One-body Density Matrix Research Articles

Effects of short-range correlations in(e,e′p)reactions and nuclear overlap functions

A study of the effects of short-range correlations over the ${(e,e}^{\ensuremath{'}}p)$ reaction for low missing energy in closed-shell nuclei is presented. We use correlated quasihole overlap functions extracted from the asymptotic behavior of the one-body density matrix, containing central correlations of Jastrow type, up to first order in a cluster expansion, and computed in the very high asymptotic region, up to 100 fm. The method to extract the overlap functions is checked in a simple shell model, where the exact results are known. We find that the single-particle wave functions of the valence shells are shifted to the right due to the short-range repulsion by the nuclear core. The corresponding spectroscopic factors are reduced by only a few percent with respect to the shell model. However, the ${(e,e}^{\ensuremath{'}}p)$ response functions and cross sections are enhanced in the region of the maximum of the missing momentum distribution due to short-range correlations.

High-momentum tail in the Tonks gas under harmonic confinement

We use boson–fermion mapping to show that the single-particle momentum distribution in a one-dimensional gas of hard point-like bosons (Tonks gas) inside a harmonic trap decays as p −4 at large momentum p. The relevant integrals expressing the one-body density matrix are evaluated for small numbers of particles in a simple Monte Carlo approach to test the extent of the asymptotic law and to illustrate the slow decay of correlations between the matter–wave field at different points.

Variational Monte Carlo Calculations of 4He Adsorbed on Graphite

We present the results of variational Monte Carlo calculations of the solid 4He monolayer adsorbed on a corrugated graphite substrate with the shadow wave function technique. We find that the solid monolayer at the commensurate density is tightly bound to the substrate and that the presence of corrugation localizes the Helium atoms in the direction parallel to the substrate around the sites of a triangular lattice where 1/3 of the adsorption sites of graphite are occupied. We have calculated the energy per particle, the density profile of the monolayer and the static structure factor. We have also studied the system in the presence of a defect (vacancy or interstitial) and we report the results for the one-body density matrix.

Pairing correlations in an explicitly particle-number conserving approach

A method to treat pairing correlations in an explicitly particle-number conserving approach is proposed and discussed. It appears to be both tractable and reliable while using state of the art effective interactions, e.g. of the Skyrme type, for the particle–hole matrix elements. It corresponds to a highly truncated shell-model calculation approach. Its fast convergence in the particle–hole basis is due to the use of single-particle states stemming from an Hartree–Fock-like mean field defined from a one-body density matrix including self-consistently the one-body effect of the correlations. Results obtained in an non-fully consistent fashion, in that the residual interaction is mocked up by a delta interaction, are presented. They mainly confirm most of the physical properties underlying the BCS-like solutions, in particular the paramount importance of pair transfers. However in weak pairing situations, e.g. near subshell- or shell-closures, they provide a microscopic description quite at variance with the usual unpaired solutions. These features are illustrated mostly for the ground state of the rigidly deformed 178Hf nucleus and its various high- K isomeric states.

Generator coordinate method calculations of one-nucleon removal reactions on40Ca

An approach to the generator coordinate method (GCM) using Skyrme-type effective forces and Woods-Saxon construction potential is applied to calculate the single-particle proton and neutron overlap functions in ${}^{40}\mathrm{Ca}.$ The relationship between the bound-state overlap functions and the one-body density matrix has been used. These overlap functions are applied to calculate the cross sections of one-nucleon removal reactions such as ${(e,e}^{\ensuremath{'}}p),$ $(\ensuremath{\gamma},p),$ and $(p,d)$ on ${}^{40}\mathrm{Ca}$ on the same theoretical footing. A consistent description of data for the different reactions is achieved. The shapes of the experimental cross sections for transitions to the ${3/2}^{+}$ ground state and the first ${1/2}^{+}$ excited state of the residual nuclei are well reproduced by the overlap functions obtained within the GCM. An additional spectroscopic factor accounting for correlations not included in the overlap function must be applied to the calculated results to reproduce the size of the experimental cross sections.

