Shell Model Monte Carlo Research Articles

Extraction of Spectra in the Shell Model MonteCarlo Method Using Imaginary-Time Correlation Matrices.

Nuclear energy levels are usually calculated using conventional diagonalization methods in the framework of the configuration-interaction (CI) shell model but these methods are prohibited in very large model spaces. The shell model MonteCarlo (SMMC) method is a powerful technique for calculating thermal and ground-state observables of nuclei in very large model spaces, but it is challenging to extract nuclear spectra in this approach. We present a novel method to extract within SMMC low-lying energy levels for given values of a set of good quantum numbers such as spin and parity. The method is based on imaginary-time correlation matrices (ITCMs) of one-body densities that satisfy asymptotically a generalized eigenvalue problem. We validate the method in a light nucleus that allows comparison with exact diagonalization results of the CI shell-model Hamiltonian. The method is broadly applicable to quantum many-body systems in other disciplines.

State densities of heavy nuclei in the static-path plus random-phase approximation

Nuclear state densities are important inputs to statistical models of compound-nucleus reactions. State densities are often calculated with self-consistent mean-field approximations that do not include important correlations and have to be augmented with empirical collective enhancement factors. Here, we benchmark the static-path plus random-phase approximation (SPA+RPA) to the state density in a chain of samarium isotopes $^{148-155}$Sm against exact results (up to statistical errors) obtained with the shell model Monte Carlo (SMMC) method. The SPA+RPA method incorporates all static fluctuations beyond the mean field together with small-amplitude quantal fluctuations around each static fluctuation. Using a pairing plus quadrupole interaction, we show that the SPA+RPA state densities agree well with the exact SMMC densities for both the even- and odd-mass isotopes. For the even-mass isotopes, we also compare our results with mean-field state densities calculated with the finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation. We find that the SPA+RPA repairs the deficiencies of the mean-field approximation associated with broken rotational symmetry in deformed nuclei and the violation of particle-number conservation in the pairing condensate. In particular, in deformed nuclei the SPA+RPA reproduces the rotational enhancement of the state density relative to the mean-field state density.

Strong enhancement of level densities in the crossover from spherical to deformed neodymium isotopes

Understanding the evolution of level densities in the crossover from spherical to well-deformed nuclei has been a long-standing problem in nuclear physics. We measure nuclear level densities for a chain of neodymium isotopes 142,144−151Nd which exhibit such a crossover. These results represent the most complete data set of nuclear level densities to date for an isotopic chain between neutron shell-closure and towards mid-shell. We observe a strong increase of the level densities along the chain with an overall increase by a factor of ≈150 at an excitation energy of 6 MeV and saturation around mass 150. Level densities calculated by the shell model Monte Carlo (SMMC) are in excellent agreement with these experimental results. Based on our experimental and theoretical findings, we offer an explanation of the observed mass dependence of the level densities in terms of the intrinsic single-particle level density and the collective enhancement.

Rotational behavior of thermally excited 56Fe, 58Fe and 60Fe isotopes

Thermal and rotational behaviors of neutron rich fp-shell isotopes such as [Formula: see text], [Formula: see text] and [Formula: see text] were analyzed microscopically within the framework of statistical theory of hot rotating nuclei (STHRN) for the angular momentum range (0–15)[Formula: see text] at excitation energy above 4[Formula: see text]MeV. Pair-breaking phenomenon and band-crossing phenomenon of Fe isotopes were discussed with and without the inclusion of BCS pairing. The STHRN method with BCS effect was extended to the Fe isotopes to determine the critical temperature [Formula: see text], where the pair-breaking phenomenon takes place and it is found to occur below [Formula: see text][Formula: see text]MeV. The observation of band-crossing phenomenon above the [Formula: see text] was explained without considering the effect of BCS pairing. The single particle energy levels were engendered from triaxially deformed Nilsson Hamiltonian. The outcomes of the present investigation on moment of inertia (MOI), back-bending phenomenon, spin cutoff parameter show a strong evidence of band-crossing phenomenon and it eventually led to a shape transition from spherical to noncollective oblate. Moreover, attention on separation energy of protons and neutrons reveals that the neutron pairs are responsible for band-crossing. STHRN method was able to reproduce results in good agreement with experiments and comparable with other theories such as projected shellmodel (PSM) and shellmodel Monte Carlo (SMMC) method.

