R. Sartor

Off-the-energy-shell properties of the mass operator and spectral functions in nuclear matter

Abstract We investigate the off-the-energy-shell properties of the mass operator M ( k ; ω ) = V ( k ; ω ) + iW ( k ; ω ), i.e., its dependence upon the nucleon momentum k and upon the nucleon frequency ω, separately. Particular attention is paid to the dispersion relation which connects its real and imaginary parts. We limit ourselves to the first two terms of the hole-line expansion of the mass operator, namely to the Brueckner-Hartree-Fock field M 1 ( k ; ω ) and to the second-order “rearrangement” term M 2 ( k ; ω ). Most previous works only dealt with “on-shell values” obtained by setting ω equal to the root e ( k ) of an energy-momentum relation, or equivalently by setting k equal to the root k ( e ) of this energy-momentum relation. We use as input a finite-rank representation of the realistic Argonne v 14 nucleon-nucleon interaction. The Fermi momentum is set equal to 1.36 fm −1 . For momenta larger than the Fermi momentum, the calculated k -dependence of the on-shell depth V 1 ( k ; e ( k )) can be approximated by a gaussian. The corresponding nonlocality range is close to that assumed by Perey and Buck in their phenomenological analysis of scattering cross sections; it is somewhat smaller than that associated with the k -dependence of the off-shell potential V 1 ( k ; ω ) for fixed ω. The calculated ω-dependence of V 2 ( k ; ω ) is in excellent agreement with the dispersion relation which connects V 2 ( k ; ω ) to the values of W 2 ( k ; ω ′) for all ω ′ e ( k 1 ). The dispersion relation between V 1 ( k ; ω ) and W 1 ( k ; ω ′) is also investigated; in that case, caution must be exercised because the values of W 1 ( k ; ω ′) for ω′ larger than 500 MeV still play a sizeable role, and also because the dispersion relation involves a large ω-independent “background”; it is proved that the latter is equal to the Hartree-Fock potential. More generally, the dispersion relation between the real and imaginary parts of the exact mass operator involves an ω-independent background for which we derive a closede xpression analogous to but different from the Hartree-Fock potential. The ω-dependence of the spectral function S ( k ; ω ) is calculated for two typical values of k , namely 3 4 k F and 5 4 k F . Its integral over all values of ω differs from unity by only a few percent, which provides an estimate of the reliability of our approximation scheme. We study the dependence upon E A + 1 ∗ of the integrated “particle” strength located below the excitation energy E A + 1 ∗ in the system formed by adding a nucleon with momentum k to the nuclear-matter ground state. We also calculated the dependence upon E A − 1 ∗ of the integrated “hole” strength located below the excitation energy E A − 1 ∗ in the system formed by taking out a nucleon with momentum k from the nuclear-matter ground state. In the limit E A − 1 ∗ → ∞ , this integrated hole strength yields an approximation of the occupation probability of the momentum k in the ground state. We compare this result with estimates obtained from other approximate expressions which involve the partial derivative of V 1 ( k ; ω ) with respect to ω. We also evaluate the “mean removal energies” and compare them to the “quasiparticle energies,” i.e., to the energies at which the spectral function presents a maximum.

Dispersion relation approach to the mean field and spectral functions of nucleons in 40Ca

Abstract The variational moment approach is applied to the construction of the complex single-particle mean field felt by protons and neutrons in 40Ca, at negative as well as at positive energies. The results are compared with those obtained from recent dispersive optical-model analyses. At positive energies, the reliability of the mean field is checked by comparing predicted with experimental differential and polarization cross sections. At negative energies, the model predictions are compared with empirical values of the single-particle energies, spectroscopic factors, rms radii of single-particle orbits and energy distributions of the spectroscopic factors associated with spread quasiparticle excitations. In particular, the predicted spectral functions are in close agreement with empirical strength distributions recently measured by means of the 40 Ca ( d , p ), 40 Ca p , d ) and 40 Ca ( e , e ′ p ) reactions.

From scattering to very deeply bound neutrons in 208Pb: Extended and improved moment approaches

