Reid Soft Core Research Articles

Hartree-Fock Calculations of 12C Nucleus at Equilibrium and Under Static Compression

Abstract: The ground state features of the semi-doubly magic 12C nucleus (i.e. binding energy, nuclear radius, radial density distribution, and single particle energies) are estimated using ab initio calculations at equilibrium and under high static compression. Nijmegen and Reid soft-core (RSC) potentials are used as input nucleon-nucleon interactions. Within the framework of the constrained spherical Hartree-Fock approximations, we do the calculations in no-core shell-model space, which consists of six major oscillator shells (i.e. 21 single particle orbitals). The sensitivity of the ground state features of the 12C nucleus to the degree of compression and the sensitivity of the equation of state to the two potentials are investigated. We also discovered that the nuclear binding energy calculated using the Nijmegen potential is higher than that calculated using the RSC potential. When utilizing the Nijmegen potential, the curve reaches zero binding energy faster than when using the RSC potential. Besides, in the case of Nijmegen, the spectrum of single-particle energies increases more quickly than in the case of RSC potential under compression. The space between single-particle energy shells is also visible in the energy spectrum. At high compression, the radial density distribution becomes higher than that in the interior zone when the RSC potential is applied. Keywords: Nijmegen potential, RSC potential, Ab-initio calculations model, Radial density distribution, Ground state, 12C nucleus.

STATIC PROPERTIES OF THE 6Li NUCLEUS IN THE THREE-CLUSTER MODEL

This scientific work is devoted to a deep study of the structure of the weakly bound light 6Li nucleus in the framework of the three-cluster α+p+n-model. The realistic Reid soft core potential (RSC-potential) have been utilized as the nucleon-nucleon interaction. The potential with even-odd splitting in waves is used for describing the interaction of the alpha-particle and nucleons. The spectrum of low-lying energy levels of the 6Li nucleus has been calculated, and realistic wave functions of the relative motion of the clusters in this nucleus have been defined. The matrix elements and operators of the main static characteristics of the 6Li nucleus are presented. In particular, the root-mean-square charge and material radii, magnetic dipole and electric quadrupole moments of the 6Li nucleus have been calculated.

Properties of 4He Nucleus in the Mean Field Approximation with Compression

Constrained spherical Hartree–Fock approximations are used to investigate nuclear system of closed shell $$^{4}{\hbox {He}}$$ nucleus. Two potentials were used: Reid soft core (RSC) and Nijmegen potentials. Moreover, the dependence of the nuclear properties was studied for the degree of compression. It was seen that it is possible to compress the nucleus into a smaller volume. This means that nuclear equation of state becomes softer for the compressed nucleus. Also, the nucleus becomes more bounded using RSC than Nijmegen potential. The $$E_{{\mathrm{HF}}}$$ increases steeply toward zero binding energy under compression for both potentials. In the case of using Nijmegen potential, the curve increases to zero binding energy more than RSC potential. In addition to that the spectrum of single particle increases more rapidly for Nijmegen than for RSC potential under compression. For the compressed nucleus, the energy spectrum also clearly shows the gaps between single-particle energy shells. In addition, it is preserved the ordering of the energy spectrum levels and the gaps among them. Finally, except in the interior region, the radial density distribution remains constant, while it is larger with RSC than with Nijmegen potential. At large compression, it becomes larger than that in the interior region when RSC potential is used.

A femtoscopic correlation analysis tool using the Schr\xf6dinger equation (CATS)

We present a new analysis framework called “Correlation Analysis Tool using the Schrödinger equation” (CATS) which computes the two-particle femtoscopy correlation function C(k), with k being the relative momentum for the particle pair. Any local interaction potential and emission source function can be used as an input and the wave function is evaluated exactly. In this paper we present a study on the sensitivity of C(k) to the interaction potential for different particle pairs: p–p, p–mathrm {Lambda }, mathrm {K^-}–p, mathrm {K^+}–p, p–mathrm {Xi }^- and mathrm {Lambda }–mathrm {Lambda }. For the p–p Argonne v_{18} and Reid Soft-Core potentials have been tested. For the other pair systems we present results based on strong potentials obtained from effective Lagrangians such as chi EFT for p–mathrm {Lambda }, Jülich models for mathrm{K}(bar{mathrm{K}})–N and Nijmegen models for mathrm {Lambda }–mathrm {Lambda }. For the p–mathrm {Xi }^- pairs we employ the latest lattice results from the HAL QCD collaboration. Our detailed study of different interacting particle pairs as a function of the source size and different potentials shows that femtoscopic measurements can be exploited in order to constrain the final state interactions among hadrons. In particular, small collision systems of the order of 1 fm, as produced in pp collisions at the LHC, seem to provide a suitable environment for quantitative studies of this kind.

