Reid Soft Core Research Articles

Parity conservingγasymmetry inn-pradiative capture

The importance of n-p radiative capture, utilizing polarized cold neutrons, as a means of measuring the weak pion coupling constant is reviewed. Parity conserving processes of the form k_gamma*[s_n,k_n] can contribute to the s_n*k_gamma photon asymmetry in any such experiment, if the apparatus is not perfectly symmetric. For an incident laboratory neutron energy of 0.003 eV a value of A^{PC}_{gamma}=0.67*10^{-8} is obtained for two different potential models (Argonne AV14 and Nijmegen Reid93). Serving as an extreme test case, the Reid soft core potential yields 0.61*10^{-8}, close to the result of the contemporary forces. Implications for extracting the weak pion coupling constant and for monitoring the beam polarization are discussed.

Magnetic dipole properties in deformed nuclei

The magnetic dipole response in deformed nuclei is investigated. Spurious states due to the rotation of the whole nucleus are removed by restricting the residual interaction in such a way that it commutes with the total angular momentum as a ground state expectation value in the quasi-particle random phase approximation. By this one guarantees that the spurious state due to rotations lies at energy zero and that it is a solution of the quasi-particle random phase equation (QRPA) and can easily be removed. The Hamiltonian is chosen to be a deformed Saxon Woods potential with a pairing force and an isoscalar and isovector generalized quadrupole-quadrupole and spin-spin interaction. The generalized isoscalar quadrupole operator is defined as the commutator of the independent quasi-particle Hamiltonian with the total angular momentum. The restoration of rotational symmetry mentioned above determines the isoscalar coupling constant. The generalized isovector quadrupole-quadrupole force does not commute with the total angular momentum. But one can find a linear “isovector” combination of the proton and the neutron total angular momentum which produces by a commutator with the independent quasiparticle Hamiltonian an “isovector” quadrupole operator which yields an isovector quadrupole-quadrupole force which always commutes with the total angular momentum. Thus one gets no restriction of the isovector quadrupole-quadrupole force constant. We fit it to the isovector giant quadrupole resonance ( E(E2) = 130 A 1 3 MeV ). The Hamiltonian contains in addition a spin-spin force, which always commutes with the total angular momentum and therefore does not spoil the restoration of rotational symmetry. The proton-proton, neutron-neutron and proton-neutron spin force constant is determined by solving the nuclear matter problem with the Reid soft core interaction within the Brückner theory and translating the result with the help of the Migdal force into finite nuclei. This yields an almost pure repulsive isovector spin-spin force. This Hamiltonian predicts a high lying ΔN = 2 scissors mode which lies between 17 and 26 MeV. It is spread over very many 1 + states with an average spacing of 15 keV. This highlying scissors mode consists of several hundred states and the overlap with an artificially constructed scissors mode is of the order of 38%, while the low lying scissors mode which shows up as 1 + states around 3 MeV has an overlap with the scissors mode of more than 50 %. In the same energy range as the high lying scissors mode lies also the isovector giant quadrupole resonance. Even at 180°, that means at completely backward angles, the E2 is comparable with the M1 excitation. This is due to the transversal quadrupole transition probability which is not disappearing in electron scattering at 180°. The same Hamiltonian is nicely describing the low lying scissors mode around 3 MeV in the rare earth and the transuranic nuclei. It gives the energies and the reduced magnetic dipole transition probabilities. It also can reproduce the formfactors measured by inelastic electron scattering. The Hamiltonian also reproduces the spin-flip states between 5 and 9 MeV excitation energy, which can be measured by inelastic proton scattering. One obtains in this region two maxima. The lowest one around 6 MeV is mainly of isoscalar nature and is weaker than the spin flip maximum at 7 to 8 MeV, which is mainly of isovector nature. The detailed form of this two-spin flip maxima depend very sensitively on the ratio of the isoscalar and isovector strength of the spin-spin force. In conclusion one can say: There exist a high lying scissors mode, but it is strongly fragmented over several hundred 1 + states between 17 and 26 MeV and even in electron scattering at backward angles the cross section for the excitation of the isovector giant quadrupole resonance is of comparable size. Thus it will be difficult to establish experimentally the existence of this high lying scissors mode.

