New USD Hamiltonians for the sd shell

Transcription

1 PHYSICAL REVIEW C, () New USD Hamiltonians for the sd shell B. Alex Brown and W. A. Richter Department of Physics and Astronomy, and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan -, USA Department of Physics, University of the Western Cape, Private Bag X, Bellville, South Africa (Received May ; published September ) We derive new Hamiltonians for the sd shell, USDA and USDB, based on a renormalized G matrix with linear combinations of two-body matrix elements adjusted to fit a complete set of data for experimental binding energies and excitation energies for the sd-shell nuclei. These Hamiltonians provide a new level of precision for realistic sd-shell wave functions for applications to nuclear structure and nuclear astrophysics. DOI:./PhysRevC.. PACS number(s):..cs,..dr,..+n I. INTRODUCTION The USD Hamiltonian [,] has provided realistic sd-shell (d /, d /, s / ) wave functions for use in nuclear structure models, nuclear spectroscopy, and nuclear astrophysics for over two decades. It is also an essential part of the Hamiltonian used for the p-sd [] and sd-pf [ ] model spaces. Results based on the USD Hamiltonian in include those in Refs. [ ]. The USD Hamiltonian is defined by sd-shell two-body matrix elements (TBME) and three single-particle energies (SPE) given in Table I of []. The values are derived from the renormalized G matrix with modifications needed to reproduce the experimental binding energies and excitation energies for nuclei in the region A =. In this paper we present results for new USD-type Hamiltonians called USDA and USDB based on an updated set of binding energies and energy levels. Theoretical methods for calculating the effective TBME with a renormalized G matrix starting from NN interactions have evolved considerably []. However, there are several reasons why the renormalized G-matrix sd-shell (RGSD) TBME may not be accurate: (i) The perturbation expansion may not be converged []; (ii) the oscillator basis used for matrix elements and energy denominators is an approximation; and (iii) real three-body forces are required as observed for ab initio calculations of light nuclei []. Three-body forces contribute to the effective one- and two-body matrix elements when they are averaged over the nucleons in the closed core, O in our case []. When applied to nuclei with three or more valence nucleons, both effective and real three-body interactions may play a role. Thus, to obtain realistic wave functions for medium-mass nuclei there is a need to modify the effective interactions based on constraints from experimental data. The resulting set of parameters for the effective interaction may be compared with the original G-matrix input to evaluate the contributions of higher order terms and three-body interactions. The USD Hamiltonian was the culmination of about a decade of work to understand the energy levels and spectroscopic properties for sd-shell nuclei in terms of a unified Hamiltonian applied in the full sd-shell model space. The earliest applications of shell-model configuration mixing to the sd-shell were limited by computational power and had to be applied to truncations within the sd-shell space. For example, in Arima et al. [] considered the properties of A = in the model space of [d /, s / ] n. The immediate predecessors to USD were the Chung-Wildenthal particle (CWP) and hole (CWH) Hamiltonians [] that were obtained from fits to data in the lower and upper parts of the sd shell, respectively. As computational power advanced it became possible in the late s to consider nuclei in the middle of the sd shell and eventually the CW Hamiltonians were merged into the universal sd (USD) Hamiltonian. The original USD Hamiltonian was obtained from a leastsquares fit of energy data with experimental errors of. MeV or less (with most experimental errors being kev or less) from nuclei. The root-mean-square (rms) deviation between experimental and theoretical energies was about kev (for energy data with uncertainties of kev or less). The number of states that were considered for each nucleus is shown in Fig.. The largest numbers of states are for those nuclei around N = Z = and N = Z =. Although many energy levels were known in the middle of the shell, only a few were included in the USD fit owing to the computational limitations in the s. Also, since there is much more and improved data for the neutron-rich nuclei. Today the computational effort is trivial on a desktop PC it is possible to obtain a complete set of low-lying energy levels for all sd-shell nuclei in a few hours. Thus we are motivated to refine the derivation of the USD