A Method to Calculate Fission-Fragment Yields Y(Z,N) versus Proton and Neutron Number in the Brownian Shape-Motion Model. Application to calculations of U and Pu charge yields
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A Method to Calculate Fission-Fragment Yields Y ( Z, N ) versusProton and Neutron Number in the Brownian Shape-MotionModel Application to calculations of U and Pu charge yields
Peter M¨oller and Takatoshi Ichikawa Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, JapanReceived: date / Revised version: date
Abstract.
We propose a method to calculate the two-dimensional (2D) fission-fragment yield Y ( Z, N )versus both proton and neutron number, with inclusion of odd-even staggering effects in both variables.The approach is to use Brownian shape-motion on a macroscopic-microscopic potential-energy surfacewhich, for a particular compound system is calculated versus four shape variables: elongation (quadrupolemoment Q ), neck d , left nascent fragment spheroidal deformation ǫ f1 , right nascent fragment deformation ǫ f2 and two asymmetry variables, namely proton and neutron numbers in each of the two fragments. Theextension of previous models 1) introduces a method to calculate this generalized potential-energy functionand 2) allows the correlated transfer of nucleon pairs in one step, in addition to sequential transfer. In theprevious version the potential energy was calculated as a function of Z and N of the compound systemand its shape, including the asymmetry of the shape. We outline here how to generalize the model fromthe “compound-system” model to a model where the emerging fragment proton and neutron numbers alsoenter, over and above the compound system composition. PACS.
In previous investigations it has been shown that a re-alistic description of the experimentally observed fission-fragment charge distributions can be obtained by means ofrandom walks on tabulated five-dimensional (5D) potential-energy surfaces calculated for a densely spaced grid forover five million different shapes [1,2,3] . It was partic-ularly encouraging that the 70 charge-yield distributionsmeasured at GSI [4] were well reproduced, including thetransition from symmetric fission for light Th isotopes toasymmetric fission for the heavier isotopes beyond A ≈ Send offprint requests to : a Present address:
Insert the address here if needed tatively describe this feature. For example the effect hasbeen correlated with Coulomb-related quantities, such as Z /A and Z /A / , referred to as order parameters, andto pairing effects on the nuclear level density and its vari-ation with excitation energy, see Refs. [5,6,7] and refer-ences therein. Other models are based on properties ofseparated fragments and thermal equilibrium at scission.For a review see Ref. [8]. Common for these models arethat they are not based on detailed, calculated potential-energy surfaces or dynamical evolution on such surfaces.The models often also contain a substantial number of pos-tulated terms with parameters which are determined fromadjustments to observed yields. Another group of modelsdo treat dynamical evolution, in a Langevin approach. Un-til now they are based on macroscopic potential-energysurfaces. or, when shell effects are included the calcula-tions are performed for three shape variables for fairlyhigh excitation energies ( E ∗ = 20 MeV), see Ref. [9] for abrief review but with extensive references to original work.In the Brownian shape-motion (BSM) model the onlyparameters are those of a well-established macroscopic-microscopic model used to calculate the five-dimensional(5D) potential-energy surfaces [10] in which the parame- P. M¨oller and T. Ichikawa: A Method to Calculate Fission-fragment Yields Y ( Z, N ) ters have been unchanged since 2002 [11], the additionalcritical neck radius at which the mass split is frozen, aweak bias potential (but the results are fairly insensitiveboth to the magnitude of the critical neck radius and tothe bias potential strength [1,2]), and two parameters in a“suppression factor” that accounts for the decrease of theshell-plus-pairing correction with energy [3].We have recently shown that the observed magnitudeof the odd-even staggering can be directly correlated tothe excitation energy above the outer part of the calcu-lated potential-energy surface [12]. Therefore we suggestedthe BSM model could describe odd-even staggering if apotential-energy model were developed that accounts forhow the individual nascent fragment properties are ex-pressed in the calculated potential-energy surface [12].Here we propose a model for the potential-energy sur-face in which the properties of the individual fragmentsgradually emerge as the scission configuration is approachedand specify the full details this proposed model. To treatodd-even staggering we add to the potential energy, as iscustomarily done since the dawn of nuclear mass calcu-lations [13,14], a pairing contribution ∆ to emerging oddfragments. However, since we start our trajectories at theground-state shape of the even-even parent nuclei, wherethere is no odd pairing effect, only a fraction of the pair-ing delta of the fully formed final fragments is added inthe initial stages of division. This fraction grows with de-creasing neck diameter, see below for further discussionand specification.We also introduce a correlated transfer of paired nu-cleon configurations. Such correlated nucleon transfers,which are different from sequential transfers, are oftenidentified in nuclear reaction experiments, for a discussionsee the presentation in Ref. [15] and the many referencescited therein, for example Refs. [16,17].It turns out that we can incorporate our new ideas incalculations of charge yields, with minimal modificationsof existing computer codes. Our first results on chargedistributions are presented below. To describe the nuclear shape we use the three-quadratic-surface (3QS) parameterization. It was introduced almost50 years ago [18]. It is much more cumbersome to dealwith than, say, a multipole expansion such as the β pa-rameterization [19], but is used to allow a realistic descrip-tion of shapes of a fissioning nucleus up to and includingdivision of the single shape into separated fragments. Par-ticularly noteworthy is that the emerging fragment shapescan be deformed spheroids or exact spheres. The latter isof special importance because it allows the extra bindingassociated with spherical doubly-magic nuclei to be accu-rately calculated. More details are found in Ref. [18]; thediscussions of Figs. 1 and 2 there are particularly infor-mative. How we design our potential-energy calculationsin terms of this parameterization is detailed in Refs. [20,10]; here we briefly summarize a few essential details. a a a ρ zl z l z l c c Three-Quadratic-Surface Parameterization
Fig. 1.