Abstract Two new methods are developed which improve and extend the iterative moment approach to the extrapolation of the nuclear mean field from positive towards negative energy and to the prediction of various single-particle properties. These two methods still use as sole input a set of phenomenological optical-model potentials. They are improvements of the original approach because they yield more accurate predictions. They are extensions of the original approach because they provide the imaginary part of the mean field, in addition to its real part; this enables one to evaluate scattering cross sections, spectral functions and occupation probabilities, which was not possible in the previous version. These extended approaches are used to construct the neutron- 208 Pb mean field from +40 MeV down to −60 MeV. They yield practically identical results. These results are moreover extremely close to those recently obtained from a dispersive optical-model analysis of the experimental n− 208 Pb scattering cross sections. It is shown that the radial shape of the real part of the full mean field depends upon energy but remains very close to a Woods-Saxon. One of the two new methods, dubbed the variational moment approach, is well suited for the evaluation of the accuracy of the calculated Woods-Saxon parameters. If the diffuseness is set equal to 0.70 fm, the potential radius at the Fermi energy ( E F = −5.65 MeV) is found equal to (1.238±0.015)A 1 3 fm , and its volume integral per nucleon at E F to −401 ±6 MeV · fm 3 . The energy dependence of the calculated real part of the full mean field is characterized by an effective mass m∗(r; E) . The effective mass at the nuclear centre and at the Fermi energy, m∗(0; E F ) , must always be larger than the value that it takes in the Hartree-Fock approximation; this property was violated in the original iterative moment approach, but is fulfilled in both of the new methods developed here. One obtains m∗(0; E F )/m = 0.82 , in close agreement with the value found in a recent dispersive optical model analysis; in the latter, however, the quantity m∗(0; E) was infinite at several energies, while here m∗(0; E) is a smooth function of energy. The complex mean field constructed from the extended iterative moment approaches predicts n- 208 Pb cross sections which are in quite good agreement with the experimental values in an energy domain which extends up to 40 MeV. The following properties are calculated for the very deeply, deeply, weakly bound and quasibound single-particle states: energies, spreading widths, spectral functions, spectroscopic factors, occupation probabilities and root-mean-square radii. The calculated energies of the valence subshells are in close agreement with experiment. Right below the Fermi energy, the calculated occupation probability is equal to 0.85, and the spectroscopic factor to 0.73. At the bottom of the Fermi sea, the calculated occupation probability is close to 0.95. The predicted energy distributions of the strengths of the 1h 11 2 , 1g 7 2 and 1 g 9 2 deeply bound states and of the 2 h 11 2 , 1 k 17 2 and 1 j 13 2 quas good agreement with experimental evidence.

The p-40Ca and n-40Ca mean fields from the iterative moment approach

Abstract The real part V ( r ; E ) of the p- 40 Ca and n- 40 Ca mean fields is extrapolated from positive towards negative energies by means of the iterative moment approach, which incorporates the dispersion relation between the real and imaginary parts of the mean field. The potential V ( r ; E ) is the sum of a Hartree-Fock type component V HF , ( r ; E ) and a dispersive correction δV ( r ; E ); the latter is due to the coupling of the nucleon to excitations of the 40 Ca core. The potentials V ( r ; E ) and V HF ( r ; E ) are assumed to have Woods-Saxon shapes. The calculations are first carried out in the framework of the original version of the iterative moment approach, in which both the depth and the radius of the Hartree-Fock type contribution depend upon energy, while its diffuseness is constant and equal to that of V ( r ; E ). The corresponding extrapolation towards negative energies is somewhat sensitive to the detailed parametrization of the energy dependence of the imaginary part of the mean field, which is the main input of the calculation. Moreover, the radius of the calculated Hartree-Fock type potential then increases with energy, in contrast to previous findings in 208 Pb and 89 Y. A new version of the iterative moment approach is thus developed in which the radial shape of the Hartree-Fock type potential is independent of energy; the justification of this constraint is discussed. The diffuseness of the potential V ( r ; E ) is assumed to be constant and equal to that of V HF ( r ; E ). The potential calculated from this new version is in good agreement with the real part of phenomenological optical-model potentials and also yields good agreement with the single-particle energies in the two valence shells. Two types of energy dependence are considered for the depth U HF ( E ) of the Hartree-Fock type component, namely a linear and an exponential form. The linear approximation is more satisfactory for large negative energies ( E E > 50 MeV). This is explained by relating the energy dependence of U HF ( E ) to the nonlocality of the microscopic Hartree-Fock type component. Near the Fermi energy the effective mass presents a pronounced peak at the potential surface. This is due to the coupling to surface excitations of the core and reflects the energy dependence of the potential radius. The absolute spectroscopic factors of low-lying single-particle excitations in 39 Ca, 41 Ca, 39 K and 41 Sc are found to be close to 0.8. The calculated p- 40 Ca and n- 40 Ca potentials are strikingly similar, although the two calculations have been performed entirely independently. The two potentials can be related to one another by introducing a Coulomb energy shift. Attention is drawn to the fact that the extrapolated energy dependence of the real part of the mean field at large positive energy sensitively depends upon the assumed behaviour of the imaginary part at large negative energy. Yet another version of the iterative moment approach is introduced, in which the radial shape of the HF-type component is independent of energy while both the radius and the diffuseness of the full potential V ( r ; E ) depend upon E . This model indicates that the accuracy of the available empirical data is probably not sufficient to draw reliable conclusions on the energy dependence of the diffuseness of V ( r ; E ).