Hamiltonian with an updated and complete set of energy data. We are able to consider states in nuclei distributed over sd-shell nuclei as shown in Fig.. The new Hamiltonians USDA and USDB lead to a new level of precision for realistic shell-model wave functions. In the next section we discuss experimental data. In Sec. III we review the linear-combination method for the least-squares fit. In Sec. IV we present the results for the new Hamiltonians, with results for binding energies and energy levels in Sec. V. II. EXPERIMENTAL DATA The data for the new fits was obtained from the most recent compilations. In the case of A = nuclei most data for excited states are from the Supplement to Energy Levels of -//()/() - The American Physical Society

2 B. ALEX BROWN AND W. A. RICHTER PHYSICAL REVIEW C, () Proton Number Neutron Number FIG.. (Color) Number of states used for the USD Hamiltonian for each nucleus. A = nuclei by P. M. Endt (published in []), used in conjunction with the previous complete review of nuclei in this mass range []. The supplement mainly discusses new data published in the period. A useful part of Endt s supplement is the inclusion of comparisons of positive-parity states with the USD predictions. More recent data are used for O[,], O[], O[], O[], F, F[], F[,], F and F[], F[], Ne [], Na [], Na and Na [], Mg [], and Si []. The ground-state binding energies used are from the atomic mass evaluation of Audi, Wapstra, and Thibault []. These are supplemented with new results from [] for O, F, O, F, F, F, Ne, Ne, Ne, Ne, Na, and Mg. Proton Number Neutron Number FIG.. (Color) Number of states used for the USDA and USDB Hamiltonians for each nucleus. We take the binding energies relative to that of O with the Coulomb energy correction used for USD [], BE(A, Z) r = BE(A, Z) BE( O) E C (Z), () with E C (Z) =. (Z = ),. (Z = ),. (Z = ),. (Z = ),. (Z = ),. (Z = ),. (Z = ),. (Z = ),. (Z = ),. (Z = ), and. (Z = ) MeV. The Coulomb energy corrections were obtained from energy differences of isobaric analog states for the nuclei near N = Z. For the new Hamiltonian we consider states in nuclei. The distribution of these states over the sd-shell nuclei is shown in Fig. Most of the new data have been added in the middle of the sd shell (where computations are now possible) and for neutron-rich nuclei. (We include in this figure the ground states for six nuclei in the island of inversion N = and Z =, although as discussed in the following they are not included in the fit.) III. FITTING PROCEDURE The shell-model effective Hamiltonian can be written as a sum of one- and two-body operators (using similar notation to that used by Honma et al. []forthepf shell, which is based on the work of Chung and Wildenthal [,]forthesd shell): H = ɛ a ˆn n + V JT (ab; cd) ˆT JT (ab; cd), () a a b,c d JT where ˆn a is the number operator for the spherical orbit a with quantum numbers (n a,l a,j a ) and ˆT JT (ab; cd) = MTz A JMTT z (ab)a JMTTz (cd) () is the scalar two-body density operator for nucleon pairs in orbits a,b and c, d coupled to spin quantum numbers JM and isospin quantum numbers TT z. We use a simplified notation H = p x i O i, () i= where the x i stand for single-particle energies ɛ a or the two-body matrix elements V JT (ab; cd), and the operators O i stand for ˆn or ˆT, respectively. The vector x = (x,x,,x p ) defines the Hamiltonian. This Hamiltonian will have eigenvectors φ k and eigenvalues λ k that can be expressed in terms of a linear combination of the Hamiltonian x : p p λ k = φ k H φ k = x i φ k O i φ k = x i βi k, () i= where βi k = φ k O i φ k. For a given starting Hamiltonian x s, we calculate βi k and then minimize the quantity ( ) N E k χ exp λ k =, () k= σ k exp i= -

3 NEW USD HAMILTONIANS FOR THE sd SHELL PHYSICAL REVIEW C, () where Eexp k are the experimental energies and σ exp k are the associated errors. This gives ( ) ( ) χ N E k exp λ k p = ( ) x j σ k x i βi k =. () x j exp k= Without taking into account the implicit dependence of βi k on x j through φ i we obtain p linear equations for an improved Hamiltonian x : ( ) N p β k Eexp k x i βi k j p ( ) σ k = e j γ ij x i =, () exp k= i= where j =,,,p. This can be written in terms of a p p matrix N βi k G = (γ ij ) = βk j ( ) σ k () exp and a p-dimensional vector e = (e i ) = k= N k= i= i= E k exp βk i ( σ k exp ) () as G x = e. () Since G is a real symmetric matrix, it can be inverted to obtain the new Hamiltonian x = G e. () To take into account the implicit dependence of βi k on x j this procedure is repeated (iterated) until convergence. The matrix G is referred to as the error matrix [] since the diagonal elements are the square of the parameter errors and the off-diagonal matrix elements are related to the correlations between the parameters. As in most multiparameter fits, the resulting parameters x i are highly correlated. In addition, the low-lying nuclear states are much more sensitive to some linear combinations of Hamiltonian parameters than others. These issues can be quantified by diagonalizing the matrix G : D = AGA T, () or in terms of the error matrix, D = AG A T, () where D is a p-dimensional diagonal matrix with positive elements D i. The eigenvalues of the error matrix D are d i = /D i. With this definition for A, Eq.