Shape generated by the three-quadratic-surface pa-rameterization. Different colors distinguish between the shapesections generated by the three expressions in Eq. (1).
In the (3QS) parameterization the shape of the nu-clear surface is specified in terms of three smoothly joinedportions of quadratic surfaces of revolution. They are com-pletely specified [18] by ρ ( z ) = a − a c ( z − l ) , l − c ≤ z ≤ z a − a c ( z − l ) , z ≤ z ≤ l + c a − a c ( z − l ) , z ≤ z ≤ z (1)Here the left-hand surface is denoted by the subscript 1,the right-hand one by 2 and the middle one by 3. Shapes 1and 2 are spheroids, for which c is the semi-symmetry axislength, a is the semi-transverse axis length, and l specifiesthe location of the center of the spheroid. The middle bodymay be a spheroid or a hyperboloid of one sheet, for which c is imaginary. At the left and right intersections of themiddle surface with the end surfaces the value of z is z and z , respectively. Surfaces 1 and 2 are also referred to asend bodies and, alternatively, nascent fragments. A shapegenerated by the parameterization in Eq. (1) is shown inFig. 1.In our calculations we use five shape parameters: elon-gation in terms of quadrupole moment Q , left and rightfragment spheroidal deformations ǫ f1 and ǫ f2 , neck diam-eter d and mass asymmetry α g . These five parameterscompletely specify the shape for which the potential en-ergy is calculated and have been extensively discussed inRef. [10]. Also, they completely exhausted the shape spaceavailable to this parameterization (except, trivially, for in-cluding larger values in each of the five shape parameters).For our studies here we need to revisit the asymmetry vari-able α g which is directly connected to the asymmetry ofnuclear shape (and the separated fragments): α g = a c − a c a c + a c . (2) . M¨oller and T. Ichikawa: A Method to Calculate Fission-fragment Yields Y ( Z, N ) 3
This is equivalent to α g = M c1 − M c2 M c1 + M c2 (3)where M c1 and M c2 are the volumes inside the end-bodyquadratic surfaces, were they completed to form closed-surface spheroids and “c” in the notation is to clarifywe are referring to quantities of the compound single-shape system. To avoid the introduction of a large numberof equivalent concepts we use somewhat interchangeably,“mass”, “volume” and nucleon number A .Our random-walk tracks end at a point where the neckradius of the nuclear shape is quite well developed, namelyat 2.5 fm (compare to the radius of a spherical Pu nu-cleus which is 7.2 fm), so the shape has almost reachedthe configuration of separated fragments. The neck radiusis actually smaller than the radius of O for which the ra-dius is 2.92 fm, and which is the lightest nucleus to whichwe have applied the macroscopic-microscopic method. Al-though M c1 and M c2 for shapes with well-developed neckscan be expected to be close to the final fragment masseswe cannot directly compare M c1 and M c2 to the observedfission fragment masses M s1 and M s2 (where the superscript“s” indicates we refer to the separated fragments) becausethe former do not quite sum up to the total nuclear vol-ume or mass A . However, by scaling M c1 and M c2 so thattheir sum adds up to the total mass number A , we candirectly relate the “mass asymmetry” of the compound-nucleus shape to the observed heavy and light fragmentmasses. We obtain trivially M s1 = r s M c1 = A α g M s2 = r s M c2 = A − α g r s = AM c1 + M c2 where r s is a scaling factor for a nucleus with A nucle-ons. The scaling is equivalent to a assigning the mass inthe neck region to the left and right nascent fragmentsin proportion to the respective volumes of these nascentfragments, were they completed to closed spheroids. Theamount of matter involved is in the range of 10–20 nu-cleons. It is obvious how the same definitions apply tothe proton and neutron numbers. The symbols Z and N are used for proton and neutron numbers of the fissioningcompound system. We then have: Z s2 = Z − Z s1 and N s2 = N − N s1 (5)Therefore we usually use only Z s1 and N s1 when we explic-itly refer to the number of fragment protons and neutrons,the other fragment proton and neutron numbers are thenalso specified. To calculate the yield as a function of bothfragment proton and neutron number obviously requiresthat the