Pauli distorted double folded potential

A new method to incorporate the Pauli principle into the double folding approach to the nucleus-nucleus potential is proposed. The description of the exchange terms at the level of the quasiclassical one-body density matrix is used. It is shown that in order to take into account the Pauli blocking properly, a redefinition of the density matrices of the free isolated nuclei must be done. A solution to the self-consistent incorporation of the Pauli blocking effects in the mean-field nucleus-nucleus potential is obtained in the Thomas-Fermi approximation.

Heavy isotope6He:Properties of bulk system and of clusters

We have theoretically revisited the heavy isotope of He, ${}^{6}\mathrm{He}.$ It was considered many years ago as a possible superfluid system, but its short lifetime makes any experiment rather difficult and theoretical predictions are not very favorable. We have applied the shadow wave-function technique to the study of bulk ${}^{6}\mathrm{He}$ and we have also studied clusters of ${}^{6}\mathrm{He}.$ We confirm that the ground state is solid but the energy difference between the two phases is rather small and the solid order is also inhibited in clusters of many hundreds of particles. From a study of the off-diagonal one-body density matrix we find that Bose-Einstein condensation is present in these clusters. Thus clusters of ${}^{6}\mathrm{He}$ offer the unique opportunity to study the evolution from a superfluid to a solid driven by size effect and the properties of a highly defected quantum solid.

Momentum flux density, kinetic energy density, and their fluctuations for one-dimensional confined gases of noninteracting fermions

We present a Green's function method for the evaluation of the particle density profile and of the higher moments of the one-body density matrix in a mesoscopic system of N Fermi particles moving independently in a linear potential. The usefulness of the method is illustrated by applications to a Fermi gas confined in a harmonic potential well, for which we evaluate the momentum flux and kinetic energy densities as well as their quantal mean-square fluctuations. We also study some properties of the kinetic energy functional E_{kin}[n(x)] in the same system. Whereas a local approximation to the kinetic energy density yields a multi-valued function, an exact single-valued relationship between the density derivative of E_{kin}[n(x)] and the particle density n(x) is demonstrated and evaluated for various values of the number of particles in the system.

One-body density matrix, natural orbits, and quasihole states in16Oand40Ca

The one body density matrix, momentum distribution, natural orbits and quasi hole states of 16O and 40Ca are analyzed in the framework of the correlated basis function theory using state dependent correlations with central and tensor components. Fermi hypernetted chain integral equations and single operator chain approximation are employed to sum cluster diagrams at all orders. The optimal trial wave function is determined by means of the variational principle and the realistic Argonne v8' two-nucleon and Urbana IX three-nucleon interactions. The correlated momentum distributions are in good agreement with the available variational Monte Carlo results and show the well known enhancement at large momentum values with respect to the independent particle model. Diagonalization of the density matrix provides the natural orbits and their occupation numbers. Correlations deplete the occupation number of the first natural orbitals by more than 10%. The first following ones result instead occupied by a few percent. Jastrow correlations lower the spectroscopic factors of the valence states by a few percent (~1-3%) and an additional ~8-12% depletion is provided by tensor correlations. It is confirmed that short range correlations do not explain the spectroscopic factors extracted from (e,e'p) experiments. 2h-1p perturbative corrections in the correlated basis are expected to provide most of the remaining strength, as in nuclear matter.