() is equivalent to D y = c, where y = A x and c = A e, which has the simple solution y i = c i d i. () The least-squares fit is thus reformulated in terms of uncorrelated linear combinations. The orthogonal parameters y i are linear combinations of the Hamiltonian parameters x i, with associated errors d i. It is clear from Eq. () that, for large d i, the values of y i are strongly affected by a small change in the data c i. Therefore, such linear combinations are only poorly determined by a given set of data. We can separate poorly determined linear combinations from well-determined ones by setting a certain criterion on the magnitude of the corresponding eigenvalues d i. The fitting procedure can be modified as follows. We start with the best available Hamiltonian x s. From the fit, mutually independent linear combinations y i are determined according to Eq. (). At the same time linear combinations of the starting Hamiltonian x s are also obtained from y s = A x s. Then new linear combinations y a are defined by adopting only welldetermined values of y i and using the starting values for the rest, yi a = y i (d i δ) + yi s (d i >δ), () where δ is taken to be a suitable value for the criterion. The number of well-determined linear combinations is denoted by N d. A new Hamiltonian is obtained from x a = A y a, which is used for the next iteration to obtain x b = A y b from yi b = y i (d i δ) + yi s (d i >δ). () This procedure is iterated until convergence. This linear-combination (LC) method for finding the welldetermined uncorrelated linear combinations of Hamiltonian parameters was used by Arima et al. [] forthed / -s / model space and by Macfarlane for the p shell []. The LC method together with the iteration method was used by Chung and Wildenthal to determine the original USD Hamiltonian [,], which was the starting point of the present work. The LC method was used in the pf shell to obtain the FPMI [] and GXFP [,] Hamiltonians. More recently, it has been used in for the (f /,p /,p /,g / ) model-space description of the Ni isotopes and N = isotones []. IV. RESULTS FOR THE FIT The calculations for the wave functions, energies, occupation numbers, and scalar two-body transition densities were carried out with OXBASH []. One iteration took about hours on a desktop PC. For the data set we consider all ground-state binding energies and all positive-parity energy levels for sd-shell nuclei. The first criterion for inclusion of the nth J π level in the fit is whether or not the J π for all lower states are known. Generally this means that we cannot consider states above the energy where one level has an unknown J π.this usually occurs starting at E x = MeV. The second criterion is the energy at which the experimental level density for agivenj π becomes suddenly higher than the theoretical level density. This is a signature of intruder states. There is a well-defined region of nuclei with N = and Z = where the difference between the experimental and theoretical ground-state binding energies is much larger than the rms average. These nuclei are in the island of inversion that requires the explicit extension of the pf -shell orbits to the model space []. Ground and excited states for these six nuclei are not included in the fit. In the end we will show the comparison of all experimental levels to all theoretical levels. With this selection we are able to consider states in nuclei with errors of less than. MeV (with most experimental errors being kev or less). The uncertainty used in Eq. () is the experimental uncertainty σ exp folded -