Delta excitations in compressed finite nuclei. II. Nucleon-nucleon interaction dependence.

Spherical Hartree-Fock calculations at equilibrium and under radial compression are presented for $^{40}\mathrm{Ca}$ and $^{90}\mathrm{Zr}$ in a model space of seven major oscillator shells. We employ an effective baryon-baryon interaction which includes the \ensuremath{\Delta}(1236) resonance explicitly. The nucleon-nucleon (N-N) interactions are based on the Nijmegen and Reid soft core potentials. The results indicate a significant dependence on the choice of the N-N interaction at large nuclear compression. We found mixing between nucleons and \ensuremath{\Delta}'s even at equilibrium in $^{90}\mathrm{Zr}$ with either choice of potentials, but not in $^{40}\mathrm{Ca}$. Substantial amounts of the excitation energy induced by compression goes to the creation of \ensuremath{\Delta} excitations. \textcopyright{} 1996 The American Physical Society.

Investigation of a matrix technique to produce an effective hamiltonian as applied to the p-shell

Abstract Several aspects of a technique for calculating effective interactions, which avoids the perturbation expansion while eliminating the effects of unlinked graphs, are investigated. In the course of this investigation the effective p-shell interactions are calculated for the Reid Soft Core, the Paris, and the coordinate space Bonn interactions. The convergence properties of two approaches to summing the folded diagram series, that of Krenciglowa and Kuo and that of Suzuki and Lee are compared. The convergence properties of the latter, in the cases studied, are found to be superior. We find that removing the influence of unlinked graphs has tittle effect on excitation energies and wave functions and may be characterized by a mass dependent shift in the binding energy. In addition we note that these calculated excitation energies agree quite well with experiment.

Complex Kohn Variational Principle for Two-Nucleon Bound-State and Scattering with the Tensor Potential

Complex Kohn variational principle is applied to the numerical solution of the fully off-shell Lippmann-Schwinger equation for nucleon-nucleon scattering for various partial waves including the coupled 3S1 -3D1 channel. Analytic expressions are obtained for all the integrals in the method for a suitable choice of expansion functions. Calculations with the partial waves 1S0 , 1P1, 1D2 , and 3S1 -3D1 of the Reid soft core potential show that the method converges faster than other solution schemes not only for the phase shift but also for the off-shell t matrix elements. We also show that it is trivial to modify this variational principle in order to make it suitable for bound-state calculation. The bound-state approach is illustrated for the 3S1 -3D1 channel of the Reid soft-core potential for calculating the deuteron binding, wave function, and the D state asymptotic parameters.

Trinucleon system with Reid soft core potential by hyperspherical harmonics expansion method

The hyperspherical harmonics expansion method has been used to solve the trinucleon system where the nucleons interact through Reid soft core potential. The binding energy of triton, both for extreme adiabatic approximation and uncoupled adiabatic approximation, and theS-state,S′-state andD-state probabilities are calculated and compared with the results by the Faddeev method.

Convergence of triton asymptotic wave function for hyperspherical harmonics expansion with two nucleon Reid soft core potential

The asymptotic normalization constants (ANC) ${\mathit{C}}_{0}$ and ${\mathit{C}}_{2}$ of the triton have been calculated by the hyperspherical harmonics expansion method with the Reid soft core potential (no three body force). The results do not agree with the corresponding calculations by the Faddeev method, when only a few hyperspherical partial waves are included. However Schneider's convergence theorems on hyperspherical expansion allow one to extrapolate the results for a large number of partial waves and then they agree fairly well with the Faddeev results. This indicates that even though the hyperspherical expansion for the asymptotic wave function is very slow, a convergent and reliable wave function is attained by extrapolation of a relatively small-sized calculation.

An Efficient Numerical Algorithm for Calculation of Matrix Elements of Trinucleon System with Realistic Potentials

A new method of evaluation of matrix elements, involving integration of a very rapidly oscillating function, has been developed. The proposed procedure is accurate, efficient, and fast and has been applied to the calculation of potential matrix elements of the trinucleon system with Reid soft core potential in hyperspherical harmonics expansion method.

