Localized , and Single-Particle Potentials in Finite Nuclei Calculated with Quark-Model Baryon-Baryon Interactions
Localized single-particle potentials for all octet baryons, , , , and , in finite nuclei, C, O, Si, Ca, Fe, and Zr, are calculated using the quark-model baryon-baryon interactions. -matrices evaluated in symmetric nuclear matter in the lowest order Brueckner theory are applied to finite nuclei in local density approximation. Non-local potentials are localized by a zero-momentum Wigner transformation. Empirical single-particle properties of the nucleon and the hyperon in nuclear medium have been known to be explained semi-quantitatively in the LOBT framework. Attention is focused in the present consideration on predictions for the and hyperons. The unified description for the octet baryon-baryon interactions by the SU quark-model enables us to obtain less ambiguous extrapolation to the and sectors based on the knowledge in the sector than other potential models. The mean field is shown to be weakly attractive at the surface, but turns to be repulsive inside, which is consistent with the experimental evidence. The hyperon s.p. potential is also attractive at the nuclear surface region, and inside fluctuates around zero. Hence hypernuclear bound states are unlikely. We also evaluate energy shifts of the and atomic levels in Si and Fe, using the calculated s.p. potentials.
pacs:21.30.Fe, 21.65.-f, 21.80.+a
One of the salient features of atomic nuclei is the success of the description of their properties by a single-particle (s.p.) picture. Although the nucleon-nucleon interaction is known to be strongly repulsive at the short range part, which was once conveniently described by a hard-core, the nucleon single-particle potential is well represented by a well-behaved local potential of the Woods-Saxon form. The theoretical base of understanding this circumstance in view of the singular two-body interaction was provided by the Brueckner theory in 1950’s (1); (2); (3). The progress of the density-dependent Hartree-Fock (DDHF) description of nuclear bulk properties followed in 1970’s (4); (5), introducing some phenomenological adjustment for -matrices in the Brueckner theory.
The mean field picture seems to hold also for hyperons in nuclei. For the hyperon, the potential properties have been known from light to heavy nuclei from formation and spectroscopy experiments (6). Experimental studies of the and hyperons in nuclear medium properties are now in progress. Because direct hyperon-nucleon scattering experiments are not readily available, the properties of the hyperon s.p. potentials are a valuable source of hyperon-nucleon interactions. This case, we have to resort to an effective interaction theory to relate s.p. properties of the hyperon embedded in nuclei with the character of hyperon-nucleon interactions.
In this paper we develop a method to obtain local potentials for octet baryons in finite nuclei with using full non-local -matrix elements in nuclear matter, starting from the baryon-baryon bare interactions. The calculation of single-particle properties in nuclear matter can provide the basic information about the baryons in nuclear medium derived from the bare interaction. Nevertheless, it is instructive to explicitly calculate the s.p. potential in finite nuclei starting from two-body baryon-baryon interactions and compare them with the empirical ones. It is not even obvious whether the shape represented by the Woods-Saxon form which has been established both for the nucleon and the hyperon mean fields is suitable for the and the hyperons. The straightforward folding of the two-body effective interaction in momentum space provides a non-local potential in a nucleus. The non-locality also comes from the exchange character of the basic interaction. In order to make a comparison with empirical data, it is meaningful to define a local potential by some localization procedure. We employ in this paper a zero-momentum Wigner transformation method based on the WKB localization approximation (7).
It is necessary for a predictive discussion about hyperon s.p. properties in nuclei to use octet baryon-baryon bare interactions as reliable as possible. With little experimental information except for the interaction, the construction of the interactions in the strangeness and sectors is not simple, although some constraints are imposed by the flavor symmetry. The typical potential model has been developed in a one-boson-exchange potential (OBEP) picture by the Nijmegen group. The early parametrization with the hard-core in the 1970s (8); (9) has been successively revised by adjusting parameters in the soft core version (10); (11) and introducing new terms (13). There are now a number of sets of parameters, reflecting ambiguities due to the lack of experimental data. Although the description for the interaction seems to be under control, there are various uncertainties in the sector. For the hyperon, namely the strangeness sector, the situation is no better.