4 B. ALEX BROWN AND W. A. RICHTER PHYSICAL REVIEW C, () quadratically with a theoretical error that is close to the rms value obtained in the best fit for which we take σ th =.MeV: (σ k ) = ( σexp k ) ( ) + σ k. th () We include ground-state binding energies in the fit directly in terms of their calculated E and β values. Excited states are included in the fit by taking the differences E E gs and β β gs. This allows for states in exotic nuclei to be included where the E E gs is known more accurately (for example, from γ -ray transitions) than E gs. As was done for USD, the SPE are taken to be constant (mass independent). One could add some mass dependence to the SPE, but it has little effect on the rms since it can be compensated by a change in the TBME (in particular in the monopole TBME combinations). One can define the effective SPE by the addition of the monopole linear combinations of the TBME [] tothe original SPE. The effective SPE are well defined by the data, but the division into their separate SPE and monopole TBME parts cannot be made on the basis of sd-shell data (i.e., it must come from a theoretical model). As was done for USD, we employ a mass dependence of the two-body matrix elements of the form ( ) p V JT (ab; cd)(a) = V JT(ab; cd)(a = ), () A with p =.. This accounts qualitatively for the mass dependence expected from the evaluation of a medium-range interaction with harmonic-oscillator radial wave functions. With hω = A /, the matrix elements for a delta-function interaction scale analytically as A / and the matrix elements of the Coulomb interaction scale analytically as A /.The rms of the fit has a shallow minimum near p =.. The minimum is shallow because the mass dependence can be partly compensated by a change in the fitted values of the TBME. The poorly determined linear combinations of Hamiltonian parameters were always constrained to the values obtained from the renormalized G matrix applied to the sd shell (RGSD). The calculations were based on the Bonn-A NN potential and include diagrams up to third order as well as folded diagrams. The TBME are given in Table of []. For the present work the original USD Hamiltonian was used for the first iteration. Then the equations were iterated with N d = well-determined (varied) linear combinations with the remaining poorly determined linear combinations set to the RGSD values. This was continued until the energies and x values converged to the level of about kev. This rather highly constrained Hamiltonian is called USDA. In Fig. we show the eigenvalues D i of the fit matrix for USDA. A physically interesting quantity from the fit is the rms deviation between experimental and theoretical energies, rms = N ( ) E k. N exp Eth k () k= In Fig. we show this rms deviation (solid line) as a function of the number of varied linear combinations N d TABLE I. Comparison of sd shell TBME for T = (in MeV). v(a,b,c,d; JT) = V JT (ab; cd)(a = ). The orbits are labeled by = s /,= d /,and= d /. Matrix element USD RGSD USDA USDB v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... v(;).... obtained for the final iteration of USDA (with the actual USDA corresponding to the results obtained at N d = ). The individual contributions to Eq. () are shown in Fig. The rms deviation of the fitted and RGSD two-body matrix elements as a function of N d are shown by the points connected by a dashed line in Fig. The motivation for linear combinations can be seen in Fig. One observes a plateau in the rms energy deviation of about kev between and linear combinations. Beyond this there is a gradual drop until about linear combinations with an rms deviation of kev that does not significantly decrease going out to the full set of combinations. Thus we also derive another Hamiltonian called USDB by varying linear combinations of parameters. Three more iterations were required for convergence. As seen in the histogram in the upper right-hand corners of Figs. and the scatter between experiment and theory for USDB is systematically reduced compared to that for USDA. We include in Figs. and six data points not included in the fit those for the ground states of, Ne,, Na, and, Mg to show the large deviation for these nuclei in the island of inversion. The TBME for A = are given in Tables I and II for T = and T =, respectively. The TBME for other A values are given by Eq. (). The single- -

5 NEW USD HAMILTONIANS FOR THE sd SHELL PHYSICAL REVIEW C, () TABLE II. Comparison of sd shell TBME for T = (in MeV). v(a,b,c,d; JT) = V JT (ab; cd)(a = ). The orbits are labeled by = s /,= d /,and= d /. Matrix element USD RGSD USDA USDB v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... v( ; ).... eigenvalue linear combination number FIG.. Eigenvalues D i of the fit matrix. Thus we have two new Hamiltonians: a conservative one, USDA, which is closest to RGSD and gives a good but not the best fit to the data, and another, USDB, that differs more from RGSD but gives a best fit to the data. Calculation of other observables such as spectroscopic factors, moments, and γ - and β-decay rates with both USDA and USDB will give an estimate of the theoretical errors attributable to the