Angular Momentum of Fission Fragments from Microscopic Theory
LLLNL-JRNL-818187
Angular Momentum of Fission Fragments from Microscopic Theory
P. Marevi´c, N. Schunck, J. Randrup, and R. Vogt
1, 3 Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Physics and Astronomy Department, University of California, Davis, CA 95616, USA (Dated: January 12, 2021)The angular momentum of fragments has a significant impact on neutron and photon emission inthe fission process, yet it is still poorly understood. In this Letter, we provide the first microscopiccalculations of angular momentum distributions in fission fragments for a wide range of fragmentmasses. For the benchmark case of
Pu, we show that the angular momentum of the fragments islargely determined by the nuclear shell structure and deformation, and that heavy fragments there-fore generally carry less angular momentum than light fragments. Simulations with the completeevent fission model
FREYA show that the microscopic distributions modify the predicted number ofemitted photons. This work is an important advance toward modeling the fission process based oninputs from microscopic theory.
Introduction. —
Nuclear fission plays an essential rolein fundamental and applied science, but even eighty yearsafter its discovery [1, 2] the microscopic foundation ofthe phenomenon is yet to be fully understood [3, 4]. Ina simple picture, fission can be viewed as the shape evo-lution of a charged liquid drop that gradually deformswhile surmounting multiple potential barriers and finallyassumes an extremely deformed shape at scission [5]. Fis-sile nuclei predominantly split into two fission fragments(FFs) that move apart due to the mutual Coulomb re-pulsion. Simultaneously, FFs relax toward their equilib-rium shapes and then deexcite by sequential emission ofneutrons and photons. The angular momentum (AM) ofFFs has an important impact on this deexcitation pro-cess, causing the neutron emission to be anisotropic andaffecting the number of emitted photons significantly [6–10]. Therefore, it is important to have a robust micro-scopic framework for calculating AM of FFs over a widerange of fragment masses. However, such a framework iscurrently missing, and the most widely used fission re-action models at the moment rely on AM distributionsthat are obtained from either statistical or semi-classicalmethods [10–15].Currently, nuclear density functional theory (DFT) isthe only fully quantum-mechanical approach capable ofdescribing many facets of the fission phenomenon [4, 16].Starting from global effective interactions and generalmany-body techniques [17], DFT models were success-fully employed in studies of spontaneous fission half-lives[18], FF mass and charge distributions [19–25], and en-ergy sharing among the FFs [26–29]. The few attemptsat describing angular momentum of FFs within DFTmodels elucidated the fundamental role of the quantalorientation pumping mechanism [30] and of nuclear de-formation [31] in generating the AM of FFs. However,these studies contained a number of limiting approxima-tions, not least of which is the strong assumption thatFFs can be represented as the ground states of isolatednuclei with the same number of nucleons. Furthermore, they were restricted to either even values of AM in a fewchosen nuclides [31] or to the specific scenarios of equalsharing of AM among the FFs [30]. Consequently, theywere not able to provide a quantitative framework forpredicting AM distributions over the wide range of FFsthat is typically observed in experiments [32, 33].In this Letter, we report the first microscopic predic-tions of angular momentum distributions in fission frag-ments for a wide range of fragment masses. Starting froma large set of scission configurations, we use projectiontechniques to extract the AM distributions of 26 differ-ent fragmentations of
Pu. We find that the AM ofFFs is largely determined by the underlying shell struc-ture and deformation of FFs at scission. As a result,the heavy FFs on average carry less angular momentumthan the light FFs. By adapting the fission simulationmodel
FREYA to reproduce the microscopic AM distribu-tions for each pair of FFs, we are able to determine theeffect on the subsequent neutron and photon emission,thus paving the way toward fission modeling based oninput data from microscopic theory.
