Microscopic Calculation of Fission Product Yields with Particle Number Projection
LLLNL-JRNL-818368
Microscopic Calculation of Fission Product Yields with Particle Number Projection
Marc Verriere ∗ and Nicolas Schunck Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, California 94551, USA
David Regnier
CEA, DAM, DIF, 91297 Arpajon, France andUniversit´e Paris-Saclay, CEA, LMCE, 91680 Bruy`eres-le-Chˆatel, France (Dated: February 5, 2021)Fission fragments’ charge and mass distribution is an important input to applications ranging frombasic science to energy production or nuclear non-proliferation. In simulations of nucleosynthesis orcalculations of superheavy elements, these quantities must be computed from models, as they areneeded in nuclei where no experimental information is available. Until now, standard techniques toestimate these distributions were not capable of accounting for fine-structure effects, such as the odd-even staggering of the charge distributions. In this work, we combine a fully-microscopic collectivemodel of fission dynamics with a recent extension of the particle number projection formalism toprovide the highest-fidelity prediction of the primary fission fragment distributions for the neutron-induced fission of
U and
Pu. We show that particle number projection is an essential ingredientto reproduce odd-even staggering in the charge yields and benchmark the performance of variousempirical probability laws that could simulate its effect. This new approach also enables for the firsttime the realistic determination of two-dimensional isotopic yields within nuclear density functionaltheory.
I. INTRODUCTION
A predictive theory of nuclear fission has been a long-standing challenge of nuclear science that has gained re-newed interest in recent years [1]. While fission is afascinating problem on its own, as it involves the large-amplitude collective dynamics of a strongly-interactingquantum many-body system, it also plays an importantrole in both fundamental science and technological appli-cations. For example, fission is a primary decay mecha-nism of superheavy elements [2] and plays a crucial role inthe rapid neutron capture process at the origin of heavyelements in the universe [3]. Progress in these disciplinesrequires accurate and precise fission data, such as thedistribution and full characteristics (charge, mass, exci-tation energy, spin, level density, etc.) of the fragmentsformed during the process. However, the ensemble of allthese fissioning nuclei covers a vast area of the nuclearchart. While precise data are available on experimen-tally accessible nuclei, information on very short-lived,neutron-rich systems out of reach of experimental facili-ties must come from theoretical predictions.There are many approaches geared toward describ-ing low-energy neutron-induced fission reactions [4–12].Among them, collective models have been particularlysuccessful in predicting fission fragment distributions [10,13–20]. These models are based on the identification ofa few collective variables driving the fission process, thecalculation of a potential energy surface in the resultingcollective space, and the explicit time-dependent simula-tion of collective motion on top of these surfaces [4]. Ba-sic fission fragment properties such as proton- or neutron- ∗ [email protected] number are mapped to the fissioning nucleus’ characteris-tics, such as its deformation. Until now, particle numberestimates were obtained by simply integrating the den-sity of particles in the prefragments. This local averagingmade it impossible to predict fine structure effects suchas the odd-even staggering in the fission fragments chargedistributions.In the seminal work of Refs. [21, 22], the authors intro-duced a new method based on particle-number projectiontechniques to predict particle transfer in heavy-ion reac-tions in the context of time-dependent density functionaltheory. In [23], this method was applied to calculate thedispersion in particle number for the most probable scis-sion configuration of Pu(n,f) fission and showed thatodd-even staggering naturally emerged. By further com-bining particle-number projection in the fission fragmentswith a strongly-damped random walk on semi-classicalpotential energy surfaces, the authors of [24] showed thatit is possible to predict odd-even staggering but also thecharge polarization of the fission fragments distribution.This paper aims to combine particle-number projectionin fission fragments with a quantum-mechanical theory oflarge-amplitude collective dynamics to predict the uncor-related mass and charge fission fragment distributions be-fore prompt emission. We investigate the role of projec-tion for reproducing odd-even staggering effects in frag-ment distributions and discuss how various phenomeno-logical probability distributions could approximate exactresults. We also present the first two-dimensional iso-topic yields predicted with such a microscopic approachand their evolution as a function of excitation energy.In Section II, we describe in detail our theoret-ical framework. It includes a short reminder onthe time-dependent generator coordinate method underthe Gaussian overlap approximation with Hartree-Fock- a r X i v : . [ nu c l - t h ] F e b Bogoliubov generator states, a comprehensive presenta-tion of the method used to extract fission yields from thetime evolution of a collective wave packet, and a discus-sion of the various methods to estimate particle numberdispersion in prefragments. Section III contains the re-sults of our calculations for the two important cases of
U(n,f) and
Pu(n,f) low-energy fission, focusing onthe impact of particle number projection and the evolu-tion of the yields as a function of excitation energy.
II. THEORETICAL FRAMEWORK
Our goal is to predict the initial, or primary, fissionfragment mass and charge distributions before promptemission of neutrons and gammas. These quantities aredetermined by populating scission configurations by solv-ing a collective Schr¨odinger-like equation in the collectivespace spanning nuclear deformations. This method, pre-sented in Sec. II A, relies on (i) defining a set of collectivevariables and (ii) a basis of many-body generator statescalculated with the constrained Hartree-Fock-Bogoliubovmethod. They are used both in the definition of the po-tential energy surface and of the collective inertia. InSec. II B, we present how fission yields can be extractedby combining the population of scission configurationsand an estimate of particle number distributions at eachconfiguration.
A. Collective Dynamics
Our model of fission dynamics is based on the time-dependent generator coordinate method (TDGCM) un-der the Gaussian overlap approximation (GOA) withHartree-Fock-Bogoliubov (HFB) generator states [25].
1. The Time-Dependent Generator Coordinate Method
The central assumption of the TDGCM is the possi-bility to decompose the many-body wave function | Φ( t ) (cid:105) ,at each time t , as a superposition of many-body states | φ ( q ) (cid:105) . It reads | Φ( t ) (cid:105) = (cid:90) d q | φ ( q ) (cid:105) f ( q , t ) , (1)where q labels the collective degrees of freedom associ-ated, in our case, with average deformations of the fis-sioning system. The complex-valued weights f ( q , t ) canbe explicitly determined by solving the non-local Hill-Wheeler-Griffin (HWG) equation [25, 26]. However, suchan approach is extremely time-consuming. Consequently,we adopt the Gaussian Overlap Approximation (GOA) totransform the HWG equation into a local, Schr¨odinger-like equation, i (cid:126) ∂∂t g ( q , t ) = H coll ( q ) g ( q , t ) , (2) where g ( q , t ) is an unknown complex-valued functionthat encodes the collective dynamics. The collectiveHamiltonian H coll ( q ) is defined as H coll ( q ) = − (cid:126) (cid:112) γ ( q ) ∇ q (cid:112) γ ( q ) B ( q ) ∇ q + V ( q ) . (3)In this expression, γ ( q ) is the metric of the collectivespace, B ( q ) is the inertia tensor and V ( q ) = E ( q ) − ε ( q ),where E ( q ) is the HFB energy of the fissioning systemat point q and ε ( q ) is a sum of zero-point-energy correc-tions. All these quantities are extracted from the states | φ ( q ) (cid:105) and a nuclear energy functional that encodes thenucleon-nucleon interactions; see [4, 27, 28] for additionaldetails.To ensure that we can describe asymmetric fission,where the existence of light and heavy fragments can bemapped to parity-breaking shapes of the fissioning nu-cleus at scission, we will work in a two-dimensional col-lective space spanned by the generic vector q = ( q , q ),where the first degree of freedom, q , is the average ofthe quadrupole moment operator ˆ Q on the generatorstate | φ ( q ) (cid:105) , and the second one, q , is the average of theoctupole moment operator ˆ Q on that same state, q = (cid:10) φ ( q ) (cid:12)(cid:12) ˆ Q (cid:12)(cid:12) φ ( q ) (cid:11) (cid:104) φ ( q ) | φ ( q ) (cid:105) q = (cid:10) φ ( q ) (cid:12)(cid:12) ˆ Q (cid:12)(cid:12) φ ( q ) (cid:11) (cid:104) φ ( q ) | φ ( q ) (cid:105) . (4)In practical calculations, an additional constraint on thedipole moment operator ˆ Q must be added to fix theposition of the center-of-mass of the nucleus.The initial collective wave-packet g ( q , t = 0) is con-structed from the eigenstates g extra k ( q ) with eigenvalues E k of the static GCM+GOA equation in an extrapolatedfirst potential well, as explained in [29]. It reads g ( q , t = 0) ∝ (cid:88) k exp (cid:20) ( E k − ¯ E ) σ (cid:21) g extra k ( q ) . (5)The parameter σ is a free parameter of the model thatcontrols the energy spread of the wave packet. Given σ ,we adjust ¯ E to set the initial wave packet’s energy relativeto the non-extrapolated Potential Energy Surface (PES)defined by V ( q ). The wave-packet is absorbed at the bor-ders of the deformation domain by adding an imaginaryterm in Eq. (2) to avoid spurious reflections [16].