previous 5D model (which always assumed thatthe proton to neutron ratios in both fragments were equalto the proton to neutron ratio in the compound system) be generalized to 6D so that the yield Y ( Z s1 , N s1 , Z s2 , N s2 ),which is a function of both proton and neutron asymme-try is obtained. We also realize that to describe odd-eveneffects requires that we space the grids we will use in termsof integer spacing of Z s1 and N s1 in some fashion which wewill now introduce. The method to calculate the two-dimensional fission-fragmentyield Y ( Z, N ) function that we now introduce is perhapsmost transparently explained by occasionally referring tospecific aspects of the computer code used to calculate, inthe macroscopic-microscopic method, potential-energies asfunctions of shape. In our current potential-energy modeland code we obtain a total potential energy for a specificcompound nucleus (
Z, N ) and a specific “shape” as a sumof a macroscopic energy (given by a liquid-drop type ex-pression) and a microscopic shell-plus-pairing correction: E pot ( Z, N, shape) = E mac ( Z, N, shape)+ E prots+p ( Z, ( N ) , shape) (6)+ E neuts+p (( Z ) , N, shape)where E mac ( Z, N, shape) is the macroscopic energy and E prots+p is the proton shell-plus-pairing correction and thefinal term the neutron shell-plus-pairing correction. In acalculation for a specific compound system and a specificshape these shell-plus-pairing corrections can trivially beindividually tabulated. To obtain, say, the proton micro-scopic correction single-particle levels are calculated nu-merically in a folded-Yukawa single-particle potential witha functional form derived from the nuclear shape [21,19].From these levels the shell correction is obtained by useof the Strutinsky method [22,23] and the pairing correc-tion through, in our case, the Lipkin-Nogami method, asdetailed in [24]. Thus, the proton microscopic correctionis independent of neutron number, except that the poten-tial radius and depth depend on both proton and neutronnumber, therefore we have used the notation ( N ) and ( Z )to indicate a weak dependence. But for small variationsof neutron number the effect on the proton microscopic correction is small and can be neglected, an importantobservation that we will make use of later. To show thisinsensitivity we have calculated the proton and neutronshell-plus-pairing corrections for the ground-ground stateshape of Hs for the single-particle fields appropriatefor this nucleus. We find for the proton and neutron shell-plus-pairing corrections -3.7023 and -5.3715 MeV, respec-tively. When we do the calculation for single-particle fieldsappropriate for Cn we find that the proton and neu-tron shell-plus-pairing corrections are -3.6305 and -5.3353MeV respectively. So when we change the proton numberby 4 units and implement the corresponding effect on theneutron single-particle field the neutron shell correctionchanges by only 0.0362 MeV. The change in the proton P. M¨oller and T. Ichikawa: A Method to Calculate Fission-fragment Yields Y ( Z, N ) shell-plus-pairing correction is larger because proton fieldchanges more than the neutron field when we change theproton number.In the fission potential-energy code the shapes that canbe studied are described in terms of the three quadraticsurfaces of revolution: two end spheroidal sections and amiddle region that near scission is a hyperboloid of revo-lution [21] as discussed in the previous section, see Eq. (1)and Fig. 1. In the computer code the equations for theseshapes need to be specified; five independent shape param-eters are required for the shape specification. Historicallythe shape parameters α and σ were used [21]. But to moredirectly visualize the shape, we now characterize the shapein terms of five equivalent shape parameters: quadrupolemoment Q , related to the elongation of the shape, neckdiameter d , left fragment spheroidal deformation ǫ f1 , rightfragment spheroidal deformation ǫ f2 , and mass asymme-try α g . The transformations from these parameters to theparameters of the quadratic functions that generate theshapes in the code are very lengthy and non-linear. Theyare described in Ref. [10]. For our discussion here we needto know that α g = M c1 − M c2 M c1 + M c2 = M s1 − M s2 M s1 + M s2 (7)Rather than