Ground State Properties of Closed Shell <sup>4</sup>He Nucleus under Compression

This paper addresses the effects of nuclear properties at different potentials. Reid Soft Core (RSC) and Nijmegen potentials were used to study the nuclear system of 4He nucleus. It has been examined with and without large compression. Moreover, the constrained spherical Hartree-Fock (CSHF) approximations are used as a major tool of analysis. The dependence of the ground state properties was investigated to the degree of compression. It was noticed that it is possible to compress the nucleus to a smaller volume and the nucleus becomes more bounded using RSC than Nijmegen potential. It was also shown that the spectrum of single particle levels increases more rapidly for Nijmegen than RSC potential under compression. Finally, the radial density distribution remains constant, except in the interior region, while it is larger with RSC than Nijmegen potential. At large compression, the radial density distribution becomes larger than that in the interior region when RSC potential is used.

The (p,2p) reaction in finite range relativistic distorted-wave impulse approximation

The [Formula: see text] reaction on [Formula: see text] at incident proton energy of 300[Formula: see text]MeV is examined in the formalism of finite-range relativistic distorted-wave impulse approximation (FR-RDWIA). In comparison to conventional t-matrix model of Love–Franey, a new form of nucleon–nucleon t-matrix effective interaction is derived at 300[Formula: see text]MeV using Reid soft core potentials for isotopic spin one and taking into account the finite-range effects in the [Formula: see text] interaction at knockout vertex. In comparison to the conventional finite range nonrelativistic and relativistic formalism, the present formalism with a new version of [Formula: see text] t-matrix is effectively reproducing the shape of cross-section energy distributions for [Formula: see text], [Formula: see text] and [Formula: see text] states for asymmetric angle pair of [Formula: see text]–[Formula: see text]. Discrepancies between the experimental cross-section data and finite range theoretical calculations at [Formula: see text][Formula: see text]MeV are reasonably resolved in the present approach. Without any adjustable parameter of bound state, the obtained spectroscopic factors are in reasonably good agreement with the relativistic and nonrelativistic theoretical predictions by [Formula: see text], [Formula: see text] and [Formula: see text] analysis.

Delta Excitation in the Compressed Finite Nucleus <sup>90</sup>Zr

Constrained spherical Hartree-Fock (CSHF) calculations under radial compression are presented for 90Zr in a model space consisting of nine major oscillator shells. An effective baryon-baryon interaction which includes the Δ resonances is used. The nucleon-nucleon (N-N) interaction is Reid Soft Core (RSC) potential. The sensitivity of the results to the choice model space is examined. It is found that the nuclear system becomes more compressible when the model space is increased. The radial density and the number of Δs are decreased by increasing model space. The results suggest that the behavior of single particle energies is independent of the model space.

The effect of Fermi momentum cutoff on the binding energy of closed shell nuclei in the LOCV framework

The ground state binding energies of the light symmetric closed-shell nuclei, i.e., 4He, 12C, 16O and 40Ca and the heavy asymmetric ones, i.e., 48Ca, 90Zr and 120Sn are calculated in the harmonic oscillator (HOS) basis, by imposing the relative Fermi momentum cutoff of two point-like interacting nucleons on the density dependent average effective interactions (DDAEI). The DDAEI are generated through the lowest order constrained variational (LOCV) method calculations for the asymmetric nuclear matter with the operator and the channel dependent type bare nucleon-nucleon potentials, such as the Argonne \(Av_{18}^{j_{\max } = 2}\) and the Reid soft core, Reid68, interactions. In the framework of harmonic oscillator shell model, the cutoff is imposed by defining the maximum value of the relative quantum numbers (RQNmax) in two ways: (1) The RQNmax of the last shell and (2) the RQNmax of each shell, in the ground state of the nucleus. It is shown that present results on the binding energies and the root means square radius are closer to the corresponding experimental data than, our previous works with the same DDAEI potentials, but without the cutoff constraint. However, for the light symmetric nuclei, the second scheme gives less binding energy and larger root mean square radius compare to the first one. While the situation is reversed for the heavier nuclei.

