Shell-model study of the isotonic chain with a realistic effective hamiltonian
We have performed shell-model calculations for the even- and odd-mass isotones, focusing attention on low-energy states. The single-particle energies and effective two-body interaction have been both determined within the framework of the time-dependent degenerate linked-diagram perturbation theory, starting from a low-momentum interaction derived from the CD-Bonn nucleon-nucleon potential. In this way, no phenomenological input enters our effective Hamiltonian, whose reliability is evidenced by the good agreement between theory and experiment.
pacs:21.60.Cs, 21.30.Fe, 27.60.+j
The region has long been the subject of shell-model studies owing to the strong doubly magic character of Sn. In this context, several calculations have been performed employing purely phenomenological two-body effective interactions Sarkar01. Clearly, a main issue in the study of this region is that it provides the opportunity to investigate the effect of adding protons to a doubly-magic core over a large number of nuclei (from mass number A=133 to A=152). Comprehensive studies of this kind were conducted some twenty years ago in Refs. Andreozzi90 and Wildenthal90. In the former a simple pairing Hamiltonian and empirical single-particle (SP) energies were used, while in the latter the two-body matrix elements and single-particle energies were determined simultaneously by a least squares fit to the ground-state and excited-state energies drawn from all experimentally studied nuclei.
Starting in the mid 1990s, however, shell-model calculations employing effective interactions derived from realistic nucleon-nucleon () potentials have been performed in the region, which have generally yielded very good results Andreozzi97; Covello97; Holt97; Suhonen98; Covello02. Some of these studies Covello97; Holt97; Suhonen98 have focused attention on the evolution of the low-energy properties of the isotones as a function of , showing that the observed behavior is well reproduced by the theory. In Refs. Holt97; Suhonen98 the shell-model results were also compared with those of a quasiparticle random-phase approximation calculation leading to the conclusion that the low-lying states are equally well described by these two approaches.
A main merit of the above mentioned realistic shell-model calculations is, of course, that no adjustable parameter appears in the matrix elements of the two-body effective interaction. In all of them, however, the single-particle energies have been taken from experiment. Actually, the calculational techniques used in these studies are based on the time-dependent degenerate linked-diagram perturbation theory Kuo90, which provides an effective shell-model hamiltonian, , containing both one- and two-body components. Usually, however, the one-body components, which represent the theoretical SP energies, are subtracted from and replaced by those obtained from the experimental spectra of nuclei with one valence nucleon Shurpin83. This procedure has indeed led to a very good description of nuclear properties in different mass regions Coraggio09.
Recently, we have performed fully realistic shell-model calculations employing both theoretical SP energies and two-body interactions in studies of and shell nuclei Coraggio05c; Coraggio07b. In these studies we have renormalized the high-momentum repulsive components of the bare potential by way of the so-called approach Bogner01; Bogner02, which provides a smooth potential preserving exactly the onshell properties of the original up to a cutoff . The effective hamiltonian has then been derived within the framework of the time-dependent degenerate linked-diagram perturbation theory. In both papers Coraggio05c; Coraggio07b a cutoff momentum fm was employed. This corresponds to the laboratory energy MeV, which is the inelastic threshold of the scattering. The results of these studies compare well with experiment and other shell-model calculations Cohen65; Richter06.
On these grounds, we have found it interesting to carry out a similar shell-model study of the chain, focusing attention on the low-lying states of both even- and odd-mass isotones. In this case, the SP energies and two-body effective interaction are determined from a derived from the high-precision CD-Bonn potential Machleidt01b with a cutoff momentum fm. This value of , which is somewhat larger than that used in our studies of the light nuclei, is needed to obtain a reasonable description of the experimental SP spectrum of the medium-heavy nucleus Sb. This is mainly due to the fact that, at variance with the model spaces for light nuclei, in this case the 50-82 major shell contains the intruder state, whose relative energy turns out to be strongly sensitive to the value of . More precisely, for small values of this state lies far away from the other SP levels of the 50-82 shell, while the correct structure of this shell is restored when increasing the value of Coraggio08b.
