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HNPS Advances in Nuclear Physics

Two body density matrix of model nuclear matter

E. Mavromanatis, Μ. Petraki, J. W. Clark

Published Dec 5, 2019• 0 citations
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Abstract

A lowest-cluster-order variational calculation of the half-diagonal two-body density matrix ρ2(r1,r2,r’1) and the corresponding generalized momentum distribution n(p.Q) is performed for three representative models of nuclear matter containing central correlations. Dynamical correlations produce significant deviations from the results for a noninteracting Fermi gas. Calculations axe in progress that include higherorder cluster corrections as well as state-dependent correlations

DOI
10.12681/hnps.2376

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A lowest-cluster-order variational calculation of the half-diagonal two-body density matrix ρ2(r1,r2,r’1) and the corresponding generalized momentum distribution n(p.Q) is performed for three representative models of nuclear matter containing central correlations. Dynamical correlations produce significant deviations from the results for a noninteracting Fermi gas. Calculations axe in progress that include higherorder cluster corrections as well as state-dependent correlations
Valuable information on the correlation structure of the nuclear medium is stored in the generalized momentum distribution n(p,Q), the Fourier transform of the half-diagonal two-body density matrix ρ_{2η}(r_1,r_2,r'). In this paper, we present a numerical calculation of n(p,Q) for two Jastrow-correlated models of symmetrical nuclear matter based on the structural decomposition of n(p,Q) derived by Ristig and Clark and on a Fermi-hypernetted-chain procedure. Results exhibit significant departures from the ideal Fermi gas case in certain kinematic domains; this behaviour indicates the strong short-range correlations present in these models. Nevertheless, such deviations are less prominent than in earlier low- cluster-order calculations. The results are also used to judge the quality of Silver's approximation for n(p,Q).
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.
We present a summary of on-going calculations that address the static and dynamic structure of nuclear matter. Specific projects include (i) evaluation of the density-density response function and corresponding dynamic structure factor, based on the correlated random-phase approximation (CRPA_I) and generalizations of this method, and (ii) low-order variational calculation of the reduced two-body density matrix and corresponding generalized momentum distribution. The numerical applications involve the model interaction V2.
The half-diagonal two-body density matrix ${\ensuremath{\rho}}_{2h}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{{1}^{\ensuremath{'}}})$ is a key quantity in theoretical descriptions of nucleon knockout reactions at intermediate energies, due to its central role in treatments of the propagation and final-state interactions of ejected nucleons in the nuclear medium. This quantity is calculated for a simple model of the ground state of infinite symmetrical nuclear matter, based on a Jastrow-correlated wave function and a Fermi hypernetted-chain analysis. The dependence of ${\ensuremath{\rho}}_{2h}$ on the variables ${r}_{{11}^{\ensuremath{'}}}=|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{{1}^{\ensuremath{'}}}|,$ ${r}_{12}=|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}|,$ and the angle $\ensuremath{\theta}$ between ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{{1}^{\ensuremath{'}}}$ and ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$ is investigated in some detail. Significant departures from ideal Fermi gas behavior in certain domains reflect the importance of short-range correlations and their interplay with statistical correlations. The quality of approximations to ${\ensuremath{\rho}}_{2h}$ proposed by Gersch et al., Rinat, and Silver is assessed by comparison with the Fermi hypernetted-chain results.
The microscopic study of the two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$,${\mathbf{r}}_{2}^{\ensuremath{'}}$) initiated for uniform Bose fluids in an earlier paper is continued for the Fermi case. We present formal results on the structure of the generalized momentum distribution n(p,q)=${\mathrm{\ensuremath{\Sigma}}}_{\mathbf{k}\mathbf{\^{}}}$〈\ensuremath{\Psi}|${\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}+\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$${\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}\mathrm{\ensuremath{-}}\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$apˆ{\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}}apˆ​akˆ{\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}}akˆ​|\ensuremath{\Psi}〉, and its Fourier inverse ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$,r2{\mathbf{r}}_{2}r2​)\ensuremath{\equiv}${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$), based on a variational ground-state wave function of Jastrow-Slater form. The structural relations are inferred from the cluster expansions of these objects, from the asymptotic condition relating ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) to the particle density and the one-body density matrix ${\mathrm{\ensuremath{\rho}}}_{1}$(r1{\mathbf{r}}_{1}r1​,${\mathbf{r}}_{1}^{\ensuremath{'}}$), and from formal diagrammatic connections with the Bose problem. The two-body density-matrix elements ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) are thereby expressed in closed form in terms of certain sums of irreducible cluster diagrams. Some of these diagram sums are familiar from the analogous theory of the one-body density matrix; all can be evaluated quantitatively by solving a set of Fermi-hypernetted-chain (FHNC) equations. Upon invoking the sequential relation between ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) and ${\mathrm{\ensuremath{\rho}}}_{1}$(r1{\mathbf{r}}_{1}r1​,${\mathbf{r}}_{1}^{\ensuremath{'}}$), the corresponding result for the generalized momentum distribution n(p,q) effects a resolution into contributions from various scattering processes occurring in the many-body medium, specified by form factors that are susceptible to FHNC evaluation. This decomposition is comparable to that derived earlier for the Bose-fluid ground state but is complicated by contributions from exchange scattering and by a dynamically dressed Pauli kinematic correction. Silver has proposed a simple expression for the generalized momentum distribution n(p,q), a function which plays an essential role in his theory of final-state effects in deep-inelastic neutron scattering from the helium liquids. Based on the present microscopic treatment, the quality of Silver's estimate is assessed for the case of normal liquid He3{\mathrm{He}}^{3}He3, by evaluating the necessary distribution functions and form factors within the FHNC/0 approximation.