discussing the asymmetry in terms of nu-cleon number A we can use the above concepts and discussthe charge asymmetry; we have earlier assumed that theproton to neutron ratio is the same in both fragments. Wenow develop an approach to treat different ratios, that isif the proton and neutron numbers are Z s1 and N s1 in oneof the fragments and Z − Z s1 and N − N s1 in the otherfragment we will treat Z s1 N s1 = ZN = Z − Z s1 N − N s1 = Z s2 N s2 (8)It follows from previously that when we discuss the asym-metry in terms of proton number we can write α g = Z s1 − Z s2 Z s1 + Z s2 (9)When we calculate the potential energy for the compoundsystem that should correspond to specific (in our case inte-ger) separated-fragment charge numbers we use Eq. (9) todefine the asymmetry α g of the corresponding compound-nucleus shape for which we calculate proton shell-plus-pairing corrections and macroscopic energies which areneeded in our model. We need additional terms to describehow the macroscopic energy changes when we allow differ-ent proton to neutron ratios in the two fragments, mainlya symmetry-energy effect. We will discuss how to obtainthis effect below. Correspondingly we can define the asym-metry α g for neutrons so that it corresponds to integersplits of neutron number and tabulate the calculated neu-tron shell-plus-pairing corrections. Below we specify howthese results serve as the starting point to obtain the po-tential energy for ratios between the proton and neutronnumbers that are different in the two fragments. We discussed above why the shell-plus-pairing correc-tions for protons and neutrons can be calculated indepen-dently of each other, to a very high degree of accuracy.Therefore we can write E pot ( Z, N, Q , d, ǫ f1 , ǫ f2 , Z s1 , N s1 ) = E mac ( Z, N, Q , d, ǫ f1 , ǫ f2 , Z s1 , N s1 )+ E prots+p ( Z, N, Q , d, ǫ f1 , ǫ f2 , α g ( Z s1 )) (10)+ E neuts+p ( Z, N, Q , d, ǫ f1 , ǫ f2 , α g ( N s1 ))+ E odd Therefore, to obtain the total shell-plus-pairing correc-tions for any fragment split ( Z s1 , N s1 ) we calculate and tab-ulate the proton shell-plus-pairing corrections for a gridin α g corresponding to integer Z s1 (and the corresponding Z s2 ), obtained from Eq. (9). We calculate the neutron shell-plus-pairing correction for a different spacing in α g corre-sponding to integer spacing in N s1 . Thus the 6-dimensionalshell-plus-pairing correction for any mass split ( Z s1 , N s1 ) isthe sum of two 5-dimensional functions.To obtain E mac ( Z, N, Q , d, ǫ f1 , ǫ f2 , Z s1 , N s1 ) we proceedin several steps. First, when we run the code to calculateand tabulate the proton shell-plus-pairing correction forinteger values of Z s1 we also tabulate the macroscopic en-ergy. It will then be obtained for non-integer values N t1 = N × Z s1 Z (11)of N s1 , because the asymmetry variable α g was chosen tocorrespond to integer values of Z s1 .Thus we have tabulated E mac ( Z, N, Q , d, ǫ f1 , ǫ f2 , Z s1 , N t1 ) (12)where we need to remember that here N t1 corresponds to anon-integer value because the asymmetry of the shape waschosen to yield integer Z s1 . The superscript “t” stands for“tabulated”. We need this tabulated value as one term inthe macroscopic energy-model expression we now develop.But we need to obtain the macroscopic energy for (sev-eral different) integer N s1 . It would be difficult to obtainsuch macroscopic energies by developing a model that in-tegrated across the compound nuclear shape for a config-uration with variable proton and neutron densities acrossthe shape. But we now pose that we will get a sufficientlyaccurate model by considering changes in the macroscopicenergy relative to the tabulated macroscopic energy wediscussed in Eq. (12). These changes are mainly due tochanges in the symmetry energies, with considerably smallercontributions from other effects. We can obtain those bysuitable consideration of changes in the macroscopic en-ergy of separated fragments. These we calculate as changes in sum of the macroscopic energy of separated sphericalfragments. Therefore we calculate the sum of the spheri-cal macroscopic energies for two separated nuclei for thespecific fixed Z s1 (related to Eq. (12)) and for a number of . M¨oller and T. Ichikawa: A Method to Calculate Fission-fragment Yields Y ( Z, N ) 5
Table 1.