Recoil-corrected continuum shell model calculations for11B(p,n)11C,11B(p,p′)11B*,12C(e,e′x)

Background: The recoil-corrected continuum shell model provides coupled-channels solutions for bound and unbound wave functions from realistic effective interactions. The wave functions are antisymmetric and contain no spurious components because the calculations are performed in the center-of-mass system.Purpose: This model has now been extended to include 1$\ensuremath{\hbar}\ensuremath{\omega}$ excitations in the structure of $p$-shell target (residual) nuclei, hence allowing 0$s$-shell knockout processes. Several reactions involving the ${}^{12}$C compound system are investigated to demonstrate the utility of the model.Methods: The states of ${}^{11}$B and ${}^{11}$C are constructed in the nonspurious 0$\ensuremath{\hbar}\ensuremath{\omega}$ plus 1$\ensuremath{\hbar}\ensuremath{\omega}$ model space. An interaction, fitted to Cohen and Kurath (8-16) plus Reid soft core $g$-matrix elements, is employed. One nucleon is coupled to these states to create a basis for the bound and scattering states for ${}^{12}$C.Results: Calculated elastic and inelastic cross sections agree well with available data. The calculated transverse response at high momentum transfer is lower than that extracted from data. Although significant, meson exchange currents are not sufficient to give agreement with data. Likewise inclusion of 0$s$-shell knockout is not sufficient to provide agreement. The high-energy octupole resonance appears at low momentum transfer and an energy of 106/${A}^{1/3}$.Conclusions: The model should provide meaningful predictions for states near the proton drip line via the ($p$,$n$) reaction. Coupled-channels solutions are necessary for describing ${}^{12}$C($e$,${e}^{\ensuremath{'}}x$) at low momentum transfer. Lack of strength at low energy and momentum transfer in optical model calculations of ${}^{12}$C($e$,${e}^{\ensuremath{'}}x$) is at least partly attributable to the omission of giant resonances. Data for (${\ensuremath{\pi}}^{+}$,${\ensuremath{\pi}}^{+\ensuremath{'}}p$)/(${\ensuremath{\pi}}^{\ensuremath{-}}$,${\ensuremath{\pi}}^{\ensuremath{-}\ensuremath{'}}p$) could verify this conclusion. Lack of strength in the transverse response may be attributable to recoil terms which are omitted in most calculations.

Nucleon-Nucleon t-matrix Effective Interaction for (p, 2p) Reactions

In (p, 2p) reactions the p-p interaction is the main knockout interaction which acts in its full glory. Recent finite range (p, 2p) calculations used a parametrized spin-isospin dependent finite range tNN (r) - effective interaction operator which is a function of the distance between the nucleons alone. In the present work we have evaluated the nucleon-nucleon (N-N) t-matrix effective interaction using the Reid soft core N-N interaction potential. In this we find that the proper t- matrix effective interaction is not only a function of spin and isospin of the interacting nucleons but is a strong function of the relative orbital angular momentum, L also. The radial behaviour of the tL (r) is also found to be very unconventional in that it is found to go to zero at r = 0 while in the conventional usage the t(r) is taken to peak at r = 0 because of its representation in terms of Yukawa functions. This very different behaviour of our calculated N-N t-matrix effective interaction operator is expected to make large changes in the Finite Range (FR-DWIA) predictions for (p, 2p) reaction cross sections. Besides this it is hoped that the marked differences one obtains in the behaviour of these tL(r)'s from different realistic N-N interactions through their resulting fits obtained for the (p, 2p) reactions will provide us a tool to select the proper N-N interaction out of the many available in literature.

Resonances-Excitation Calculation Studies Investigation of Δ(3, 3) in Ground State of 90Zr Cold Finite Heavy Nucleus at Equilibrium and Under Large Compression

A non-relativistic microscopic mean field theory of finite nuclei is investigated where the nucleus is described as a collection of nucleons and delta resonances. The ground state properties of 90Zr nucleus have been investigated at equilibrium and large amplitude compression using a realistic effective baryon-baryon Hamiltonian based on Reid Soft Core (RSC) potential. The sensitivity of the ground state properties is studied, such as binding energy, nuclear radius, radial density distribution, and single particle energies to the degree of compression. It is found that the most of increasing in the nuclear energy generated under compression is used to create the massive Δ particles. For 90Zr nucleus under compression at 2.5 times density of the normal nuclear density, the excited nucleons to Δ's are increased sharply up to 14% of the total number of constituents. This result is consistent with the values extracted from relativistic heavy-ion collisions. The single particle energy levels are calculated and their behaviors under compression are examined too. A good agreement between results with effective Hamiltonian and the phenomenological shell model for the low lying single-particle spectra is obtained. A considerable reduction in compressibility for the nucleus, and softening of the equation of state with the inclusion of the Δ's in the nuclear dynamics are suggested by the results.