2. Generator States
We assume that all the many-body states | φ ( q ) (cid:105) of thePES are fermionic Bogoliubov vacuums. These productstates of quasiparticles are defined as | φ ( q ) (cid:105) ≡ (cid:89) µ ˆ β ( q ) µ | (cid:105) , (6)where ˆ β ( q ) µ = (cid:88) k (cid:0) U ( q ) ∗ kµ ˆ a k + V ( q ) ∗ kµ ˆ a † k (cid:1) . (7)The operators ˆ a k and ˆ a † k are the ladder operators as-sociated with an arbitrary single-particle basis, and thecomplex numbers U ( q ) kµ and V ( q ) kµ define the quasi-particlestate µ at point q . They are obtained through standardconstrained HFB calculations with the constraints q ofEq. (4) on the collective variables.This work is a follow-up study of [16, 30, 31]. There-fore, we make the exact same choices concerning thenuclear energy functional: the nuclear part of the to-tal energy is computed from the Skyrme energy densityfunctional (EDF) with the Skm * [32] parametrization; thepairing part of it is calculated from a surface-volume,density-dependent pairing force locally adjusted to re-produce the odd-even staggering of Pu, with a cut-offof 60 MeV; the Coulomb exchange part is computed atthe Slater approximation. The HFB equation is solvedby expanding the solutions in the stretched harmonic os-cillator basis; see [30] for numerical details. Practicalcalculations were performed with the
HFODD solver [33].
3. Collective Inertia
It is well-known that the GCM theory does not re-produce the asymptotic limit in the case of translationalmotion when considering only a collective subspace oftime-even generator states q [34]. In principle, this limi-tation can be remedied by doubling the number of degree-of-freedom via the introduction of conjugate momenta p for each collective variable q [35]. However, it has notbeen attempted in realistic calculations yet and remainsto be verified [1]. Instead, it has become customary infission calculations to use the adiabatic time-dependentHartree-Fock-Bogoliubov (ATDHFB) approach to esti-mate the collective inertia, even though it breaks the in-ternal consistency of the theory [25]. Better methodsare slowly becoming available [36, 37]. Since this workfocuses on the impact of particle number projection tech-niques, we will simply use the GCM inertia at the crank-ing approximation (only the diagonal part of the RPAmatrix is considered and time-odd terms are neglected)with derivatives computed locally (“perturbative” crank-ing); see [4] for additional details. B. Fission Fragments Distributions
In practical applications, fission models that rely onthe calculation of a potential energy surface (e.g., thesemi-classical random walk and Langevin approaches orthe fully-microscopic TDGCM) cannot correctly describethe separation of the fissioning nucleus into two frag-ments that are simultaneously completely separated anddo not interact anymore (other than through Coulomb),yet also excited. Instead, quantities such as fission frag-ment distributions are computed just before scission,which is defined, somewhat arbitrarily, based on severalpossible criteria; see discussions in [4, 38].
1. Population of Scission Configurations
Following common practice, we define scission config-urations through the average value of the operator, ˆ Q N ,counting the number of nucleons in the neck. It is de-fined, as in Ref. [39], byˆ Q N = e − ( z − z N ) /a . (8)This expression contains two parameters. The neck lo-cation, z N , is the point between the two prefragmentsalong the z -axis of the intrinsic reference frame wherethe local density is minimum. The position of the neckdefines two prefragments: a point ( x, y, z ) belongs to the“left” fragment if z < z N and to the “right” fragmentif z ≥ z N . The dispersion, a N , is chosen to be equal toone nucleon. By convention, we consider that a config-uration is located past scission, i.e., corresponds to twofully separated fragments, if q N ≡ (cid:10) φ ( q ) (cid:12)(cid:12) ˆ Q N (cid:12)(cid:12) φ ( q ) (cid:11) ≤ q scissN , (9)where q scissN is a parameter. This allows us to define thescission configurations as the set of all the states | φ ( q ) (cid:105) such that (i) q N > q scissN , and (ii) at least one of theirneighbor is located past scission.In our two-dimensional calculations, the set of scissionconfigurations can be parameterized by a single coordi-nate ξ . The population of scission configurations is thenobtained by integrating the flux density φ ( ξ, t ), whichreads in this specific case φ ( ξ, t ) = J ( q ( ξ ) , t ) d q d ξ − J ( q ( ξ ) , t ) d q d ξ , (10)where J = ( J , J ) is the current as defined in [16]. Weassume that the probability to exit through the point q ( ξ ) is proportional to the time-integrated flux density,defined as F ( ξ ) = lim t →∞ (cid:90) τ = tτ =0 d τ φ ( ξ, τ ) . (11)
2. Expression of the yields
Previous studies using the TDGCM framework to pre-dict fission fragment yields accounted for the width ofthe particle-number probability distribution at a singlescission configuration, as well as a possible experimen-tal mass resolution (to compare with experimental pre-neutron yields), by convoluting the time-integrated fluxdensity of Eq. (11) with a Gaussian function [16, 17, 19,40, 41]. The mean of such a Gaussian is typically setto the average number of particles in the right fragmentat the scission configuration | φ ( q ( ξ )) (cid:105) , while its standarddeviation is a free parameter. The major novelty of thiswork is the determination of the fission fragment massand charge yields, Y ( Z f , A f ) by computing the exactprobabilities for each fragments Z f , A f using a particle-number projection-based technique rather than a convo-lution with a Gaussian.We seek to calculate the probability P ( Z R , N R | Z, N )to measure Z R charges and N R neutrons in the rightfragment at scission given a compound system with Z protons and N neutrons. In our approach, it is given by P ( Z R , N R | Z, N ) ∝ (cid:90) d ξ F ( ξ ) P ( Z R , N R | Z, N, q ( ξ )) , (12)where P ( Z R , N R | Z, N, q ( ξ )) is the probability, whose de-termination is presented in Sec. II B 3, associated with aright fragment having Z R protons and N R neutrons, onlyconsidering the part of the compound system’s state at q ( ξ ) having Z protons and N neutrons. Once this quan-tity is inserted in Eq. (12) and after a proper normaliza-tion, we recover the fission fragments yields, in percent,assuming that the system splits in two fragments Y ( Z f , N f ) = 100 × (cid:2) P ( Z R = Z f , N R = N f | Z, N )+ P ( Z R = Z − Z f , N R = N − N f | Z, N ) (cid:3) . (13)We could use the same procedure to determine one-dimensional yields. For example, the charge distributionswould read Y ( Z f ) = 100 × (cid:2) P ( Z R = Z f | Z, N )+ P ( Z R = Z − Z f | Z, N ) (cid:3) , (14)where the probability P ( Z R | Z, N ), associated with Z R protons in the right fragment at scission given Z and N ,is P ( Z R | Z, N ) ∝ (cid:90) d ξ F ( ξ ) P ( Z R | Z, N, q ( ξ )) , (15)and the marginalized probability P ( Z R | Z, N, q ) is givenby P ( Z R | Z, N, q ) = N (cid:88) n =0 P ( Z R , N R = n | Z, N, q ) . (16)However, all the distributions of interest can alternativelybe obtained directly from Y ( Z f , N f ) through the relations Y ( Z f , A f ) = Y ( Z f , N f = A f − Z f ) (17) Y ( Z f ) = N (cid:88) n =0 Y ( Z f , N f = n ) (18) Y ( A f ) = (cid:88) n =0 Y ( Z f = A f − n, N f = n ) . (19)Therefore, the essential quantity to compute the fissionyields is the probability distribution P ( Z R , N R | Z, N, q ).