Using the spin-flavor SU quark-model, the Kyoto-Niigata group (14); (15); (16); (17); (18) have developed a unified description for the interactions between full octet baryons. In this model, the interaction is constructed as the Born kernel in the framework of the resonating group method for the three-quark clusters, the short range part of which is composed of the effective one-gluon exchange mechanism. The basic SU symmetry provides a specific framework to the interactions between octet baryons and the Pauli principle respected on the quark level in addition brings about a characteristic structure to them. Incorporating effective meson exchange potentials between quarks, namely the scalar, pseudo-scalar, and vector mesons exchanges, the model is able to account the scattering data as accurately as other modern potential models.
Parameters of the SU quark-model by the Kyoto-Niigata group are mostly fixed in the and sectors, and the uncertainties in the extension to the and channels are limited. In fact, the definite predictions such as the smallness of the spin-orbit interaction and the overall repulsive nature of the -nucleus s.p. potential in nuclei have been supported by the experiments afterwords. Therefore it is interesting to examine the whole prediction of this potential for s.p. properties of all the octet baryons in nuclear medium. In particular, concrete predictions are presented for the hyperon. We use the most recent quark-model potential fss2 (17); (18) in this paper. The parameter set includes no adjustable parameter for the tuning afterward. The original interaction as a Born kernel has an inherent energy-dependence. Recently the method to eliminate the energy-dependence was developed (19). We actually use this renormalized version of the fss2 potential.
We present, in Sec. II, basic expressions of the method to evaluate localized , , , and s.p. potentials in a finite nucleus. We first discuss numerical results of these s.p. potentials in nuclear matter to represent basic characters of the -matrices of the quark-model baryon-baryon interactions. Calculated results in finite nuclei, C, O, Si, Ca, Fe, and Zr, are shown in Sec. IV. The energy shift and the width of the and atomic levels in Si and Fe are studied in Sec. V on the basis of the s.p. potential obtained in Sec. IV. Summary is given in Sec. VI.
Ii Localized single-particle potential in a finite nucleus
We calculate a baryon single-particle potential in a finite nucleus which is defined by folding or -matrix elements in nuclear matter with respect to nucleon occupied states through the local density approximation. It has been a traditional method for the microscopic study of bulk properties of nuclei to construct density-dependent two-body local interaction based on the -matrices in nuclear matter and apply the effective interaction to mean field calculations for finite nuclei. Avoiding this procedure, we directly fold -matrix elements to obtain non-local s.p. potentials and localize them. In the DDHF calculations, some phenomenological adjustments are introduced to reproduce the properties of the well-known nuclei. The purpose of the present paper is not to accomplish the reproduction of empirical values, but examine overall implications of the unified description of the baryon-baryon bare interaction by the quark model potential fss2 (18) for hyperon s.p. potentials in finite nuclei. In this section, we derive a basic expression for the s.p. potential, by introducing some approximations and a localized method by the zero-momentum Wigner transform.
ii.1 direct term
First we consider the following direct term contribution. The wave function denotes the nucleon s.p. wave function of the nucleus with the orbital angular momentum and the total angular momentum , and the dummy wave function of the baryon for which the potential is calculated is denoted by . The average over the -component of the total angular momentum, with , means that we assume the spherical symmetry from the beginning. We do not write the isospin indices for simplicity in the following derivation, but recover them in the final expression.
The effective baryon-baryon interaction in a coordinate space is supposed to be related to the -matrix in momentum space by
Here each momentum has the following relation: , , , . The mass of the baryon for which the s.p. potential is evaluated is denoted by and the mass of the nucleon in the target nucleus by . The -matrix is evaluated in symmetric nuclear matter by solving the baryon-channel coupling Bethe-Goldstone equations
with the suffix specifying the pair of a baryon and a nucleon by . The Pauli exclusion operator is treated in the standard angle-average approximation. The explicit expression may be found in ref. (20). is a s.p. energy of the baryon in nuclear matter. We employ the continuous choice for the energy denominator in Eq. (3). That is, is defined self-consistently by the following definition of the s.p. potential .
where and . The prescription for the starting energy in the local density approximation is explained in the next section.