Nuclear shell model with delta excitations

Working in an oscillator basis, we define the nuclear Hamiltonian in a no-core model space to consist of an effective nucleon-nucleon interaction obtained with Brueckner theory from the Reid Soft Core interaction, a Coulomb term, nucleondelza transition potentials, and delta-delta interaction terms. This Hamiltonian is suitable for direct diagonalization and will account for interaction effects usually attributed to three-body, four-body, etc., forces arising from the virtual excitation of deltas. A natural outcome of this approach is a nuclear shell-model involving single particle states which are primarily nucleon in character and others, at higher energies, which are primarily delta in character. By performing spherical Hartree Fock calculations with this “realistic baryon Hamiltonian” we have found that the ground states of 16O, 40Ca and 56Ni are nearly pure nucleonic states while 2 Sn has about 10% of one delta in the ground state. For an estimate of how the delta degree of freedom is excited as one goes away from the ground state, we have performed spherical Hartree-Fock calculations with a radial constraint to compress the nucleus. The delta degree of freedom is gradually populated as the nucleus is compressed. Our results suggest that inclusion of the delta in the nuclear dynamics could lead to a significant softening of the nuclear equation of state.

Relativistic light-cone approach to elastic electron-deuteron scattering

The connection between light-cone dynamics and field theory is analyzed in detail within the zero-range approximation, with particular attention to the so-called “angular condition”. New and detailed calculations for the elastic form factors of the deuteron are presented for the Bonn, Paris, Reid Soft-Core, Argonne V14 and one-pion-exchange potentials with various choices of the electromagnetic form factors of the nucleon.

Hyperspherical harmonics approach to the trinucleon system with Reid soft core potential: Calculation of geometrical structure coefficients

We present the full set of equations for the solution of the trinucleon problem by the hyperspherical harmonics expansion (HHE) method where nucleons interact via the Reid soft core (RSC) potential. The coupling potential matrix elements are expressed in terms of geometrical structure coefficients (GSC) and potential multipoles (PM). Introduction of GSC greatly simplifies the calculation of the potential matrix and makes the numerical algorithm efficient. A method for calculating all the twelve independent sets of GSC needed, by using the completeness property of the Jacobi polynomials has been presented. A convenient sum rule for each set of GSC has also been derived and precision of the calculated GSC has been checked by the sum rule. Such calculations of GSC are efficient and fast, in view of the complexity of the HHE equations.

Are the low-lying isovector 1 + states scissors vibrations?

At the Technische Hochschule in Darmstadt the group of Richter and coworkers found in 1983/84 in deformed rare earth nuclei low-lying isovector 1 + states. Such states have been predicted in the generalized Bohr-Mottelson model and in the interacting boson model no. 2 (IBA2). In the generalized Bohr-Mottelson model one allows for proton and neutron quadrupole deformations separately. If one includes only static proton and neutron deformations the generalized Bohr-Mottelson model reduces to the two rotor model. It describes the excitation energy of these states in good agreement with the data but overestimates the magnetic dipole transition probabilities by a factor 5. In the interacting boson model (IBA2) where only the outermost nucleons participate in the excitation the magnetic dipole transition probability is only overestimated by a factor 2. The too large collectivity in both models results from the fact that they concentrate the whole strength of the scissors vibrations into one state. A microscopic description is needed to describe the spreading of the scissors strength over several states. For a microscopic determination of these scissors states one uses the Quasi-particle Random Phase Approximation (QRPA). But this approach has a serious difficulty. Since one rotates for the calculation the nucleus into the intrinsic system the state corresponding to the rotation of the whole nucleus is a spurious state. The usual procedure to remove this spuriosity is to use the Thouless theorem which says that a spurious state created by an operator which commutes with the total hamiltonian (here the total angular momentum, corresponding to a rotation of the whole system) produces the spurious state if applied to the ground state. It says further the energy of this spurious state lies at zero excitation energy (it is degenerate with the ground state) and is orthogonal to all physical states. Thus the usual approach is to vary the quadrupole-quadrupole force strength so that a state lies at zero excitation energy and to identify that with the spuríous state. This procedure assumes that a total angular momentum commutes with a total hamiltonian. But this is not the case since the total hamiltonian contains a deformed Saxon-Woods potential. Thus one has to take care explicitly that the spurious state is removed. This we do in our approach by introducing Lagrange multipliers for each excited states and requesting that these states are orthogonal to the spurious state which is explicitly constructed by applying the total angular momentum operator to the ground state. To reduce the number of free parameters in the hamiltonian we take the Saxon-Woods potential for the deformed nuclei from the literature (with minor adjustments) and determine the proton-proton, neutron-neutron and the proton-neutron quadrupole force constant by requesting that the hamiltonian commutes with the total angular momentum in the (QRPA) ground state. This yields equations fixing all three coupling constants for the quadrupole-quadrupole force allowing even for isospin symmetry violation. The spin-spin force is taken from the Reid soft core potential. A possible spin-quadrupole force has been taken from the work of Soloviev but it turns out that this is not important. The calculation shows that the strength of the scissors vibrations are spread over many states. The main 1 + state at around 3 MeV has an overlap of the order of 14 % of the scissors state. 50% of that state are spread over the physical states up to an excitation energy of 6 MeV. The rest is distributed over higher lying states. The expectation value of the many-body hamiltonian in the scissors vibrational state shows roughly an excitation energy of 7 MeV above the ground state. The results also support the experimental findings that these states are mainly orbital excitations. States are not very collective. Normally only a proton and neutron particle-hole pair are with a large amplitude participating in forming these states. But those protons and neutrons which are excited perform scissors type vibrations.

