A systematic global investigation of differential charge radii has been performed within the CDFT framework for the first time. Theoretical results obtained with conventional covariant energy density functionals and the separable pairing interaction of Tian et al. [Phys. Lett. B 676, 44 (2009)] are compared with experimental differential charge radii in the regions of the nuclear chart in which available experimental data crosses the neutron shell closures at $N=28,50,82$, and 126. The analysis of absolute differential radii of different isotopic chains and their relative properties indicate clearly that such properties are reasonably well described in model calculations in the cases when the mean-field approximation is justified. However, while the observed clusterization of differential charge radii of different isotopic chains is well described above the $N=50$ and $N=126$ shell closures, it is more difficult to reproduce it above the $N=28$ and $N=82$ shell closures because of possible deficiencies in the underlying single-particle structure. The impact of the latter has been evaluated for spherical shapes and it was shown that the relative energies of the single-particle states and the patterns of their occupation with increasing neutron number have an appreciable impact on the evolution of the $\delta {\langle r^2\rangle }^{N,N^{\prime }}$ values. These factors also limit the predictive power of model calculations in the regions of high densities of the single-particle states of different origin. It is shown that the kinks in the charge radii at neutron shell closures are due to the underlying single-particle structure and due to weakening or collapse of pairing at these closures. The regions of the nuclear chart in which the correlations beyond mean field are expected to have an impact on charge radii are indicated; the analysis shows that the assignment of a calculated excited prolate minimum to the experimental ground state allows us to understand the trends of the evolution of differential charge radii with neutron number in many cases of shape coexistence even at the mean-field level. It is usually assumed that pairing is a dominant contributor to odd-even staggering (OES) in charge radii. Our analysis paints a more complicated picture. It suggests a new mechanism in which the fragmentation of the single-particle content of the ground state in odd-mass nuclei due to particle-vibration coupling provides a significant contribution to OES in charge radii.

The isobaric multiplet mass equation (IMME) is known to break down in the first $T=2,A=32$ isospin quintet. In this work we combine high-resolution experimental data with state-of-the-art shell-model calculations to investigate isospin mixing as a possible cause for this violation. The experimental data are used to validate isospin-mixing matrix elements calculated with newly developed shell-model Hamiltonians. Our analysis shows that isospin mixing with nonanalog $T=1$ states contributes to the IMME breakdown, making the requirement of an anomalous cubic term inevitable for the multiplet.

Low-energy fission of ${}^{234,235}\mathrm{U}$ and ${}^{237,238}\mathrm{Np}$ radioactive beams, provided by the Fragment Separator (FRS) of the GSI Helmholtzzentrum für Schwerionenforschung facility (GSI), has been studied using the Reactions with Relativistic Radioactive Beams / Studies on Fission with Aladin (R3B/SOFIA) setup. The latter allows us, on an event-by-event basis, to simultaneously identify, in terms of their mass and atomic numbers, the fissioning nucleus in coincidence with both fission fragments after prompt-neutron emission. This article reports new results on elemental, isotonic, isobaric, and isotopic yields. Moreover, the high accuracy of our data allowed us to study in detail proton even-odd staggering, from elemental yields; neutron excess, from isotopic yields; and total prompt-neutron multiplicity, from the difference of masses of the fissioning nucleus and fission fragments. These results are then compared to previous experimental data in order to probe how these fission observables change as function of the excitation energy and atomic and neutron numbers of the compound nucleus.

We study the deuteron production via the deuteron pion and nucleon catalysis reactions, $\pi pn\leftrightarrow \pi d$ and $Npn\leftrightarrow Nd$, by employing stochastic multiparticle reactions in the hadronic transport approach SMASH for the first time. This is an improvement compared to previous studies, which introduced an artificial fake resonance $d^{\prime }$ to simulate these $3\leftrightarrow 2$ reactions as a chain of $2\leftrightarrow 2$ reactions. The derivation of the stochastic criterion for multiparticle reactions is presented in a comprehensive fashion and its implementation is tested against an analytic expression for the scattering rate and the equilibrating particle yields in box calculations. We then study Au $+$ Au collisions at $\sqrt{s_{NN}}=7.7$ GeV, where we find that multiparticle collisions substantially reduce the time required for deuterons to reach partial chemical equilibrium with nucleons. Subsequently, the final yield of $d$ is practically independent from the number of $d$ at particlization, confirming the results of previous studies. The mean transverse momentum and the integrated elliptic flow as a function of centrality are rather insensitive to the exact realization of the $2\leftrightarrow 3$ reactions.