The straightforward calculation of Eq. (1) needs much computational effort and is not fruitful to obtain a physical insight for baryon properties in nuclei by starting from the bare baryon-baryon interactions. We introduce two simplifying approximations. One is the spin-average in taking the sum of the matrix elements, which means that we take the average over the spin orientation:
where the -matrix is decomposed to partial waves and stands for the Legendre polynomial with specifying the orbital angular momentum. The other simplification is the following replacement.
This corresponds to the zero-momentum Wigner transformation of the non-local potential. That is, we set for the Wigner transformation of the non-local potential .
Results shown in Sect. IV for nucleons and lambdas, for which we know what s.p. potentials potentials are expected in -matrix calculations with bare and interactions in various studies in literature, implies that the zero-momentum approximation works well. More direct confirmation of the reliability of this approximation will be presented elsewhere (21).
To evaluate Eq. (1) with the above simplification it is convenient to use the Fourier transform of the s.p. wave function ,
where is a nodal quantum number and the Fourier transformation of the radial wave function is defined as
After carrying out some integrations and taking angular momentum recouplings, we obtain the final expression as follows.
where and are defined by and as
ii.2 Exchange term
We also have to consider the exchange term contribution, which is familiar for the nucleon through the antisymmetrization of the wave function. For hyperons, such terms appear in association with the exchange character of the hyperon-nucleon interaction, which is realized by the strange meson exchange in the OBEP description. Denoting the space-exchange operator and the spin-exchange operator by and , respectively, and specifying the even and odd components of the interaction under the space-exchange, the matrix element of the interaction is written as
where the relation is used. The first term was treated in the previous subsection as a direct term contribution, and the second term is considered in this subsection. The effective interactions in the direct and exchange contributions should be treated as such a combination of the even and odd parts in each spin and isospin channels, though the isospin dependence is disregarded in the above expression because the inclusion of it in the final expression is simple.
In this case we define and as
Introducing the same simplifying approximations as in the direct term, we obtain
where and are defined by and by
These and are obtained by changing the sign of in and of Eqs. (11) and (12). It is easy to see that the difference of the expressions of and is only the factor . Thus, recovering the isospin degrees of freedom, we obtain
In the above expression. is the isospin of the baryon for which the s.p. potential is considered, and is its -component. The index denotes the proton or neutron in the target nucleus.
The Eq. (19) defines a s.p. potential to give
It is noted that the potential does not have - and -dependences due to approximations introduced in the derivation.
Iii Single-particle potentials in Symmetric Nuclear Matter
Before discussing baryon s.p. potentials in finite nuclei, we show s.p. potentials in nuclear matter at various Fermi momenta, fm, with the quark-model potential fss2 (18). This potential is defined as a Born kernel of the RGM description of the interaction between the three-quark clusters. We use the energy-independent renormalized version of the fss2 potential (19). The details of the -matrix calculation for hyperons in nuclear matter are reported in ref. (20). It has been known that the LOBT saturation curve in ordinary nuclear matter does not reproduce the empirical saturation property. Although the curve obtained by the potential fss2 with the continuous choice for intermediate spectra almost goes through the empirical saturation point of fm and MeV, the energy minimum appears at fm and MeV. Nevertheless the LOBT calculation provides a useful starting point and meaningful information for the baryon s.p. potentials in nuclear medium in the microscopic studies based on the bare baryon-baryon interactions. Missing effects in the LOBT , such as con-
tributions from higher order diagrams and three-body forces are now semi-quantitatively understood in the nucleon sector (22).
Real and imaginary parts of the calculated s.p. potentials for , , , and in symmetric nuclear matter are shown in Figs. 1 and 2 as a function of the momentum . These are the results after the self-consistency for the starting energy being reached. The Fermi momentum is chosen approximately in a step of one tenth of the normal nucleon density . These densities are used as the discretized points of the density in the local density approximation for considering finite nuclei. Below fm, the nuclear matter -matrix calculation becomes unstable due to the appearance of a bound state in the channel. Because we expect little relevance of this phenomenon to ground states of finite nuclei, the instability is not inspected further. In the case that the effective interactions at low density below fm is needed in the local density approximation for finite nuclei, we use the -matrices at fm.
As for the nucleon, the result is very similar to those by other realistic potentials. The depth of the s.p. potential at the normal density is considerably larger than the magnitude of the standard Woods-Saxon potential, which is MeV. It has been known that the rearrangement potential reduces the strength by about MeV.
