Single-nucleon Momentum Distribution Research Articles

Imprints of high-momentum nucleons in nuclei on hard photons from heavy-ion collisions near the Fermi energy

The short-range correlation (SRC) induced by the tensor force in the isosinglet neutron-proton interaction channel leads to a high-momentum tail (HMT) in the single-nucleon momentum distributions n(k) in nuclei. Owing to the remaining uncertainties about the tensor force, the shape of the nucleon HMT may be significantly different from the dilute interacting Fermi gas model prediction $n(k) \sim1/k^4$ similar to the HMT in cold atoms near the unitary limit. Within an isospin- and momentum-dependent Boltzmann-Uehling-Uhlenbeck transport model incorporating approximately the nucleon HMT, we investigate hard photon emissions in $^{14}$N+$^{12}$C and $^{48}$Ca+$^{124}$Sn reactions at beam energies around the Fermi energy. Imprints of different shapes of the HMT on the energy spectrum, angular distribution and transverse momentum spectrum of hard photons are studied. While the angular distribution does not carry any information about the shape of the nucleon HMT, the energy spectra and especially the mid-rapidity transverse momentum spectra of hard photons are found to bare strong imprints of the shapes of nucleon HMTs in the two colliding nuclei.

Examination of an isospin-dependent single-nucleon momentum distribution for isospin-asymmetric nuclear matter in heavy-ion collisions

Within a transport model using nucleon momentum profiles as the input from a parameterized isospin-dependent single-nucleon momentum distribution, with a high momentum tail induced by short-range correlations, we employ \(^{197}\)Au + \(^{197}\)Au collisions at 400 MeV/nucleon to examine the effects of the short-range correlations on the pion and flow observables in probing the nuclear symmetry energy. We investigate how reliable this isospin-dependent single-nucleon momentum distribution is and determine the corresponding parameter settings. Apart from the significant effects of the short-range correlations on the pion and flow observables that are observed, we also find that the theoretical simulations of the \(^{197}\)Au + \(^{197}\)Au collisions with this momentum distribution using two sets of parameters, extracted from the experimental analysis and the self-consistent Green’s function prediction, can reproduce the neutron elliptic flows of the FOPI-LAND experiment and the \(\pi ^-/\pi ^+\) ratios of the FOPI experiment, respectively, under the symmetry energy setting in a particular range. Therefore, we conclude that this parameterized isospin-dependent single-nucleon momentum distribution is reliable for isospin-asymmetric nuclear matter. Correspondingly, two sets of parameters extracted from both the experimental analysis and the self-consistent Green’s function prediction cannot be excluded according to the available experimental information at present.

Isospin composition of the high-momentum fluctuations in nuclei from asymptotic momentum distributions

The variations of short-range correlations (SRC) across nuclei can be quantified in an approximately model-independent fashion in terms of the so-called SRC scaling factors. We provide predictions for the SRC scaling factors for both bound protons and bound neutrons across the nuclear periodic table and determine the contribution of the different nucleon pair combinations to this quantity. Thereby we employ the low-order correlation operator approximation (LCA) to compute the SRC contribution to the single-nucleon momentum distribution. We do this for a sample of fifteen nuclei from He to Pb thereby gaining access to the evolution of the SRC scaling factor with the nuclear mass and the neutron-to-proton ratio. For the studied sample of fifteen nuclei, the total SRC scaling factor is in the range 4.05-5.14. The computed SRC scaling factors in the LCA are in line with those of ab-initio calculations and with measured quantities from SRC-sensitive inclusive electron-scattering data. In asymmetric nuclei there are sizeable differences between the SRC scaling factors for protons and neutrons. It is suggested that this phenomenon may impact the variations of the magnitude of the European muon collaboration (EMC) effect across nuclei. Our results corroborate the finding that SRC physics can be qualitatively understood by universal principles that build on local modifications of mean-field wave functions of nucleon pairs.

The isospin and neutron-to-proton excess dependence of short-range correlations

We provide a systematic study of the isospin composition and neutron-to-proton (NZ) ratio dependence of nuclear short-range correlations (SRC) across the nuclear mass table. We use the low-order correlation operator approximation (LCA) to compute the SRC contribution to the single-nucleon momentum distributions for 14 different nuclei from A=4 to A=208. Ten asymmetric nuclei are included for which the neutrons outnumber the protons by a factor of up to 1.54. The computed momentum distributions are used to extract the pair composition of the SRC. We find that there is a comprehensive picture for the isospin composition of SRC and their evolution with nucleon momentum. We also compute the non-relativistic kinetic energy of neutrons and protons and its evolution with nuclear mass A and NZ. Confirming the conclusions from alternate studies it is shown that the minority species (protons) become increasingly more short-range correlated as the neutron-to-proton ratio increases. We forge connections between measured nucleon-knockout quantities sensitive to SRC and single-nucleon momentum distributions. It is shown that the LCA can account for the observed trends in the data, like the fact that in neutron-rich nuclei the protons are responsible for an unexpectedly large fraction of the high-momentum components.