Extrapolation from positive to negative energy of the Woods-Saxon parametrization of the n-208Pb mean field

Abstract The real part V ( r ); E ) of the nucleon-nucleus mean field is assumed to have a Woods-Saxon shape, and accordingly to be fully specified by three quantities: the potential depth U v ( E ), radius R V ( E ) and diffuseness a v ( E ). At a given nucleon energy E these parameters can be determined from three different radial moments [ r q ] v = (4 π / A ) ∝ V ( r ; E ) r q d r . This is useful because a dispersion relation approach has recently been developed for extrapolating [ r q ] V ( E ) from positive to negative energy, using as inputs the radial moments of the real and imaginary parts of empirical optical-model potentials V ( r ; E ) + iW ( r ; E ). In the present work, the values of U v ( E ), R v ( E ) and a v ( E ) are calculated in the case of neutrons in 208 Pb in the energy domain −20 E r q ] V ( E ) for q = 0.8, 2 and 4. It is found that both U V ( E ) and R v ( E ) have a characteristic energy dependence. The energy dependence of the diffuseness a a ( E ) is less reliably predicted by the method. The radius R V ( E ) increases when E decreases from 40 to 5 MeV. This behaviour is in agreement with empirical evidence. In the energy domain −10 MeV E R V ( E ) is predicted to decrease with decreasing energy. The energy dependence of the root mean square radius is similar to that of R V ( E ). The potential depth U v slightly increases when E decreases from 40 to 15 MeV and slightly decreases between 10 and 5 MeV; it is consequently approximately constant in the energy domain 5 E U v increases linearly with decreasing E in the domain −10 MeV E r q ] v ( E ) for E > 0 and the parametrization [ r q ] w ( E ) of the energy dependence of the radial moments of the imaginary part of the empirical optical-model potentials. In the energy domain −10 MeV E V ( r ; E ) yields good agreement with the experimental single-particle energies; the model thus accurately predicts the shell-model potential ( E E > 0). In the dispersion relation approach, the real part V ( r ; E ) is the sum of a Hartree-Fock type contribution V HF ( r ; E ) and of a dispersive contribution ΔV ( r ; E ). The latter is due to the excitation of the 208 Pb core. The dispersion relation approach enables the calculation of the radial moment [ r q ] ΔV ( E ) from the parametrization [ r q ] w ( E ): several schematic models are considered which yield algebraic expressions for [ r q ] Δ V ( E ). The radial moments [ r q ] HF ( E ) are approximated by linear functions of E . When in addition, it is assumed that V HF ( r ; E ) has a Woods-Saxon radial shape, the energy dependence of its potential parameters ( U HF , R HF , a HF ) can be calculated. Furthermore, the values of ΔV ( r ; E ) can then be derived. It turns out that ΔV ( r ; E ) is peaked at the nuclear surface near the Fermi energy and acquires a Woods-Saxon type shape when the energy increases, in keeping with previous qualitative estimates. It is responsible for the peculiar energy dependence of R V ( E ) in the vicinity of the Fermi energy.

Single-particle potential and quasiparticle properties of protons in 208Pb

Abstract The real part V(r; E) of the proton-208Pb mean field is approximated by a Woods-Saxon potential with diffuseness equal to 0.70 fm. Its depth Uv(E) and radius Rv(E) are extrapolated from positive to negative energy by using a previously developed iteration procedure based on the dispersion relation which connects the real and imaginary parts of the mean field. It is shown that these extrapolated values are quite stable with respect to changes of the main input of the calculation, namely a set of empirical optical-model potentials. Near the Fermi energy the potential radius Rv(E) has a characteristic energy dependence which is qualitatively similar to that recently found in the case of neutrons in 208Pb. However, interesting differences exist. For nucleon energies larger than 10 MeV, the mean field felt by protons has approximately the same root mean square radius as that felt by neutrons. In contradistinction, the root mean square radius of the proton potential at the Fermi energy is sizeably smaller than that of the neutron potential. Furthermore, the potential depth Uv(E) approximately has a linear energy dependence from −20 MeV to +50 MeV in the case of protons, while in the case of neutrons it displays a plateau at small energy. The calculated potential V(r; E) is the sum of a Hartree-Fock type contribution VHF(r; E) and of a dispersive contribution ΔV(r; E) which is due to excitations of the 208Pb core. If VHF(r; E) is assumed to have a Woods-Saxon shape, ΔV(r; E) is found to be surface peaked, even at energies as large as 30 MeV at which in the case of neutrons ΔV(r; E) has a Woods-Saxon shape. The proton single-particle energies calculated from the full potential V(r; E) are in good agreement with the empirical values, while those obtained from the Hartree-Fock type component VFH(r; E) alone are too widely spaced. The energy dependence of V(r; E) is characterized by the effective mass m ∗ (r; E) , whose dependence upon r and E is calculated. At the nuclear centre, the effective mass is almost independent of energy in contrast to the neutron case; this difference is ascribed to the Coulomb barrier. At the nuclear surface, m ∗ displays a sharp enhancement peak for E close to the Fermi energy, as in the neutron case. The spectroscopic factors of single-particle excitations in 207Tl and 209Bi are calculated from the ratio between m ∗ and its Hartree-Fock type approximation; they are in fair agreement with empirical values deduced from recent electron inelastic scattering data.

