Most atomic nuclei are deformed with a quadrupole shape described by its overall strength ${\beta }_2$ and triaxiality $\gamma$. The deformation can be accessed in high-energy heavy ion collisions by measuring the collective flow response of the produced quark-gluon plasma to the eccentricity ${\varepsilon }_2$ and the density gradient $d_{\perp }$ in the initial state. Using an analytical estimate and a Glauber model, I show that the variances $\langle {\varepsilon }_2^2\rangle$ or $\langle {(\delta d_{\perp }/d_{\perp })}^2\rangle$ and skewnesses $\langle {\varepsilon }_2^2\delta d_{\perp }/d_{\perp }\rangle$ or $\langle {(\delta d_{\perp }/d_{\perp })}^3\rangle$ have a simple analytical form of $a^{\prime }+b^{\prime }{\beta }_2^2$ and $a^{\prime }+[b^{\prime }+c^{\prime }cos\left(3\gamma \right)]{\beta }_2^3$, respectively. From these, I constructed several normalized skewnesses to isolate the $\gamma$ dependence from that of ${\beta }_2$ and show that the correlations between a normalized skewness and a variance can constrain simultaneously ${\beta }_2$ and $\gamma$. Assuming a linear relation with elliptic flow $v_2$ and mean-transverse momentum $\left[p_{\mathrm{T}}\right]$ of final-state particles, $v_2\propto {\varepsilon }_2$ and $\delta \left[p_{\mathrm{T}}\right]/\left[p_{\mathrm{T}}\right]\propto \delta d_{\perp }/d_{\perp }$, similar conclusions are also expected for the variances and skewnesses of $v_2$ and $\left[p_{\mathrm{T}}\right]$, i.e., $a+b{\beta }_2^2$ for $\langle v_2^2\rangle$ and $\langle {(\delta \left[p_{\mathrm{T}}\right]/\left[p_{\mathrm{T}}\right])}^2\rangle$ and $a+[b+ccos\left(3\gamma \right)]{\beta }_2^3$ for $\langle v_2^2\delta \left[p_{\mathrm{T}}\right]/\left[p_{\mathrm{T}}\right]\rangle$ or $\langle {(\delta \left[p_{\mathrm{T}}\right]/\left[p_{\mathrm{T}}\right])}^3\rangle$. My findings motivate a dedicated system scan of high-energy heavy ion collisions at RHIC and LHC to measure triaxiality of atomic nuclei: one first determines the coefficients $b$ and $c$ by collisions of isobaric near prolate nuclei, $cos\left(3\gamma \right)\approx 1$, and near oblate nuclei, $cos\left(3\gamma \right)\approx -1$, with known ${\beta }_2$ values, followed by collisions of other species of interest with similar mass number. The $({\beta }_2,\gamma )$ values for this species can be inferred directly from the measured variance and skewness observables from these collisions. The results demonstrate the unique opportunities offered by high-energy collisions as a tool to perform interdisciplinary nuclear physics studies.

Nuclear properties across the chart of nuclides are key to improving and validating our understanding of the strong interaction in nuclear physics. We present high-precision mass measurements of neutron-rich Fe isotopes performed at the TITAN facility. The multiple-reflection time-of-flight mass spectrometer (MR-ToF-MS), achieving a resolving power greater than $600000$ for the first time, enabled the measurement of ${}^{63–70}\mathrm{Fe}$, including first-time high-precision direct measurements ($\delta m/m\approx {10}^{-7}$) of ${}^{68–70}\mathrm{Fe}$, as well as the discovery of a long-lived isomeric state in ${}^{69}\mathrm{Fe}$. These measurements are accompanied by both mean-field and ab initio calculations using the most recent realizations which enable theoretical assignment of the spin-parities of the ${}^{69}\mathrm{Fe}$ ground and isomeric states. Together with mean-field calculations of quadrupole deformation parameters for the Fe isotope chain, these results benchmark a maximum of deformation in the $N=40$ island of inversion in Fe and shed light on trends in level densities indicated in the newly refined mass surface.

Observables from nuclear and high-energy experiments can be degraded by detector performance and/or methodology in extracting the observables, such as of the final-state characteristics of heavy-ion collisions in relation to a coarsely estimated reaction-plane direction. We propose the use of deblurring methods, such as in optics, to correct for observable degradation. Our main focus is the restoration of triple-differential particle distributions in heavy-ion collisions. We demonstrate that these could be extracted from collision measurements following the Richardson-Lucy deblurring method from optics. We illustrate basic features of the restoration methodology in a schematic model assuming either ideal or more realistic particle detection. The inferred three-dimensional (3D) distributions for collisions may be easier to interpret in terms of collision dynamics and sought properties of bulk matter than the currently employed Fourier coefficients, that combine information from different azimuthal angles relative to the reaction plane.