3. Particle Number Projection
In this work, we estimate the distribution of probabil-ity P ( Z R , N R | Z, N, q ) based on particle-number projec-tion techniques. Particle-number projection (PNP) wasoriginally introduced to restore the number of particles insuperfluid systems that spontaneously break the particlenumber symmetry [27, 42–45]. Following Refs [21–23], weuse PNP to estimate the probability P ( Z R , N R | Z, N, q ).For a fissioning nucleus described by the many-body state | φ ( q ) (cid:105) , this value can be interpreted as the probability tomeasure Z f protons and N f neutrons in the right frag-ment in the component of | φ ( q ) (cid:105) with Z protons and N neutrons.We do not consider isospin mixing in this work, i.e., | φ ( q ) (cid:105) = | φ ( q ) (cid:105) neut . ⊗ | φ ( q ) (cid:105) prot . . (20)Thus, the probabilities P ( Z R , N R | Z, N, q ) can be de-composed accordingly, P ( Z R , N R | Z, N, q ) = P ( N R | N, q ) × P ( Z R | Z, q ) . (21)The probability distributions P ( Z R | Z, q ) and P ( N R | N, q ) both derive from a double projectionof the scission configuration | φ ( q ) (cid:105) P ( Z R | Z, q ) = (cid:10) φ ( q ) (cid:12)(cid:12) ˆ P (R)p ( Z R ) ˆ P p ( Z ) (cid:12)(cid:12) φ ( q ) (cid:11)(cid:10) φ ( q ) (cid:12)(cid:12) ˆ P p ( Z ) (cid:12)(cid:12) φ ( q ) (cid:11) . (22)The operator ˆ P p ( Z ) projects the state onto the states’eigenspace with a good total proton number Z . We im-plemented this projector in its standard gauge angle in-tegral form ˆ P p ( Z ) = 12 π (cid:90) π d θ e iθ ( ˆ Z − Z ) . (23)Similarly, the operator ˆ P (R)p ( Z R ) projects onto theeigenspace of states with a good number of protons Z R inthe right half-space. We recall that the right half-spacecorresponds to the set of spatial coordinates whose com-ponent along the z -axis is greater than z N , the positionof the neck. Its expression readsˆ P (R)p ( Z R ) = 12 π (cid:90) π d θ e iθ ( ˆ Z R − Z R ) , (24)where ˆ Z R counts the number of protons in the right half-space. This operator can be expressed from the protoncreation (ˆ a (p) † ( r , σ )) and annihilation (ˆ a (p) ( r , σ )) opera-tors asˆ Z R = (cid:88) σ = ↓ , ↑ (cid:90) x (cid:90) y (cid:90) + ∞ z = z N d r ˆ a (p) † ( r , σ )ˆ a (p) ( r , σ ) . (25)We define the projectors on the neutrons number, inthe full space ( ˆ P n ( N )) as well as in the right half-space( ˆ P (R)n ( N R )) in a similar way, and we follow the same pro-cedure to calculate the neutron probability P ( N R | N, q ) = (cid:10) φ ( q ) (cid:12)(cid:12) ˆ P (R)n ( N R ) ˆ P n ( N ) (cid:12)(cid:12) φ ( q ) (cid:11)(cid:10) φ ( q ) (cid:12)(cid:12) ˆ P n ( N ) (cid:12)(cid:12) φ ( q ) (cid:11) . (26)In our calculations, we determine the numerator ofEqs. (22) and (26) using the gauge angle integrals basedon a Fomenko quadrature [46] using 41 integration points.Instead of explicitely calculating the corresponding de-nominator, we directly normalize the distributions using (cid:10) φ ( q ) (cid:12)(cid:12) ˆ P p ( Z ) (cid:12)(cid:12) φ ( q ) (cid:11) = (cid:88) z (cid:10) φ ( q ) (cid:12)(cid:12) ˆ P (R)p ( z ) ˆ P p ( Z ) (cid:12)(cid:12) φ ( q ) (cid:11) (27) (cid:10) φ ( q ) (cid:12)(cid:12) ˆ P n ( N ) (cid:12)(cid:12) φ ( q ) (cid:11) = (cid:88) n (cid:10) φ ( q ) (cid:12)(cid:12) ˆ P (R)n ( n ) ˆ P n ( N ) (cid:12)(cid:12) φ ( q ) (cid:11) . (28) III. APPLICATION
This section summarizes our results for the thermalneutron-induced fission of
U and
Pu. In Sec. III A,we discuss the static fission properties related to the po-tential energy surfaces and the properties of prefragmentsat scission. In Sec. III B, we show the primary fissionfragments mass and charge yields obtained by combin-ing the TDGCM+GOA collective dynamics with PNP.We then compare the projected yields with results ob-tained from assuming several analytical distributions inSec. III C, and assuming different criteria for the scissionline in Sec. III D. Finally, the evolution of both the two-dimensional yields and their one-dimensional reductionsas a function of the energy of the incident neutron isillustrated in Sec. III E.
A. Static properties
While the potential energy surfaces used in this workhave already been presented in other publications – [30]for
Pu and [47] for
U – we recall them for the sakeof completeness. Figure 1 thus shows the total potentialenergy V ( q ) entering the collective Schr¨odinger equationof Eq. (2) for both Pu (top) and
U (bottom) as afunction of the axial quadrupole and octupole moments.The PES is characterized by a ground-state at around Q ≈
30 b, a fission isomer at Q ≈
85 b, and the mainlarge fission valley opening up beyond the second bar-rier. Note that triaxiality is included in this calculation,but plays a role only near the first barrier; see [30] fora discussion. Additional details about the resolution ofthe HFB equation, such as the characteristics of the har-monic oscillator basis or the convention for the multipoleoperators used as constraints can be found in that samereference.
200 400 600 Q [b] Q [ b / ] Pu, SkM*
200 400 600 Q [b] Q [ b / ] U, SkM*
Figure 1. Potential energy surface for
Pu (top) and
U(bottom) as a function of the axial quadrupole and axial oc-tupole moments.
An HFB generator state | φ ( q ) (cid:105) is associated with eachpoint of the PES of Fig. 1. The scission configurationsare defined based on the procedure outlined in Sec. II B.For each state in the scission region, we identify the neckposition z N , which is used to compute the average charge,number of neutron, and mass, according to (cid:68) ˆ Z R (cid:69) = (cid:90) + ∞−∞ d x (cid:90) + ∞−∞ d y (cid:90) + ∞ z N d z ρ p ( r ) , (29) (cid:68) ˆ N R (cid:69) = (cid:90) + ∞−∞ d x (cid:90) + ∞−∞ d y (cid:90) + ∞ z N d z ρ n ( r ) , (30) (cid:68) ˆ A R (cid:69) = (cid:90) + ∞−∞ d x (cid:90) + ∞−∞ d y (cid:90) + ∞ z N d z ρ ( r ) , (31)where ρ n and ρ p are respectively the neutron and protondensity distributions and ρ = ρ n + ρ p is the isoscalar(total) density. Since the HFB wave function is not aneigenstate of the operator of Eq. (25) counting the num-ber of particles in the right fragment, the fluctuations (cid:104) ˆ Z (cid:105) and (cid:104) ˆ A (cid:105) are non-zero and could be in principlecomputed as well.Projection techniques give us a much more completeview of the content in particle number in the fragments.For each scission configuration of the two PES we per-formed the projection on the total number of protonsand neutrons as well as on the number of protons andneutrons in the right half-space following Eq. (22). Theprojection on the fragment particle numbers was onlyperformed for eigenvalues Z R and N R close to the meanvalues defined by Eq. (29), where the probability is notnegligible. In practice we considered in each scission con-figuration the set (cid:98)(cid:104) ˆ Z R (cid:105)(cid:99) − ≤ Z R < (cid:98)(cid:104) ˆ Z R (cid:105)(cid:99) + 20 and (cid:98)(cid:104) ˆ N R (cid:105)(cid:99) − ≤ N R < (cid:98)(cid:104) ˆ N R (cid:105)(cid:99) + 20, hence a total of 41projected values for protons and for neutrons. The no-tation (cid:98) X (cid:99) indicates the integer part of the real number X .Figure 2 presents an example of these probabilities dis-tributions for the case of a few scission configurations in U, the locations q of which are indicated in the legend.More specifically, it shows the probability P ( Z R | Z, q ) asa function of the charge number Z R for the fixed value Z = 92 of the total number of proton in the fissioningnucleus. These results highlight two important featuresbrought about by projection: (i) the curves are not nec-essarily symmetric around the mean value and (ii) therecan be odd-even staggering effects, as in the case of theconfiguration ( q , q ) = (325 b ,
40 b / ). These twofeatures are absent by construction when considering em-pirical dispersion laws such as Gaussian folding.
