A.R. Bodmer

Relativistic Calculation of Nuclear Matter and the Nuclear Surface

Relativistic mean field Hartree and Thomas-Fermi calculations are made for infinite and semiinfinite symmetric nuclear matter. Cubic and quartic scalar meson self-interactions are included. The strength of these self-interactions, the scalar and vector meson coupling constants, and the scalar meson mass are considered as parameters chosen to fit the empirical properties of nuclear matter and of the nuclear surface. Acceptable fits imply large self-interactions. No abnormal nuclear matter solutions are found. The best fit of the nuclear surface yields a compressibility coefficient of 150 ± 50 MeV. The Hartree and TF results are in good agreement for larger surface thicknesses.

Relativistic mean field theory of nuclei with a vector meson self-interaction

Abstract The inclusion of an isoscalar vector-meson quartic self-interaction (VSI) in the relativistic mean field theory of nuclei is discussed. We consider mostly nuclear matter, and in particular the equation of state (EOS) at T = 0. A VSI softens the EOS; in particular a moderately strong VSI gives an EOS similar to those obtained from nonrelativistic many-body calculations. Estimates of the spin-orbit splitting in light nuclei indicate that this splitting is not sensitive to the strength of the VSI and will depend mostly on the effective mass M ∗ . The energy dependence of the optical potential gives M ∗ /M ∼- 0.6 independent of the VSI, and consistent with that obtained from the spin-orbit splitting.

Relativistic mean field theory for nuclei

Abstract The phenomenology of relativistic mean field theory of nuclei is discussed for general scalar potential functions U ( φ ) with a minimum at zero scalar field φ = 0. The cubic-plus-quartic scalar self-interaction model U 34 is discussed as an important special case. We obtain general conditions on U required by the saturation properties including the incompressibility coefficient K . The effective mass M ∗ is uniquely related to the vector coupling. The pathologies of the U 34 model are discussed and modifications are proposed which cure the model when the quartic term is negative as phenomenologically required. It is argued that the equation of state (at T = 0) is effectively independent of the form of U if the parameters are adjusted to a given K, M ∗ , and that the phenomenology of finite nuclei, in particular the spin-orbit splitting in light nuclei, determines M ∗ ≈ 0.6 M essentially independently of U . The resulting (stiff) e.o.s. is then determined within narrow limits independently of the form of U .

Binding energies of λλ hypernuclei and the λλ interaction

Abstract The binding energies of the ΛΛ hypernuclei 10ΛΛBe and 6ΛΛHe are calculated variationally with a 2α + 2Λ and with an α + 2Λ model, respectively. For 10ΛΛBe the integrations were made with Monte Carlo techniques while for 6ΛΛHe direct numerical methods were used. A wide range of phenomenological ΛΛ potentials based on meson-exchange models was considered. An approximately universal linear relation between the calculated values of B ΛΛ ( 10 ΛΛ Be ) and of B ΛΛ ( 6 ΛΛ He ) is obtained. For the experimental value of B ΛΛ ( 10 ΛΛ Be ) = 17.7 ± 0.1 MeV this relation predicts a much too small value of B ΛΛ ( 6 ΛΛ He ) , well below the lower limit of the quoted experimental value of 10.9 ± 0.6 MeV. For ΛΛ potentials with repulsive cores the relations, for a given B ΛΛ ( 10 ΛΛ Be ) , between the low-energy ΛΛ scattering parameters a, r0 and the intrinsic range b are both approximately universal and independent of the detailed potential shape. For the experimental value of B ΛΛ ( 10 ΛΛ Be ) the preferred values are 2.5 ≲ −a ≲ 3.5 fm, 2.6 ≲ r0 ≲ 3.1 fm. The large and negative values of a correspond to conditions not too far from a bound 1S0 ΛΛ state and could indicate a 6-quark dibaryon state with the same quantum numbers and above the ΛΛ threshold.

