During the period of our last NSF grant, we have developed and implemented 9methods for exact solution of the nuclear shell model based on Monte Carlo techniques. Shell Model Monte Carlo (SMMC) methods have overcome the severe limitations of the conventional diagonalization techniques and now allow complete 0 calculations with realistic interactions for nuclei, a task which appeared impossible only a couple of years ago. During the last year we have applied the SSMC method to several interesting and important problems, many of them representing ``first-of'' calculations. A Physics Reports, which reviews and summarizes the method and highlights its applications, is nearly completed.
We systematically study the gross properties of even-even and N=Z nuclei with A=48-64 using SMMC methods. Our calculations account for all 0 omega configurations in the pf-shell and employ the modified Kuo-Brown interaction KB3. We find good agreement with data for masses and total B(E2) strengths, the latter employing effective charges e_p=1.35e and e_n=0.35e. The calculated total Gamow-Teller strengths agree consistently with the B(GT_+)-values deduced from (n,p) data if the shell model results are renormalized by 0.64, as has already been established for sd-shell nuclei. The present calculations therefore suggest that this renormalization (often related to g_A=1 in the nuclear medium) is universal. With the conventional renormalization of the spin operator, our calculation also reproduces the relative B(M1) strengths for those nuclei in this mass range for which data exist. The SMMC approach has been benchmarked against the results of conventional diagonalization calculations for nuclei with A=48-52. A manuscript discussing this work will be published in the August 1995 issue of the Physical Review C.
We have used the SMMC approach to study the thermal properties of ^54Fe with the Brown-Richter interaction within the complete pf model space. We observe that the BCS-like proton and neutron pairing gaps, which determine the properties of even-even nuclei in the ground state and at low excitations, vanish in a rather small temperature interval near T = 1 MeV. Related to this pairing phase transition, the moment of inertia of the nucleus increases drastically, while the B(M1) strength partially unquenches. It is noteworthy that the total Gamow-Teller strength remains strongly quenched, even at temperatures above the phase transition. We have interpreted this behaviour as evidence that the isoscalar proton-neutron correlation in nuclei persist to higher temperatures. This conjecture has been confirmed in detailed subsequent studies (see II.3.a8). This work has been published in Physical Review Letters 74 (1995) 2909.
Donati et al. (Phys. Rev. Lett. 72 (1994) 2835) have proposed that the nuclear symmetry energy increases with temperature. If correct, electron capture in a supernova would be increasingly hindered with increasing temperature, resulting in a larger electron-to-baryon ratio and a larger mass of the homologous core than is commonly believed. As a consequence, the shock wave, formed at the surface of the homologous core, would need less energy to explode the star.
We have tested this conjecture in a series of SMMC studies in the iron-nickel mass region. If shell and pairing effects can be ignored, the empirical Bethe-Weizsäcker formula suggests that an increase of the symmetry energy with temperature should be reflected in an increase of the difference of excitation energies between isobars with temperature. We have therefore studied the difference of excitation energies for several pairs of even-even isobars as a function of temperature. These calculations do not confirm a systematic increase of the symmetry energy with temperature. A manuscript discussing this work has been accepted for publication in Physics Letters B.
We have performed SMMC calculations for nuclei with A 100-140 within the full 50-82 shell for both protons and neutrons. The interaction is determined solely by self-consistency and odd-even mass differences, and includes both monopole and quadrupole pairing and the collective quadrupole interaction. The methods are illustrated for ^124Sn, ^128Te and ^124Xe. We calculate shape distributions, moments of inertia, and pairing correlations as functions of temperature and angular velocity. Our calculation is the first microscopic evidence of the softness of nuclei in this region and gives the correct systematics of excitation energies, deformations, and E2 intensities. Interpretation within the context of the Interacting Boson Model indicates that the protons in ^124Xe tend towards an O(6) symmetry, while the neutrons are intermediate between O(6) and SU(3). A manuscript describing this work has been submitted to Physical Review Letters.