30 35 40 45 50Charge number in the right fragment Z R . . . . . . . P r o b a b ili t y ( Z R | Z , q ) q = (325 , q = (330 , q = (340 , q = (350 , q = (375 , q = (375 , q = (380 , q = (430 , q = (500 , Figure 2. Probability P ( Z R | Z, q ) of having Z R protons inthe right fragment given the total number of protons Z as afunction of Z R . We show such a probability distribution fora set of scission configurations q in U. The deformations q are in b ( q ) and b / ( q ), where b = barn. B. Fragment distributions
In this section, we show the primary mass andcharge distributions for the two important cases of thelow-energy, neutron-induced fission of
U and
Pu.Throughout this section, FELIX calculations were per-formed with the GCM collective inertia, the metric γ ( q )and the GCM zero point energy correction, and the fron-tier was defined by q scissN = 4 .
0. In Eq. (5), the initialcollective wave packet had a width σ = 0 . t = 2 × − zs and we always simulatedthe dynamics up to a time t = 20 zs. For the two actinides under consideration, there areconsiderably fewer measurements on charge distributions Y ( Z ) than on mass distributions Y ( A ). Yet charge dis-tributions are important tests for theory since they canexhibit an odd-even effect, namely, the yield of even- Z elements is higher than that of odd- Z [48–52]. Note thatan experimental determination of such an effect requiresa sufficiently good resolution in the detection of the frag-ment charge. Typically, for the case of neutron-inducedfission of U at thermal energy, odd-even effects werenot reported in [53]. According to the authors, the exper-imental technique used as well as the hypothesis assumedin the data analysis could smooth out the odd-even struc-tures in this work. On the contrary, the experiment [54]claims a resolving power of Z/ ∆ Z = 45 (full-width 1/10maximum) for the charge Z = 40 at the maximum ofthe light peak. With such resolution, the resulting yieldspresent a strong odd-even staggering effect. Such effectwas also observed for various fissioning systems in thehigh precision measurements based on inverse kinemat-ics reactions at higher excitation energies [55–57].In Fig. 3, we show the charge distributions for theneutron-induced fission Pu(n,f) and
U(n,f) reac-tions, at an excitation energy of 1 MeV above the firstfission barrier. For these fissile isotopes it should corre-spond to an incident neutron energy of E n (cid:39) Pu.In the case of
U, it is worth noting that our calcula-tions do not predict odd-even effects in the charge yields.We will discuss this in more details in Sec. III D.In Fig. 4, we show the mass distributions for thesesame isotopes. The agreement with data is not as good,especially in the case of
U(n,f). However, we mustbear in mind that these experimental primary mass yieldsdo not have a perfect mass resolution. The mass of the(primary) fragments is typically deduced from the mea-surement of the kinetic energy of the fission fragmentsin ionisation chambers. This technique implies a massresolution of roughly FWHM=5.5 mass units for A [67].Therefore, we also show in Fig. 4 the results of the cal-culation when accounting for such a mass resolution.In both Figs. 3 and 4, we compare the theoretical calcu-lations with PNP with the standard method of Gaussianfolding. Note that the width σ of the folding must bedifferent for protons and for masses as the overall pro-ton density is smaller than the total one. As a result,most of the (theoretical) uncertainty on the proton num-ber can be attributed to pairing effects in each prefrag-ment. As seen in Fig. 2, the average dispersion aroundthe mean value is of the order of 1.6 indeed. In contrast,neutrons are much less localized in the prefragments andcontribute rather significantly to the overall neck betweenthe prefragments [30, 68, 69]. The number of particlesin the neck used to define scission configuration in ourcalculations, q scissN = 4 .
0, is larger than the typical dis-persion in particle number one could expect from pairing
30 35 40 45 50 55 60 65Charge01020 Y i e l d ( % ) Pu Gaussian σ = 1 .
30 35 40 45 50 55 60 65Charge01020 Y i e l d ( % ) U Gaussian σ = 1 . Figure 3. Charge distribution Y ( Z ) in Pu (top) and
U(bottom) as a function of Z . The yields obtained with PNPare compared to a Gaussian convolution of the raw flux with σ = 1 . effects and thus dominates. This justifies fixing σ = 4for the Gaussian folding of mass yields. It is worth men-tioning that for this reference calculation at q scissN = 4 . Y ( Z, A ). Suchdistributions are especially important when simulatingthe deexcitation of fission fragments [70]. The resultsfor the particular case of
Pu are presented in Fig. 5.To our knowledge, no experimental data can be directlycompared to these pre-neutron emission yields. To givesome reference point, we therefore the represent the two-
80 90 100 110 120 130 140 150 160Mass0510 Y i e l d ( % ) Pu Gaussian σ = 4 .
80 90 100 110 120 130 140 150 160Mass051015 Y i e l d ( % ) U Gaussian σ = 4 . Figure 4. Mass distribution Y ( A ) in Pu (top) and
U(bottom) as a function of A . Results obtained after PNPare compared to a Gaussian convolution of the raw flux with σ = 4 . dimensional independent yields (after the emission of theprompt neutrons) evaluated in the JEFF-3.3 evaluateddata library [71]. We show that the PNP approach pre-dicts a significant width for the light and heavy peaks ofthe two-dimensional fission yields. This width is largerthan the data of JEFF-3.3. Currently, it is not clearwhether this reduction of the width could be totally ex-plained by the prompt neutron emission.From Y ( Z f , A f ) we can extract the charge polarization∆ Z f which measures the deviation of the most probablecharge in a fragment of given mass to the UnchangedCharge Distribution (UCD) approximation [72],∆ Z f ( A f ) = ¯ Z f ( A f ) − A f ZA , (32)
This work (primary)JEFF-3.3 (independent)
Figure 5. Two-dimensional isotopic fission yields Y ( Z, A )for the reaction
Pu(n,f) for an incoming neutron energyof E n = 1 . with ¯ Z f ( A f ) = (cid:80) Z f Z f × Y ( Z f , A f ) (cid:80) Z f Y ( Z f , A f ) . (33)For the thermal neutron induced-fission induced on Pu, a large corpus of experimental data agree on acharge polarization of the order of -0.5 charge unit forthe heavy fragments [73, 74]. We find in this work a con-sistent value of -0.58 on average on the fragments in theheavy peak ( A f ∈ [130 ,
35 40 45 50 55 60Charge1 . . . . . ¯ N f ( Z f ) / Z f Pu Caamano (2015) E n ’ . E n = 2 . E n = 2 . Figure 6. Average neutron excess of fragments produced inthe fission of
Pu. PNP results at E n = 2 . tron excess defined as¯ N f ( Z f ) Z f = (cid:80) N f N f /Z f × Y ( Z f , N f ) (cid:80) N f Y ( Z f , N f ) . (34)We compare in Fig. 6 the predictions of the TDGCMdynamics with PNP to the experimental data obtainedwith the VAMOS spectrometer [76]. The experimentaldata comes from an inverse kinematic experiment withan equivalent incident neutron energy of (cid:39) C. Comparison with various analyticaldistributions
Many studies of fission fragment distributions in theframework of the TDGCM+GOA approximate the un-certainty on particle number at scission, or equivalently,particle transfer effects, by simple Gaussian folding ofthe mean value of particles for each scission configura-tion [16–19, 41, 79]. This corresponds to setting thequantity P ( Z R , N R | Z, N, q ( ξ )) involved in Eq. (12) tofollow a Gaussian distribution as discussed in Sec.III C.In the work of [80], this convolution of the flux was alsoobtained from a random neck rupture mechanism.In this section, we compare fission distributions ob-tained through the direct calculation of the projectedmass and charge of the fragments with the ones ob-tained assuming analytic probability distributions for P ( Z R , N R | Z, N, q ). For each distribution, the mean µ is set to the average number of particles (neutrons orprotons) in the fragments. Note that this slightly differsfrom our previous works [16, 18] where µ was chosen tobe the closest integer to the average number of particles.The first analytical distribution we consider is theGaussian distribution used in earlier works and also usedto add a mass resolution in the fission fragment distri-bution before comparing with experimental data [81].While the mean of this distribution is specified by theaverage number of particles, the standard deviation σ isa free parameter. The contribution of a scission configu-ration is then P ( Z R | Z, N, q ) = 12 erf (cid:18) Z R + − µ ( q ) √ σ (cid:19) −