Inclusive electron-nucleus cross section within the self-consistent Green's function approach

We compute inclusive electron-nucleus cross sections using ab initio spectral functions of $^4$He and $^{16}$O obtained within the Self Consistent Green's Function approach. The formalism adopted is based on the factorization of the spectral function and the nuclear transition matrix elements. This allows to provide an accurate description of nuclear dynamics and to account for relativistic effects in the interaction vertex. Our calculations use a saturating chiral Hamiltonian in order reproduce the correct nuclear sizes. When final state interactions for the struck particle are accounted for, we find nice agreement between the data and the theory for the inclusive electron-$^{16}$O cross section. The results lay the foundations for future applications of the Self Consistent Green's Function method, in both closed and open shell nuclei, to neutrino data analysis. This work also presents results for the point-proton, charge and single-nucleon momentum distribution of the same two nuclei. The center of mass can affect these quantities for light nuclei and cannot be separated cleanly in most ab initio post-Hartree-Fock methods. In order to address this, we developed a Metropolis Monte Carlo calculation in which the center of mass coordinate can be subtracted exactly from the trial wave function and the expectation values. We gauged this effect for $^4$He by removing the center of mass effect from the Optimal Reference State wave function that is generated during the Self Consistent Green's Function calculations. Our findings clearly indicate that the residual center of mass contribution strongly modifies calculated matter distributions with respect to those obtained in the intrinsic frame. Hence, its subtraction is crucial for a correct description of light nuclei.

Single- and two-nucleon momentum distributions for local chiral interactions

We present quantum Monte Carlo calculations of the single- and two-nucleon momentum distributions in selected nuclei for $A\le16$. We employ local chiral interactions at next-to-next-to-leading order. We find good agreement at low momentum with the single-nucleon momentum distributions derived for phenomenological potentials. The same agreement is found for the integrated two-nucleon momentum distributions at low relative momentum $q$ and low center-of-mass momentum $Q$. We provide results for the two-nucleon momentum distributions as a function of both $q$ and $Q$. The large ratio of $pn$ to $pp$ pairs around $q=2\,\rm fm^{-1}$ for back-to-back $(Q=0)$ pairs is confirmed up to $^{16}$O, and results are compatible with those extracted from available experimental data.

Interplay of short-range correlations and nuclear symmetry energy in hard-photon production from heavy-ion reactions at Fermi energies

Within an isospin- and momentum-dependent transport model for nuclear reactions at intermediate energies, we investigate the interplay of the nucleon-nucleon short-range correlations (SRC) and nuclear symmetry energy $E_{sym}(\rho)$ on hard photon spectra in collisions of several Ca isotopes on $^{112}$Sn and $^{124}$Sn targets at a beam energy of 45 MeV/nucleon. It is found that over the whole spectra of hard photons studied, effects of the SRC overwhelm those due to the $E_{sym}(\rho)$. The energetic photons come mostly from the high-momentum tails (HMT) of single-nucleon momentum distributions in the target and projectile. Within the neutron-proton dominance model of SRC based on the consideration that the tensor force acts mostly in the isosinglet and spin-triplet nucleon-nucleon interaction channel, there are equal numbers of neutrons and protons, thus a zero isospin-asymmetry in the HMTs. Therefore, experimental measurements of the energetic photons from heavy-ion collisions at Fermi energies have the great potential to help us better understand the nature of SRC without any appreciable influence by the uncertain $E_{sym}(\rho)$. These measurements will be complementary to but also have some advantages over the ongoing and planned experiments using hadronic messengers from reactions induced by high-energy electrons or protons. Since the underlying physics of SRC and $E_{sym}(\rho)$ are closely correlated, a better understanding of the SRC will in turn help constrain the nuclear symmetry energy more precisely in a broad density range.

Variational calculation of the ground state of closed-shell nuclei up to A=40

Variational calculations of ground-state properties of $^4$He, $^{16}$O, and $^{40}$Ca are carried out employing realistic phenomenological two- and three-nucleon potentials. The trial wave function includes two- and three-body correlations acting on a product of single-particle determinants. Expectation values are evaluated with a cluster expansion for the spin-isospin dependent correlations considering up to five-body cluster terms. The optimal wave function is obtained by minimizing the energy expectation value over a set of up to 20 parameters by means of a nonlinear optimization library. We present results for the binding energy, charge radius, one- and two-body densities, single-nucleon momentum distribution, charge form factor, and Coulomb sum rule. We find that the employed three-nucleon interaction becomes repulsive for $A\geq16$. In $^{16}$O the inclusion of such a force provides a better description of the properties of the nucleus. In $^{40}$Ca instead, the repulsive behavior of the three-body interaction fails to reproduce experimental data for the charge radius and the charge form factor. We find that the high-momentum region of the momentum distributions, determined by the short-range terms of nuclear correlations, exhibit a universal behavior independent of the particular nucleus. The comparison of the Coulomb sum rules for $^4$He, $^{16}$O, and $^{40}$Ca reported in this work will help elucidate in-medium modifications of the nucleon form factors.