α-nucleus optical potential in the double-folding model

The double-folding formalism for the $\ensuremath{\alpha}$-nucleus optical potential is revised to study the exchange effects and density dependence of the effective nucleon-nucleon $(\mathrm{NN})$ interaction in detail. A realistic density dependent M3Y interaction, based on the G-matrix elements of the Paris NN potential, has been used in the folding calculation. The local approximation for the nonlocal one-body density matrix in the calculation of the exchange potential was tested by using the harmonic oscillator representation of the nonlocal density matrices of the $\ensuremath{\alpha}$-particle and target nucleus. The inclusion of a realistic density dependence into the effective NN interaction was shown to be vital for a correct description of the refractive $\ensuremath{\alpha}$-nucleus scattering data. A high sensitivity of the density distributions of the $\ensuremath{\alpha}$-particle and target nucleus to the shape of the $\ensuremath{\alpha}$-nucleus potential was found, which can be used in the folding analysis to test various density models for the $\ensuremath{\alpha}$-particle and target as well as to choose the most realistic approximation for the overlap density in the dinuclear system. Our results also stress the importance of $\ensuremath{\alpha}$-nucleus scattering experiment in the nuclear structure study.

Quasicondensate and superfluid fraction in the two-dimensional charged boson gas at finite temperature

The Bogoliubov - de Gennes equations are solved for the Coulomb Bose gas describing a fluid of charged bosons at finite temperature. The approach is applicable in the weak coupling regime and the extent of its quantitative usefulness is tested in the three-dimensional fluid, for which diffusion Monte Carlo data are available on the condensate fraction at zero temperature. The one-body density matrix is then evaluated by the same approach for the two-dimensional fluid with e^2/r interactions, to demonstrate the presence of a quasi-condensate from its power-law decay with increasing distance and to evaluate the superfluid fraction as a function of temperature at weak coupling.

Bose-Einstein condensation in trapped bosons: A variational Monte Carlo analysis

Several properties of trapped hard-sphere bosons are evaluated using variational Monte Carlo techniques. A trial wave function composed of a renormalized single-particle Gaussian and a hard-sphere Jastrow function for pair correlations is used to study the sensitivity of condensate and noncondensate properties to the hard-sphere radius and the number of particles. Special attention is given to diagonalizing the one-body density matrix and obtaining the corresponding single-particle natural orbitals and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single-particle orbital with the highest occupation. The effect of interaction on other quantities such as the ground-state energy, the mean radial displacement, and the momentum distribution is calculated as well. Results are compared with mean-field theory in the dilute limit.

Erratum: Semiclassical distorted wave model with Wigner transform of one-body density matrix [Phys. Rev. C60, 064605 (1999)

Erratum: Semiclassical distorted wave model with Wigner transform of one-body density matrix [Phys. Rev. C<b>60</b>, 064605 (1999)

One-body density matrix and momentum distribution ins−pands−dshell nuclei

Analytical expressions of the one- and two-body terms in the cluster expansion of the one-body density matrix and momentum distribution of the $s\ensuremath{-}p$ and $s\ensuremath{-}d$ shell nuclei with $N=Z$ are derived. They depend on the harmonic oscillator parameter b and the parameter $\ensuremath{\beta}$ which originates from the Jastrow correlation function. These parameters have been determined by least squares fit to the experimental charge form factors. The inclusion of short-range correlations increases the high momentum component of the momentum distribution $n(\mathbf{k})$ for all nuclei we have considered while there is an A dependence of $n(\mathbf{k})$ both at small values of k and the high momentum component. The A dependence of the high momentum component of $n(\mathbf{k})$ becomes quite small when the nuclei ${}^{24}\mathrm{Mg},$ ${}^{28}\mathrm{Si},$ and ${}^{32}\mathrm{S}$ are treated as $1d\ensuremath{-}2s$ shell nuclei having the occupation probability of the $2s$ state as an extra free parameter in the fit to the form factors.

Extended Thomas–Fermi approximation to the one-body density matrix

The one-body density matrix is derived within the Extended Thomas–Fermi approximation. This has been done starting from the Wigner–Kirkwood distribution function for a non-local single-particle potential. The links between this new approximation to the density matrix with former approaches available in the literature are widely discussed. The semiclassical counterpart of the Hartree–Fock energy at the Extended Thomas–Fermi level is also obtained in the case of a non-local one-body Hamiltonian. The semiclassical binding energies and root mean square radii are compared with those obtained using the Strutinsky averaged method. The full Hartree–Fock values are compared with those obtained using the Kohn–Sham scheme based on the different approximations to the density matrix considered in the text. Numerical applications are performed using the Gogny, Brink–Boeker and BDM3Y1 ∗ effective interactions.