Δ-Resonances in Ground State Properties of 2040Ca Spherical Cold Finite Nucleus at Equilibrium and under Compression

The ground state properties of the spherical nucleus 40Ca have been investigated by using constrained spherical Hartree–Fock (CSHF) approximation at equilibrium and under high radial compression in a six major shells. The effective baryon-baryon interaction that includes the Δ(1236) resonance freedom degrees to calculate nuclear properties is used. The nucleon-nucleon (N-N) interaction is based on Reid soft core (RSC) potential. The results of calculations show that much of increase in the nuclear energy generated under compression is used to create the massive Δ particles. The number of Δ's can be increased to about 2.1% of constituents of nucleus when nuclear density reaches about 1.34 times of normal density. The single particle energy levels are calculated and their behavior under compression is also examined. A good agreement has been found between current calculations and phenomenological shell model for low lying single-particle spectra. The gap between shells is very clear and L-S coupling become stronger as increasing the static load on the nucleus. The results show a considerable reduction in compressibility when freedom degrees of Δ's are taken into account. It has been found that the total nuclear radial density becomes denser in the interior and less dense in the exterior region of nucleus. The surface of nucleus becomes more and more responsive to compression than outer region.

The LOCV nuclear matter calculation and the magnetic susceptibility of neutron matter

The lowest order constrained variational (LOCV) calculation for asymmetrical nuclear matter is revisited by using Reid93, Av 18 , etc., potentials. It is shown that our previous two-body cluster energy formulations need some modifications when they switch from the old Reid soft core interaction to the Argonne type potentials which have the operator basis structure. The modified equations increase the saturation energy of nuclear matter by ∼ 5 MeV for Av 18 interaction and it is in agreement with that of Reid68 potential. Similar to our 1979 work, the new formulation is used to calculate the magnetic susceptibility of neutron matter (MSNM). The predicted MSNM is in agreement with the others methods such as the Brueckner–Hartree–Fock (BHF) for the Av 14 and Av 18 potentials, but as before, it is considerably larger than the similar calculation (BHF) performed by Haensel with Reid68 interaction. While no evidence of ferromagnetic phase is found with these potentials, the magnetic phase transition is observed for Reid93 interaction.

The one-body momentum distribution of nuclear matter at finite temperature

The one-body momentum distribution of nuclear matter at finite temperature, n ( k ) , is evaluated by using the cluster expansion theory for the occupation probability. The lowest order constrained variational (LOCV) method at finite temperature is used to calculate the correlation functions. The input nucleon–nucleon interactions are the Nijmegen- Reid93 and A v 18 as well as Reid68 and Δ - Reid soft core, phenomenological potentials. The gap in n ( k ) at the Fermi surface is found to be about 97 per cent comparing to 1.0 for the noninteracting Fermi gas model at zero temperature and it is shown to decrease by increasing the temperature. It is also demonstrated that the high-momentum tail of n ( k ) gets larger as we increase the temperature and the application of different interactions do not have dramatic effect on its behavior.

State-dependent calculation of three-body cluster energy for nuclear matter and the validity of the lowest order constrained variational formalism

It is shown that the method of lowest order constrained variational (LOCV) which is based on the cluster expansion theory is a reliable many-body technique to calculate the nuclear matter equation of state. In this respect, the state dependent correlation functions and the effective interactions which have been produced by the LOCV calculation with the Reid and {delta}-Reid soft core interactions are used to estimate the size of higher order cluster terms such as the effect of three-body cluster energy on the nuclear matter ground state energy. Finally it is shown that the LOCV normalization constraint plays a major role in the convergence of the cluster expansion and the result of LOCV calculation can be as good as more sophisticated approaches which go beyond lowest order.

A review of <sup>20</sup>Ne structure in a full microscopic self-consistent shell–model calculation with tensor correlations

A set of single-particle energies together with a set of two-body matrix- elements derived in a self –consistent manner from the Reid soft–core potential are used to calculate the energy levels of 20 Ne. We used a harmonic oscillator wave function folded with two-body correlation functions in our calculation. It is found that the calculated spectra agree very well with experiment and the best available shell-model calculations by other workers. As a result we have demonstrated that it is possible to calculate the spectroscopy of nuclei microscopically and self-consistently in such a way that both the single –particle energies and the effective two-body interactions are derived from the same procedure. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 63-68

Polarization transfer and spin response functions of theH2(p⃗,n⃗)reaction at345MeV