sd-shell Hamiltonian as well as determine whether or not USDB is in fact superior to USDA. The original USD and RGSD TBME are compared on the right-hand side of Fig.. USD deviates the most from RGSD with a -kev rms in the TBME perhaps because of the smaller data set used for its determination.. particle energies for (d /, s /, d / )are(.,.,.) MeV for USD, (.,.,.) MeV for USDA, and (.,.,.) MeV for USDB. The fitted and input RGSD (G matrix) TBME are shown in Fig.. The USDA TBME are similar to the RGSD with an rms deviation between the TBME of kev (see Fig. ). Compared to the maximum magnitude of MeV for the TBME, this represents only a % rms difference. However, this specific % change is critical for obtaining accurate binding energies and spectra. The USDB Hamiltonian, where parameters are varied, is compared with RGSD in middle panel of Fig. ; the result is a -kev rms difference for the TBME (% of the largest). When the rms sum in Eq. () is restricted to the original set of energy data used for the original USD interaction ( data points with σ k of kev or less) the result is and kev for USDA and USDB, respectively. Thus the added data do not worsen the quality of the fit, but they do provide more constraints on the determination of the TBME. rms deviation (MeV) number of adjusted linear combinations FIG.. The rms deviations as a function of the number of fitted linear combinations for the final USDA iteration. The points connected by a line show the deviation between experimental and theoretical sd-shell energies. The points connected by a dashed line show the rms deviation between the two-body matrix elements for the USD and RGSD Hamiltonians. -

6 B. ALEX BROWN AND W. A. RICHTER PHYSICAL REVIEW C, () USDA USDB deviation (MeV) -.. deviation (MeV) -.. O F Ne Na Mg Al Si P S Cl O F Ne Na Mg Al Si P S Cl - - deviation (MeV) deviation (MeV) - data number FIG.. Deviations between experimental and theoretical energies for USDA. The ground-state binding energies are plotted on the top panel and the excitation energies on the bottom. The inset shows a histogram of the deviations binned over -kev intervals. The monopole linear combinations of TBME are among the most important for the fit: (J + )V JT (ab; ab) V ab,t = J (J + ) J. () They combine with the SPE to determine how the effective SPE evolve as a function of N and Z. The monopole combinations for the various interactions are compared in Fig.. The values for USDA and USDB are essentially the same. They are also close to the original USD except for the s / -s / and d / -d / T = terms being kev smaller for USD. The T = monopole terms are large and attractive with a general agreement between USD and RGSD in the overall pattern. The d / -d / T = monopole term is the largest. This term with its large tensor interaction component is responsible for the change in shell structure as a function of N and Z in the sd shell []. This strong tensor contribution is important for the changes in shell structure for all nuclei [,]. All of the USD Hamiltonians deviate from the RGSD values by as much as MeV for both T = and T =. For USDB the A = single-particle energies are.,., and. MeV for d /, s /, and d /, respectively, to be compared with experimental values of. and. for d / and s /.Thed / particle strength in O lies above the neutron-decay threshold and can only be observed in - data number FIG.. Same as Fig. except for the USDB Hamiltonian. particle-transfer reactions. The O(d,p) O reaction has only been analyzed [] for the lowest and strongest / + state at. MeV corresponding to a d / SPE of.. =. MeV. There are states seen at higher energy in these reactions that have not been analyzed that could contain fragmented d / strength, and the spectra above MeV were not measured. Thus the position of the centroid d / strength in O is not known. For USDB the A = single-hole energies, [BE( Ca) BE( Ca)], are.,., and. MeV for d /, s /, and d / respectively, to be compared with experimental values of. and. for d / and s /.Thed / particle strength in K is fragmented over many states in the energy range of. MeV [] with a centroid value of about. MeV corresponding to a neutron-hole SPE of.. =. MeV. Experiment and calculation are in good agreement. RGSD TBME (MeV) USDA TBME (MeV) USDB TBME (MeV) USD TBME (MeV) FIG.. Comparison of the fitted and RGSD TBME. -