Background: The Pauli exclusion principle plays a crucial role as a building block of many-body quantal systems comprised of fermions. It also induces a “Pauli repulsion” in the interaction between di-nuclear systems. It has been shown in [Phys. Rev. C 95, 031601(R) (2017)] that the Pauli repulsion widens the nucleus-nucleus potential barrier, thus hindering sub-barrier fusion.

Purpose: To investigate the proton and neutron contributions to the Pauli repulsion, both in the bare potential neglecting shape polarization and transfer between the reactants, as well as in the dynamical potential obtained by accounting for such dynamical rearrangements.

Methods: As the basis of our study we utilize the Pauli kinetic energy (PKE) obtained by studying the nuclear localization function (NLF). Recently this approach has been generalized to incorporate all of the dynamical and time-odd terms present in the nuclear energy density functional. This approach is employed in the density constrained frozen Hartree-Fock (DCFHF) and in the density constrained time-dependent Hartree-Fock (DC-TDHF) microscopic methods.

Results: The PKE spatial distribution shows that a repulsion occurs in the neck between the nuclei when they first touch. Inside the barrier, neutrons can contribute significantly more to the Pauli repulsion in neutron-rich systems. Dynamical effects tend to lower the Pauli repulsion near the barrier. Proton and neutron dynamical contributions to the PKE significantly differ inside the barrier for asymmetric collisions, which is interpreted as an effect of multinucleon transfer.

Conclusions: The PKE is shown to make a significant contribution to nuclear interaction potentials. Protons and neutrons can play very different roles in both the bare potential and in the dynamical rearrangement. Further microscopic studies are required to better understand the role of transfer and to investigate the effect of pairing and deformation.

Background: The recent announcement by the PREX collaboration of an unanticipated thick neutron skin in ${}^{208}\mathrm{Pb}$ ($R_{\mathrm{skin}}^{208}$) has challenged the understanding of neutron-rich matter in the vicinity of nuclear saturation density. Whereas earlier constraints indicate that the symmetry energy is relatively soft, the PREX-2 result seems to suggest the opposite.

Purpose: To confront constraints on the symmetry energy obtained from measurements of the electric dipole polarizability against those informed by the PREX-2 measurement of $R_{\mathrm{skin}}^{208}$ and by the correlations that it entails.

Methods: Covariant energy density functionals informed by the properties of finite nuclei are used to compute the electric dipole response of ${}^{48}\mathrm{Ca}, {}^{68}\mathrm{Ni}, {}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$. The set of functionals used in this work are consistent with experimental data, yet are flexible enough in that they span a wide range of values of $R_{\mathrm{skin}}^{208}$.

Results: It is found that theoretical predictions of the electric dipole polarizability that are consistent with the PREX-2 measurement systematically overestimate the corresponding values extracted from the direct measurements of the distribution of electric dipole strength.

Conclusions: The neutron skin thickness of ${}^{208}\mathrm{Pb}$ extracted from parity violating electron scattering and the electric dipole polarizability measured in photoabsorption experiments are two of the cleanest experimental tools used to constrain the symmetry energy around nuclear saturation density. However, the recent value of $R_{\mathrm{skin}}^{208}$ that suggests a fairly stiff symmetry energy stands in stark contrast to the conclusions derived from the electric dipole polarizability. At present, I offer no solution to this dilemma.

We propose a novel method to measure the neutron-neutron scattering length using the ${}^6\mathrm{He}(p,p\alpha )nn$ reaction in inverse kinematics at high energies. The method is based on the final-state interaction (FSI) between the neutrons after the sudden knockout of the $\alpha$ particle. We show that the details of the neutron-neutron relative-energy distribution allow for a precise extraction of the $s$-wave scattering length. We present the state of the art in regard to the theory of this distribution. The distribution is calculated in two steps. First, we calculate the ground-state wave function of ${}^6\mathrm{He}$ as a $\alpha nn$ three-body system. For this purpose we use Halo effective field theory, which also provides uncertainty estimates for the results. We compare our results at this stage to model calculations done with the computer code face. In a second step we determine the effects of the $nn$ FSI using the $nn$ t-matrix. We compare these FSI results to approximate FSI approaches based on standard FSI enhancement factors. While the final distribution is sensitive to the $nn$ scattering length, it depends only weakly on the effective range. Throughout we emphasize the impact of theoretical uncertainties on the neutron-neutron relative-energy distribution, and discuss the extent to which those uncertainties limit the extraction of the neutron-neutron scattering length from the reaction ${}^6\mathrm{He}(p,p\alpha )nn$.