Reaction-matrix calculation of the Λ-particle binding in nuclear matter

Abstract The two principal procedures for obtaining the phenomenological Λ well depth D from the experimental separation energies B Λ are discussed. A value D ∼ 30 ± 2 MeV, with an upper limit D ≲ 35 MeV, is consistent with the results obtained from both procedures. A reactionmatrix calculation of D is made for central ΛN potentials. This is exact in the g -matrix approximation which gives the term proportional to the density ρ in an expansion in powers of p . To calculate the g -matrix the reference-spectrum method is used in the Kallio-Day form. The free kinetic energies are used for the unoccupied states. The difference between the singleparticle energies of the bound and excited states of the Λ involves a gap which is identified with the well depth. This leads to a self-consistency condition for the determination of D . The nucleon-spectrum parameters are taken from the calculations of Sprung et al . The angle-average approximation for the exclusion-principle operator, appropriate to a Λ in nuclear matter, is examined and found to be excellent. For hard-core potentials, a perturbation expansion in the strength of the attractive tail is in general inadequate and, especially for potentials of short range, it is necessary to obtain the g -matrix exactly for the complete potentials. For given lowenergy scattering parameters, the presence of a sizable hard core very substantially reduces D , The p-state contribution to D can be quite large, especially for the phenomenological potentials with hard cores and with larger intrinsic ranges. Thus for Herndon and Tangs potentials which fit the binding energies of the A = 3 and 4 hypernuclei as well as the Λp scattering data and which have rather large hard cores, one can get close to the phenomenological value of D by assuming zero p-state interaction. With some p-state interaction — about half of that for the s-state — reasonable suppression of the ΛN-σN coupling in nuclear matter can give agreement with the phenomenological depth. The well depth is found to depend fairly linearly on ρ over a large range and the Λ effective mass can be as small as 0.65 M Λ .

The α-cluster hypernuclei 5ΛHe, 6ΛΛHe, 9ΛBe, 10ΛΛBe and hypernuclear interactions☆

Variational calculations for 6ΛΛHe, 9ΛBe and 10ΛΛBe have been made with nαα-nΛΛ cluster models. Various αΛ potentials are considered, all of which reproduce BΛ(5ΛHe). Some of the VαΛ are obtained from an αΛ model by a folding procedure using effective ΛN and ΛNN forces; realistic VαΛ are also obtained from 4N-Λ Monte Carlo variational calculations of 5ΛHe. These MC VαΛ include many-body effects which give a central hump even with only Λ N forces. 9ΛBe is overbound by about one MeV with realistic αα and αΛ potentials, even with reasonable estimates of ΛN exchange effects which give a reduction ≅0.5 MeV. Repulsive ΛNN forces which can account for the overbinding of 5ΛHe give rise to a repulsive ααΛ potential which brings the calculated BΛ(9ΛBe) into good agreement with the experimental energy. The ΛΛ interaction strengths are obtained from 6ΛΛHe and 10ΛΛBe for a number of ΛΛ potential shapes. The strengths obtained from 10ΛΛBe underbind 6ΛΛHe by more than one MeV for all our ΛΛ and αΛ potentials. The ΛΛ interaction obtained from the well established ΛΛ10Be event is found to be quite strongly attractive, comparable to the 1S0 NN interaction without OPE, with correspondingly large negative ΛΛ scattering lengths of ≅ −4 to −5 fm.

ΛN-ΣN coupling for ΛN scattering and for the Λ-particle binding in nuclear matter

Abstract The effect of coupling of the ΛN to the (virtual) ∑N channel on the Λ-particle binding in nuclear matter (the Λ-well depth) is studied with perturbation theory and especially in the reaction-matrix approach with use of the g -matrix approximation. The g -matrix is calculated self-consistently by use of the Kallio-Day version of the reference-spectrum method, the free kinetic energies being assumed for the unoccupied single-particle states. A large variety of interactions is considered including in particular, hard-core interactions with (static) one-pion exchange (OPE) and ϱp-meson exchange coupling potentials. The couplings have both central and tensor components. A meaningful measure of the suppression of the coupling in nuclear matter is obtained in perturbation theory by considering the effective nonlocal central potentials that represent the (s-state) effect of the coupling for nuclear matter and for ΛN scattering. For hard-core interactions we use one-channel central ΛN potentials with the same hard core and the same low-energy scattering characteristics as the coupled-channel interactions to define the “unsuppressed” well depths appropriate to the effect of the coupling for scattering. The suppression δD of the well depth arises from the effect of the exclusion principle and of the spectrum gaps (binding-energy effects) on the coupling. δD depends strongly on the details of the interaction and is relatively much larger for central than for tensor couplings and is less for shorter ranges of the same type of coupling potential. Furthermore, δD depends on the “unsuppressed” (i.e., intrinsic) strength of the coupling. For the OPE coupling the tensor component is dominant and the central components contribute only very slightly; reasonable Λ∑πn coupling constants can give large values of δD (as much as 15 MeV). For the ϱ-exchange coupling the central components for both spin states, S = 0 and 1, can give quite large δD (possibly as much as 5 MeV for S = 0) in spite of the short range. Interesting possibilities arise from possible destructive interference between the OPE and ϱ-exchange tensor couplings because the interference for nuclear matter differs from that for scattering, since the OPE coupling is more strongly suppressed than the ϱ-exchange one. The possibility of 15 MeV suppression implies that agreement of the calculated with the phenomenological depth (≈ 30 MeV) can be obtained for reasonable hard-core radii (≈ 0.4 fm) and for an appreciable p-state interaction of about half the strength of the s-state one.