A better understanding of the quenching of the GT strength might be had from studies of the double-magic nucleus ^100Sn, which has recently been observed for the first time at GSI and Ganil. As the large shell gap between the occupied (g_9/2)-orbital and the other orbitals in the (0 g-1d-2s) shell (Delta E 6 MeV) suppresses correlations involving the higher orbitals, there should be little hindrance of the transformation of valence (g_9/2)-protons into (g_7/2)-neutrons by the GT_+ operator. Thus, there should be little quenching of the GT_+-strength.
While a measurement for ^100Sn has not been feasible so far, a surprisingly large and as yet theoretically unexplained strong quenching of the GT_+ strength has been observed in neighbouring N=50 isotones. The SMMC allows now, for the first time, calculations of the GT strength for the N=50 isotones in the complete (0g-1d-2s) shell. Preliminary results, obtained with a (0g-1d-2s) interaction supplied by T. Kuo, indicate quenching factors (q=4.8.3, 3.5.1, 2.64.1, 3.66.1, 2.71.1 for ^94Ru, ^96Pd, ^96Cd, ^98Cd, and ^100Sn), which are significantly larger than in truncated shell model or QRPA calculations. Our results are to be compared with the experimental quenching factors of q=7.2^+0.6_-0.7, 4.6^+1.7_-1.2, and 4.1^+1.0_-0.8 for ^94Ru, ^96Pd and ^98Cd. Future SMMC calculations will explore the interaction dependence of these quenching factors.
The shape of a nucleus is strongly dependent on its temperature, rotational frequency, and mass. In particular, nuclei in the rare-earth region exhibit interesting shape fluctuations as one varies the above quantities. Although several studies of the rare-earth nuclei have been performed using mean-field techniques or within the Interacting Boson Model, the SMMC allows microscopic investigations of the interplay between shape fluctuations and the residual interaction. Our calculations are performed within the complete (2s1d0g_7/20h_11/2) and (2p1f0g_9/2i_13/2) shells for protons and neutrons, respectively, using a residual interaction of pairing-plus-quadrupole form. A pilot study of the mid-shell nucleus ^170Dy revealed the particularly important effect of pairing on the various shape transitions (Phys. Lett. B317, 275 (1993)). We are currently extending this study to the other even-even Dy-isotopes, systematically exploring the ground state shapes along this isotope chain. Future SMMC calculations will investigate shape transitions in these isotopes as a function of temperature and rotational frequency. This work will form the basis for a Ph.D. thesis of J. White.
The second-order weak process (Z,A)(Z+2,A)+2e^-+2 is an important ``background'' to searches for the lepton-number violating neutrinoless mode, (Z,A)(Z+2,A)+2e^-. The calculation of the nuclear matrix element for these two processes is a challenging problem in nuclear structure, and has been done in a full 0 model space for only the lightest of several candidates, ^48Ca.
SMMC methods are being applied to calculate 2\; beta beta matrix elements in very large model spaces. The required matrix elements can be derived from observables like
F(;)= Tr[e^-(--)H
G^ G^ e^- HGe^-HG]
Tre^- H\;,
where G is the Gamow-Teller beta-decay operator. When beta and tau are large, setting tau_1=0 gives the nuclear matrix element involved in the usual closure approximation, while integrating over tau_1 gives the required intermediate-state energy denominator and hence the exact matrix element.
For both the exact matrix element and the closure approximation, our method has been benchmarked successfully against direct diagonalization results for ^48Ca. We are currently calculating the 2nu\; beta beta matrix element for ^76Ge, which has the strongest experimental constraints on this rare decay mode. Our calculations are performed within the complete (1pf_5/2g_9/2) model space and employ a residual interaction supplied by T. Kuo. In closure approximation, our calculated matrix elements is significantly smaller than results obtained previously in a truncated shell model calculation. Preliminary results for the exact matrix elements are encouraging. A manuscript discussing our calculations is in preparation for Physical Review Letters. This work will form the basis for a Ph.D. thesis by P. B. Radha, expected in June, 1996.