Effective Two-Nucleon Interaction in Asymmetric Nuclear Matter

On the basis of the G·matrix calculation with the Reid soft core potential, the effective two· nucleon interaction ( V) of the central force type in asymm~tric nuclear matter is constructed. It is expressed in the five· range Gaussian and depends on the two· nucleon state, the density and the asymmetry parameter (a). This effective interaction is expected to be useful in the study on the properties of neutron star matter where protons admix with neutrons. The proton fraction of cold neutron star matter is investigated as an example and the importance of the a-dependence of V is stressed. Possible approximation for the a-dependence of the potential energy is also discussed.

Effects of tensor interaction in p-n breakup states on deuteron elastic scattering

The Reid soft core potential is employed for p-n continuum states in virtual-breakup components of deuteron elastic scattering. Compared to the previous calculation with central gaussian potentials, the new calculation produces small but non-negligible corrections to tensor analyzing powers and a remarkable change in a particular linear combination of polarization transfer coefficients, suggesting the latter to be a useful source of information on nuclear forces.

Numerical quasiparticle representation of the nucleon-nucleonTmatrix

In this paper we develop a practical ``hybrid'' numerical representation of the nucleon-nucleon T matrix. Part of the expression contains nonseparable terms which are easily calculated, and the rest consists of a separable representation of small rank in terms of Weinberg states. The method rests on the observation that when a set of positive-energy Weinberg states is used to obtain a separable representation of the potential V then the residue \ensuremath{\Delta}V, due to the basis-set truncation, has very special properties: (1) The contribution \ensuremath{\Delta}T to the T matrix due to \ensuremath{\Delta}V is identical to the undistorted T matrix for \ensuremath{\Delta}V alone, ${\mathit{T}}_{\mathrm{\ensuremath{\Delta}}}$, i.e., the usual Moeller distortion factors in the two-potential formula are unity in this case. (2) A perturbative-iterative treatment of ${\mathit{T}}_{\mathrm{\ensuremath{\Delta}}}$ in powers of \ensuremath{\Delta}V is found to be equivalent to the finite-rank representation of operators of the type T, T-V, T-V-${\mathit{VG}}_{0}$V, and so on. This equivalence has both practical and theoretical implications. On the one hand, it provides a reliable method for calculating the T matrix and for analyzing the corresponding accuracy properties. On the other hand, a connection is established between each order of the quasiparticle method and the different variational principles which underlie the finite-rank representation of operators such as T, T-V, T-V-${\mathit{VG}}_{0}$V, etc. Numerical examples are provided for two different nucleon-nucleon singlet potentials (Reid soft core and Malfliet-Tjon). In the Malfliet-Tjon case, for instance, two Weinberg states are found to be sufficient in order to give an accuracy of 0.1% for the calculation of T-V-${\mathit{VG}}_{0}$V, while for T-V and T the same two states give an accuracy of 1% and 10%, respectively, in an interval of 6 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$ around the on-shell point.