12 erf (cid:18) Z R − − µ ( q ) √ σ (cid:19) . (35)The hypothesis that the dispersion in charge number Z R in the fission fragments is minimal leads to the sec-ond distribution considered in this paper, the most lo-cal distribution of mean µ . The domain of this distri-bution is the set of positive integers. The only inte-gers associated with a non-zero probability are (cid:98) µ (cid:99) and (cid:98) µ (cid:99) + 1, respectively having the probabilities 1 − x and x , where x = µ − (cid:98) µ (cid:99) . Consequently, the contribution P ( Z R , N R | Z, N, q ) reads P ( Z R | Z, N, q ) = − x if X f = (cid:98) µ ( q ) (cid:99) x if X f = (cid:98) µ ( q ) (cid:99) + 10 otherwise . (36)If we model scission as a set of independent and iden-tically distributed draws where each nucleons is put inthe left or right fragment according to a coin tossing, theprobability distribution follows a Binomial law P ( Z R | Z, N, q ) = (cid:18) XX f (cid:19) ( µ/X ) X f [1 − ( µ/X )] X − X f . (37)Finally, the law of rare events suggests that the moreasymmetrical fission is, the better the binomial law canbe approximated by the fourth and last distribution con-sidered in this work, a Poisson distribution whose contri-bution at each point q is P ( Z R | Z, N, q ) = µ X f e − µ X f ! . (38)The results are summarized in Fig. 7 for the chargeyields; the mass yields are qualitatively similar. ThePoisson and Binomial distributions are considerably toobroad. In contrast, both the local discrete and Gaussian
30 35 40 45Charge01020 Y i e l d ( % ) Pu LocalBinomialPoissonGaussian σ = 1 . Figure 7. Light peak of the charge distribution Y ( Z )in Pu as a function of Z for different analytic forms for P ( Z R , N R | Z, N, q ) as well as from PNP. We only show thecurve for the light fragment, since the whole distribution iscompletely symmetric with respect to Z/
2; see text for addi-tional details. distribution do a decent job at reproducing the width ofthe charge distribution, although the local discrete distri-bution is too narrow leading to overestimating the heightof the peaks. The fact that a Gaussian convolution repro-duces quite well the PNP is consistent with the conclu-sion of Ref. [82] although they were obtained in a differentcontext where the projection subvolume is spherical.
D. Evolution of Yields as Function of FrontierDefinition
One of the well-known limitations of adiabatic ap-proaches to fission is the dependency of results on thedefinition of scission configurations [4, 68, 83]. This prob-lem is especially relevant in self-consistent HFB calcula-tions of PES: the large computational cost often leads tolimiting the number of active collective variables – two inour case, which leads to discontinuities in the PES [84].Very often, scission configurations are in fact associatedwith such a major discontinuity. In Fig. 1, the line thatseparates the colored area from the white background at q >
300 b is the visual representation of such a discon-tinuity, where the expectation value q N of the Gaussianneck operator drops from 3 – 6 down to values smallerthan 0.1. It is especially important to quantify the im-pact of the definition of the frontier on charge distri-butions as recent work suggest the odd-even staggeringeffect may only manifest itself at rather small values ofthe neck between the two prefragments [24].In Fig. 8, we thus show the evolution of the chargedistribution in Pu as a function of the criterion q scissN used to define scission configurations, as explained inSec. II B 1. As expected, there is no evidence of odd-evenstaggering for large values of q N . For 3 . ≤ q N ≤ .
0, thepeak does not really change very significantly. The first0
30 35 40 45Charge01020 Y i e l d ( % ) Pu q sciss N = 2 . q sciss N = 2 . q sciss N = 3 . q sciss N = 4 . q sciss N = 6 . q sciss N = 8 . Figure 8. Light peak of the fragment charge distribution Y ( Z ) for Pu at E n = 1 . q scissN . hints of odd-even staggering only appear at q N ≤ . Z . While this result isencouraging, we should point out that values of q N < E. Evolution of Yields as Function of ExcitationEnergy
Within our approach, we can estimate the primary fis-sion fragment mass and charge distributions Y ( A ; E n )and Y ( Z ; E n ) for different values of the incoming neutronenergy E n . To this purpose, we make the approximationthat all the excitation that the compound nucleus ac-quires after absorbing the neutron is of collective nature.This implies that varying incident neutron energies E n translate simply into varying mean energies of the collec-tive wave packet; see Eq. (5). We are very aware that thisapproximation is rather strong and probably not entirelyvalid. A more accurate treatment of collective dynamicswould require including quasiparticle excitations in theTDGCM formalism along the lines of [85], or adopting afinite-temperature approach [31, 41]. The former strat-egy faces considerable computational challenges, whilethe latter is not formally well-defined: the very conceptof a GCM wave function made of a superposition of ketsrequires generalization at finite temperature; see, e.g.,[86]. Pending such future developments and bearing inmind the limitations of our approach, we can still providea useful reference point for future comparisons.In Fig. 9, we thus show the evolution of the full, two-dimensional isotopic fragment distribution as a functionof incident neutron energy. As expected from statisti- cal mechanics – at “infinite” excitation energy, all frag-mentations should become equiprobable – the distribu-tion Y ( Z, A ; E n ) includes an ever larger set of fragmen-tations as the incident neutron energy increases from E n = 0 . E n = 10 . E n > . A L ≈
80 get more and morepopulated. A more careful analysis of these maps showthat the centroid of the distribution for the light frag-ment shifts toward lower Z L and lower A L : at E n = 0 . Z L , A L ) = (39 . , . Z L , A L ) = (37 . , . E n = 10 . Zr ( Z L = 40, N L = 60) at E n = 0 . Sr( Z L = 37 . A L = 95 . E n = 10 . Pu for various incident neutron energies inthe range E n ∈ [0 .
5; 10] MeV. These results are com-pared to the experimental data of D. Ramos et al. [56]obtained in inverse kinematics. This experiment lever-ages the transfer reaction C( U , Pu) Be to pro-duce the compound Plutonium system with an averageexcitation energy of 10.7 MeV and a standard deviationof 3.0 MeV. This energy corresponds to an incident neu-tron energy of E n = E ∗ − S n (cid:39) . E n = 4 MeV compare well with the experimental dataof Ref. [87]. We still notice a strong overestimation of thepredicted charge yields in the symmetric valley.Finally, Fig. 11 shows the evolution of the neutron ex-cess of Eq. (34) as a function of incident neutron energy.We see that the addition of the excitation energy to thesystem tends to flatten the structure of the neutron ex-cess. These predictions are qualitatively consistent withthe observed fission of Cf at high excitation energy( E ∗ = 43 MeV) [76] as well as with the interpretation ofshell effects being smoothed out as the energy increases. IV. CONCLUSION
In this paper, we reported the first microscopic cal-culation of fission fragment distribution within theTDGCM+GOA framework where the number of parti-cles in fission fragment was extracted by direct particlenumber projection. While using Gaussian folding of par-ticle number at scission is a good approximation of theexact PNP result, it still requires specifying the width.Even guided by consideration about pairing fluctuationsor the size of the neck, this parameter remains somewhatarbitrary: PNP techniques allow eliminating it entirelyand thereby obtaining more realistic distributions.When comparing with experimental data for the two1 Z f E n = 0 . MeV E n = 1 . MeV Z f E n = 2 . MeV E n = 3 . MeV Z f E n = 4 . MeV E n = 6 . MeV
80 90 100 110 120 130 140 150 160 A f Z f E n = 8 . MeV
80 90 100 110 120 130 140 150 160 A f E n = 10 . MeV − − Fragment probability (%)
Figure 9. Isotopic yields Y ( Z f , A f ) for the reaction Pu(n,f) for different incoming neutron energy E n using PNP in thefragments. Symmetric fission becomes more probable with increasing excitation energy. standard cases of U and
Pu, we find that the agree-ment on both charge and mass yields is satisfactory, es-pecially considering the rather large uncertainties on theprimary fragment distributions, which translates itselfinto a large experimental mass resolution for the massyields, and somewhat conflicting datasets for the chargeyields. Our analysis of the two-dimensional isotopicyields recovers the excepted charge polarization alreadyat the level of the primary yields, i.e., before any evapo-ration of neutrons. Many recent measurements seem to confirm the existence of an odd-even staggering in thecharge distributions. Our calculations with small necksizes for scission configurations seem to confirm this, al-though the resolution of the potential energy surface isnot sufficient to draw firm conclusions.The evolution of the yields as a function of incidentneutron energy, which in our framework translates intocollective excitation energy of the fissioning nucleus, iscompatible with what is expected from statistical me-chanics and observed experimentally: the peaks of the2
30 35 40 45Charge01020 Y i e l d ( % ) Pu Ramos (2018) E n ’ . E n = 1 . E n = 4 . E n = 7 . E n = 10 . MeV
Figure 10. Light peak of the fragment charge distribution Y ( Z ) for Pu obtained with different values of initial energyof the compound nucleus. Results with PNP are compared tothe experimental data from Ref. [56].
