Z frag >
57 and Z frag <
43. In an at-tempt to understand the origins of these sharp structuresand different global trends, we have sought for possiblecorrelations with other structure properties, namely pro-ton separation energies of: i) fissioning nuclei along scis-sion lines, and ii) each nascent fragment. No clear cutcorrelation is found. However the structures observed inthe ∆ Z frag values for both nuclides seem to coincide withthe variations of the fragment pairing energies E pair (p) as can be seen when comparing the plots in the upperand bottom panels in Figs. 17 and 18. H. Total kinetic energy
1. Distance between fragments
The total kinetic energy (TKE) of a given fragmenta-tion can be estimated from the formula E TKE = e Z H Z L d ch , (10)where e is the electron charge, Z H ( Z L ) the charge ofthe heavy (light) fragment, and d ch the distance betweenfragment centers of charge at scission. The distance d ch deduced from our calculations is plotted as a function offragment mass for Th,
Fm,
Fm and
Fm in E p a i r ( p ) ( M e V )
30 35 40 45 50 55 60 65 70Z frag (u)-1-0.500.51 ∆ Z fr a g ( u ) Fm FIG. 18: Same as Fig. 17 for
Fm.
120 130 14014161820 d c h (f m ) Th 140 160 18014161820
Fm140 160 180A frag (u)14161820 d c h (f m ) Fm 140 160 180A frag (u)14161820 Fm FIG. 19: Distances between nascent fragment centers ofcharge calculated as functions of fragment mass for
Th and − Fm.
Fig. 19. For all considered nuclei, the distance betweenfragment centers of charge at scission falls in the range d ch = 14 −
20 fm. The distance d cm between centers ofmass has also been calculated. The difference δd between d cm and d ch appear rather small: δd ≃ .
08 fm in
Thand δd ≃ .
05 fm in − Fm.
2. Total kinetic energy for Th The TKE values of
Th fission fragments are plot-ted as functions of fragment mass in Fig. 20. The dotsrepresent the result of Eq. (10) whereas the solid linefollows the experimental data of Ref. [40, 51] obtainedin electro-magnetic induced fission measurements. Onenotices that theoretical results present many more struc-2
70 80 90 100 110 120 130 140 150 160A frag (u)140160180200 T K E ( M e V ) Th theo.
Th exp. a bc d ef g hi j
FIG. 20:
Th. TKE values of nascent fission fragments as afunction of fragment mass. Comparison between predictions(solid dots) and data [40, 51]. tures than do experimental data. This difference may beexplained from the fact that the experimental measure-ments correspond to an excitation energy of the fissioningnucleus of the order of 11 MeV, whereas formula (10) isvalid only for low energy fission. As well known, an in-crease in the fission energy smooths out kinetic energydistribution. In particular the kinetic energy in the sym-metric mass region increases [52] which explains why ex-perimental TKE display only a very shallow minimumfor A frag = A/ A frag ≃
132 ( Z frag ≃
54) and A frag ≃
94 ( Z frag ≃ th ∼
169 MeV, is found close to the ex-perimental mean value (TKE) exp = 167 . ± .
3. Total kinetic energy for − Fm.
The TKE values of − Fm fission fragments areplotted in Fig. 21 as functions of fragment mass. TheTKE curves look rather similar in all three isotopes.They display a sharp peak reaching TKE ≃
250 MeVfor symmetric fission. These features are characteristicof compact scission, where the fissioning system gives riseto nearly spherical fragments separated by a small dis-tance. Furthermore, we also observe that the full width
80 90 100 110 120 130 140 150 160 170 180A frag (u)180200220240260 T K E ( M e V ) Fm Fm Fm FIG. 21: (Color online) − Fm. TKE values of nascentfission fragments as functions of fragment mass.
130 135 140 145 150 155 160A frag (u)160180200220240260 T K E ( M e V ) FIG. 22:
Fm. TKE values of fission fragments as func-tions of fragment mass. Results of theoretical calculations(dots) are displayed with pre-neutron-emission data (contourdiagram). The solid line represents the experimental averageTKE [54]. ∆ TKE at half maximum is narrowing from ∆
TKE = 20 uto ∆
TKE = 14 u in going from
Fm to
Fm.Theoretical TKE in
Fm (black points) are comparedwith experimental ones [54] in Fig. 22. Calculated resultsare found in very good agreement with the average TKEdata for asymmetric fission ( A frag = 138 − A frag < d ch is toosmall for scission configurations close to symmetry. Thisremark is consistent with the fact that for the same frag-mentations, ν frag values calculated from deformation en-ergies is underestimated (see Fig. 15). These under- and3over-estimations are interpreted as due to our study, re-stricted to the ( q , q ) deformations, which favors com-pact scission. Accessing elongated fission configurationsfor nearly-symmetric fragmentations of Fm impliesthat at least three collective coordinates must be con-sidered in constrained HFB calculations.