Spherical fragment macroscopic energies and theirsums for various fragmentations of the compound system
Uleading to fragment charges 52/40; only the neutron fragmen-tations vary. the columns labeled E f1 and E f2 correspond thefirst and second term of line two in Eq. (13). The lowest sum isobtained with the proton to neutron ratio as equal as possiblein the two fragments and to that of the compound system. Theline corresponding to this division is indicated by a “C” in thelast column. Z s1 N s1 E f1 Z s2 N s2 E f2 E f1 + E f2
52 96 -15.95 40 48 -84.661 -100.60852 94 -26.35 40 50 -87.387 -113.74152 92 -36.10 40 52 -88.671 -124.77052 90 -45.16 40 54 -88.592 -133.75152 88 -53.51 40 56 -87.222 -140.73152 86 -61.12 40 58 -84.626 -145.74852 84 -67.97 40 60 -80.868 -148.83952 82 -74.03 40 62 -76.004 -150.032C52 80 -79.26 40 64 -70.088 -149.34852 78 -83.64 40 66 -63.170 -146.80752 76 -87.12 40 68 -55.298 -142.42252 74 -89.68 40 70 -46.515 -136.19952 72 -91.28 40 72 -36.862 -128.14352 70 -91.87 40 74 -26.377 -118.25052 68 -91.42 40 76 -15.097 -106.51552 66 -89.87 40 78 -3.055 -92.926 different N ν : E sepmac ( Z s1 , N ν , Z − Z s1 , N − N ν ) = E sphmac ( Z s1 , N ν ) + E sphmac ( Z − Z s1 , N − N ν ) (13)No odd-particle pairing effects should be included here;those are treated as discussed below. This function is tab-ulated for Z s1 = 52 in Table 1 for fission of U. Wenote that in this integer-spaced grid the minimum energyoccurs for a split where the sum of
Z/N ratios in thetwo fragments is as close as possible to 2 × Z/N of thecompound nucleus. The line corresponding to this split isindicated by a “C” at the very right.We pose that the macroscopic energy for any fragmentdivision ( Z s1 , N s1 ) in the fissioning system is given as E mac ( Z, N, Q , d, ǫ f1 , ǫ f2 , Z s1 , N s1 ) = E mac ( Z, N, Q , d, ǫ f1 , ǫ f2 , Z s1 , N t1 )+ E sepmac ( Z s1 , N ν , Z − Z s1 , N − N ν ) − E sepmac ( Z s1 , N t1 , Z − Z s1 , N − N t1 ) (14)where the last term is calculated by interpolation in thetable corresponding to Eq. (13). As a specific examplewe discuss a fragment division where the charge split is52/40. Then N t1 = 81 .
39. We tabulate the sum in Eq.(13) as the right column in Table 1 and plot this sumfor the specific charge division in our example in Fig. 2. ∆ E
90 88 86 84 82 80 78 76 74 N t Heavy-Fragment Neutron Number N
54 56 58 60 62 64 66 68 70 Light-Fragment Neutron Number N − − − − − − E ne r g y E m a c ( , N ν ) + E m a c ( , − N ν ) ( M e V ) Fig. 2.
Sum of separated-fragment macroscopic energies.
As an example that we can now, in our model, calcu-late the macroscopic energy for any ( Z s1 , N s1 ). To illustratethis we continue to discuss our specific example. Equation14 is a complete specification of the method. Suppose wewant to calculate the macroscopic energy for the partic-ular case E mac (92 , , Q , d, ǫ f1 , ǫ f2 , , N s1 = 64the second term is given by the energy at the upper hori-zontal line and the third term by the energy at the lowerhorizontal line in Fig. 2. Thus in this example we obtainthe macroscopic energy as a sum of the tabulated macro-scopic energy plus ∆E indicated in Fig. 2.One may argue that since some of the fissioning shapesinvolve deformed nascent fragments the terms in Eq. (13)should be evaluated for the corresponding deformed shapes.But for the differences we consider here it only makes aminuscule difference. Let us choose Zr and
Zr in Ta-ble 1 as an illustrative example. The energy difference forthe spherical shapes ∆E sph tabulated is ∆E sph = − . − ( − . .
865 (15)We have evaluated the macroscopic energies for deformedshapes with ǫ = 0 .
5, which is the largest deformation foremerging fragments in our specified deformation grid, seeRef. [10] and obtain ∆E def = − . − ( − . .