Isospin dependence of nucleon effective masses in neutron-rich matter

In this talk, we first briefly review the isospin dependence of the total nucleon effective mass $M^{\ast}_{J}$ inferred from analyzing nucleon-nucleus scattering data within an isospin dependent non-relativistic optical potential model, and the isospin dependence of the nucleon E-mass $M^{\ast,\rm{E}}_{J}$ obtained from applying the Migdal-Luttinger theorem to a phenomenological single-nucleon momentum distribution in nuclei constrained by recent electron-nucleus scattering experiments. Combining information about the isospin dependence of both the nucleon total effective mass and E-mass, we then infer the isospin dependence of nucleon k-mass using the well-known relation $M^{\ast}_{J}=M^{\ast,\rm{E}}_{J}\cdot M^{\ast,\rm{k}}_{J}$. Implications of the results on the nucleon mean free path (MFP) in neutron-rich matter are discussed.

Nucleon effective E-mass in neutron-rich matter from the Migdal–Luttinger jump

The well-known Migdal-Luttinger theorem states that the jump of the single-nucleon momentum distribution at the Fermi surface is equal to the inverse of the nucleon effective E-mass. Recent experiments studying short-range correlations (SRC) in nuclei using electron-nucleus scatterings at the Jefferson National Laboratory (JLAB) together with model calculations constrained significantly the Migdal-Luttinger jump at saturation density of nuclear matter. We show that the corresponding nucleon effective E-mass is consequently constrained to $M_0^{\ast,\rm{E}}/M\approx2.22\pm0.35$ in symmetric nuclear matter (SNM) and the E-mass of neutrons is smaller than that of protons in neutron-rich matter. Moreover, the average depletion of the nucleon Fermi sea increases (decreases) approximately linearly with the isospin asymmetry $\delta$ according to $\kappa_{\rm{p}/\rm{n}}\approx 0.21\pm0.06 \pm (0.19\pm0.08)\delta$ for protons (neutrons). These results will help improve our knowledge about the space-time non-locality of the single-nucleon potential in neutron-rich nucleonic matter useful in both nuclear physics and astrophysics.

Symmetry energy of cold nucleonic matter within a relativistic mean field model encapsulating effects of high-momentum nucleons induced by short-range correlations

Significant progress has been made recently in constraining the isospin-dependent parameters characterizing the SRC (short-range correlation)-modified single-nucleon momentum distribution in neutron-rich nucleonic matter using both experimental data and microscopic model calculations. Using the constrained single-nucleon momentum distribution in a nonlinear relativistic mean field (RMF) model, we study the equation of state (EOS) of asymmetric nucleonic matter (ANM), especially the density dependence of nuclear symmetry energy $E_{\rm{sym}}(\rho)$. Firstly, as a test of the model, the average nucleon kinetic energy extracted recently from electron-nucleus scattering experiments using a neutron-proton dominance model is well reproduced by the RMF model incorporating effects of the SRC-induced high momentum nucleons, while it is significantly under predicted by the RMF model using a step function for the single-nucleon momentum distribution as in free Fermi gas (FFG) models. Secondly, the kinetic symmetry energy of quasi-nucleons is found to be $E^{\rm{kin}}_{\rm{sym}}(\rho_0)=-16.94\pm13.66\,\rm{MeV}$ which is dramatically different from the prediction of $E^{\rm{kin}}_{\rm{sym}}(\rho_0)\approx 12.5$ MeV by FFG models at nuclear matter saturation density $\rho_0=0.16\,\rm{fm}^{-3}$. Thirdly, comparing the RMF calculations with and without the high momentum nucleons using two sets of model parameters both reproducing identically all empirically constraints on the EOS of symmetric nuclear matter (SNM) and the symmetry energy of ANM at $\rho_0$, the SRC-modified single-nucleon momentum distribution is found to make the $E_{\rm{sym}}(\rho)$ more concave around $\rho_0$ by softening it significantly at both sub-saturation and supra-saturation densities, leading to an isospin-dependent incompressibility of ANM in better agreement with existing experimental data.