A creation operator for a proton pair with angular momentum J and projection M can be defined as
A^JM = {1+j,j'}^-1/2 [ a^j a^j' ]_JM, where a^jm creates a proton with quantum numbers j,m. Analogue definitions can be used for creation operators for neutron pairs and isoscalar and isovector proton-neutron pairs. The eigen spectrum of the matrix langle M A^JM A_JM is an indicator of the pair content (for given J and isospin) of a nucleus. In particular, the presence of a pair condensate in a correlated ground state will be signaled by the largest eigenvalue for a given J being much larger than the other eigenvalues. This expectation is confirmed by SMMC studies of the ^54-58Fe ground states, where all of the proton pairing strength in the 0^+ channel is essentially found in one large eigenvalue. While the neutrons in the semi-magic nucleus ^54Fe show only little additional coherence beyond the mean-field, neutron pairing exhibits two large eigenvalues in the two other isotopes.
To investigate the isospin and temperature dependence of the pair correlations we have performed a series of SMMC calculations for ^54,56,58Fe and ^56Cr. In these studies we interpret the difference between the sum of eigenvalues of the pair matrix in the SMMC and corresponding mean-field calculations as a measure of the pairing correlations for given J,T.
As expected, we find that J=0^+ pairing between like nucleons dominates at low temperatures. At T 1 MeV these correlations vanish. We find that the moment of inertia is correlated with the J=0^+ pairing. The B(M1) strength unquenches in two steps: with the vanishing of the 0^+ proton pairing the orbital part unquenches, while the spin-part is related to the isoscalar neutron-proton correlations and unquenches when the latter vanish at higher temperatures. The quenching of the Gamow-Teller strength is correlated with the isoscalar proton-neutron pairing. A manuscript describing this work is in preparation.
Electron capture on nuclei in the ``iron'' region plays an essential role in the early stage of the supernova collapse. For many of these nuclei Gamow-Teller (GT) transitions contribute significantly to the electron capture rate. Due to insufficient experimental information, the GT transition rates have so far been treated only qualitatively in presupernova collapse simulations, assuming the total GT_+ strength to reside in a single resonance whose energy relative to the daughter ground state has been parametrized phenomenologically; the total GT_+ strength has been taken from the single particle model. Recent (n,p) experiments, however, show that the GT_+ strength is fragmented over many states, while the total strength is significantly quenched compared to the single particle model. (A recent update of the GT rates for use in supernova simulations assumed a constant quenching factor of 2.)
Reliable theoretical estimates of the GT_+ strengths in the ``iron'' mass range are now possible through SMMC calculations. After the well-established renormalization by the factor of 0.64 (see II.3.a1), these calculations reproduce the GT_+ strength distribution in the daughter nucleus well. As electron capture in the presupernova collapse is usually on odd nuclei, we have tested our method for the nuclei ^51V and ^59Co, whose GT_+ strength distributions are known from (n,p) experiments. For presupernova conditions, the electron capture rate, calculated from the theoretical GT_+ strength distribution, agrees typically within a factor of 2 with that obtained from the experimental data. We are currently calculating the electron capture rates for nuclei, whose GT_+ strength is not known experimentally.
The ability to perform multiple shell (multi-hbar omega) calculations will greatly enhance our ability to solve a number of outstanding problems in nuclear structure physics and nuclear astrophysics. Among these are the effects of core polarization of ``closed'' shells upon valence nucleons and the effect of forbidden weak transitions (e.g. e-capture, nu-scattering, etc.) upon the evolution of Type II supernovae. The primary difficulty in performing multi-hbar omega calculations is the separation of the true nuclear structure information from spurious center-of-mass motion. We are currently investigating several possible routes to first, gauge the importance of the spuriousity, and second, successfully remove it from the calculations.