846 (16)
P. M¨oller and T. Ichikawa: A Method to Calculate Fission-fragment Yields Y ( Z, N ) Thus, although the absolute energies change considerablythe effect on their differences is completely negligible, only0.019 MeV.Finally we need to account for odd pairing effects inthe emerging fragments. When the fragments are fully de-veloped with zero neck radius of the compound system acommon assumption is that for odd-odd splits the extraodd contribution to the energy should be E odd = 2 × ∆ where ∆ is the pairing-gap parameter, chosen as ∆ = 1 . E odd = 2 × ∆ × ( B W − k odd Z s1 , Z s2 E odd = 0 even Z s1 , Z s2 (17)where, with the choice k = 1, ( B W − k is the shapedependence of the Wigner term in our potential-energymodel. The shape factor B W is 1 for a shapes with no neckand increases continuously, as the neck develops, to 2 forseparated fragments. It is necessary to postulate such ashape dependence because the macroscopic energy of twoseparated fragments contains two Wigner terms, the orig-inal system only one. Without such a shape dependencea discontinuity of the order of 10 MeV would occur atscission of actinide nuclei. That is, if we calculated theenergy for a deforming nucleus up to the scission pointwe would at scission obtain a 10 MeV lower energy thanif we calculated the energy of two approaching separatedfragments. A pedagogical figure illustrating this and thenecessity of this shape dependence is in Ref. [11], Fig. 1.Since we need a realistic potential-energy surface in thescission region we do need to consider these issues (whichhave in many investigations been ignored). The compre-hensive discussion of the shape-dependence of the Wignerterm in Ref. [25] carries directly over to how the effect ofthe pairing ∆ increases as the neck becomes more narrow.There is no known derivation of the Wigner shape depen-dence so it is just postulated, but with consideration ofits limiting behavior [25]. The power constant k , whichwe introduced here to allow some sensitivity studies, gov-erns how early in the division process the character of thetwo fragments causes the “second” Wigner term, or in ourcase, the odd pairing effect, to manifest itself. Below wewill present sensitivity studies on the shape dependenceand on the magnitude of the pairing delta. A calculation of the complete (2D) Y ( Z, N ) yield distri-bution based on the above model would lead to muchmore complex calculations compared to our current 5Dimplementation [1,2,3], because of its 6D nature and as-sociated vastly increased storage requirements, But, as atest of the above approach, we can study many of its as-pects by looking at the odd-even staggering in charge dis-tributions without treating the neutrons separately. We
I-2 I-1Elongation Index I I+1 I+23637383940 F r ag m en t C ha r ge Z Next Track-Point Candidates
Fig. 3.
Two-dimensional slice in the 5D space schematicallyillustrating possible candidate points for the next trajectorystep. The smaller square with gray-filled circles inside, indi-cates the limit originally chosen for next step candidate points.The circles partially filled with gray correspond to transfer ofpaired configurations. obviously then have to calculate the potential energy forfield asymmetries α g that correspond to integer spacingof the proton number Z s1 . For Pu, for each combinationof the other 4 shape variables [10], we calculate the po-tential energies for asymmetries corresponding to chargesplits 47/47, 46/48, 45/49 . . . , and accordingly for otherelements. This corresponds to “averaging” or “summing”over the neutron variable, a procedure we assume has lim-ited effects on the charge distributions obtained relative tosumming over N a calculated complete 2D Y ( Z, N ) dis-tribution. Thus, we obtain in strict analogy with our pre-vious results, on a discreet grid the 5D potential-energymatrix E pot ( I , I , I , I , I ) where I corresponds to thequadrupole moment (or elongation), I to the neck radius, I to spheroidal deformations of one of the emerging frag-ments, I to spheroidal deformations of the other emerg-ing fragment, I to charge asymmetries Z s1 /Z s2 , where thecharge numbers are integers. However, since the calculatedenergy, E comp , does not contain contributions from pair-ing effects in the emerging fragments we add the shape-dependent odd enhancement according to Eq. (17). It turnsout very few modifications of the random walk code arerequired for this calculation.In the BSM model we find the yield distributions bygenerating paths through the potential-energy matrix asfollows. We usually start the path at the ground state(black dot in the schematic 2D Fig. 3). We then selectrandomly one of the 242 surrounding points (circles withgray interior in Fig. 3, only 8 of them in this schematicrepresentation) as a candidate for the next point on thetrack. Suppose the energy difference between this pointand the current point is ∆V . Then, if ∆V < ∆V > P = exp( − ∆V /T ) where T is the temperature; fulldetails are in Refs. [1,2]. When the critical neck radius . M¨oller and T. Ichikawa: A Method to Calculate Fission-fragment Yields Y ( Z, N ) 7 exp. th. with pair transfer ∆ = E * = U
30 40 50 60 Fragment Charge Number Z f C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th.,2 ∆ added for odd Z ∆ = (n,f), E * = U C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th., no odd (n,f), E * = U C ha r ge Y i e l d Y ( Z f ) ( % ) Fig. 4.
Calculated and measured charge yields for neutron-induced fission of
U. The top panel represents the originalmodel, the middle panel has 2 × ∆ added to the potential energyin the matrix locations corresponding to odd charge splits. Inthe bottom panel steps corresponding to two-proton changesin asymmetry are allowed. c = 2 . U, U, Pu, and photon-induced fission with energies centeredaround 11 MeV for
U in Figs. 4, 5, 7, and 6 respec-tively. The experimental data for the (n,f) reactions are ∆ = γ ,f), E * = U
30 40 50 60 Fragment Charge Number Z f C ha r ge Y i e l d Y ( Z f ) ( % ) ∆ = ∆ added for odd Z ( γ ,f), E * = U C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th., no odd ( γ ,f), E * = U C ha r ge Y i e l d Y ( Z f ) ( % ) Fig. 5.