35 40 45 50 55 60Charge1 . . . . . ¯ N f ( Z f ) / Z f Pu UCD E n = 1 . E n = 4 . E n = 6 . E n = 10 . MeV
Figure 11. Evolution of the neutron excess of the primary fis-sion fragments for the reaction
Pu(n,f) according to E n . Ifthe UCD were satisfied, all these quantities would be constantequal to 1.55. distribution widen and symmetric fission increases. Wenote that the shell effects visible on the predicted chargepolarization tends to vanish with increasing energy. ACKNOWLEDGMENTS
This work was supported in part by the NUCLEISciDAC-4 collaboration DE-SC001822 and was per-formed under the auspices of the U.S. Department ofEnergy by Lawrence Livermore National Laboratory un-der Contract DE-AC52-07NA27344. Computing supportcame from the Lawrence Livermore National Laboratory(LLNL) Institutional Computing Grand Challenge pro-gram. [1] M. Bender, R. Bernard, G. Bertsch, S. Chiba,J. Dobaczewski, N. Dubray, S. A. Giuliani, K. Hagino,D. Lacroix, Z. Li, P. Magierski, J. Maruhn,W. Nazarewicz, J. Pei, S. P´eru, N. Pillet, J. Ran-drup, D. Regnier, P.-G. Reinhard, L. M. Robledo,W. Ryssens, J. Sadhukhan, G. Scamps, N. Schunck,C. Simenel, J. Skalski, I. Stetcu, P. Stevenson, S. Umar,M. Verriere, D. Vretenar, M. Warda, and S. ˚Aberg,Future of nuclear fission theory, J. Phys. G: Nucl. Part.Phys. , 113002 (2020).[2] A. Baran, M. Kowal, P.-G. Reinhard, L. M. Robledo,A. Staszczak, and M. Warda, Fission barriers and proba-bilities of spontaneous fission for elements with z ≥ , 442 (2015).[3] M. R. Mumpower, P. Jaffke, M. Verriere, and J. Randrup,Primary fission fragment mass yields across the chart ofnuclides, Phys. Rev. C , 054607 (2020). [4] N. Schunck and L. M. Robledo, Microscopic theory ofnuclear fission: a review, Reports on Progress in Physics , 116301 (2016).[5] D. G. Madland and J. Nix, New model of the averageneutron and proton pairing gaps, Nuclear Physics A ,1 (1988).[6] J. Berger, M. Girod, and D. Gogny, Time-dependentquantum collective dynamics applied to nuclear fission,Computer Physics Communications , 365 (1991).[7] Y. Tanimura, D. Lacroix, and S. Ayik, Microscopicphase-space exploration modeling of Fm spontaneousfission, Phys. Rev. Lett. , 152501 (2017).[8] A. Bulgac, P. Magierski, K. J. Roche, and I. Stetcu,Induced fission of
Pu within a real-time microscopicframework, Phys. Rev. Lett. , 122504 (2016).[9] P. M¨oller, A. Sierk, T. Ichikawa, and H. Sagawa, Nuclearground-state masses and deformations: FRDM(2012),Atomic Data and Nuclear Data Tables , 1 (2016).[10] H. Goutte, J. F. Berger, P. Casoli, and D. Gogny, Micro-scopic approach of fission dynamics applied to fragmentkinetic energy and mass distributions in U, Phys. Rev.C , 024316 (2005).[11] J.-F. Lemaˆıtre, S. Goriely, S. Hilaire, and J.-L. Sida, Fullymicroscopic scission-point model to predict fission frag-ment observables, Phys. Rev. C , 034612 (2019).[12] K.-H. Schmidt, B. Jurado, C. Amouroux, and C. Schmitt,General description of fission observables: GEF modelcode, Nuclear Data Sheets , 107 (2016), special Issueon Nuclear Reaction Data.[13] P. M¨oller, D. G. Madland, A. J. Sierk, and A. Iwamoto,Nuclear fission modes and fragment mass asymmetries ina five-dimensional deformation space, Nature , 785(2001).[14] P. M¨oller, J. Randrup, and A. J. Sierk, Calculated fissionyields of neutron-deficient mercury isotopes, Phys. Rev.C , 024306 (2012).[15] P. M¨oller and J. Randrup, Calculated fission-fragmentyield systematics in the region 74 ≤ Z ≤
94 and 90 ≤ N ≤ , 044316 (2015).[16] D. Regnier, N. Dubray, N. Schunck, and M. Verri`ere,Fission fragment charge and mass distributions in Pu( n, f ) in the adiabatic nuclear energy density func-tional theory, Phys. Rev. C , 054611 (2016).[17] H. Tao, J. Zhao, Z. P. Li, T. Nikˇsi´c, and D. Vretenar,Microscopic study of induced fission dynamics of Thwith covariant energy density functionals, Phys. Rev. C , 024319 (2017).[18] D. Regnier, N. Dubray, and N. Schunck, From asym-metric to symmetric fission in the fermium isotopeswithin the time-dependent generator-coordinate-methodformalism, Phys. Rev. C , 024611 (2019).[19] J. Zhao, J. Xiang, Z.-P. Li, T. Nikˇsi´c, D. Vretenar, and S.-G. Zhou, Time-dependent generator-coordinate-methodstudy of mass-asymmetric fission of actinides, Phys. Rev.C , 054613 (2019).[20] M. R. Mumpower, P. Jaffke, M. Verriere, and J. Randrup,Primary fission fragment mass yields across the chart ofnuclides, Phys. Rev. C , 054607 (2020).[21] C. Simenel, Particle transfer reactions with the Time-Dependent Hartree-Fock theory using a particle num-ber projection technique, Phys. Rev. Lett. , 192701(2010).[22] G. Scamps, C. Simenel, and D. Lacroix, Superfluid dy-namics of Fm fission, Phys. Rev. C , 011602(R)(2015).[23] M. Verriere, N. Schunck, and T. Kawano, Number ofparticles in fission fragments, Phys. Rev. C , 024612(2019).[24] M. Verriere and M. R. Mumpower, Improvements tothe macroscopic-microscopic approach of nuclear fission(2020), arXiv:2008.06639 [nucl-th].[25] M. Verriere and D. Regnier, The time-dependent gener-ator coordinate method in nuclear physics, Frontiers inPhysics , 233 (2020).[26] Verri`ere, Marc, Dubray, No¨el, Schunck, Nicolas, Regnier,David, and Dossantos-Uzarralde, Pierre, Fission descrip-tion: First steps towards a full resolution of the time-dependent hill-wheeler equation, EPJ Web Conf. ,04034 (2017).[27] P. Ring and P. Schuck, The Nuclear Many-Body Problem ,Texts and Monographs in Physics (Springer, 2004). [28] H. Krappe and K. Pomorski,