We are concerned mainly with the prediction of the quark-model potential fss2 (18) for hyperon s.p. potentials. The strength of the attractive s.p. potential in normal nuclear matter shown in Fig. 1 is almost 45 MeV, which is again larger than the empirically known value of around 30 MeV. At the low density the potential is shallower. However, as will be shown in the next section, the depth of the calculated s.p. potential in finite nuclei, taking into account the finite geometry and the effects of the non-diagonal properties of the , seems to be dictated by the potential depth at the normal nuclear density. We can expect that the rearrangement effects give rise to a repulsive contribution of the order of 10 MeV for the mainly through the energy dependence of the -matrix.
Nuclear matter calculations using the early version of the Kyoto-Niigata SU quark-model potential, FSS, predicted a repulsive s.p. potential (20). Results shown in Fig. 2 are obtained by the most recent quark-model parameterization, fss2 (18). The potential at is definitely repulsive of about 15 MeV at normal density. This repulsion chiefly comes from the strong repulsive contribution of the state in the isospin channel, which is naturally predicted by the quark-model as the consequence of the Pauli principle on the quark level. The interaction in the with channel is also repulsive. These repulsive contributions overwhelm the attractive contributions from the with and the with channels. The width of the hyperon in nuclear medium is related to the imaginary strength of the s.p. potential by . is seen in Fig. 2 to be more than 30 MeV at normal density.
The s.p. potential in symmetric nuclear matter predicted by fss2 is weakly attractive as is shown in Fig. 2. As the momentum increases, the magnitude of the attraction is seen to become larger at the low momentum region of fm. The momentum dependence may be characterized by the effective mass. To obtain a rough estimation of it, we parameterize the potential by . In this case the effective mass at is obtained by
Calculated s.p. potentials give at fm and at fm.
The elastic and inelastic cross-section measurements at low energy by Ahn et al. (23) indicate that the width of a s.p. state in nuclear medium is MeV. Although it is uncertain at which energy this number should be compared with the calculated imaginary strength, the small imaginary strength of the s.p. potential at the low momentum region given in Fig. 2 is in accord with the empirical small width of the in nuclear medium.
It is encouraging to see that the results for and hyperons agree at least qualitatively with empirical indications so far obtained.
Iv Results in Finite Nuclei
We apply the calculational method presented in Sect. II to from light to medium-heavy nuclei: C, O, Si, Ca, Fe, and Zr. Nucleon density distributions are prepared by density-dependent Hartree-Fock calculations using the Campi-Sprung G-0 force (5). Profiles of the point nucleon density distribution which is a sum of the neutron and proton densities are shown in Fig. 3.
Nuclear matter -matrices are used in finite nuclei by the local density approximation. At the position where the s.p. potential is evaluated the local Fermi momentum is defined by the correspondence . The -matrices calculated in nuclear matter with this Fermi momentum are used in Eq. (19). In actual calculations, -matrix calculation is carried out only for the Fermi momenta shown in Figs. 1 and 2. At each position , the Fermi momentum which is closest to among these twelve values is chosen. As explained in Sect. II, for small densities below fm, namely the total density fm, we always use fm. In homogeneous matter the s.p. potential is determined by the matrix element with the zero momentum transfer, namely diagonal () components of . In finite nuclei non-diagonal components of the matrices also contribute to the s.p. potential.
The starting-energy dependence of the -matrix plays an important role in the LOBT. The prescription of the starting-energy as the sum of s.p. energies of the two baryons considered means that certain higher-order diagrams are included. Hence the self-consistency between the s.p. energy which is defined by the -matrix and the -matrix which depends on the starting-energy is required. Calculations in nuclear matter shown in Sect. II are the results with this consistency achieved. In the case that the -matrix in nuclear matter is applied to a finite nucleus, however, there is no simple way to treat the starting-energy dependence. We introduce an ad hoc prescription to use an s.p. potential value at the median momentum : for the nucleon and for the hyperons.