In this paper we model low-lying states of atomic nuclei in the nucleon-pair approximation of the shell model, using three approaches to select collective nucleon pairs: the generalized seniority scheme, the conjugate gradient method, and the Hartree-Fock approach. We find the collective pairs obtained from the generalized seniority scheme provide a good description for nearly spherical nuclei, and those from the conjugate gradient method or the Hartree-Fock approach work well for transitional and deformed nuclei. Our NPA calculations using collective pairs with angular momenta 0, 2, and 4 (denoted by $SDG$ pairs) reproduce the nuclear shape evolution in the $N=26$ isotones, ${}^{46}\mathrm{Ca}, {}^{48}\mathrm{Ti}, {}^{50}\mathrm{Cr}$, and ${}^{52}\mathrm{Fe}$, and yield good agreement with full configuration-interaction calculations of low-lying states in medium-heavy transitional and deformed nuclei: ${}^{44-48}\mathrm{Ti}, {}^{48}\mathrm{Cr}, {}^{50}\mathrm{Cr}, {}^{52}\mathrm{Fe}, {}^{60-64}\mathrm{Zn}, {}^{64,66}\mathrm{Ge}, {}^{84}\mathrm{Mo}$, and ${}^{108-112}\mathrm{Xe}$. Finally, using the $SDGI$-pair approximation we describe low-lying states of ${}^{112,114}\mathrm{Ba}$, cases difficult to reach by conventional configuration-interaction methods.

During nuclear fission, a heavy nucleus splits into two rotating fragments. The associated angular momentum is large, yet the mechanism of its generation and its dependence on the mass of fragments remain poorly understood. In this Letter, we provide the first microscopic calculations of angular-momentum distributions in fission fragments for a wide range of fragment masses. For the benchmark case of ${}^{239}\mathrm{Pu}(n_{\text{th}},f)$, we find that the angular momentum of the fragments is largely determined by the nuclear shell structure and deformation, and that the heavy fragments therefore typically carry less angular momentum than their light partners. We use the fission model freya to simulate the emission of neutrons and photons from the fragments. The dependence of the angular momenta on fragment mass after the emission of neutrons and statistical photons is linear for the heavy fragments and either constant or weakly linear for the light fragments, consistent with the universal sawtooth pattern suggested by recent experimental data. Finally, we observe that using microscopic angular-momentum distributions modifies the number of emitted photons significantly.

Background: The neutron distribution of neutron-rich nuclei provides critical information on the structure of finite nuclei and neutron stars. Parity violating experiments—such as PREX and CREX—provide a clean and largely model-independent determination of neutron densities. Such experiments, however, are challenging and expensive, which is why sound statistical arguments are required to maximize the information gained.

Purpose: We introduce a new framework, the “transfer function formalism,” aimed at uncertainty quantification, model selection, and experimental design in the context of neutron densities.

Methods: The transfer functions (TFs) are built analytically by expressing the linear response of the objective function (e.g., ${\chi }^2$) to small perturbations of the data. Using the TF formalism, we are able to analyze the expected overall uncertainty—quantified in terms of bias and variance—of the mean square radius and interior density of ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$.

Results: Using relativistic mean field models as a proxy for the weak-charge density—and assuming that a total of five measurements could be performed on the weak form factor of ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$—we identify the optimal models and experimental locations that minimize the uncertainty in the extraction of the radius and interior density. We also explore the use of the TF formalism to understand the influence of prior distributions for the model parameters, as well as the optimization of model hyperparameters not constrained by the data.

Conclusions: We establish how the choice of experimental locations and the model that is used can have a significant impact on the final uncertainties of the extracted quantities of interest. For challenging experiments such as CREX and PREX, a proper quantification of such uncertainties is critical. We have demonstrated how the TF formalism provides several advantages for this type of analysis.

In this work we present the first steps towards benchmarking isospin symmetry breaking in ab initio nuclear theory for calculations of superallowed Fermi $\beta$ decay. Using the valence-space in-medium similarity renormalization group, we calculate $b$ and $c$ coefficients of the isobaric multiplet mass equation, starting from two different Hamiltonians constructed from chiral effective field theory. We compare results to experimental measurements for all $T=1$ isobaric analog triplets of relevance to superallowed $\beta$ decay for masses $A=10$ to $A=74$ and find an overall agreement within approximately 250 keV of experimental data for both $b$ and $c$ coefficients. A greater level of accuracy, however, is obtained by a phenomenological Skyrme interaction or a classical charged-sphere estimate. Finally, we show that evolution of the valence-space operator does not meaningfully improve the quality of the coefficients with respect to experimental data, which indicates that higher-order many-body effects are likely not responsible for the observed discrepancies.