Theory of Nuclear Fission (Springer, 2012).[29] D. Regnier, N. Dubray, M. Verriere, and N. Schunck,FELIX-2.0: New version of the finite element solver forthe time dependent generator coordinate method withthe gaussian overlap approximation, Computer PhysicsCommunications , 180 (2018).[30] N. Schunck, D. Duke, H. Carr, and A. Knoll, Descriptionof induced nuclear fission with Skyrme energy function-als: Static potential energy surfaces and fission fragmentproperties, Phys. Rev. C , 054305 (2014).[31] N. Schunck, D. Duke, and H. Carr, Description of inducednuclear fission with Skyrme energy functionals. II. Finitetemperature effects, Phys. Rev. C , 034327 (2015).[32] J. Bartel, P. Quentin, M. Brack, C. Guet, and H.-B.H˚akansson, Towards a better parametrisation of Skyrme-like effective forces: A critical study of the SkM force,Nucl. Phys. A , 79 (1982).[33] N. Schunck, J. Dobaczewski, W. Satu(cid:32)la, P. Baczyk,J. Dudek, Y. Gao, M. Konieczka, K. Sato, Y. Shi,X. Wang, and T. Werner, Solution of the skyrme-hartree–fock–bogolyubov equations in the cartesian de-formed harmonic-oscillator basis. (viii) hfodd (v2.73y):A new version of the program, Computer Physics Com-munications , 145 (2017).[34] M. Baranger and M. Veneroni, An adiabatic time-dependent Hartree-Fock theory of collective motion infinite systems, Ann. Phys. , 123 (1978).[35] K. Goeke and P.-G. Reinhard, The generator-coordinate-method with conjugate parameters and the unificationof microscopic theories for large amplitude collective mo-tion, Ann. Phys. , 249 (1980).[36] M. Matsuo, T. Nakatsukasa, and K. Matsuyanagi, Adi-abatic Selfconsistent Collective Coordinate Method forLarge Amplitude Collective Motion in Nuclei with Pair-ing Correlations, Prog. Theor. Phys. , 959 (2000).[37] N. Hinohara, T. Nakatsukasa, M. Matsuo, and K. Mat-suyanagi, Gauge-Invariant Formulation of the AdiabaticSelf-Consistent Collective Coordinate Method, Prog.Theor. Phys. , 451 (2007).[38] W. Younes, D. M. Gogny, and J.-F. Berger, A Micro-scopic Theory of Fission Dynamics Based on the Gener-ator Coordinate Method , Lecture Notes in Physics, Vol.950 (Springer International Publishing, Cham, 2019).[39] M. Warda, J. L. Egido, L. M. Robledo, and K. Pomorski,Self-consistent calculations of fission barriers in the Fmregion, Phys. Rev. C , 014310 (2002).[40] D. Regnier, N. Dubray, and N. Schunck, From asym-metric to symmetric fission in the fermium isotopeswithin the time-dependent generator-coordinate-methodformalism, Phys. Rev. C , 024611 (2019).[41] J. Zhao, T. Nikˇsi´c, D. Vretenar, and S.-G. Zhou, Micro-scopic self-consistent description of induced fission dy-namics: Finite-temperature effects, Phys. Rev. C ,014618 (2019).[42] H.-J. Mang, The self-consistent single-particle model innuclear physics, Phys. Rep. , 325 (1975).[43] J.-P. Blaizot and G. Ripka, Quantum Theory of FiniteSystems (The MIT Press, Cambridge, 1985).[44] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Self-consistent mean-field models for nuclear structure, Rev.Mod. Phys. , 121 (2003).[45] N. Schunck, Energy Density Functional Methods forAtomic Nuclei. , IOP Expanding Physics (IOP Publish- ing, Bristol, UK, 2019).[46] V. N. Fomenko, Projection in the occupation-numberspace and the canonical transformation, J. Phys. A: Gen.Phys. , 8 (1970).[47] N. Schunck, Z. Matheson, and D. Regnier, Microscopiccalculation of fission fragment mass distributions at in-creasing excitation energies, in Proceedings of the 6th In-ternational Workshop on Compound-Nuclear Reactionsand Related Topics CNR*18 , Springer Proceedings inPhysics, edited by J. Escher, Y. Alhassid, L. A. Bern-stein, D. Brown, C. Fr¨ohlich, P. Talou, and W. Younes(Springer Berlin Heidelberg, 2020).[48] B. Ehrenberg and S. Amiel, Independent Yields of Kryp-ton and Xenon Isotopes in Thermal-Neutron Fission of
U. Observation of an Odd-Even Effect in the ElementYield Distribution, Phys. Rev. C , 618 (1972).[49] S. Amiel and H. Feldstein, Odd-even systematics in neu-tron fission yields of U and
U, Phys. Rev. C ,845 (1975).[50] G. Mariolopoulos, C. Hamelin, J. Blachot, J. P. Boc-quet, R. Brissot, J. Cran¸con, H. Nifenecker, and C. Ris-tori, Charge distributions in low-energy nuclear fissionand their relevance to fission dynamics, Nucl. Phys. A , 213 (1981).[51] J. P. Bocquet and R. Brissot, Mass, energy and nuclearcharge distribution of fission fragments, Nucl. Phys. A , 213 (1989).[52] F. G¨onnenwein, On the notion of odd-even effects in theyields of fission fragments, Nucl. Instrum. Methods inPhys. Res. A , 405 (1992).[53] W. Reisdorf, J. P. Unik, H. C. Griffin, and L. E. Glen-denin, Fission fragment K X-ray emission and nuclearcharge distribution for thermal neutron fission of U, U, Pu and spontaneous fission of
Cf, Nucl. Phys.A , 337 (1971).[54] W. Lang, H.-G. Clerc, H. Wohlfarth, H. Schrader, andK.-H. Schmidt, Nuclear charge and mass yields for235U(nth, f) as a function of the kinetic energy of thefission products, Nuclear Physics A , 34 (1980).[55] K.-H. Schmidt, S. Steinh¨auser, C. B¨ockstiegel, A. Grewe,A. Heinz, A. R. Junghans, J. Benlliure, H.-G. Clerc,M. de Jong, J. M¨uller, M. Pf¨utzner, and B. Voss, Rel-ativistic radioactive beams: A new access to nuclear-fission studies, Nucl. Phys. A , 221 (2000).[56] D. Ramos, M. Caama˜no, F. Farget, C. Rodr´ıguez-Tajes,L. Audouin, J. Benlliure, E. Casarejos, E. Clement,D. Cortina, O. Delaune, X. Derkx, A. Dijon, D. Dor´e,B. Fern´andez-Dom´ınguez, G. de France, A. Heinz,B. Jacquot, A. Navin, C. Paradela, M. Rejmund,T. Roger, M.-D. Salsac, and C. Schmitt, Isotopic fission-fragment distributions of U 238 , Np 239 , Pu 240 , Cm244 , and Cf 250 produced through inelastic scattering,transfer, and fusion reactions in inverse kinematics, Phys-ical Review C , 10.1103/PhysRevC.97.054612 (2018).[57] A. Chatillon, J. Ta¨ıeb, H. Alvarez-Pol, L. Audouin,Y. Ayyad, G. B´elier, J. Benlliure, G. Boutoux,M. Caama˜no, E. Casarejos, D. Cortina-Gil, A. Ebran,F. Farget, B. Fern´andez-Dom´ınguez, T. Gorbinet,L. Grente, A. Heinz, H. T. Johansson, B. Jurado,A. Keli´c-Heil, N. Kurz, B. Laurent, J.-F. Martin, C. No-ciforo, C. Paradela, E. Pellereau, S. Pietri, A. Proc-hazka, J. L. Rodr´ıguez-S´anchez, H. Simon, L. Tassan-Got, J. Vargas, B. Voss, and H. Weick, Experimentalstudy of nuclear fission along the thorium isotopic chain: From asymmetric to symmetric fission, Phys. Rev. C ,054628 (2019).[58] C. Wagemans, E. Allaert, A. Deruytter, R. Barth´el´emy,and P. Schillebeeckx, Comparison of the energy and masscharacteristics of the Pu(n th ,f) and the Pu(sf) frag-ments, Phys. Rev. C , 218 (1984).[59] P. Geltenbort, F. G¨onnenwein, and A. Oed, Precisionmeasurements of mean kinetic energy release in thermal-neutron-induced fission of 233U, 235U and 239Pu, Radi-ation Effects , 57 (1986), publisher: Taylor & FrancisGroup.[60] P. Schillebeeckx, C. Wagemans, A. J. Deruytter, andR. Barth´el´emy, Comparative study of the fragments’mass and energy characteristics in the spontaneous fus-sion of 238Pu, 240Pu and 242Pu and in the thermal-neutron-induced fission of 239Pu, Nuclear Physics A ,623 (1992).[61] K. Nishio, Y. Nakagome, I. Kanno, and I. Kimura, Mea-surement of Fragment Mass Dependent Kinetic Energyand Neutron Multiplicity for Thermal Neutron InducedFission of Plutonium-239, J. Nucl. Sci. Technol. , 404(1995), tex.ids: nishioMeasurement1995.