Method. —
We first determine a set of scission config-urations by solving the Hartree-Fock-Bogoliubov (HFB)equations with the
HFBTHO package [34], using the SkM*parameterization of the Skyrme energy functional [35]and a mixed volume-surface contact pairing force [36].We impose constraints on the values of the axially-symmetric quadrupole ( q ) and octupole ( q ) moments[37], corresponding to the elongation and the mass asym-metry of the nuclear shape, respectively. In addition, weconstrain the expectation value of the neck operator ( q N ),which estimates the number of particles in a thin neckconnecting the two fragments [38]. This approach en-ables us to explore a three-dimensional q ≡ ( q , q , q N )hypersurface in collective space and to generate a largeset of scission configurations [39]. For the benchmarkcase of Pu, we considered a total of 1482 configura-tions | Φ( q ) (cid:105) with neck values 1 . ≤ q N ≤ . a r X i v : . [ nu c l - t h ] J a n upplemental Material [40] for more details on technicalaspects of the HFB calculation and on the properties ofscission configurations.The scission configurations have axially-symmetricdensities that are dumbbell-shaped and can readily bedivided into heavy ( z < z N ) and light ( z > z N ) FFs,where z N locates the minimum of the density. By adapt-ing standard symmetry restoration techniques [41–43] tothe case of FFs, angular momentum distributions of theheavy ( F = H ) and the light fragment ( F = L ) for eachconfiguration | Φ( q ) (cid:105) can be calculated as | a FJ ( q ) | = (cid:90) β (cid:104) Φ( q ) | ˆ R Fy ( β ) | Φ( q ) (cid:105) , (1)where (cid:82) β ≡ ( J + ) (cid:82) π dβ sin β d J ∗ ( β ) denotes integrationover the rotational angle β with Wigner matrix elements d J ( β ) = P J (cos β ) [44] ( P J is Legendre polynomial oforder J ) as weights, and ˆ R Fy ( β ) = exp( − iβ ˆ J Fy ) is therotation operator for fragments. The angular momentumoperators ˆ J Fy are defined within the spatial region S F containing each fragment [45, 46]. They are computedfrom the associated kernels, J Fy ( r , σ ) = Θ F ∗ ( z − z N ) J y ( r , σ )Θ F ( z − z N ) , (2)where J y ( r , σ ) = L k ( r ) + S k ( σ ) corresponds to the usualangular momentum operator that depends on the spa-tial coordinates r ≡ ( r ⊥ , φ, z ) and the spin coordinate σ ,Θ H ( z − z N ) = 1 − H ( z − z N ), Θ L ( z − z N ) = H ( z − z N ),and H ( z ) is the Heaviside step function [47]. The centerof mass of each fragment is located at r F CM = (0 , , z F CM ).Therefore, we take r → r − r F CM in Eq. (2) to determinethe angular momentum with respect to the center of massof each fragment. To ensure a proper convergence of in-tegrals in Eq. (1) for all J values, N β = 60 rotationalangles are taken into account. In addition, we emphasizethat | Φ( q ) (cid:105) are expanded in a basis which is not closedunder rotation [48]. Therefore we must employ the re-cently introduced technique of symmetry restoration inincomplete bases [49] to evaluate Eq. (1).The number of particles in FFs can be calculated asan integral of the total density over the subspace S F containing each fragment. In this sense, our calcula-tion represents a mapping of a set of collective variables q in the compound nucleus ( Z, A ) on a set of chargesand masses ( Z F ( q ) , A F ( q )) in the two fragments. Thesecharges and masses satisfy Z H ( q ) + Z L ( q ) = Z and A H ( q ) + A L ( q ) = A but are generally not integers. Inprinciple, one should combine the outlined method withthe particle number projection (PNP) in FFs [23, 50, 51],a formidable task that is yet to be undertaken. To extractthe desired quantities for integer numbers of particles ineach fragment, we instead perform a Gaussian Processinterpolation [52, 53] over a subset of scission configura-tions with both the proton and the neutron number inthe vicinity of the target values. Finally, we briefly note that, due to the axial symmetry, we cannot extract odd J if q F →
0. To account for these rare cases, we solveEq. (1) for even J , interpolate for odd J , and normalizethe entire distribution. We verified that this proceduresmooths the distributions of q F → Results. —
Starting from a set of 1482 scission con-figurations, we determined the AM distributions | a FJ | of26 different integer fragmentations of Pu in the massrange 126 ≤ A H ≤ Pu( n th ,f) [32, 33]. We note thatour collective space q typically enables us to extract onlyone Z value per each A value: without PNP, obtain-ing a broader charge dispersion would require enlargingthe collective space by including triaxial deformations orisospin-breaking variables. Furthermore, the initial setwas comprised of a wide range of quadrupole and oc-tupole deformation parameters of the FFs, β F ( q ) ≤ . | β F ( q ) | ≤ .