7 NEW USD HAMILTONIANS FOR THE sd SHELL PHYSICAL REVIEW C, () monopole interaction (MeV) T= j j = T= G matrix USD USDA USDB Proton Number FIG.. The sd-shell monopole interactions. The solid lines are for USDA and USDB, the crosses are for USD, and filled circles for the RGSD. The USDB neutron-hole SPE for O (an unbound nucleus) are.,., and. MeV for d /, s /, and d /, respectively. The positive value for the highest hole state gives the marginally unbound nature of O. Note that the spin-orbit splitting is still large in this very neutron-rich nucleus. Also note the large SPE gaps for both d / -s / and d / -d / that are related to the observed shell gaps in the doubly-magic nuclei O and O, respectively []. The USD proton particle SPE for O [BE( F) BE( O)] are.,., and. MeV for d /, s /, and d /, respectively (which includes the Coulomb correction of. MeV). The protons are of course deeply bound in O. The stability of F is determined by the neutron separation energy. With USDB, Fis.MeVmore bound than F, in agreement with its observed stability. The experimental binding energy difference is. ±. MeV. (The large error in F and F precludes them from having any influence in the fit and it is not clear whether these nuclei are in the island of inversion.) V. RESULTS FOR BINDING ENERGIES AND SPECTRA For the binding energies, it is instructive to compare the results of the original USD with the updated set of data as shown in Fig.. Most of the differences lie in the nominal rms range of kev. The binding energies predicted for the neutron-rich fluorine (Z = ) isotopes turn out to be larger than experiment by up to. MeV. This indicates that the USD TBME involving the d / orbital in this mass region are incorrect. The oxygen isotopes (Z = ) beyond N = are known to be unbound [ ] and the neutron-decay properties are not yet measured. The USD predicts that these nuclei are unbound except for O, which is bound by. MeV with USD. Thus both neutron-rich oxygen and fluorine are too tightly bound with USD. In contrast, the binding energies predicted for Z = and N = are Neutron Number FIG.. (Color) Difference between the experimental and theoretical (USD) ground-state binding energies. A positive value indicates that experiment is more bound than theory. larger than experiment by up to. MeV. This indicates that the USD TBME are incorrect or that other shell-model configurations dominate the wave functions. The binding energy difference for USDB is shown in Fig. (The results for USDA are similar.) The fluorine problem for USD is corrected by the new USDA and USDB Hamiltonians. In addition, all of the oxygen isotopes are unbound with USDA and USDB. However, the Z =, N = difference cannot be corrected, confirming the intruder state (island-of-inversion) interpretation for these nuclei. Although, F appear to lie outside of the island of inversion, they Proton Number Neutron Number FIG.. (Color) Difference between the experimental and theoretical (USDB) ground-state binding energies. A positive value indicates that experiment is more bound than theory. -

8 B. ALEX BROWN AND W. A. RICHTER PHYSICAL REVIEW C, () -. experiment Al -. theory -. experiment Mg -. theory FIG.. Comparison of experimental and USDB theoretical levels (T = ) for Al. See the text for details. have large experimental errors (.. MeV) and they were thus not included in the fit. The improved USDA and USDB results for the ground-state binding energies of the neutron-rich fluorine and oxygen isotopes with N are related to the increased energy for the effective d / SPE. We note that this particular problem with USD was corrected when it was applied to the sd-pf model space []. Spectra were calculated for all sd-shell nuclei for both USDA and USDB and compared to experiment. The compete set of comparisons can be found on the Web []. As an example, we show the results for A =. The spectrum for the positive-parity states in Al with T = isshownin Fig.. Experiment is shown on the left-hand side with lines that indicate the J value for the known positive-parity states. The lowest J value is labeled. Levels with an unknown J π assignment are shown by the shortest lines, with the first such level for Al lying at about MeV. The theoretical levels are shown on the right-hand side, also indicated by lines of different length for the J value. The experimental and theoretical levels joined by a line in the middle are those included in the fit. The slope of these lines shows the difference between experiment and theory for the excitation energy. The ground-state binding energies are shown at the bottom. Starting at about MeV we cannot make a level to level assignment between theory and experiment owing to the incomplete experimental information. From MeV the experimental and theoretical level densities are similar; but there are too many levels of unknown spin-parity to make definitive associations between experiment and theory. Above MeV in Al the experimental level density becomes higher, indicating the onset of intruder