Classical-equations-of-motion calculations of multiplicities for high-energy collisions of 20Ne+20Ne

Abstract Nucleon multiplicity distributions have been obtained as a function of impact parameter b for 20 Ne + 20 Ne at a laboratory energy of 800 MeV/nucleon with classical-equations-of-motion calculations. The average multiplicity decreases-quite rapidly with b , but there is a quite wide spread of multiplicities for a given b . This indicates, at least for equal-mass, light nuclei, that the multiplicity can only determine the impact parameter with rather poor resolution.

An alpha-deuteron-lambda model of the hypernucleus 7ΛLi

Abstract The hypernucleus 7 Λ Li is studied with a three-body model consisting of the Λ and an undistorted α-particle and deutron. The interactions between the constituents are obtained by making use of the basic α-N and Λ-N interactions. However, the strengths of these are adjusted to give the observed binding energies of the two isolated pairs α-d (i.e., 6 Li) and α-Λ (i.e., 5 Λ He). The strength of the d-Λ interaction is not fixed but is determined as a function of the binding energy of 7 Λ Li by our calculations for 7 Λ Li. For 6 Li a 2s relative α-d wave function is used. The 1s α-d component ψ 1s is assumed to be entirely spurious both for 6 Li and 7 Λ Li. in Variational calculations for 7 Λ Li are made with flexible trial wave functions and with particular effort given to reducing the amplitude of ψ 1s as much as possible. In achieving this, some interesting new features arise in our variational calculation. Our results confirm the indications of a previous two-body model study that there is a strong α-Λ correlation together with only a slight distortion of the 6 Li core by the Λ. They also indicate that comparing (as a function of the range of the Λ-N interaction) the strength of the Λ-d interaction obtained from 3 Λ H with that from 7 Λ Li could give information about the range of the Λ-N interaction. However, no conclusive results could be obtained in this respect since although we obtain the major part of the binding energy of 7 Λ Li there is nevertheless a sufficient discrepancy to indicate that our model is not completely adequate. In particular, the probable need to consider correlations of the Λ with the individual nucleons of the deuteron suggests the use of a four-body, α-n-p-Λ model for further studies of 7 Λ Li.

Reaction-matrix calculations of the Λ-particle binding in nuclear matter for central ΛN potentials

Abstract Extensive results, obtained in the G -matrix approximation for a wide range of central ΛN Yukawa potentials both with hard cores (HC) and soft repulsive cores, are presented for the Λ well depth (binding energy of a Λ-particle in nuclear matter) and related quantities. The HC results, in particular, span a wide range of scattering lengths a , effective ranges r 0 and hard-core radii c , sufficient to include any potentials likely to be proposed. The s-state well depth D s is to a good approximation determined by a , r 0 and the s-state correlation volume (wound integral) κ s NM which characterizes the core in an effectively shape-independent way. The depth D s decreases with κ s NM for given a , r 0 . A successful search was made for a “scattering” parameter κ s SC which also characterizes the core in a shape-independent way but which is determined only by (low-energy) s-wave scattering calculations; κ s SC is the (zero-energy) scattering correlation volume up to the separation distance d . The parameterization of D s in terms of a , r 0 and κ s SC is then effectively universal for local potentials, permitting one to obtain D s for any reasonable local potential by use only of s-wave scattering calculations. The p-wave depth D p is generally quite large with only a moderate dependence on c for given a and r 0 but a strong dependence on r 0 . The p-wave correlation volume is not a useful shape-independent parameter for D p . The d-wave depth is quite small although not negligible. Limited results are obtained for the dependence on density. The implications of a comparison of the calculated and the phenomenological well depths for the ΛN interaction are discussed.