The results with the quark-model potential fss2 (18) in the energy-renormalized form are shown in Figs. 49. The charge state of the baryon specified by in Eq. (19) is set to be . Comments on the calculated s.p. potential of each baryon are given in the following.
iv.1 Nucleon s.p. potential
The shape of the calculated nucleon potential is seen to follow the density distribution, and the depth is MeV which corresponds to the s.p. potential in nuclear matter at the normal density. It is well known that the straightforward application of the LOBT starting from realistic interactions overestimate the attractive nucleon-nucleus potential. To compare the calculated potential with the empirical one, we need to include the so-called rearrangement potential. The repulsive strength is known to be MeV (5). If this contribution is taken into account, the resulting potential becomes closer to the phenomenological potential of the Woods-Saxon form.
It is noted that it is still a remaining problem in nuclear physics to understand nuclear bulk properties in a fully microscopic way on the basis of the realistic interactions including higher-order correlations, three-body forces, and other possible medium effects.
iv.2 hyperon s.p. potential
The state has a similar character to the state in the spin-flavor SU symmetry, although there is small admixture of a completely quark Pauli forbidden component. Similarly, the interaction in the channel resembles that in the channel with a smaller magnitude by a factor of , although there is an important difference that the pion exchange is absent. Thus it is probable that the -nucleus potential is attractive, about a half of the -nucleus potential in magnitude. Looking at the density-dependence of the s.p. potential in nuclear matter, we should expect a similar rearrangement potential as in the nucleon case. The addition of the hyperon to nuclear medium does not change directly the nucleon density and hence the nucleonic Pauli effect. The rearrangement effect for the hyperon originates, in the LOBT, mainly from the energy-dependence of the and -matrices. If we assume a repulsive rearrangement potential of the order of 10 MeV, calculated results shown in Figs. 4 9 correspond well to the empirical -nucleus potential in the Woods-Saxon form with the depth of about 30 MeV.
iv.3 hyperon s.p. potential
The experimental information has been limited for the s.p. potential in nuclei. Because the state in a nucleus is expected to have a large width due to the strong conversion process, it is unlikely to observe clear peak structure in the -formation spectra. Nevertheless, results from the early experiments of inclusive spectra (24) measured at CERN were interpreted as indicating that the -nucleus potential is moderately attractive. The discovery of the bound He+ and H+ systems (25) and the theoretical consideration by Harada et al. (26) showed that the attraction in the channel should be attractive enough. Another important source of the -nucleus interaction is the energy shift and the width of atomic orbits extracted from the X-ray data. Batty, Friedman, and Gal (27) analyzed the data to conclude that the -nucleus potential changes its sign toward higher density region in a nucleus from the attractive potential at the surface region. Dabrowski (28) analyzed the BNL experiment of spectrum on Be (29) in a plane wave model and conjectured that the potential is repulsive of the order of MeV. Recent experimental data with better accuracy of inclusive spectra measured at KEK (30) was reported to suggest that the -nucleus potential is strongly repulsive, the strength being more than 100 MeV. Several theoretical analyses carried out later (31); (32) confirmed the repulsive nature of the s.p. potential, but the height may be a few 10 MeV.
On the theoretical side, interaction models admit large uncertainties. Most parameter sets of the Nijmegen hyperon-nucleon OBEP potential (8); (9); (10); (12) predict an attractive -nucleus potential in nuclear matter, except the model F (33); (34); (35). On the contrary, the strong repulsive character in the channel is inherent in the quark-model description (18); (36). Thus the Kyoto-Niigata quark-model potential predicts an overall repulsive s.p. potential in nuclear matter. Results given in Figs. 49 show the consequence of this property to finite nuclei. At higher density region inside a nucleus the -nucleus potential is repulsive of 1020 MeV. The overall repulsive nature of the -nucleus potential has been deduced by the analyses of formation inclusive spectra (30); (31); (32). Beyond the surface region the potential becomes attractive. It is interesting to see that the radial dependence indicated by the analyses of atomic data (27) is actually derived by the microscopic calculation using the quark-model bare interaction with no phenomenological adjustment.