The dependence of the results on would obviously vanish when complementing the effective two-body interaction with different three- and higher-body components for each value of . However, it is at present a very hard task, from the computational point of view, to take into account two- and three-body forces on an equal footing in a third-order perturbative calculation for medium- and heavy-mass nuclei. Therefore, the value of the cutoff may be considered as a parameter that we have chosen so as to reproduce satisfactorily the experimental SP spectrum. For the sake of completeness, it should be mentioned that the trend of our calculated SP energies suggests that a larger value of the cutoff could even improve the agreement with experiment. However, this would amount to include higher-momentum components, which would rapidly deteriorate the convergence properties of the perturbative expansion used to derive the effective shell-model hamiltonian. In the following section we show that our choice of is a reasonable compromise to assure both good perturbative properties and quality of the results.
The paper is organized as follows. In Sec. II we give a brief outline of our calculations, focusing attention on the convergence properties of the effective hamiltonian. Sec. III is devoted to the presentation and discussion of our results, while some concluding remarks are given in Sec. IV.
Ii Outline of calculations
Our goal is to derive an effective hamiltonian for shell-model calculations in the proton shell, which is the standard model space to describe the spectroscopic properties of isotones. Within the framework of the shell model, an auxiliary one-body potential is introduced in order to break up the hamiltonian for a system of nucleons as the sum of a one-body term , which describes the independent motion of the nucleons, and a residual interaction :
Once has been introduced, a reduced model space is defined in terms of a finite subset of ’s eigenvectors. The diagonalization of the many-body hamiltonian (1) in an infinite Hilbert space, that it is obviously unfeasible, is then reduced to the solution of an eigenvalue problem for an effective hamiltonian in a finite space.
In this paper, we derive by way of the time-dependent perturbation theory Kuo90. Namely, is expressed through the Kuo-Lee-Ratcliff (KLR) folded-diagram expansion in terms of the vertex function -box, which is composed of irreducible valence-linked diagrams Kuo71; Kuo81. We include in the -box one- and two-body Goldstone diagrams through third order in . The folded-diagram series is summed up to all orders using the Lee-Suzuki iteration method Suzuki80.
The hamiltonian contains one-body contributions, whose collection is the so-called -box Shurpin83. As mentioned in the Introduction, in realistic shell-model calculations it is customary to use a subtraction procedure, so that only the two-body terms of , which make up the effective interaction , are retained while the SP energies are taken from experiment. In this work, we have adopted a different approach employing SP energies obtained from the -box calculation. In this regard, it is worth pointing out that, owing to the presence of the term in , -insertion diagrams arise in the -box. In our calculation we use the harmonic oscillator (HO) potential, , and take into account all -insertion diagrams up to third order. The oscillator parameter is MeV, according to the expression Blomqvist68 for .
Let us now outline the approach Bogner01; Bogner02 to the renormalization of . The repulsive core contained in is smoothed by integrating out the high-momentum modes of down to a certain cutoff . This integration is carried out with the requirement that the deuteron binding energy and phase shifts of are preserved by , which is achieved by the following -matrix equivalence approach. We start from the half-on-shell matrix for
where denotes the principal value and , and stand for the relative momenta. The effective low-momentum matrix is then defined by
where the intermediate state momentum is integrated from 0 to the momentum space cutoff and . The above matrices are required to satisfy the condition
The above equations define the effective low-momentum interaction , and it has been shown Bogner02 that their solution is provided by the KLR folded-diagram expansion Kuo71; Kuo90 mentioned before. In addition to the preservation of the half-on-shell matrix, which implies preservation of the phase shifts, this preserves the deuteron binding energy, since eigenvalues are preserved by the KLR effective interaction. For any value of , the can be calculated very accurately using iteration methods. Our calculation of is performed by employing the method proposed in Andreozzi96, which is based on the Lee-Suzuki similarity transformation Suzuki80.
As mentioned in the Introduction, our has been derived from the CD-Bonn potential with a cutoff momentum fm, and for protons the Coulomb force has been explicitly added to .
A discussion of the convergence properties of is now in order. For the sake of clarity, we first consider the two-body matrix elements (TBME) of and then the SP energies.