states (e.g., wave functions that contain the excitation of two or more nucleons out of the p shell or into the pf shell). Generally toward the beginning and end of the sd shell where the sd-shell level density is smaller, the intruder states are more obvious in the low-lying spectra. For example, in Otwo + states appear below MeV compared to only one in the theory. The structure of this extra + in terms of p-shell excitations is FIG.. Comparison of experimental and USDB theoretical levels for Mg. See the text for details. well known []. In the upper part of the sd shell we find for example the lowest + state in Si experimentally at. MeV and theoretically at. MeV. The lowest + state is interpreted as the neutron p-h configuration that becomes the ground state in the island-of-inversion nuclei Mg and Ne []. The spectrum for Mg(showninFig.) is another nucleus for which there is much well-established experimental data. All positive-parity levels are well matched between experiment and theory up to. MeV. Above this point there are levels with unknown spin and only a few high-spin levels can be matched. Above about. MeV the experimental level density becomes higher than that of theory. For Na (shown in Fig. ) there are few experimental data with spin assignments to compare to theory. Recent results for Na have been obtained from the C( C,d) -. experiment Na -. theory FIG.. Comparison of experimental and USDB theoretical levels for Na. The theoretical ground state is + with the first excited + at kev. See the text for details. -

9 NEW USD HAMILTONIANS FOR THE sd SHELL PHYSICAL REVIEW C, () TABLE III. Gamow-Teller (GT) β-decay properties for Ne ( + ) Na ( + ). The B(GT) values include an overall quenching factor of. []. Experiment USD USDA USDB E x (MeV) B(GT) E x (MeV) B(GT) E x (MeV) B(GT) E x (MeV) B(GT)..() () () reaction []. The low-lying quadruplet is now established with the fourth state being +, in agreement with theory. In the region MeV levels are predicted and levels are seen in experiment, but still most levels do not have an experimental spin assignment. The highest spin of + predicted at. MeV is seen at. MeV in experiment. The + states can be assigned from the β decay of Ne; the results are shown in Table III. It was noted [,] that the USD Hamiltonian predicted that four + states should be seen in the β decay of Ne [], but only three were observed in experiment []. The USDA and USDB results in Table III show that the fourth + state has a very small B(GT) value (much smaller than the USD value), indicating that it would be too weak to be observed. As noted in [] several levels in the region of MeV have γ -decay properties consistent with spin-parity +. A future project will be to apply the new Hamiltonians to the γ -decay properties of Na as well as other sd-shell nuclei. (The + states from [] were not included in the fit owing to the disagreement in the number of + states that should be observed.) For Ne (shown in Fig. ) there are little experimental data. The ground-state and first excited + state energies are in good agreement with theory. The + state found in experiment at. MeV is significantly lower than the theoretical state at. (.) MeV with USDA (USDB), indicating that this is an intruder state perhaps associated with the (sd) n (pf ) configuration that becomes the ground-state configuration in Ne. The transitional role of Ne of having nearly degenerate (sd) n and (sd) n (pf ) configurations has been noted [], but the experimental ground-state binding energy for Ne turns out to be in good agreement with all of the USD Hamiltonians. For F (shown in Fig. ) the ground-state spin and energy are in agreement with theory but nothing is known about excited states. There are two levels below the neutron separation energy of MeV. The + level should be isomeric and may β decay. The γ decay of the + level at. MeV (with USDB) to the ground state may correspond to the. MeV γ -ray decay reported in []. Levels for Oare shown in Fig.. The ground state of O is bound to one-neutron decay by. (.) MeV with USDA (USDB), but it is unbound to two-neutron decay by. (.) MeV, in agreement with the experimental observation that it is unstable [,]. This unique situation for a dineutron decay would be interesting to observe experimentally. The predicted neutron-decay properties of O will be the subject of a separate paper. For neutron-rich nuclei with Z =, the increase in the d / effective single-particle energy needed to improve the ground-state properties of nuclei with N is reflected in improvements in the spectra for excited states related to -. experiment Ne -. theory -. F -. experiment theory FIG.. Comparison of experimental and USDB theoretical levels for Ne. See the text for details. FIG.. Comparison of experimental and USDB theoretical levels for F. See the text for details. -