5, where β Fλ = (4 π ) / (3 A F R λF ) q Fλ , q Fλ are the multipole moments [54] of the FFs, and R F = 1 . A / F fm. We used the same Gaussian Processprocedure to extract these parameters for the integer-valued FFs [40]. We obtained FFs with a wide range ofquadrupole deformations and confirmed the findings ofRefs. [28, 55] that FFs are octupole-deformed at scission.In Fig. 1, we show the calculated average values ofAM for (a) the heavy FFs and (b) the light FFs in the A − Z plane. The average values J F are calculated as J F ( J F + 1) = (cid:80) J =0 J ( J + 1) | a FJ | . In addition, the areaof the circles is proportional to the quadrupole deforma-tion parameter β F of FFs calculated at scission. Theheavy FFs display a wide range of angular momenta andquadrupole deformations that appear to be strongly cor-related. The minimal value J H ≈ . (cid:126) is found for Sn,at a proton shell closure and one neutron away from thedoubly-magic
Sn nucleus. We note that the latter isnot a part of our set of 26 fragmentations. At the otherend, the rare-earth
Pr is highly deformed and on aver-age carries substantial angular momentum ( J H ≈ . (cid:126) ).The average values around the most likely fragmentation( A H /A L = 136 / − (cid:126) . On theother hand, we find that the light FFs are substantiallymore deformed and therefore on average carry more an-gular momentum, typically 7 − (cid:126) . This result is atodds with predictions of the widely used phenomenolog-ical models such as FREYA [10–12] and
CGMF [13–15]. Inthese models, the AM of the FFs are calculated usinggeneric moments of inertia that do not include structureor deformation effects. The corresponding moments scalewith mass, I H > I L , and the heavy fragment is generallyfavored, J H > J L . Shell effects play an essential role inthe fission process, from modifying the structure of fis-sion barriers [56] to enabling asymmetric fragmentations2
25 130 135 140 145 150 155 A Z H = 0.1 H = 0.5 (a) average angular momentum (in )
85 90 95 100 105 110 115 A Z L = 0.1 L = 0.5 (b) average angular momentum (in ) FIG. 1. Average angular momentum of (a) the heavy fragments and (b) the light fragments in
Pu( n th ,f). The circles havean area proportional to the quadrupole deformation parameter β F extracted for the fragments at scission. and stabilizing octupole deformation in fission fragments[54]. Here, we demonstrate that they also determine theangular momentum of fission fragments, hindering theirvalues significantly in the vicinity of the doubly magic Sn nucleus.To shed more light on the effect of the underlying shellstructure on the angular momentum of FFs, in the leftpanel of Fig. 2, we display the AM distributions of fournuclides in the vicinity of the Z = 50 and N = 82 magicnumbers. Sn, with N = 78, is four neutrons from thedouble shell closure and has an average angular momen-tum J H ≈ . (cid:126) . The situation drastically changes as weapproach N = 82. In particular, Sn and
Sn carry anangular momentum of only J H ≈ . (cid:126) and J H ≈ . (cid:126) ,respectively. However, changing only one neutron to aproton leads to another abrupt change since Sb isno longer a closed-shell nucleus and carries as much as J H ≈ . (cid:126) . It is remarkable that HFB ground states ofall three Sn isotopes have β = 0 and Sb has β ≈ δ functions, δ J . This highlights a lim-itation of simpler microscopic models that describe fis-sion fragments with the corresponding ground-state wavefunctions at zero temperature [30, 31].The angular momentum distributions of the mostlikely fragmentations in Pu and
U have also beenstudied very recently [57] within the framework of time-dependent density functional theory (TDDFT) for super-fluid systems [28, 29]. In the context of nuclear fission,TDDFT is currently the only microscopic framework thatcan describe spatially well-separated, highly-excited FFswhose large deformations acquired at scission have re-laxed to their equilibrium values. On the other hand,the TDDFT approach can only account for a very narrowwindow of masses around the most likely fragmentation [29], preventing its application to the entire range of FFsrelevant in fission. Such an approach is therefore comple-mentary to the present model, which can describe a widerange of FFs, with the caveat that these mostly cold frag-ments are still connected by a thin neck. Comparing theirrespective predictions might thus offer a valuable insightinto the microscopic mechanism of generation and evo-lution of angular momentum in FFs. In the right panelof Fig. 2, we display the AM distributions of the heavyand light FFs for the most likely ( A H /A L = 136 / Pu, which can be directly comparedto Fig. 1 in Ref. [57]. The average angular momentum ofthe heavy FF in our calculation, J H ≈ . (cid:126) , matches thevalue reported in [57]. In addition, J L ≈ . (cid:126) is onlymarginally higher than their reported value 9 . . (cid:126) ,where the uncertainty stems from considering differentinitial conditions in the TDDFT calculation. This is re-markable, bearing in mind that the two models deal withFFs at different points in time and the correspondingdeformations are therefore also different. It might indi-cate that most of the angular momentum imparted toFFs during thermal-neutron-induced fission is generatedat scission as a consequence of a large deformation ofthe nuclear shape. The TDDFT model further propa-gates the corresponding configurations in time, allowingconversion of the deformation energy into internal ex-citations. This effectively increases the temperature ofthe FFs, often assumed to be an adjustable parameter inphenomenological models. However, the angular momen-tum content of FFs from scission to their relaxed shapemight in fact remain rather stable as a function of timedue to the canceling effects of decreasing deformation andincreasing temperature. Extending the TDDFT calcula-tions to other fragmentations or to earlier times duringthe relaxation period could therefore shed more light on3 J | a J | (a) Sn Sn Sn Sb J | a J | (b) I Nb FIG. 2. Angular momentum distributions of fission fragments in
Pu( n th ,f). Panel (a) shows distributions for four heavyfragments in the vicinity of the Z = 50 and N = 82 magic numbers. Symbols represent calculated integer- J points andthe lines are obtained by a simple spline interpolation. Shaded areas represent 2 σ (95%) confidence intervals stemming fromthe Gaussian Process interpolation. Panel (b) shows distributions for the heavy and the light fragment of the most probable( A H /A L = 136 / the microscopic origin of angular momentum in FFs.Finally, we employed the complete event fission model FREYA to assess the impact of the microscopic AM distri-butions on neutron and photon multiplicities.