Empirical and theoretical investigation of the average potential of nucleons in 40Ca and 208Pb

Abstract From compilations of the empirical values of the complex mean field V ( r ; E ) + iW ( r ; E ) felt by protons and neutrons in 40 Ca and 208 Pb, it is concluded that the radial moments ∫ V ( r ; E ) r q d r and ∫ W ( r ; E ) r q d r are fairly well determined by the experimental data, the favoured values of the exponent being q ≈ 0.4–0.8 in the case of the real part V and q ≈ 1.2 ± 0.4 in the case of the imaginary part W . For a given value of q , the energy dependence of the radial moment of V can be obtained from that of W by means of a dispersion relation. The latter satisfactorily accounts for the anomalous energy dependence of the radial moments of V at low energy. It predicts that this anomaly is accompanied by a rapid energy dependence of the radial shape of V , in keeping with a recent phenomenological analysis of the scattering cross sections of neutrons by 208 Pb.

Surface and volume contributions to real and imaginary parts of the heavy-ion optical potential

Abstract Starting from the Feshbach expression for the optical potential, explicit formulae for the real and the imaginary parts of the optical potential between two heavy ions (HIs) are obtained. They are each composed of a volume and a surface term. The contributions to the volume term are calculated in two nuclear Fermi liquids which flow through each other starting from the realistic Reid soft core nucleon-nucleon (NN) interaction. Since the Fermi surface is formed by two spheres one obtains a complex Brueckner reaction matrix which is approximated by a complex, effective local interaction. It is used in a fully antisymmetrized double folding procedure to obtain the volume terms of the optical potential between the two HIs. The surface contributions are directly calculated in the collision of the two finite HIs. The collective surface vibrations (3 − octupole state and 2 + , 4 + ( T = 0) giant resonances for the 16 O− 16 O collision) are included as intermediate states. This yields especially an imaginary contribution at the surface which reduces the transparency found with the volume terms alone. The method is applied to 16 O− 16 O scattering at 83 and 332 MeV laboratory energy. The local approximations to the real and imaginary parts obtained in this way agree well with phenomenological fits. The surface terms improve the agreement of the differential cross section at 80 MeV where experimental data are available.

On the self-consistency requirement in the low density expansion of the optical potential in nuclear matter

Abstract We critically discuss the choice of the auxiliary potential U which is introduced in the low density expansion of the mass operator M ( k , w ). This choice is related to the analytical properties of M ( k , w ) in the complex w -plane and we take due account of momentum conservation in the intermediate states appearing in the diagrams associated with M ( k , w ). We also provide a computation of the one-hole line, rearrangement and renormalization contributions to the optical potential, of the hole state spectral function and of the momentum distribution in nuclear matter. We use a real auxiliary potential which is self-consistent up to the order considered here, i.e. which takes into account the rearrangement and the renormalization corrections. Rearrangement is treated rigorously. The dependence of the obtained results on the choice of the nucleon-nucleon interaction, namely the Hamman-Ho Kim one in our calculation, is discussed.

Energy dependence of the global properties of the empirical nucleon-nucleus potential for 40Ca, 132Sn and 208Pb

Abstract At positive energy the empirical nucleon-nucleus potential is identified with the real part of the optical-model potential, while at negative energy it is required to yield the empirical single-particle energies. It is assumed that the potential has a Woods-Saxon shape with depth U and radius R. Compilations of empirical values of the product URq are performed in the case of neutrons and protons in 40Ca, 132Sn and 208Pb. Three values are considered for the exponent q, namely q = 1.4, 2 and ≈ 3. It is concluded that, in the energy domain which ranges from − 20 MeV to + 60 MeV, the experimental data mainly determine the product URq with q ≈ 1.4. From compilations of URq, rather accurate information is derived on the energy dependence of the effective-mass ratio m ∗ m . It is found that, for E close to the Fermi energy, m ∗ m is approximately equal to 0.8 for protons and neutrons in 40Ca and to 1.1 for protons and neutrons in 132Sn and 208Pb. The ratio m ∗ m abruptly drops to a small value (≈ 0.3) when E becomes equal to a few MeV; it then slowly increases and reaches about 0.7 when E is equal to a few tens of MeV.