IV. CONCLUSION
In this work, large scale HFB calculations using theGogny D1S force have been performed in order to in-vestigate structure properties of
Th and − Fm atscission and the characteristics of fission fragments alongscission lines. Scission configurations are first analyzedassuming that axial quadrupole and octupole collectivecoordinates play a major role in fission. We have foundfrom our constrained HFB calculations that the scissionmechanism depends on which heavy nuclide and frag-mentation are considered. This mechanism may displayeither a smooth or an abrupt character in the ( q , q )plane. The former property means that the potential en-ergy of the fissioning system changes smoothly over de-formation from outer saddle to scission and beyond whereCoulomb repulsion takes place between fission fragments.This scission property is found for the Fm symmetricfragmentations. For asymmetric fragmentations in allnuclei of present interest, there is a sudden drop in po-tential energies whenever scission takes place. To accom-modate with these contrasted properties, post-scissionpoints are defined for matter densities present in the neckthat are weaker than ρ = 0 .
06 fm − . With this criterion,scission lines are calculated, and the fragmentations de-termined assuming sharp cuts across the necks.Properties of fission fragments and correlations withproperties of fissioning systems along scission lines havebeen discussed. These comprise potential and deforma- tion energies, quadrupole and octupole deformations, to-tal kinetic energies, prompt neutron emissions, deviationfrom unchanged charge distribution, and energy parti-tioning. All these properties reflect either shell and/orpairing contents of potential energies of both fission frag-ments and fissioning nuclei, in particular for multipole de-formations, neutron multiplicities, and total kinetic ener-gies. Predictions are found in reasonably good agreementwith experimental data for total kinetic energy ( Th,
Fm) and prompt neutron multiplicity (
Fm) of fis-sion fragments.The present microscopic analysis shows that the struc-ture of the two-dimensional ( q , q ) potential energysurface in Th is similar to those previously calculatedin U and Pu. The different behavior with respect to scis-sion found in Fm isotopes and the fact that symmetricelongated configurations do not appear in our descriptionmay indicate that a collective space with more that twodimensions is needed to describe scission configurationsand fragment properties in these nuclei. Preliminary in-vestigations show that other heavy actinides probablyalso require an enlargement of the dimension of the col-lective space used. In view of the encouraging resultsobtained so far, in particular in light actinides, it seemsworth attempting to extend the present static calcula-tion to three dimensions or even more. Of course, adescription of fission observables such as fragment massand kinetic energy distributions will also require to ex-tend the two-dimensional dynamical model employed inUranium [20] to higher dimensions.
ACKNOWLEDGMENTS
We gratefully acknowledge D. Gogny and J.-F. Bergerfor stimulating and enlightening discussions. [1] K.-H. Schmidt et al., Nucl. Phys.
A665 , 221 (2000).[2] M. Warda, J.L. Egido, L.M. Robledo, and K. Pomorski,Phys. Rev.
C66 , 014310 (2002).[3] M. Warda, J.L. Egido, L.M. Robledo, and K. Pomorski,Phys. At. Nucl. , 1178 (2003).[4] M. Warda, K. Pomorski, J.L. Egido, and L.M. Robledo,Int. J. Mod. Phys. E13 , 169 (2004).[5] L. Bonneau and P. Quentin,
Proceedings of the Third In-ternational Workshop on Nuclear Fission and Fission-Product Spectroscopy, Cadarache , edited by H. Goutte,H. Faust, G. Fioni and D. Goutte (AIP, Melville, NewYork), p. 77 (2005).[6] A. Staszczak, J. Dobaczewski, and W. Nazarewicz,Int. J. Mod. Phys.
E14 , 395 (2005).[7] L. Bonneau, Phys. Rev.
C74 , 014301 (2006).[8] R. Brandt, S.G. Thompson, R.C. Gatti, and L. Phillips,Phys. Rev. , 2617 (1963).[9] W. John, E.K. Hulet, R.W. Lougheed, and J.J.Wesolowski, Phys. Rev. Lett. , 45 (1971). [10] J.P. Balagna, G.P. Ford, D.C. Hoffman, and J.D. Knight,Phys. Rev. Lett. , 145 (1971).[11] K.F. Flynn, E.P. Horwitz, C.A.A. Bloomquist, R.F.Barnes, R.K. Sjoblom, P.R. Fields, and L.E. Glendenin,Phys. Rev. C5 , 1725 (1972).[12] R.M. Harbour, K.W. MacMurdo, D.E. Troutner, andM.V. Hoehn, Phys. Rev. C8 , 1488 (1973).[13] R.C. Ragaini, E.K. Hulet, R.W. Lougheed, and J. Wild,Phys. Rev. C9 , 399 (1974).[14] K.F. Flynn, J.E. Gindler, R.K. Sjoblom, and L.E. Glen-denin, Phys. Rev. C11 , 1676 (1975).[15] K.F. Flynn, J.E. Gindler, and L.E. Glendenin, Phys. Rev.
C12 , 1478 (1975).[16] J.E. Gindler, K.F. Flynn, L.E. Glendenin, and R.K.Sjoblom, Phys. Rev.
C16 , 1483 (1977).[17] E.K. Hulet, R.W. Lougheed, J.H. Landrum, J.F. Wild,D.C. Hoffman, J. Weber, and J.B. Wilhelmy, Phys. Rev.
