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arXiv:nucl-th/0012048 · Physical Review C

One-body density matrix, natural orbits, and quasihole states in 16O and 40Ca

Adelchi Fabrocini, Giampaolo Co'

Published Mar 21, 2001• 29 citations
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Abstract

The one-body density matrix, momentum distribution, natural orbits, and quasihole states of ${}^{16}\mathrm{O}\mathrm{}$ and ${}^{40}\mathrm{Ca}\mathrm{}$ 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 ${v}_{8}^{\ensuremath{'}}$ two-nucleon and Urbana IX three-nucleon interactions. The correlated ${}^{16}\mathrm{O}$ momentum distribution is in good agreement with the variational Monte Carlo results and shows the well-known enhancement at large momentum values with respect to the independent-particle model. A similar behavior is found in ${}^{40}\mathrm{Ca}.$ The relative importance of the different types of correlations (mainly Jastrow and tensor) on the momentum distribution appears to be similar in the nuclei and in nuclear matter. Diagonalization of the density matrix provides the natural orbits and their occupation numbers. Correlations deplete the occupation number of the first natural orbital by more than $10%.$ The orbitals following the first one result instead, occupied by a few percent, or less. The single particle overlap functions and the spectroscopic factors are computed in the correlated model for both nuclei and compared with previous estimates. Jastrow correlations lower the spectroscopic factors of the valence states by a few percent $(\ensuremath{\sim}1--3%)$ with respect to unity. An additional $\ensuremath{\sim}8--12%$ depletion is provided by spin-isospin tensor correlations. It is confirmed that a variational treatment of short-range correlations does not explain the spectroscopic factors extracted from ${(e,e}^{\ensuremath{'}}p)$ experiments. Such an approach corresponds to the zeroth order of the correlated basis function theory and two-hole one-particle perturbative corrections in the correlated basis are expected to provide most of the remaining strength, as in nuclear matter.