It is remarked that in the present evaluations we apply -matrices in symmetric nuclear matter to finite nuclei without separating neutron and proton density distributions. However, in heavy nuclei, e.g. Zr in our calculations, in which neutron and proton distributions are visibly different, we should take care of the isospin-dependence when solving the Bethe-Goldstone equation. In that case, the repulsive contribution to the s.p. potential from the channel becomes more predominant.
iv.4 hyperon s.p. potential
For the s.p. potential, the experimental information has been more scarce than the . The hyperon is formed in reaction on nuclei with small production rates. There has been no concrete evidence of the hypernuclear bound state. The existing experimental data of the formation spectrum (37); (38); (39) has suggested that the feels attractive potential in nuclei, the depth of which is not so large, MeV. Our results shown in Figs. 49 with the quark-model potential fss2 (18) show that the -nucleus potential is weakly attractive at and beyond the nuclear surface region, which is similar to the -nucleus potential. Toward the inside of the nucleus the s.p. potential tends to be repulsive and oscillates around zero with the magnitude of about 10 MeV. It is not possible to simulate the potential shape by a single Woods-Saxon form. No hypernuclear bound state is expected from such a weak potential. The situation does not change even if the actual potential strength differs by a factor of 2 or so. In that case the level shift of the atomic orbit should be a valuable source of the information about the -nucleus interaction. This subject is addressed in the next section.
The evaluated -matrices include full baryon-channel couplings, namely the possible -- or -- couplings. It is helpful to use equivalent interactions in low-momentum space to check the character of the interaction and the effect of the baryon-channel coupling in each spin and isospin state. Inspecting the matrix elements in ref. (40), we see that the effective interaction from the fss2 in the channels both in and are repulsive. The interaction is very weak, and the interaction is attractive for which the -- coupling is responsible. It turns out that the net -wave contribution is small and thus the attractive -wave contribution plays an important role to make the s.p. potential to be attractive at the surface region.
V Energy shift and width of atomic orbit
|real part||imaginary part|
|real part||imaginary part|
The level shift and the width of atomic orbits are a valuable source of the information on the -nucleus strong interaction. The analyses by Batty, Friedman, and Gal (27) indicated that the -nucleus potential is attractive at the surface region, but at higher density region in a nucleus the potential turns to be repulsive. The radial dependence of the calculated s.p. potential shown in the previous section agrees with this. Therefore it is instructive to explicitly evaluate the energy shift and the width of atomic orbits with the calculated potential. As will be shown below, the result is consistent with the experimental data. This indicates that the microscopic calculation with the quark-model fss2 is reliable in the channel. Thus, it is interesting to extend the level shift calculation to atomic orbits. The experimental data should be available in near future, because the first measurement of atomic X rays from Fe target is proposed (43) to be performed at J-PARC. The theoretical prediction provides a guiding information for this experiment.
We consider Si and Fe for explicit evaluations of the level shift of the atomic orbit. We first fit the shape of the calculated s.p. potential using the Woods-Saxon form. For Si a sum of three Woods-Saxon shapes is used and for Fe a sum of two Woods-Saxon shapes and one derivative of the Woods-Saxon shape is assigned. Parameters are given in Table I. It is noted that the imaginary parts are also given to illustrate the order of the magnitude of the absorptive strength, intending to demonstrate that the imaginary potential is about one order of magnitude smaller than the one. However, actual numbers should not be taken very seriously because nuclear matter calculations tend to overestimate the imaginary strength as the calculations (41) of nucleon optical model potential indicates. In addition, the prescription to use the -matrices at fm for all the densities below fm probably leads to the overestimation of the imaginary strength at the surface region. It is also remarked that localized imaginary potential through the zero-momentum Wigner transformation may become positive at some points.
Results of the level shift and the width for the - and -atomic levels on Si and the - and -atomic levels on Fe are given in Table II, where stands for the Coulomb bound state energy without the -nucleus strong interaction. When the real part of the s.p. potential is taken into account, the energy of the orbit on Si is shifted downward by 222 eV. To investigate the contribution of the absorptive effect, we do not use the calculated potential given in Table I. The imaginary potential is rather strong as explained above. The magnitude of the level shift and the width depends non-linearly on the strength of the imaginary potential. Hence we use an phenomenological imaginary potential to discuss the level shift of the atomic orbits of the . We add an imaginary potential in the single Woods-Saxon form used in ref. (27), namely and with the depth of MeV. In that case, we obtain eV and eV, which well correspond to the experimental values of eV and eV. This result indicates that the real part of the s.p. potential calculated microscopically in the LOBT starting from the two-body quark-model potential fss2 is reasonable.