The importance of the spurious components of the wave functions decreases as the size of the nucleus ( A) increases. To gauge the importance of the center-of-mass upon observables as a function of A, we are currently performing direct diagonalization studies, where there is a well-defined prescription for removing the spurious states by adding a large multiple of the center-of-mass Hamiltonian, H_CM, to the true nuclear Hamiltonian. Simultaneously we are also implementing the same method to remove the spurious states in the Monte Carlo code. Numerical stability of the latter requires that the multiplier of H_CM not be too large. We are currently benchmarking the Monte Carlo results against direct diagonalization to determine if we can remove the spuriousity completely, or the extent to which we can do so. Preliminary results for observables like the Gamow-Teller or B(E2) strengths look encouraging. Once these studies are complete we will begin to use the code to study multi-hbar omega problems, such as those mentioned above.
Neutron-proton correlations are expected to play an important role in the ground states of N=Z nuclei. To explore these correlations we have performed SMMC calculations of the N=Z nuclei in the mass range A=48-56 within the complete pf-shell using the modified Kuo-Brown interaction. For these nuclei the pair matrix (see II.3.a9) in all three isovector 0^+ channels essentially exhibits only one large eigenvalue, which can be conveniently interpreted as a measure for the pairing strength in these nuclei. As the even-even N=Z nuclei have T=0, the expectation values of A^ A are identical in all three isovector 0^+ channels. For the odd-odd N=Z nuclei, which have T=1 ground states, we find the 0^+ proton-neutron pairing strength significantly larger than the 0^+ pairing between like nucleons.
We have then performed SMMC calculations of N Z nuclei in the A=60-74 mass region within the complete (0f_5/2,1p,0g_9/2) space, again focussing on the 0^+ pair correlations. Again we find strong enhancement of proton-neutron pairing for N=Z nuclei when compared to neighbouring N=Z+2,Z+4, nuclei. Additionally we observe a strong correlation between the quadrupole moment and the J=1,T=0 proton-neutron pairing channel. The work is currently being written up for submission to Physical Review Letters.
The solution of many-nucleon problems by direct diagonalization becomes untractable for modest sizes of active particles involved due to the combinatorial growth of the number of basis states required. One route to circumvent this problem is offered by the Monte Carlo shell model developed by our group in recent years (see II.3.a). Very recently Otsuka and collaborators (Physical Review Letters .. (1995) ) proposed a novel method for solving interacting quantum many-body systems. This approach diagonalizes the many-body Hamiltonian in an ``optimally'' chosen basis generated using SMMC sampling from a trial state. The method has been applied to a boson system and was found to give promising results.
We have applied the method to many-body fermion systems and have compared its performance to the standard truncation scheme technique developed for solving the nuclear shell model. As a warm-up problem we studied ^46Ti with the model space restricted to the f_7/2 shell. We find that a basis of 200 determinants (out of about 1200 present in the model space) converges the energy to within 200 keV of the exact answer. However, the lowest energy eigenstate has still some 20% admixture from excited states, which we monitored by calculating the expectation value of J^2. We have performed calculations with unordered determinants and with determinants ordered according to their energy expectation values; the latter strategy does not significantly improve the convergence.
We have then studied ^48Cr within the complete pf-shell using the modified KB3 interaction. This model space contains about 12 10^6 determinants, while we have restricted the hybrid Monte Carlo diagonalization to at most some 100 determinants. In this case the energy converged to roughly the same value as that obtained in the lowest order of the conventional truncation scheme based on particle-hole number, which for ^48Cr corresponds to a basis of about 400 determinants. Upon closer inspection, we find that the truncation scheme technique shows a logarithmic convergence with the number of basis states. Although we expect that the hybrid Monte Carlo diagonalization method has a similar convergence behavior, we are planning to explore other possibilities by which the convergence can be improved. Our preliminary studies, however, indicate that the hybrid Monte Carlo diagonalization method is not likely to improve significantly on existing techniques for solving interacting many-fermion systems.