We present a detailed account of neutron capture experiments of astrophysical relevance on ${}^{78,80,84,86}\mathrm{Kr}(n,\gamma )$ reactions at the border between weak and main $s$ process. The experiments were performed with quasi-Maxwellian neutrons from the Liquid-Lithium Target (LiLiT) and the mA-proton beam at 1.93 MeV (2–3 kW) of the Soreq Applied Research Accelerator Facility (SARAF). The setup yields high-intensity $\approx 40$ keV quasi-Maxwellian neutrons (3–5 $\times {10}^{10} n$/s) closely reproducing the conditions of $s$-process stellar nucleosynthesis. A sample of 100 mg of atmospheric, pre-nuclear-age Kr gas contained in a Ti spherical shell was activated in the LiLiT neutron field. The abundances of long-lived Kr isotopes (${}^{81,85g}\mathrm{Kr}$) were measured by atom counting via atom trap trace analysis (ATTA) at Argonne National Laboratory and low-level counting (LLC) at University of Bern. This work is the first measurement of a nuclear cross section using atom counting via ATTA. The activities of short-lived Kr isotopes (${}^{79,85m,87}\mathrm{Kr}$) were measured by $\gamma$-decay counting with a high-purity germanium detector. Maxwellian-averaged cross sections for $s$-process thermal energies are extracted. By comparison to reference values, our nucleosynthesis network calculations show that the experimental cross sections have a strong impact on calculated abundances of krypton and neighboring nuclides, in some cases improving agreement between theory and observations.

We discover new, hitherto unknown, shadow poles in Brune's alternative parametrization of $R$-matrix theory [Brune, Phys. Rev. C 66, 044611 (2002)]. Where these poles are, and how many there are, depends on how one continues $R$-matrix operators to complex wave numbers (especially the shift $S$ and penetration $P$ functions). This has little consequence for the exact $R$-matrix formalism (past the last energy threshold), as we show one can still always fully reconstruct the scattering matrix with only the previously known alternative parameters (poles and corresponding resonance widths), for which there were as many poles as the number of levels $N_{\lambda }$. However, we generalize the alternative parametrization to the Reich-Moore formalism and show that the choice of continuation is now critical as it changes the alternative parameters values (poles and residue widths are now complex). In order to establish nuclear libraries with alternative parameters, the nuclear community will thus have to decide what convention to adopt. We argue in favor of analytical continuation (against the legacy of Lane and Thomas approach) in the second article [P. Ducru, B. Forget, V. Sobes, G. Hale, and M. Paris, Phys. Rev. C 103, 064609 (2021)] of this trilogy on pole parametrizations of $R$-matrix theory, on which we build to establish the windowed multipole representation or $R$-matrix cross sections in the third and last article [P. Ducru, A. Alhajri, I. Meyer, B. Forget, V. Sobes, C. Josey, and J. Liang, Phys. Rev. C 103, 064610 (2021)]. We observe the first evidence of shadow poles in the alternative parametrization of $R$-matrix theory in the isotope ${}^{134}\mathrm{Xe}$ spin-parity group $J^{\pi }=1/2^{(-)}$ and show how they indeed depend on the choice of continuation to complex wave numbers.