We conduct a comprehensive survey of the shape parameter space of the nuclear pasta phases in neutron star crusts by conducting three-dimensional $\mathrm{Hartree-Fock}+\mathrm{BCS}$ calculations. Spaghetti, waffles, lasagna, bicontinuous phases and cylindrical holes occupy local minima in the resulting constant-pressure Gibbs energy surfaces, implying multiple geometries coexist at a given depth. Notably, the bicontinuous phase, in which both the neutron gas and nuclear matter extend continuously in all dimensions appears over a large depth range. Our results support the idea that nuclear pasta is a glassy system. At a characteristic temperature, of order ${10}^8\text{–}{10}^9\mathrm{K}$, different phases may become frozen into domains whose sizes we estimate to be 1–50 times the lattice spacing and over which the local density and electron fraction can vary. Above this temperature, very little long-range order exists and matter is an amorphous solid. Electron scattering off domain boundaries may contribute to the disorder resistivity of the pasta phases. Annealing of the domains may occur during cooling; repopulating of local minima during crustal heating might lead to temperature-dependent transport properties in the deep crust layers. We identify four regions distinguished by whether pasta is the true ground state, and whether the pasta structure allows delocalization of protons. The whole pasta region can occupy up to 70% of the crust by mass and 25% by thickness, and the layer in which protons are delocalized could occupy 45% of the crust mass and 15% of its thickness.

We perform first-principle calculations of electron-nucleus scattering on ${}^3\mathrm{He}$ and ${}^3\mathrm{H}$ using the Green's function Monte Carlo method and two approaches based on the factorization of the final hadronic state: the spectral-function formalism and the short-time approximation. These three methods are benchmarked among each other and compared to the experimental data for the longitudinal and transverse electromagnetic response functions of ${}^3\mathrm{He}$, and the inclusive cross sections of both ${}^3\mathrm{He}$ and ${}^3\mathrm{H}$. Since these three approaches are based on the same description of nuclear dynamics of the initial target state, comparing their results enables a precise quantification of the uncertainties inherent to factorization schemes. At sufficiently large values of the momentum transfer, we find an excellent agreement of the Green's function Monte Carlo calculation with experimental data and with both the spectral-function formalism and the short-time approximation. We also analyze the relevance of relativistic effects, whose inclusion becomes crucial to explain data at high momentum and energy transfer.

Background: Abundance anomalies in some globular clusters, such as the enhancement of potassium and the depletion of magnesium, can be explained in terms of an earlier generation of stars polluting the presently observed ones. It was shown that the potential range of temperatures and densities of the polluting sites depends on the strength of a few critical reaction rates. The ${}^{30}\mathrm{Si}(p,\gamma ){}^{31}\mathrm{P}$ reaction has been identified as one of these important reactions.

Purpose: The key ingredient for evaluating the thermonuclear ${}^{30}\mathrm{Si}(p,\gamma ){}^{31}\mathrm{P}$ reaction rate is the strength of the resonances which, at low energy, are proportional to their proton width. Therefore, the goal of this work is to determine the proton widths of unbound ${}^{31}\mathrm{P}$ states.

Method: States in ${}^{31}\mathrm{P}$ were studied at the Maier-Leibnitz-Laboratorium using the one-proton ${}^{30}\mathrm{Si}({}^3\mathrm{He},d){}^{31}\mathrm{P}$ transfer reaction. Deuterons were detected with the high resolution one quadrupole three dipoles (Q3D) magnetic spectrometer. Angular distribution and spectroscopic factors were extracted for 27 states, and proton widths and resonance strengths were calculated for the unbound states.

Results: Several ${}^{31}\mathrm{P}$ unbound states have been observed for the first time in a one-proton transfer reaction. Above 20 MK, the ${}^{30}\mathrm{Si}(p,\gamma ){}^{31}\mathrm{P}$ reaction rate is now entirely estimated from the observed properties of ${}^{31}\mathrm{P}$ states. The reaction rate uncertainty from all resonances other than the $E_r^{\text{c.m.}}=149$ keV resonance has been reduced down to less than a factor of 2 above that temperature. The unknown spin and parity of the $E_r^{\text{c.m.}}=149$ keV resonance dominates the uncertainty in the rate in the relevant temperature range.

Conclusion: The remaining source of uncertainty on the ${}^{30}\mathrm{Si}(p,\gamma ){}^{31}\mathrm{P}$ reaction rate comes from the unknown spin and parity of the $E_r^{\text{c.m.}}=149$ keV resonance which can change the reaction rate by a factor of 10 in the temperature range of interest.