C21 , 966 (1980).[18] P. M¨oller, J.R. Nix, and W.J. Swiatecki, Nucl. Phys. A469 , 1 (1987).[19] S. ´Cwiok, P. Rozmeij, P. Sobiczewski, and Z. Patyk,Nucl. Phys.
A491 , 281 (1989).[20] H. Goutte, P. Casoli, J.-F. Berger, and D. Gogny,Phys. Rev.
C71 , 024316 (2005).[21] S. Lemaire, P. Talou, T. Kawano, M.B. Chadwick, andD.G. Madland, Phys. Rev.
C72 , 024601 (2005).[22] S. Lemaire, P. Talou, T. Kawano, M.B. Chadwick, andD.G. Madland, Phys. Rev.
C73 , 014602 (2006).[23] N.V. Kornilov, F.-J. Hambsch, and A.S. Vorobyev,Nucl. Phys.
A789 , 55 (2007).[24] P. Ring and P. Schuck,
The Nuclear Many Body Problem (Springer-Verlag, New York, 1980), p. 267.[25] J. Decharg´e and D. Gogny, Phys. Rev.
C21 , 1568 (1980).[26] J.-F. Berger, M. Girod, and D. Gogny,Comp. Phys. Comm. , 365 (1991).[27] G. Bertsch, M. Girod, S. Hilaire, J.-P. Delaroche,H. Goutte, and S. P´eru, ArXiv preprint nucl-th/0701037 (2007), Phys. Rev. Lett. (in press).[28] J.-P. Delaroche, M. Girod, H. Goutte, and J. Libert,Nucl. Phys. A771 , 103 (2006).[29] J.-F. Berger, Ph.D. thesis, Centre d’Orsay, Universit´eParis-Sud (1985).[30] J.-F. Berger and D. Gogny, Nucl. Phys.
A428 , 23c(1984).[31] L. Bonneau, P. Quentin, and I.N. Mikhailov,
Proceedingsof the Third International Workshop on Nuclear Fissionand Fission-Product Spectroscopy, Cadarache , edited byH. Goutte, H. Faust, G. Fioni and D. Goutte (AIP,Melville, New York), p. 297 (2005).[32] L. Bonneau, P. Quentin, and I.N. Mikhailov, Phys. Rev.
C75 , 064313 (2007).[33] E.K. Hulet, J.F. Wild, R.J. Dougan, R.W. Lougheed,J.H. Landrum, A.D. Dougan, P.A. Baisden, C.M. Hen-derson, and R.J. Dupzyk, Phys. Rev.
C40 , 770 (1989).[34] P. M¨oller, D.G. Madland, A.J. Sierk, and A. Iwamoto,Nature (London) , 785 (2001).[35] M.G. Mustafa and R.L. Ferguson, Phys. Rev.
C18 , 301 (1978).[36] F. G¨onnenwein, Nucl. Phys.
A654 , 855c (1999).[37] D.C. Hoffman, J.B. Wilhelmy, J. Weber, and W.R.Daniels, Phys. Rev.
C21 , 972 (1980).[38] D.C. Hoffman, Nucl. Phys.
A502 , 21c (1989).[39] P. Ring and P. Schuck,
The Nuclear Many Body Problem (Springer-Verlag, New York, 1980), p. 252.[40] K.-H. Schmidt, J. Benlliure, and A.R. Junghaus, Nucl.Phys.
A693 , 169 (2001).[41] S. Hilaire and M. Girod, (2007).[42] G. Simon, Ph.D. thesis, Centre d’Orsay, Universit´e Paris-Sud (1990).[43] A. Ruben and H. M¨arten, Z. Phys. A Atomic Nuclei ,237 (1990).[44] H.-H. Knitter, U. Brosa, and C. Budtz-Jørgensen,
TheNuclear Fission Process , edited by C. Wagemans (CRCPress, Boca Raton, FL), p. 497 (1991).[45] D.G. Madland, Nucl. Phys.
A772 , 113 (2006).[46] C. Budtz-Jørgensen and H.H. Knitter, Nucl. Phys.
A490 , 307 (1988).[47] J.E. Gindler, Phys. Rev.
C19 , 1806 (1979).[48] A.C. Wahl, R.L. Ferguson, D.R. Nethaway, D.E. Trout-ner, and K. Wolfsberg, Phys. Rev. , 1112 (1962).[49] J.P. Bocquet and R. Brissot, Nucl. Phys.
A502 , 213c(1989).[50] F. G¨onnenwein,
The Nuclear Fission Process , edited byC. Wagemans (CRC Press, Boca Raton, FL), p. 400(1991).[51] C. B¨ockstiegel, Ph.D. thesis, Technischen Universit¨atDarmstadt (1997).[52] S. Pomm´e, E. Jacobs, M. Piessens, D. De Frenne, K. Per-syn, K. Govaert, and M.-L. Yoneama, Nucl. Phys.
A572 ,237 (1994).[53] K.-H. Schmidt et al., Nucl. Phys.
A685 , 60c (2001).[54] D.C. Hoffman, G.P. Ford, J.P. Balagna, and L.R. Veeser,Phys. Rev.