Fragmentation of 9 GeV/ c deuterons in the region of proton transverse momenta of 0.5–1 GeV/ c and the deuteron wave function at small distances

Momentum spectra of protons emitted at angles of 0.103, 0.139 and 0.157 rad in fragmentation of 9 GeV/ c deuterons on 1H, 2H and 12C nuclei have been measured at the JINR synchrophasotron by using a magnetic spectrometer. It is found that the shape of the high-momentum parts of the spectra is defined only by the deuteron structure and the mechanism of deuteron-nucleon interaction and does not depend on a target. The contributions to the differential cross section of the 1H(d, p)X reaction have been determined, which are conditioned by elastic and inelastic scatterings of deuteron nucleons on a proton, stripping, their interference, and also the excitation of the pion degrees of freedom. In the framework of the light front dynamics, calculations of the proton spectra with the deuteron structure functions for the Paris, Reid soft core, and Bonn potentials have been performed; a better agreement with the experiment is achieved with the deuteron wave function for the Paris potential. A conclusion is made that the nucleons in a deuteron keep their individuality up to relative motion momenta of ∼ 1 GeV/ c.

Circular polarization of γ-quanta in the reaction np → d γ with polarized neutrons

We calculate the photon circular polarization at the capture np → d γ for polarized thermal neutrons. The calculations are performed both in the zero-range approximation and with the wave functions for the Reid soft core potential. The contribution to the circular polarization, which is of the same origin as the nonadditivity of the nucleon magnetic moments, has been taken into account in two ways. The first approach is based on the known value of the deuteron magnetic moment. In the second we have obtained relativistic corrections for the magnetic moment operator for the two-nucleon system, and using them calculate both the deuteron magnetic moment and the circular polarization in the neutron radiative capture independently.

Nuclear force with an ultra short range attraction

The authors propose an ultra short range attraction in the nucleon-nucleon potential to imitate some qualitative features of the quark model. A modified version of the Reid soft core potential is presented. It is proved to be capable of describing the physical observables of the deuteron as well as the original Reid soft core potential. The probability of the six-quark configuration of the two-nucleon system can be defined and a good estimate for this probability can be obtained.

Potential model calculations of parity violation in proton-proton scattering.

The proton-proton parity-violating analyzing power is calculated at laboratory energies up to 1000 MeV using a parity-violating weak meson-exchange potential with form factors and coupling constants consistent with the nonrelativistic Bonn potential. A real distorted-wave Born-approximation formulation of the weak scattering amplitude is introduced using real standing waves with distortions by Coulomb and strong interactions. This formulation includes a new and simplified expression for the S matrix in terms of strong and real weak amplitudes. The form-factor contributions to the weak potential significantly reduce the overall magnitude of the analyzing power, but do not qualitatively affect the relative energy or angular dependence. Results using, alternatively, Paris, Reid soft-core, Hamada-Johnston, and pure Coulomb distorted waves with the same weak potential are compared. The angular and total analyzing powers are calculated with long-range Coulomb contributions for scattering and transmission experiments. The long-range Coulomb effects can be removed, leading to a reduced analyzing power \^A that depends only on short-range forces.

Relativistic calculation of the deuteron quadrupole and magnetic moments.

The deuteron quadrupole and magnetic moments are calculated assuming light-front dynamics. The advantage of this approach is that conventional bound-state wave functions and empirical nucleon form factors can be used to construct eigenfunctions of the four-momentum and the spin, together with a representation of the electromagnetic current, in such a way that the current-density operator and the wave functions transform consistently under a unitary representation of the Poincar\'e group. The sensitivity of our model to the choice of the deuteron bound-state wave functions is tested by using Reid soft core, Argonne ${v}_{14}$, Paris, and Nijmegen nucleon-nucleon interactions, as well as three Bonn potentials. The exact results increase both quadrupole and magnetic moments by small amounts compared to the nonrelativistic values. Expansions in powers of the nucleon velocity are found to be unreliable.