[62] C. Tsuchiya, Y. Nakagome, H. Yamana, H. Moriyama,K. Nishio, I. Kanno, K. Shin, and I. Kimura, Simultane-ous Measurement of Prompt Neutrons and Fission Frag-ments for Pu(nth,f), J. Nucl. Sci. Technol. , 941(2000).[63] R. M¨uller, A. A. Naqvi, F. K¨appeler, and F. Dickmann,Fragment velocities, energies, and masses from fast neu-tron induced fission of U, Phys. Rev. C , 885 (1984),publisher: American Physical Society.[64] G. Simon, J. Trochon, F. Brisard, and C. Signarbieux,Pulse height defect in an ionization chamber investi-gated by cold fission measurements, Nuclear Instrumentsand Methods in Physics Research Section A: Acceler-ators, Spectrometers, Detectors and Associated Equip-ment , 220 (1990).[65] S. Zeynalov, V. I. Furman, F. J. Hambsch, M. Florec,V. Y. Konovalov, V. A. Khryachkov, and Y. S. Zamy-atnin, Investigation of mass-TKE distributions of fissionfragments from the U-235(n,f)- reaction in resonances,in Proceedings of the 13th International Seminar on In-teraction of Neutrons with Nuclei (ISINN-13) - NeutronSpectroscopy, Nuclear Structure, Related Topics , Vol. 13(Joint Institute for Nuclear Research, Russia, 2006) pp.351–359.[66] C. Romano, Y. Danon, R. Block, J. Thompson, E. Blain,and E. Bond, Fission fragment mass and energy distribu-tions as a function of incident neutron energy measuredin a lead slowing-down spectrometer, Phys. Rev. C ,014607 (2010), publisher: American Physical Society.[67] A. G¨o¨ok, F.-J. Hambsch, S. Oberstedt, and M. Vidali,Prompt neutrons in correlation with fission fragmentsfrom U(n,f), Phys. Rev. C , 044615 (2018), pub-lisher: American Physical Society.[68] W. Younes and D. Gogny, Nuclear Scission and QuantumLocalization, Phys. Rev. Lett. , 132501 (2011).[69] J. Sadhukhan, C. Zhang, W. Nazarewicz, andN. Schunck, Formation and distribution of fragments inthe spontaneous fission of Pu, Phys. Rev. C , 061301(2017).[70] B. Becker, P. Talou, T. Kawano, Y. Danon, and I. Stetcu,Monte Carlo Hauser-Feshbach predictions of prompt fis-sion γ rays: Application to n th + U, n th + Pu, and Cf (sf), Phys. Rev. C , 014617 (2013).[71] A. J. M. Plompen, O. Cabellos, C. De Saint Jean,M. Fleming, A. Algora, M. Angelone, P. Archier,E. Bauge, O. Bersillon, A. Blokhin, F. Cantargi, A. Cheb-boubi, C. Diez, H. Duarte, E. Dupont, J. Dyrda, B. Eras-mus, L. Fiorito, U. Fischer, D. Flammini, D. Foligno,M. R. Gilbert, J. R. Granada, W. Haeck, F.-J. Hambsch,P. Helgesson, S. Hilaire, I. Hill, M. Hursin, R. Ichou,R. Jacqmin, B. Jansky, C. Jouanne, M. A. Kellett,D. H. Kim, H. Kim, I. Kodeli, A. J. Koning, A. Y.Konobeyev, S. Kopecky, B. Kos, A. Krasa, L. C. Leal,N. Leclaire, P. Leconte, Y. O. Lee, H. Leeb, O. Litaize,M. Majerle, J. Marquez Damian, F. Michel-Sendis,R. W. Mills, B. Morillon, G. Noguere, M. Pecchia,S. Pelloni, P. Pereslavtsev, R. J. Perry, D. Rochman,A. Roehrmoser, P. Romain, P. Romojaro, D. Roubtsov,P. Sauvan, P. Schillebeeckx, K. H. Schmidt, O. Serot,S. Simakov, I. Sirakov, H. Sjostrand, A. Stankovskiy,J. C. Sublet, P. Tamagno, A. Trkov, S. van der Marck,F. Alvarez-Velarde, R. Villari, T. C. Ware, K. Yokoyama,and G. Zerovnik, The joint evaluated fission and fusionnuclear data library, JEFF-3.3, Eur. Phys. J. A , 181(2020).[72] A. C. Wahl, Systematics of fission-product yields, LosAlamos Report LA-13928 (2002).[73] C. Schmitt, A. Guessous, J. P. Bocquet, H. G. Clerc,R. Brissot, D. Engelhardt, H. R. Faust, F. G¨onnenwein,M. Mutterer, H. Nifenecker, J. Pannicke, C. Ristori, andJ. P. Theobald, Fission yields at different fission-productkinetic energies for thermal-neutron-induced fission of239Pu, Nuclear Physics A , 21 (1984).[74] A. Bail, O. Serot, L. Mathieu, O. Litaize, T. Materna,U. K¨oster, H. Faust, A. Letourneau, and S. Panebianco,Isotopic yield measurement in the heavy mass region for Pu thermal neutron induced fission, Phys. Rev. C ,034605 (2011), publisher: American Physical Society.[75] W. N¨orenberg, Theory of mean primary charge distribu-tion in low energy fission of even-even nuclei, Z. Physik , 246 (1966).[76] M. Caama˜no, Characterization of the scission point fromfission-fragment velocities, Phys. Rev. C , 034606(2015).[77] H. Naik, S. P. Dange, R. J. Singh, and S. B. Manohar, Systematics of charge distribution studies in low-energyfission of actinides, Nuclear Physics A , 143 (1997).[78] H. Naik, S. P. Dange, and A. V. R. Reddy, Charge distri-bution studies in the odd-Z fissioning systems, NuclearPhysics A , 1 (2007).[79] W. Younes and D. Gogny, Fragment Yields Calculated ina Time-Dependent Microscopic Theory of Fission , Tech.Rep. LLNL-TR-586678 (Lawrence Livermore NationalLaboratory (LLNL), Livermore, CA, 2012).[80] A. Zdeb, A. Dobrowolski, and M. Warda, Fission dynam-ics of Cf 252, Phys. Rev. C , 054608 (2017).[81] P. Jaffke, P. M¨oller, P. Talou, and A. J. Sierk, Hauser-feshbach fission fragment de-excitation with calculatedmacroscopic-microscopic mass yields, Phys. Rev. C ,034608 (2018).[82] D. Lacroix and S. Ayik, Counting statistics in finite fermisystems: Illustrations with the atomic nucleus, Phys.Rev. C , 014310 (2020).[83] K. T. R. Davies and J. R. Nix, Calculation of moments,potentials, and energies for an arbitrarily shaped diffuse-surface nuclear density distribution, Phys. Rev. C ,1977 (1976).[84] N. Dubray and D. Regnier, Numerical search of disconti-nuities in self-consistent potential energy surfaces, Com-puter Physics Communications , 2035 (2012).[85] R. Bernard, H. Goutte, D. Gogny, and W. Younes, Mi-croscopic and nonadiabatic Schr¨odinger equation derivedfrom the generator coordinate method based on zero- andtwo-quasiparticle states, Phys. Rev. C , 044308 (2011).[86] K. Dietrich, J.-J. Niez, and J.-F. Berger, Microscopictransport theory of nuclear processes, Nucl. Phys. A ,249 (2010).[87] D. Ramos, M. Caama˜no, F. Farget, C. Rodr´ıguez-Tajes,L. Audouin, J. Benlliure, E. Casarejos, E. Clement,D. Cortina, O. Delaune, X. Derkx, A. Dijon, D. Dor´e,B. Fern´andez-Dom´ınguez, G. de France, A. Heinz,B. Jacquot, A. Navin, C. Paradela, M. Rejmund,T. Roger, M.-D. Salsac, and C. Schmitt, Isotopic fission-fragment distributions of U, Np,
Pu,
Cm, and
Cf produced through inelastic scattering, transfer, andfusion reactions in inverse kinematics, Phys. Rev. C97