FREYA [10–12] is a Monte Carlo model that generates large samplesof complete fission events, providing the full kinematicinformation for the two product nuclei and the emittedneutrons and photons in each event. Inputs to
FREYA areprimarily based on experimental data, though a certaindegree of modeling is necessary. In particular,
FREYA ob-tains the AM of the FFs from a statistical sampling ofthe dinuclear wriggling and bending modes, employingan adjustable spin temperature T S [9, 10]. As it turnsout, for each fragmentation considered, it is possible toadjust T S such that the AM distribution generated by FREYA closely matches the one extracted microscopically.Consequently, we can determine the neutron and photonobservables resulting from the microscopic AM distribu-tions. For several selected fragmentations, we generatedtwo hundred thousand fission events and compared thedefault
FREYA results with those associated with the mi-croscopic AM distributions. We find that the neutronmultiplicities depend rather weakly on the AM distribu-tion, in agreement with the previously reported obser-vation that the rotation of FFs influences the directionof emitted neutrons but not their number [10]. On theother hand, the number of emitted photons is signifi-cantly modified, as can be seen from Table I. In partic-ular, the microscopic calculation predicts lower J H val-ues around the most likely fragmentation and especiallyin the Sn region, leading to lower photon multiplici-ties. The trend reverses at larger and smaller A H , wheremore pronounced FF deformations lead to larger photonmultiplicities. The large number of microscopic fragmen- tations generated for this study is still insufficient to al-low a full-fledged FREYA simulation that could be directlycompared with experimental data on photon multiplici-ties, which range between N γ = 6 . ± .
35 [58] and N γ = 7 . ± . TABLE I. Total average photon multiplicities N γ for severalfragmentations of Pu calculated with
FREYA , based on themicroscopic DFT or default
FREYA
AM distributions. Massyields Y ( A H ) and charge yields Y ( Z H ) (each normalized to100 for the heavy fragment only) used in FREYA [11] are alsolisted as estimates of the relative importance of each fragmen-tation.( Z H , A H ) Y ( Z H ) Y ( A H ) N γ (DFT) N γ (Default)(50 , . .
04 11 .
25 8 . , . .
85 5 .
17 7 . , . .
03 7 .
86 8 . , . .
09 6 .
36 7 . , . .
38 9 .
21 7 . Conclusion. —
This work represents the first attemptto extract the angular momentum distributions for alarge number of fission fragments from a fully microscopictheory. Our calculations reveal an important impact ofthe underlying shell structure on the angular momentaof fission fragments, hindering the ability of heavy frag-ments around
Sn to carry angular momentum. Theydemonstrate the importance of using a predictive theoryto compute properties that can vary significantly fromone fragment to the next and they could thus providevaluable guidance for nuclear data evaluation. Our treat-4ent may also be useful for predicting the decay of frag-ments in very neutron-rich nuclei, such as those involvedin nucleosynthesis processes, where experimental data areunavailable.We thank A. Bulgac for critical reading of themanuscript. P. M. and N. S. acknowledge stimulatingdiscussions with M. Verri`ere and D. Regnier. This workwas performed under the auspices of the U.S. Departmentof Energy by Lawrence Livermore National Laboratoryunder Contract DE-AC52-07NA27344. J. R. acknowl-edges support from the Office of Nuclear Physics in theU.S. Department of Energy under Contracts DE-AC02-05CH11231. Computing support for this work came fromthe Lawrence Livermore National Laboratory (LLNL) In-stitutional Computing Grand Challenge program. [1] O. Hahn and F. Strassmann. ¨Uber den Nachweis unddas Verhalten der bei der Bestrahlung des Urans mit-tels Neutronen entstehenden Erdalkalimetalle.
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Cf.
Phys. Rev. C , 7:1173–1185, 1973., 7:1173–1185, 1973.