First generation, or Population III, stars have a different evolution than those of later generations owing to their initial primordial abundance composition. Most notably, the lack of carbon, oxygen, and nitrogen means that primordial massive stars must rely on the less efficient $p\text{−}p$ chains, thereby requiring the star to contract to reach temperatures high enough to eventually trigger $3\alpha$ reactions. Even small amounts of the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reactions begin feeding the CNO mass range and enable the CNO cycle to generate energy, but this occurs at higher temperature compared to later stellar generations. It is currently controversial if the observed enhanced abundances of Ca in the most metal-poor stars could be a result of the high temperature H-burning conditions in the first massive stars. The level of this enrichment depends on the hot breakout path from the CNO cycles via the ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$ reaction. In this work, the rates of both the ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$ and competing ${}^{19}\mathrm{F}(p,\alpha ){}^{16}\mathrm{O}$ reactions are re-evaluated using the phenomenological $R$-matrix approach, simultaneously considering several ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$, ${}^{19}\mathrm{F}(p,\alpha ){}^{16}\mathrm{O}$, and ${}^{19}\mathrm{F}(p,p){}^{19}\mathrm{F}$ data sets, to better characterize the rate uncertainties. It is found that the rate uncertainty for ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$ reaction is considerably larger than previously reported. This is the result of undetermined interferences between observed resonances, a possible threshold state, possible subthreshold states, direct capture, and background levels. Additional experimental measurements are therefore needed to determine if ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$ CNO breakout is responsible for Ca enrichment in metal-poor stars. Astrophysically, the breakout reaction revision makes it less likely that Ca observed in the most Fe-poor stars can originate in hot CNO breakout H-burning nucleosynthtesis, thereby casting doubt on the prevailing faint supernova scenario to explain the abundances observed in these stars.

We present a systematic investigation of few-nucleon systems and light nuclei using the current Low Energy Nuclear Physics International Collaboration interactions comprising semilocal momentum-space regularized two- and three-nucleon forces up to third chiral order (${\mathrm{N}}^2\mathrm{L}\mathrm{O}$). Following our earlier study utilizing the coordinate-space regularized interactions, the two low-energy constants entering the three-body force are determined from the triton binding energy and the differential cross-section minimum in elastic nucleon-deuteron scattering. Predictions are made for selected observables in elastic nucleon-deuteron scattering and in the deuteron breakup reactions, for properties of the $A=3$ and $A=4$ nuclei, and for spectra of $p$-shell nuclei up to $A=16$. A comprehensive error analysis is performed including an estimation of correlated truncation uncertainties for nuclear spectra. The obtained predictions are generally found to agree with experimental data within errors. Similarly to the coordinate-space regularized chiral interactions at the same order, a systematic overbinding of heavier nuclei is observed, which sets in for $A\sim 10$ and increases with $A$.

We present a precise measurement of the cross section, proton and ${}^3\mathrm{He}$ analyzing powers, and spin correlation coefficient $C_{y,y}$ for $p\text{−}{}^3\mathrm{He}$ elastic scattering near 65 MeV, and a comparison with rigorous four-nucleon scattering calculations based on realistic nuclear potentials and a model with $\mathrm{\Delta }$-isobar excitation. Clear discrepancies are seen in some of the measured observables in the regime around the cross section minimum. Theoretical predictions using scaling relations between the calculated cross section and the ${}^3\mathrm{He}$ binding energy are not successful in reproducing the data. Large sensitivity to the $NN$ potentials and rather small $\mathrm{\Delta }$-isobar effects in the calculated cross section are noticed as different features from those in the deuteron-proton elastic scattering. The results obtained above indicate that $p\text{−}{}^3\mathrm{He}$ scattering at intermediate energies is an excellent tool to explore nuclear interactions not accessible by three-nucleon scattering.

While Josephson-like junctions, transiently established in heavy-ion collisions (${\tau }_{\mathrm{coll}}\approx {10}^{-21}\mathrm{s}$) between superfluid nuclei—and through which Cooper-pair tunneling ($Q$-value $Q_{2n}$) proceeds mainly in terms of successive transfer of entangled nucleons—are deprived of the macroscopic aspects of a supercurrent, they display many of the special effects associated with spontaneous symmetry breaking in gauge space (BCS condensation), which can be studied in terms of individual quantum states and of tunneling of single Cooper pairs. From the results of studies of one- and two-neutron transfer reactions carried out at energies below the Coulomb barrier we estimate the value of the mean-square radius (correlation length) of the nuclear Cooper pair. A quantity related to the largest distance of closest approach for which the absolute two-nucleon tunneling cross section is of the order of the single-particle one. Furthermore, emission of $\gamma$ rays of (Josephson) frequency ${\nu }_J=Q_{2n}/h$ distributed over an energy range $\hslash /{\tau }_{\mathrm{coll}}$ is predicted.