Symmetry-projected Hartree–Fock–Bogoliubov equations

Symmetry-projected Hartree–Fock–Bogoliubov (HFB) equations are derived using the variational ansatz for the generalized one-body density matrix in the Valatin form. It is shown that the projected-energy functional can be completely expressed in terms of the HFB density matrix and the pairing-tensor. The variation of this projected energy is shown to result in HFB equations with modified expressions for the pairing-potential Δ and the Hartree–Fock field Γ. The expressions for these quantities are explicitly derived for the case of particle number projection. The numerical applicability of this projection method is studied in an exactly soluble model of a deformed single-j shell.

Semiclassical distorted wave model with Wigner transform of one-body density matrix

A semiclassical distorted wave (SCDW) model with realistic single particle wave functions in a finite range single particle potential is presented for multistep direct processes in nucleon inelastic scattering and charge exchange reactions to the continuum, making use of the Wigner transform of a one-body density matrix. The higher momentum components of target nucleons are properly taken into account in comparison with the previous SCDW model with a local density Fermi gas model. The new SCDW model still gives a simple closed-form expression for the cross section with no free adjustable parameter which allows a simple intuitive interpretation, just as in the previous SCDW model. This model is applied to the analyses of multistep direct processes of ${}^{90}\mathrm{Zr}{(p,p}^{\ensuremath{'}}x)$ reactions at incident energies of 80 and 160 MeV. The calculated double differential cross sections including up to three-step processes are compared with experimental data. The calculated cross sections at backward angles are larger than those given by the previous calculations, and the agreement with experimental data is much improved. Discussions on the mechanism of this improvement are given in terms of the momentum distribution of target nucleons.

One-particle exchange in the double-folded potential in a semiclassical approximation

The one-particle exchange in the double-folded model is analysed. The extended Thomas-Fermi approach to the one-body density matrix is used. The nucleon-nucleon force with Yukawa, Gauss and Coulomb-type form factors are considered. The energy dependence of the exchange part of the double-folded potential is investigated and a comparison with former approaches is carried out.

Linked cluster expansion for the calculation of the semi-inclusiveA(e,e′p)Xprocesses using correlated Glauber wave functions

The distorted one-body mixed density matrix, which is the basic nuclear quantity appearing in the definition of the cross section for the semi-inclusive A(e,e'p)X processes, is calculated within a linked-cluster expansion based upon correlated wave functions and the Glauber multiple scattering theory to take into account the final state interaction of the ejected nucleon. The nuclear transparency for 16O and 40Ca is calculated using realistic central and non-central correlations and the important role played by the latter is illustrated.

Damping of collective nuclear motion and thermodynamic properties of nuclei beyond mean field

The dynamical description of correlated nuclear motion is based on a set of coupled equations of motion for the one-body density matrix ϱ(11′; t) and the two-body correlation function c 2(12, 1′2′; t), which is obtained from the density-matrix hierarchy beyond conventional mean-field approaches by truncating three-body correlations. The resulting equations non-perturbatively describe particle-particle collisions (short-range correlations) as well as particle-hole interactions (long-range correlations). Within a basis of time-dependent Hartree-Fock states these equations of motion are solved for collective vibrations of 40Ca at several finite thermal excitation energies corresponding to temperatures T = 0 – 6 MeV. Transport coefficients for friction and diffusion are extracted from the explicit solutions in comparison to the solutions of the associated TDHF, VUU, Vlasov or damped quantum oscillator equations of motion. We find that the actual magnitude of the transport coefficients is strongly influenced by particle-hole correlations at low temperature which generate large fluctuations in the nuclear shape degrees of freedom. Thermodynamically, the specific heat and the entropy of the system as a function of temperature does not differ much from the mean-field limit except for a bump in the specific heat around T ⋍ MeV which we attribute to the melting of shell effects in the correlated system.