In Fig. 1, we report the diagonal TBME of as a function of the maximum allowed excitation energy of the intermediate states expressed in terms of the oscillator quanta . We have chosen the matrix elements because they are the largest ones in the entire set of TBME and play a key role in determining the relative spectra and ground-state (g.s.) properties of the even isotones. From Fig. 1 it is clear that our results have practically achieved convergence at .
As regards the order-by-order convergence, an estimate of the value to which the perturbative series should converge may be obtained by using Padé approximants. In Table 1, we compare all the -box TBME calculated at third order with those given by the Padé approximant Baker70
being the th-order contribution to the TBME in the linked-diagram expansion. From Table 1 we see that our third-order results are in good agreement with those from the Padé approximant, the differences being all in the order of few tens of keV with the exception of the diagonal matrix element for the configuration. This indicates a weak dependence of our results on higher-order -box perturbative terms.
In Fig. 2 we report the relative SP energies calculated at third order in as a function of . In the same figure, the results obtained by calculating the Padé approximant are also reported (red lines). Once again, the agreement between third-order results and those from this approximant points to a good perturbative behavior of the calculated relative SP energies. We also see that an calculation yields satisfactorily converged values for the relative SP energies.
A different situation occurs for the absolute SP energies. Actually, the calculated energy of the level relative to Sn is -4.468, -5.225, and -5.954 MeV for =11, 13, and 15, respectively. This shows that there is no sign of convergence of the absolute SP energies as a function of the number of intermediate states. Moreover, the Padé approximant of the energy for is equal to -7.262, showing that also the order-by-order convergence of the absolute SP energies is unsatisfactory. This poor perturbative behavior may be traced to the chosen value of the cutoff momentum, fm, which is substantially larger than the standard one, fm. However, the inaccuracy of the absolute SP energies affects only the g.s. energies of the nuclei considered. In fact, the calculated spectroscopic properties (energy spectra, electromagnetic transition rates) are certainly reliable, based on the good convergence of the relative SP energies and TBME.
Iii Results and discussion
We have performed calculations, using the Oslo shell-model code EngelandSMC, for the even-mass isotones up to Hf, which is the last known nucleus belonging to this chain, and for the odd-mass ones up to Ho, the last odd-mass nucleus with a well-established experimental low-lying energy spectrum. This is the longest isotonic chain approaching the proton drip line, and therefore provides an interesting laboratory to study the evolution of nuclear structure when adding pairs of identical particles.
|0.000 (-6.999)||0.000 (-9.663)|
In Table 2 our calculated SP energies are reported and compared with the experimental spectrum of Sb. We see that the latter is on the whole reasonably well reproduced by the theory. However, while the calculated positions of the and levels come very close to the experimental ones, the energy of the level is overestimated by about 0.85 MeV. As we shall discuss later in this section, the energy of the level, for which there is no experimental information, appears to be somewhat underestimated.
A strong test for our effective hamiltonian is given by the calculation of the energy spectrum of Te, since the theory of the effective interaction is tailored for systems with two valence nucleons. From Fig. 3, where the experimental nndc and calculated Te spectra are reported up to 3.5 MeV excitation energy, we see that a very good agreement is indeed obtained.
In Fig. 4, we show the calculated and experimental Audi03 g.s. energies (relative to the Sn core) per valence proton as a function of the number of valence particles of even-mass isotopes. We see that the experimental and theoretical curves are practically straight lines having the same slope, while being about 2.4 MeV apart. This discrepancy is essentially the same as that existing between the theoretical and experimental g.s. energies of Sb (see Table 2). This confirms the reliability of our SP spacings and TBME, since the pattern of the theoretical curve depends only on these quantities.