The number of particles detected in a nucleus-nucleus collision strongly depends on the impact parameter of the collision. Therefore, multiplicity fluctuations, as well as rapidity correlations of multiplicities, are dominated by impact parameter fluctuations. We present a method based on Bayesian inference which allows for a robust reconstruction of fluctuations and correlations at fixed impact parameter. We apply the method to ATLAS data on the distribution of charged multiplicity and transverse energy. We argue that multiplicity fluctuations are smaller at large rapidity than around central rapidity. We suggest simple, new analyses to confirm this effect.

In the hydrodynamic model description of heavy ion collisions, the final-state anisotropic flows $v_n$ are linearly related to the strengths of the multipole shape of the distribution of nucleons in the transverse plane, ${\varepsilon }_n$: $v_n\propto {\varepsilon }_n$. The ${\varepsilon }_n$, for $n=1,2,3,4$, are sensitive to the shapes of the colliding ions, characterized by the quadrupole ${\beta }_2$, octupole ${\beta }_3$, and hexadecapole ${\beta }_4$ deformations. This sensitivity is investigated analytically and also in a Monte Carlo Glauber model. One observes a robust linear relation, $\langle {\varepsilon }_n^2\rangle =a_n^{\prime }+b_n^{\prime }{\beta }_n^2$, for events in a fixed centrality. The $\langle {\varepsilon }_1^2\rangle$ has a contribution from ${\beta }_3$ and ${\beta }_4$, and $\langle {\varepsilon }_3^2\rangle$ from ${\beta }_4$. In ultracentral collisions, there are little cross contributions between ${\beta }_2$ and ${\varepsilon }_3$ and between ${\beta }_3$ and ${\varepsilon }_2$, but clear cross contributions are present in noncentral collisions. Additionally, $\langle {\varepsilon }_n^2\rangle$ are insensitive to nonaxial shape parameters such as the triaxiality. This is good news because the measurements of $v_2$, $v_3$, and $v_4$ can be used to constrain simultaneously the ${\beta }_2$, ${\beta }_3$, and ${\beta }_4$ values. This is best done by comparing two colliding ions with similar mass numbers and therefore nearly identical $a_n^{\prime }$, to obtain a simple equation that relates the ${\beta }_n$ of the two species. This opens up the possibility to map the shape of the atomic nuclei at a timescale $(<{10}^{-24}\mathrm{s})$ much shorter than probed by low-energy nuclear structure physics $(<{10}^{-21}\mathrm{s})$, which ultimately may provide information complementary to that obtained in the nuclear structure experiments.

Background: Deep-inelastic scattering (DIS) on the deuteron with spectator nucleon tagging represents a unique method for extracting the free neutron structure functions and exploring the nuclear modifications of bound protons and neutrons. The detection of the spectator (with typical momentum $\lesssim 100$ $\mathrm{MeV/}c$ in the deuteron rest frame) controls the nuclear configuration during the DIS process and enables a differential analysis of nuclear effects. At the future Electron Ion Collider (EIC) such measurements will be performed using far-forward detectors.

Purpose: Simulate deuteron DIS with proton or neutron tagging with the baseline EIC far-forward detector design. Quantify detector acceptance and resolution effects. Study feasibility of free nucleon structure extraction using pole extrapolation in the spectator momentum.

Methods: DIS events with proton and neutron spectators are generated using the BeAGLE Monte Carlo generator. The spectator nucleon momentum is reconstructed including effects of detector acceptance and resolution. Pole extrapolation is performed under realistic conditions. The free nucleon structure extraction is validated by comparing with the input model.

Results: Proton and neutron spectator detection is possible over the full transverse momentum range $0<p_T<100$ $\mathrm{MeV/}c$ needed for pole extrapolation. Resolution effects on the distributions before corrections are $\approx 10\%$ for proton and $\approx 30\%$ for neutron spectators. The overall accuracy of nucleon structure extraction is expected to be at the few-percent level.

Conclusions: Free neutron structure extraction through proton tagging and pole extrapolation is feasible with the baseline EIC far-forward detector design. The corresponding extraction of free proton structure through neutron tagging provides a reference point for future studies of nuclear modifications.

Inclusive electron scattering from nuclear targets has been measured to extract the nuclear dependence of the inelastic cross section (${\sigma }_A$) in Hall C at the Thomas Jefferson National Accelerator facility. Results are presented for ${}^2\mathrm{H}, {}^3\mathrm{He}, {}^4\mathrm{He}, {}^9\mathrm{B}, {}^{12}\mathrm{C}, {}^{63}\mathrm{Cu}$, and ${}^{197}\mathrm{Au}$ at an incident electron beam energy of 5.77 GeV for a range of momentum transfer from $Q^2$ = 2 to 7 (${\mathrm{GeV}/\mathrm{c})}^2$. These data improve the precision of the existing measurements of the EMC effect in the nuclear targets at large $x$ and allow for more detailed examinations of the $A$ dependence of the EMC effect.

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.