10 B. ALEX BROWN AND W. A. RICHTER PHYSICAL REVIEW C, () O -. experiment theory FIG.. Comparison of experimental and USDB theoretical levels for O. See the text for details. the d / orbit in nuclei with N. For example, the doublet of states near MeV in Ne is suggested to be on the order / + -/ + and is incorrect with the original USD [], but is now corrected with USDA and USDB (see []). Excited states of Ne were not included in the fit because the spin assignments are uncertain and because the recent experimental work [] was published after the new fit was completed. VI. SUMMARY AND CONCLUSIONS The systematic application of USDA and USDB to other observables such as moments, electromagnetic decay, and β decay will be carried out over the foreseeable future. As a consequence we will gain a better understanding of the precision and limitation of this approach to nuclear structure. We should have more precise theoretical input for astrophysical applications such as those for electron-capture [] and proton-capture cross sections [] where experimental data cannot be obtained. For this study we have used the renormalized G matrix from the Bonn-A potential as given in Table of []. The results with Bonn-B and Bonn-C, also given in this table, show that there is some sensitivity to the NN interaction. We will explore which of the modern NN interactions and renormalization methods provide the best starting point for USD-type Hamiltonians. From the introduction we repeat the reasons why the renormalized G-matrix sd-shell TBME may not be adequate: (i) The perturbation expansion may not be converged []; (ii) the oscillator basis used for matrix elements and energy denominators is an approximation; and (iii) real three-body forces are required as observed for ab initio calculations of light nuclei []. For (i) we may be able to apply the ab initio approaches used for lighter nuclei via a similarity transformation to evaluate an effective sd-shell Hamiltonian for A =. For (ii) we should explore the use of Hartree-Fock basis states in place of the harmonic oscillator for evaluation of the renormalized G matrix. For (iii) we need to find the contribution from realistic three-body forces to the effective two-body matrix elements for A =. The most important deviations to understand are the monopole interaction terms shown in Fig.. Real and effective three-body interactions in principle require an explicit three-body interaction operator for three or more valence particles. However, the average effects of these three-body terms can be evaluated from an interaction calculated for two holes in the closed shell of Ca (taking into account the three-body interaction explicitly). The USD Hamiltonians should be regarded as an average of the ab initio interactions for two particles and two holes. Eventually we may understand how much of the remaining - to -kev deviation between experiment and theory can be understood in terms of an explicit evaluation of the threebody Hamiltonian. With our updated database, new ideas can be readily incorporated into new versions of the USD-type Hamiltonians that are constrained by experimental data. ACKNOWLEDGMENTS We thank Dr. Honma for his computer program for the linear combination method fits. Support for this work was provided from U.S. National Science Foundation Grant Nos. PHY- and PHY- and by South African NRF ISL Grant GUN No.. This work is partly supported by the National Research Foundation of South Africa under Grant No.. [] B. H. Wildenthal, Prog. Part. Nucl. Phys., (). [] B. A. Brown and B. H. Wildenthal, Annu. Rev. Nucl. Part. Sci., (). [] E.K.Warburton andb.a.brown,phys.rev. C, (). [] E. K. Warburton, J. A. Becker, and B. A. Brown, Phys. Rev. C, (). [] Y. Utsuno, T. Otsuka, T. Mizusaki, and M. Honma, Phys. Rev. C, (). [] S. Nummela et al., Phys. Rev. C, (). [] A. Heger et al., Phys. Lett. B, (). [] K. Kaneko et al., Phys. Rev. C, (). [] P. Mason et al., Phys. Rev. C, (). [] M. Hasegawa and K. Kaneko, Nucl. Phys. A, (). [] A. Volya and V. Zelevinsky, Phys. Rev. Lett., (). [] V. Velazquez, J. G. Hirsch, A. Frank, J. Barea, and A. P. Zuker, Phys. Lett. B, (). [] Y. Parpottas et al., Phys. Rev. C, (). [] B. Davids et al., Phys.Rev. Lett., (), and references therein. [] V. Tripathi et al., Phys. Rev. Lett., (). [] W. Geithner et al., Phys. Rev. C, (). [] J. Y. Moon et al., Nucl. Phys. A, c (). [] R. G. T. Zegers et al., Nucl. Phys. A, c (). [] L. Weissman et al.,j.phys.g, (). -

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