In Figs. 5, 6, and 7 we report, as a function of the mass number , the experimental and calculated excitation energies of the , , and yrast states, respectively. The experimental behavior is well reproduced, expecially for the and states, for which the discrepancies do not exceed 350 keV. However, it should be noted that according to our calculations the proton subshell closure, which experimentally occurs at Gd, is not reproduced. As mentioned before, this can be traced to the theoretical position of the orbital, which appears to be too low in energy. This is confirmed by the fact Andreozzi97 that this level has to be placed at 2.8 MeV in order to reproduce the experimental energy of the at 2.150 MeV in Cs, which is predominantly of SP nature Wildenthal71. We have verified that, if the SP level is placed at 2.8 MeV, we obtain that the excitation energy in Gd raises from 1.716 MeV to 1.976 MeV, while in Dy it decreases from 1.827 to 1.758 MeV, thus reproducing the observed subshell closure.
To have a more complete test of the theory, we have also calculated the transition rates up to Sm employing an effective operator obtained at third order in perturbation theory, consistently with the derivation of . Our results are reported and compared with the experimental data in Table 3. We see that the agreement is quite good, providing evidence for the reliability of our calculated effective operator which takes into account microscopically core-polarization effects.
Let us now come to the odd-mass isotones. In Figs. 8, 9, 10, and 11 we report, as a function of the mass number , the experimental and calculated energies of the , , , and yrast states relative to the yrast state, respectively. The experimental behavior is well reproduced for the , and states. For the latter, however, most of the calculated energies are higher than the experimental ones by about 300-400 keV, which reflects the theoretical overestimation of the value of the SP energy. Less satisfactory is the comparison between the calculated and experimental behavior of the yrast states, which is a further confirmation that the theoretical level in Sb lies too low in energy. As a matter of fact, by performing a calculation within the seniority scheme up to , we have found that in Cs and La the lowest states, which are quite well reproduced by the theory, have significant seniority components. From on, the experimental levels are predominantly of SP nature, as testified by the spectroscopic factors nndc, and the corresponding calculated states, which are dominated by components, underestimate their energies.
As pointed out in the Introduction, the main purpose of this study has been to test the use of a fully realistic shell-model hamiltonian for the description of medium-heavy mass nuclei. In this context, it should be noted that, aside from adopting experimental single-particle energies, earlier calculations on the N=82 isotones differ from the present ones in other respects, like the starting potential and the renormalization procedure. For instance, while we employ the CD-Bonn potential renormalized through the procedure, in the works of Refs. Holt97; Suhonen98 use is made of the Bonn A potential and the Brueckner -matrix approach. A comparison between the results of earlier works and the present one would not therefore be very meaningful. It is worth mentioning, however, that the agreement between experiment and theory achieved in this paper is on the whole comparable with that obtained in the previous calculations employing realistic effective interactions.
Iv Concluding remarks
As discussed in detail in the Introduction, the N=82 isotonic chain has long been considered a benchmark for shell-model calculations. This has resulted in a number of theoretical works which have practically all led to a good description of low-energy properties of these nuclei, evidencing the strong doubly magic character of Sn.
A main step towards a microscopic shell-model description of nuclear structure has been the use of two-body effective interactions derived from the free nucleon-nucleon potential. This approach, in which no adjustable parameter is involved in the calculation of the TBME, has been successfully applied to the N=82 isotones in the last decade, showing the ability of realistic effective interactions to provide an accurate description of nuclear structure properties.
However, some important theoretical questions still remain open. In particular, these concern the calculation of the single-particle energies and the role of three-body correlations. In the present paper, we have tried to investigate the former issue by constructing a realistic effective shell-model hamiltonian for the isotones, where both the SP energies and TBME have been obtained starting from a derived from the CD-Bonn potential. This is a natural extension of our studies of shell nuclei Coraggio05c and oxygen isotopes Coraggio07b. Here, we have shown that our effective hamiltonian yields results which are in quite good agreement with experiment along the whole chain of the isotones. As regards the calculated single-particle energies, significant discrepancies with the experimental values occur only for the and levels. Note that for the latter no direct comparison is possible since the single-particle state is still missing in the experimental spectrum of Sb. However, we have verified that increasing the calculated value by 0.8 MeV the proton subshell closure occurs at Gd, in agreement with the experimental findings.
At this point, the question naturally arises as to how contributions from three-body forces would modify the present results, in particular as regards the single-particle energies. While this remains a major task for future investigations, we feel that the present study paves the way toward a better understanding of the microscopic foundations of the nuclear shell model.