Differential cross sections and a complete set of polarization observables have been measured in the quasielastic $^{2}\mathrm{H}(p,n)$ reaction at a bombarding energy of $345\phantom{\rule{0.3em}{0ex}}\text{MeV}$ and laboratory scattering angles of $16\ifmmode^\circ\else\textdegree\fi{}$, $22\ifmmode^\circ\else\textdegree\fi{}$, and $27\ifmmode^\circ\else\textdegree\fi{}$. The data are compared with plane-wave impulse approximation calculations employing an optimal factorization approximation. The agreement between the experimental and theoretical results validates these approximations in the present momentum- and energy-transfer regions. The experimental spin-longitudinal and spin-transverse response functions, ${R}_{L}$ and ${R}_{T}$, respectively, are deduced from the data. The obtained ${R}_{T}$ is consistent with that obtained from the quasielastic electron scattering. The theoretical calculations with the Reid soft core potential give good descriptions for ${R}_{L}$ and ${R}_{T}$, whereas the latter is slightly underestimated around the quasielastic peak.

The reaction pd→(pp)n at high momentum transfer and short-range NN properties

A recent cross section measurement of the deuteron breakup reaction pd→(pp)(0°)+n(180°), in the GeV region in a kinematics similar to backward pd elastic scattering, strongly contradicts predictions of a pd→dp model based on the one-nucleon exchange, single pN scattering and Δ excitation mechanisms, and on the wave functions of the Reid soft core and Paris NN potentials. We show within the same model that for the CD Bonn NN potential there is qualitative agreement with the data. It is attributed to a reduction of the one-nucleon exchange at energies above 1 GeV and an increase of the Δ(1232)-isobar contribution, both related to the short-range properties of the wave functions generated by this potential.

Quark Cluster Model of Baryon-Baryon Interaction

The quark cluster model of the baryon-baryon interaction is reviewed. The emphasis is on the foundation of the approach and the main features of the model. The origins of the short-range repulsion in the nuclear force and other baryonic interactions are discussed. The origin of the nuclear force is a central problem of nuclear physics. It has a long history since Yukawa 1) proposed the static one-pion exchange potential (OPEP) of the range of the pion Compton wave length. The discovery of the pion boosted both the experimental and theoretical investigation of the nuclear force. It was realized in the early stage that the short range part of the nuclear force requires further ingredients, including the two- and multi-pion exchange interactions. In order to study the shorter-range part, Taketani proposed to consider three regions of the nuclear force separately. 2) (I) The long-range part (R ≥ 2 fm) is described by OPEP. (II) The medium-range part (1 ≤ R ≤ 2 fm) is attributed to nonstatic part of the pion exchange, multipion exchanges and heavy meson exchanges. Finally, (III) the shortrange part (R ≤ 1 fm) is left for phenomenology as it is most “complicated”. The OPEP was firmly established since then and the region (II) was studied in various approaches that treat nonstatic contribution and heavy mesons. It is believed to give a strong attraction due mostly to exchange of two pions correlated in the S-wave, which is often represented by the exchange of a scalar sigma meson. It turned out that this strong attraction of the second region is crucial for the nuclear binding. The shortest range region (III) was found to have a strong repulsion as suggested from the medium-energy nucleon-nucleon ( NN ) scattering data. Thus the nuclear binding is on the balance between the attraction in the regions (I) and (II) and the repulsion in (III). Meanwhile, many phenomenological models have been constructed. Main source of information is the NN scattering observables as well as properties of the deuteron. The proposed models include the Hamada-Johnston potential, 3) the Reid soft core potential, 4) the Tamagaki potential, 5) all of which can explain the low energy observables fairly well. The real development in the study of region (II) was achieved in 1970’s resulting in two popular models of nuclear force, Bonn 6) and Paris 7) po

Relative wavefunctions, vertex functions and coupling constants for the virtual decay of 4He

The d + d, t + p and h + n relative wavefunctions and their asymptotic normalizations are considered in the framework of the generator coordinate method (GCM) and compared with ATMS (amalgamation of two-body correlation into multiple scattering processes) method which used the realistic Reid soft core interaction. The asymptotic normalization of relative wavefunctions provide various coupling constants, the cluster probability amplitude (the so-called Z 1/2-factor) and matter RMS radii. These wavefunctions are also used to obtain 4He − d − d, 4He − t − p and 4He − h − n vertex functions in the virtual decay of 4He. The extrapolation of vertex functions for negative values of q 2 upto the corresponding poles provide the vertex constants which are comparable with other estimates. It is noticed that in GCM the coupling constants C 2 for 4He − d − d vertex is less than 2 as has been obtained in the forward dispersion relation technique.