As Fig. 4, but the experimental data are for ( γ ,f) reac-tions leading to E ∗ ≈ −
14 MeV; they include contaminationfrom fission after 1n( ≈ ≈ U at E ∗ = 11 MeV. from Ref. [26], the ( γ ,f) data from Ref. [4]. The top framein each of the four figures is with the original model withno pairing effect added. The middle frames are based onBSM in the 5D potential modified according to Eq. (17).We find little staggering in the calculated curves, althoughwe implemented the “standard” explanation for odd-evenstaggering: for odd-odd splits the potential is on average2 × ∆ (for zero neck radius; less in earlier stages of thefission process) higher than the potential for even splits. P. M¨oller and T. Ichikawa: A Method to Calculate Fission-fragment Yields Y ( Z, N ) exp. th. with pair transfer ∆ = E * = U
30 40 50 60 Fragment Charge Number Z f C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th.,2 ∆ added for odd Z ∆ = E * = U C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th.,no odd (n,f), E * = U C ha r ge Y i e l d Y ( Z f ) ( % ) Fig. 6.
As Fig. 4 but for U. But it is clear that the original formulation of the BSMmodel would never be able to describe odd-even stagger-ing, even after the addition of a pairing effect to the cal-culated potential energy. To illustrate why let us look atFig. 1 and specifically at Z = 38 on the experimentalcurve. Let us assume that Z = 38 is the asymmetry ofthe current point on our evolving track. Because we onlyconsider next-neighbor gridpoints as candidate points forthe next point on the path, we have to populate Z = 39which has a very low yield, to subsequently populate thehigh-yield Z = 40 point, for example. To obtain a pro-nounced staggering with these model features is impossi- exp. th. with pair transfer ∆ = E * = Pu
30 40 50 60 Fragment Charge Number Z f C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th.,2 ∆ added for odd Z ∆ = (n,f), E * = Pu C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th.,no odd (n,f), E * = Pu C ha r ge Y i e l d Y ( Z f ) ( % ) Fig. 7.
As Fig. 4 but for
Pu. ble. But as the shape evolves in the asymmetry directionand two levels cross it is reasonable that an alternativeto breaking a pair and transferring only one proton be-tween the evolving fragments is that a paired two-protonconfiguration could be transferred in one step. We haveimplemented this possibility in the BSM model by alsoallowing Z − Z + 2 as next track-point candidates(shown as circles partially filled with gray in Fig. 3). As al-luded to above transfer reactions indicate that correlatedtransfer of nucleon pairs are common see Refs. [15,16,17].In our current treatment transfer of either a paired con-figuration or breaking of a pair and transfer of one proton . M¨oller and T. Ichikawa: A Method to Calculate Fission-fragment Yields Y ( Z, N ) 9 exp. th.,1 or 2 steps in asym, frag.def, Q ∆ = E * = Pu
30 40 50 60 Fragment Charge Number Z f C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th.,1 or 2 steps in asym, frag.def. ∆ = E * = Pu C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th, no ∆ for odd conf., but with pair trans. ∆ = E * = Pu C ha r ge Y i e l d Y ( Z f ) ( % ) Fig. 8.
Calculated charge yield for neutron-induced fission of
Pu in the BSM model for different assumptions about pair-ing and next-step grid-point candidates. In the top panel nopairing effect is added to the potential energy for odd splits, butone or two steps in asymmetry is implemented. In the middlepanel we add a pairing effect to odd splits and permit as nextstep candidates one or two grid points in asymmetry and bothof the fragment deformations. In the bottom panel we havealso allowed one or two steps in the elongation coordinate Q . can occur. Typical excitation energies at the end of theasymmetric tracks, see Ref. [12] are 7.7–11.6 MeV, andlower earlier in the shape evolution. Our consideration of exp. th. with ( B w − ∆ = E * = Pu
30 40 50 60 Fragment Charge Number Z f C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th. with ( B w − ∆ = E * = Pu C ha r ge Y i e l d Y ( Z f ) ( % ) exp. th. ∆ = E * = Pu C ha r ge Y i e l d Y ( Z f ) ( % ) Fig. 9.