Neutron star (NS) merger ejecta offer viable sites for the production of heavy $r$-process elements with nuclear mass numbers $A\gtrsim 140$. The crucial role of fission recycling is responsible for the robustness of this site against many astrophysical uncertainties. Here, we introduce improvements to our scission-point model, called SPY, to derive the fission fragment distribution for all neutron-rich fissioning nuclei of relevance in $r$-process calculations. These improvements include a phenomenological modification of the scission distance and a smoothing procedure of the distribution. Such corrections lead to much better agreement with experimental fission yields. Those yields are also used to estimate the number of neutrons emitted by the excited fragments on the basis of different neutron evaporation models. Our fission yields are extensively compared to those predicted by the GEF (general description of fission observables) model. The impact of fission on the $r$-process nucleosynthesis in binary neutron mergers is also reanalyzed. Two scenarios are considered, the first one with low initial electron fraction subject to intense fission recycling, in contrast to the second one, which includes weak interactions on nucleons. The various regions of the nuclear chart responsible for fission recycling during the neutron irradiation and after freeze-out are discussed. The contribution fission processes may have to the final abundance distribution is also studied in detail in the light of newly defined quantitative indicators describing the fission recycling, the fission seeds, and the fission progenitors. In particular, those allow us to estimate the contribution of fission to the final abundance distribution stemming from specific heavy nuclei. Calculations obtained with SPY and GEF fission fragment distributions are compared for both $r$-process scenarios.

Background: Many quantal many-body methods that aim at the description of self-bound nuclear or mesoscopic electronic systems make use of auxiliary wave functions that break one or several of the symmetries of the Hamiltonian in order to include correlations associated with the geometrical arrangement of the system's constituents. Such reference states have been used already for a long time within self-consistent methods that are either based on effective valence-space Hamiltonians or energy density functionals, and they are presently also gaining popularity in the design of novel ab initio methods. A fully quantal treatment of a self-bound many-body system, however, requires the restoration of the broken symmetries through the projection of the many-body wave functions of interest onto good quantum numbers.

Purpose: The goal of this work is threefold. First, we want to give a general presentation of the formalism of the projection method starting from the underlying principles of group representation theory. Second, we want to investigate formal and practical aspects of the numerical implementation of particle-number and angular-momentum projection of Bogoliubov quasiparticle vacua, in particular with regard of obtaining accurate results at minimal computational cost. Third, we want to analyze the numerical, computational, and physical consequences of intrinsic symmetries of the symmetry-breaking states when projecting them.

Methods: Using the algebra of group representation theory, we introduce the projection method for the general symmetry group of a given Hamiltonian. For realistic examples built with either a pseudopotential-based energy density functional or a valence-space shell-model interaction, we then study the convergence and accuracy of the quadrature rules for the multidimensional integrals that have to be evaluated numerically and analyze the consequences of conserved subgroups of the broken symmetry groups.

Results: The main results of this work are also threefold. First, we give a concise, but general, presentation of the projection method that applies to the most important potentially broken symmetries whose restoration is relevant for nuclear spectroscopy. Second, we demonstrate how to achieve high accuracy of the discretizations used to evaluate the multidimensional integrals appearing in the calculation of particle-number and angular-momentum projected matrix elements while limiting the order of the employed quadrature rules. Third, for the example of a point-group symmetry that is often imposed on calculations that describe collective phenomena emerging in triaxially deformed nuclei, we provide the group-theoretical derivation of relations between the intermediate matrix elements that are integrated, which permits a further significant reduction of the computational cost of the method. These simplifications are valid regardless of the number parity of the quasiparticle states and therefore can be used in the description of even-even, odd-mass, and odd-odd nuclei.

Conclusions: The quantum-number projection technique is a versatile and efficient method that permits to restore the symmetry of any arbitrary many-body wave function. Its numerical implementation is relatively simple and accurate. In addition, it is possible to use the conserved symmetries of the reference states to reduce the computational burden of the method. More generally, the ever-growing computational resources and the development of nuclear ab-initio methods opens new possibilities of applications of the method.

We calculate neutron multiplicities from fission fragments with specified mass numbers for events having a specified total fragment kinetic energy. The shape evolution from the initial compound nucleus to the scission configurations is obtained with the metropolis walk method on the five-dimensional potential-energy landscape, calculated with the macroscopic-microscopic method for the three-quadratic-surface shape family. Shape-dependent microscopic level densities are used to guide the random walk, to partition the intrinsic excitation energy between the two proto-fragments at scission, and to determine the number of neutrons evaporated from the fragments. The contribution to the total excitation energy of the resulting fragments from statistical excitation and shape distortion at scission is studied. Good agreement is obtained with available experimental data on neutron multiplicities in correlation with fission fragments from ${}^{235}\mathrm{U}(n_{\mathrm{th}},f$). With increasing neutron energy a superlong fission mode grows increasingly prominent, which affects the dependence of the observables on the total fragment kinetic energy.