Isospin quartic term in the kinetic energy of neutron-rich nucleonic matter

The energy of a free gas of neutrons and protons is well known to be approximately isospin parabolic with a negligibly small quartic term of only $0.45$ MeV at the saturation density of nuclear matter $\rho_0=0.16/\rm{fm}^3$. Using an isospin-dependent single-nucleon momentum distribution including a high (low) momentum tail (depletion) with its shape parameters constrained by recent high-energy electron scattering and medium-energy nuclear photodisintegration experiments as well as the state-of-the-art calculations of the deuteron wave function and the equation of state of pure neutron matter near the unitary limit within several modern microscopic many-body theories, we show for the first time that the kinetic energy of interacting nucleons in neutron-rich nucleonic matter has a significant quartic term of $7.18\pm2.52\,\rm{MeV}$. Such a large quartic term has significant ramifications in determining the equation of state of neutron-rich nucleonic matter using both terrestrial and astrophysical observables.

Stylized features of single-nucleon momentum distributions

Nuclear short-range correlations (SRC) typically manifest themselves in the tail parts of the single-nucleon momentum distributions. We propose an approximate practical method for computing those SRC contributions to the high-momentum parts. The framework adopted in this work is applicable throughout the nuclear mass table and corrects mean-field models for central, spin–isospin and tensor correlations by shifting the complexity induced by the SRC from the wave functions to the operators. It is argued that the expansion of these modified operators can be truncated to a low order. The proposed model can generate the SRC-related high-momentum tail of the single-nucleon momentum distribution. These are dominated by correlation operators acting on mean-field pairs with vanishing relative radial and angular-momentum quantum numbers. The proposed method explains the dominant role of proton–neutron pairs in generating the SRC and accounts for the magnitude and mass dependence of SRC as probed in inclusive electron scattering. It also provides predictions for the ratio of the amount of correlated proton–proton to proton–neutron pairs which are in line with the observations. In asymmetric nuclei, the correlations make the average kinetic energy for the minority nucleons larger than for the majority nucleons.

Nucleon and nucleon-pair momentum distributions inA≤12nuclei

We report variational Monte Carlo calculations of single-nucleon momentum distributions for A <= 12 nuclei and nucleon-pair and nucleon-cluster momentum distributions for A <= 8. The wave functions have been generated for a Hamiltonian containing the Argonne v18 two-nucleon and Urbana X three-nucleon potentials. The single-nucleon and nucleon-pair momentum distributions exhibit universal features attributable to the one-pion-exchange tensor interaction. The single-nucleon distributions are broken down into proton and neutron components and spin-up and spin-down components where appropriate. The nucleon-pair momentum distributions are given separately for pp and pn pairs. The nucleon-cluster momentum distributions include dp in 3He, tp and dd in 4He, alpha-d in 6Li, alpha-t in 7Li, and alpha-alpha in 8Be. Detailed tables are provided on-line for download.

Delineating effects of tensor force on the density dependence of nuclear symmetry energy

In this talk, we report results of our recent studies to delineate effects of the tensor force on the density dependence of nuclear symmetry energy within phenomenological models. The tensor force active in the isosinglet neutron roton interaction channel leads to appreciable depletion/population of nucleons below/above the Fermi surface in the single-nucleon momentum distribution in cold symmetric nuclear matter (SNM). We found that as a consequence of the high momentum tail in SNM the kinetic part of the symmetry energy Ekinsym(ρ) is significantly below the well-known Fermi gas model prediction of approximately 125(ρ/ρ0)2/3. With about 15% nucleons in the high momentum tail as indicated by the recent experiments at J-Lab by the CLAS Collaboration, the Ekinsym(ρ) is negligibly small. It even becomes negative when more nucleons are in the high momentum tail in SNM. These features have recently been confirmed by three independent studies based on the state-of-the-art microscopic nuclear many-body theories. In addition, we also estimate the second-order tensor force contribution to the potential part of the symmetry energy. Implications of these findings in extracting information about nuclear symmetry energy from nuclear reactions are discussed briefly.

Extraction of nucleon momentum distributions from inclusive electron scattering on nuclei

We address the problem of extracting single-nucleon momentum distributions n( p) from inclusive lepton scattering data. A model for these relates nuclear and nucleon structure functions (SF) through an intermediate SF f PN of a nucleus of point-particles. In addition to the asymptotic limit (AL), f PN contains, generally q-dependent, Final State Interactions (FSI) parts. In the inverse problem we employ the structure of the theory, but no numerical results, in order to separate q-dependent FSI from the AL, which depends on n( p). It appears that in the q-range of the analyzed lepton data, the FSI are only weakly q-dependent, making it virtually impossible to obtain parameters in a free fit of the parametrized components of f. Imposing we obtain n( p) for Fe and 4 He.