Study of yield sensitivity to the magnitude of the pair-ing ∆ and the onset of fragment pairing effects. Except for thevery small pairing effects on which the bottom panel is basedthe results are quite robust and similar to the standard resultin the bottom panel in Fig. 7. paired configurations is quite consistent with the resultsof Ref. [27], where in one example at 8.4 MeV excitationthe proportion of paired configurations is 36%.The calculated yields with transfers of correlated pairsallowed as next track-point candidates are in the bottompanels in Figs. 4–7. Staggering is now obvious in the cal-culated results and in close agreement with the experi-mental data. In the calculated results in Figs. 4–7 it may Y ( Z, N ) seem that the crucial generalization that we introducedto describe the odd-even staggering is not the first stepwe took, namely the addition of 2 × ∆ to the odd chargesplits, but the second step in which we permitted a changein Z of two units, corresponding to a transfer of a pairedproton configuration. But what if we had implementedthis as a first step? We have investigated this possibilityand show in the top panel of Fig. 8 the calculated chargeyield for neutron-induced fission of Pu with no pairingenergy added to the odd charge splits, but with both oneand two steps in Z permitted as next candidate points onthe random-walk trajectory. There is no odd-even stag-gering in the calculated curve which is extremely similarto the calculated yield in the top panel in Fig. 7. It is in-teresting to observe that although allowing both one andtwo steps in Z effectively corresponds to increasing thespeed of motion in asymmetry, or equivalently increasingthe distance between grid points, there is little change inthe calculated yield curve. In the middle panel of Fig. 8we have added (fractions of) 2 ∆ to the potential energiesfor odd charge splits, allow one or two steps in asymmetry,but also allow one or two steps in the fragment deforma-tion shape variables. There is little difference compared tothe bottom panel in Fig. 7. This again shows, as was ear-lier pointed out [2] that the calculated yield curves in theBSM model do not depend sensitively on most changes inthe deformation grid. In the bottom panel of Fig. 8 wehave furthermore allowed one or two steps in the elonga-tion variable Q . In this case there is a noticeable effecton the calculated yield. Now the calculated distributionis slightly wider than the experimental results. In Ref. [2]we showed that tripling the number of points in Q led toa calculated distribution that was narrower than the ex-perimentally observed one. Obviously, extreme changes inthe grid will influence the calculated results. For exampleif we were to use only three grid points in the elongationvariable we would not obtain realistic yields.Finally we study, for Pu, the sensitivity to variationsof the postulated shape dependence that governs the on-set of odd-even effects on the potential energy and to themagnitude of the pairing ∆ . The results are shown in Fig.9. The top panel shows the effect of increasing the pairing ∆ by 20% relative to our standard assumption of ∆ = 1 . ∆ is a non-issue in this first study of odd-even stagger-ing. In larger systematic studies a well-defined prescriptionshould obviously be introduced. The middle and bottompanels show the results for different forms of the shapefactor ( B W − k which governs the rate with which theeffect of the odd-even pairing ∆ in the fragments affectsthe calculated potential energy as scission is approached.The quantity B w is close to 1.5 at our selected scissionconfiguration. Therefore, in the middle panel the factor( B W − . is 0.7, so that for the odd configurations 1.4MeV is added, less earlier in the division process. The stag-gering here is only very slightly larger than in our standardcalculation in the bottom panel of Fig. 7 where 1.0 MeV isadded in the scission region. In the bottom panel of Fig. 9 the shape factor ( B W − comes out to 0.25 so that only0.5 MeV or less is added to the odd-odd divisions. Herethe staggering is much reduced, as might be expected.However from these sensitivity studies we conclude thatthe results are quite robust for reasonable variations ofassumptions about the onset of fragment character on thepotential energy as well as to the magnitude of the pairing ∆ . In summary we have shown that to describe odd-evenstaggering in the BSM model we must add simultane-ously two effects: (1) odd-even effects on the calculatedpotential-energy surface and (2) allow transfers of corre-lated paired proton configurations; they work together inthe development of odd-even staggering.As future “perspectives for the next decade” we antic-ipate that to develop more accurate descriptions we needto – implement the extensions we discuss here into a com-puter model framework and calculate the full Y ( Z, N )fission-fragment yield distributions (a two-dimensionalfunction of neutron and proton number), – treat from more basic principles how the number ofpaired configurations decrease with excitation energy(see Ref. [27] for a discussion) which should influencethe probability with which a point corresponding totwo-nucleon transfer is chosen as the next candidatepoint on the trajectory, – calculate the damping of shell effects based on actualsingle-particle structure rather than use a parameter-ized approach. – understand issues related to the deformation grid. Clearlythe calculated yields do depend on the selection of thegrid. To take an obvious example, were we to have only3 grid points in the elongation direction we would notobtain any realistic yields. However, in the extensivesensitivity studies in Ref. [2] it was shown that theyields were remarkably insensitive to many types ofgrid changes, as is also shown by the results of Figs. 8and 9 above.Discussions with A. Sierk, A. Iwamoto, and J. Ran-drup are appreciated. This work was supported by travelgrants for P.M. to JUSTIPEN (Japan-U.S. Theory Insti-tute for Physics with Exotic Nuclei) under grant numberDE-FG02-06ER41407 (U. Tennessee). This work was car-ried out under the auspices of the NNSA of the U.S. De-partment of Energy at Los Alamos National Laboratoryunder Contract No. DE-AC52-06NA25396. TI was sup-ported in part by MEXT SPIRE and JICFuS and JSPSKAKENHI Grant no. 25287065. References
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