DOI
10.1103/physrevc.63.044319

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The one-body density matrix, momentum distribution, natural orbits, and quasihole states of 16O{}^{16}\mathrm{O}\mathrm{}16O and 40Ca{}^{40}\mathrm{Ca}\mathrm{}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\ensuremath′{v}_{8}^{\ensuremath{'}}v8\ensuremath′​ two-nucleon and Urbana IX three-nucleon interactions. The correlated 16O{}^{16}\mathrm{O}16O momentum distribution is in good agreement with the variational Monte Carlo results and shows the well-known enhancement at large momentum values with respect to the independent-particle model. A similar behavior is found in 40Ca.{}^{40}\mathrm{Ca}.40Ca. The relative importance of the different types of correlations (mainly Jastrow and tensor) on the momentum distribution appears to be similar in the nuclei and in nuclear matter. Diagonalization of the density matrix provides the natural orbits and their occupation numbers. Correlations deplete the occupation number of the first natural orbital by more than 1010%.10 The orbitals following the first one result instead, occupied by a few percent, or less. The single particle overlap functions and the spectroscopic factors are computed in the correlated model for both nuclei and compared with previous estimates. Jastrow correlations lower the spectroscopic factors of the valence states by a few percent (\ensuremath∼1−−3(\ensuremath{\sim}1--3%)(\ensuremath∼1−−3 with respect to unity. An additional \ensuremath∼8−−12\ensuremath{\sim}8--12%\ensuremath∼8−−12 depletion is provided by spin-isospin tensor correlations. It is confirmed that a variational treatment of short-range correlations does not explain the spectroscopic factors extracted from (e,e\ensuremath′p){(e,e}^{\ensuremath{'}}p)(e,e\ensuremath′p) experiments. Such an approach corresponds to the zeroth order of the correlated basis function theory and two-hole one-particle perturbative corrections in the correlated basis are expected to provide most of the remaining strength, as in nuclear matter.
The natural orbitals (NOs) and natural occupation numbers (NON) of various N = Z, sp and sd shell nuclei are calculated by applying a correlated one-body density matrix (OBDM). The correlated density matrix has been evaluated by considering central correlations of Jastrow type and an approximation named factor cluster expansion. The correlation effects on NOs, NON and the Fermi sea depletion (FSD) are discussed and analyzed. In addition, an approximate expression for the correlated OBDM of the nuclear matter has been used for the evaluation of the relative momentum distribution and FSD. We found that the value of FSD is higher in closed shell nuclei compared to open shell ones and it is lower compared to the case of nuclear matter. This statement could be confirmed by relevant experimental studies.
In this work a unitary correlation operator is presented that explicitly describes the short-ranged central and tensor correlations in the nuclear many-body system. These short-ranged correlations are induced by the repulsive core and the pronounced tensor force of realistic nucleon-nucleon interactions and cannot be described by the simple many-body states of a mean-field or shell model approach. The unitary correlation operator is discussed for the Argonne V18 and the Bonn-A interactions. Applying the correlation operator onto the Hamiltonian a common effective interaction for low momenta is obtained. Calculations for 4He using the one- and two-body part of the correlated Hamiltonian compare favorably with exact many-body methods. Calculations for 16O and 40Ca which are not possible with exact methods are performed using harmonic oscillator shell model states. The observed deviation from the experimental binding energies and radii is attributed to the missing three-body forces. The correlated interaction in a basis-free representation is used in the Fermionic Molecular Dynamics model to calculate the nuclei of the p- and the sd-shell. In this model the uncorrelated many-particle state is given as a Slaterdeterminant of Gaussian wave-packets with spin and isospin degrees of freedom which allows a consistent description of spherically symmetric, intrinsically deformed and clustered nuclei.
The hypotheticai system of nucleons known as nuclear matter provides a very interesting field of study with relevance to both nuclear physics and astrophysics. The task of finding nucleon-nucleon potentials that explain the empirical properties of nuclear matter, as well as the two-body data, has been hampered by the lack of an adequate many-body theory for treating the complicated forces involved. Realistic nucleon-nucleon potentials have strong spin, isospin, tensor and spin-orbit components which induce correspondingly strong correlations in nuclear matter. In this thesis we present a variational theory designed to cope with these complicated correlations. To place the theory in perspective we give an overview of several different many-body theories and compare their predictions in the somewhat simpler systems of the 4He and 3He liquids, and for a simple central potential model of nuclear matter. The variational theory is then developed, starting with the choice of wave function. A generalized Jastrow product of two-body correlation functions containing central, spin, isospin, tensor, and spin-orbit operators is used. The correlation function is generated by a series of two-body Schrodinger equations with boundary conditions which require the wave function to heal at a distance d. The wave function is parameterized by d and by the magnitudes of the non-central correlations. Expectation values with variational wave functions of the Jastrow type may be studied conveniently with generalized Mayer diagrams. Diagram rules are given and a general diagrammatic cluster expansion is derived. It is an expansion in powers of commutators of the non-central operators, and in their absence reduces to the usual Mayer cluster expansion. Evaluation of diagrams in the cluster expansion is complicated by the presence of the non-commuting operators. Some simple rules and useful methods for calculating the contribution due to the central, spin, isospin, and tensor operators are given. Spin-orbit operators are neglected beyond this point, but the approach used is extendible to them. Large classes of diagrams contributing to an expectation value can be summed by means of integral equations. The development of single chain, hypernetted chain, and Fermi hypernetted chain (FHNC) equations for central correlations is reviewed. The chain summation methods are extended to sum single operator chains (SOC) for the non-central operators. Methods are also given for treating the leading commutator corrections in the expansi0n by simple vertex factors. The energy expectation value for potentials with six operators are evaluated using the FHNC/SOC functions, with an exact treatment of the commutators involved. The energy is found to have a minimum with respect to variations in all parameters. Results of calculations with model potentials based on the Reid, Bethe-Johnson, Hamada-Johnston, and Gammel-Thaler potentials are reported. A crude estimate of the effect of the neglected spin-orbit potentials on the nuclear matter binding energy and saturation density indicates that it could be significant. A correct treatment of the spin-orbit potentials is the next major step that should be taken.
The generalized momentum distribution n(p,Q), related to the half-diagonal two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2\mathrm{h}}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) by Fourier transformation in the variables r1{\mathbf{r}}_{1}r1​-${\mathbf{r}}_{1}^{\ensuremath{'}}$ and r1{\mathbf{r}}_{1}r1​-r2{\mathbf{r}}_{2}r2​, plays a key role in the description of final-state interactions in the nuclear medium and other strongly interacting many-body systems. The function n(p,Q) is explored for two Jastrow-correlated models of infinite nuclear matter within a Fermi hypernetted-chain procedure. Significant departures from ideal Fermi gas behavior in certain kinematic domains provide signatures of the strong short-range correlations contained in these models. However, such deviations are less prominent than in earlier calculations based on low-order cluster truncations; correspondingly, violations of the sequential relation are greatly reduced. Simple prescriptions for improved low-cluster-order approximations to n(p,Q) are suggested by analysis of the results of the Fermi hypernetted-chain evaluation. These results are also used to assess the quality of Silver's approximation n(p,Q)\ensuremath{\approx}n(p)[S(Q)-1] for the generalized momentum distribution in terms of the ordinary momentum distribution n(p) and the static structure function S(Q), with findings that have potential implications for the interpretation of data from inclusive electron scattering by nuclei at high momentum transfers.
The proton momentum and density distributions of closed shell nuclei are calculated with a model treating short-range correlations up to first order in the cluster expansion. The validity of the model is verified by comparing the results obtained using purely scalar correlations with those produced by finite nuclei Fermi hypernetted chain calculations. State dependent correlations are used to calculate momentum and density distributions of 12{}^{12}12C, 16{}^{16}16O, 40{}^{40}40Ca, and 48{}^{48}48Ca, and the effects of their tensor components are studied.
A general theory of long range correlations in the ground state of a closed shell nucleus is outlined, using a number of ideas introduced or adumbrated, but not always successfully developed, in previous work of the authors and others. These include the following: (i) Any operator can be formally expressed as a series in particle-hole (p-h) excitation and destruction operators which should converge rapidly for a suitable choice of single-particle basis at the closed shells. By means of these relations and with the help of sum rules all observables can be computed without the explicit use of wave functions. (ii) For a fixed choice of single-particle basis, the ground state energy can be computed from a variational principle in which in lowest nontrivial approximation, in consequence of (i), the parameters are amplitudes of the same type as occur in the random phase approximation (RPA). The precise relation to RPA is fully detailed. (iii) The single particle basis chosen is the natural orbital basis of Lowdin, which presents itself as the obvious choice both for formulation of the dynamics and for the algorithm of solution. (iv) Though the theory deals with pairs of fermion operators, particular attention is paid that all Pauli principle restrictions are satisfied. (v) It is verified that in the perturbation limit, known results are reproduced. The method is generally applicable to any system for which the correlations can be viewed as particle-hole excitations with respect to a reference Slater determinant.