A NEW INSIGHT INTO NEUTRINO ENERGY LOSS BY ELECTRON CAPTURE OF IRON GROUP NUCLEI IN MAGNETAR SURFACES

ABSTRACT Based on the relativistic mean-field effective interactions theory, and the Lai dong model, we discuss the influences of superstrong magnetic fields (SMFs) on electron Fermi energy, nuclear blinding energy, and single-particle level structure in magnetar surfaces. Using the Shell-Model Monte Carlo method and the Random Phase Approximation theory, we analyze the neutrino energy loss rates (NELRs) by electron capture for iron group nuclei in SMFs. First, when B 12 < 100, we find that the SMFs have a slight influence on the NELRs for most nuclides at relativistic low temperatures (e.g., T 9 = 0.233); nevertheless, the NELRs increase by more than four orders of magnitude at relativistic high temperatures (e.g., T 9 = 15.53). When B 12 > 100, the NELRs decrease by more than three orders of magnitude (e.g., at T 9 = 15.53 for 52–61Fe, 55–60Co, and 56–63Ni). Second, for a certain value of magnetic field and temperature, the NELRs increase by more than four orders of magnitude when , but as the density increases (i.e., when ), there is almost no influence on the density of NELRs. For the density around , there is an abrupt increase in NELRs when B 12 ≥ 103.5. Such jumps are an indication that the underlying shell structure has changed due to single-particle behavior by SMFs. Finally, we compare our NELRs with those of Fuller et al. (FFN) and Nabi & Klapdor-Kleingrothaus (NKK). For the case without SMFs, one finds that our rates for certain nuclei are close to about five orders of magnitude lower than FFN and NKK at relativistic low temperatures (e.g., T 9 = 1). However, at a relativistic high temperature (e.g., T 9 = 3), our results are in good agreement with NKK, but about one order of magnitude lower than FFN. For the case with SMFs, our NELRs for some iron group nuclei can be about five orders of magnitude higher than those of FFN and NKK. (Note that B 12, T 9, and ρ 7 are in units of 1012 G, 109 K, and , respectively.)

Benchmarking mean-field approximations to level densities

We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus $^{162}$Dy, representing a heavy deformed nucleus, and $^{148}$Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo (SMMC) calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are seen to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point approximation to extract the number-projected densities is not a significant source of error compared to other errors inherent to the mean-field theory. We also present an alternative formulation of the saddle-point approximation that makes direct use of an approximate particle-number projection and avoids computing the usual three-dimensional Jacobian of the saddle-point integration. We find that the pairing condensate is less amenable to approximate particle-number projection methods due to the explicit violation of particle-number conservation in the pairing condensate. Nevertheless, the Hartree-Fock-Bogoliubov theory is accurate to less than one unit of entropy for $^{148}$Sm at the neutron threshold energy, which is above the pairing phase transition.

An new estimate of electron capture of nuclides 55Co and 56Ni in magnetars

Basing on the theory of relativity in ultrastrong magnetic fields (UMFs) and Random Phase Approximation (RPA), by using the Shell-Model Monte Carlo (SMMC) method, the influence of UMFS on electron capture (EC) and the rates of electron fraction change (REFC) of nuclides 55Co and 56Ni are investigated in magnetars. Basing on the Dirac δ-function and Pauli exclusion principle, we also discuss the influence on electron Fermi energy by UMFs in magnetars. The results show that the EC rates are increased greatly and even exceed five orders of magnitude by UMFs. The REFC is decreased even more than six orders of magnitude due to UMFs.

Collective enhancement of nuclear state densities by the shell model Monte Carlo approach

The shell model Monte Carlo (SMMC) approach allows for the microscopic calculation of statistical and collective properties of heavy nuclei using the framework of the configuration-interaction shell model in very large model spaces. We present recent applications of the SMMC method to the calculation of state densities and their collective enhancement factors in rare-earth nuclei.

Recent Advances in the Application of the Shell Model Monte Carlo Approach to Nuclei

The shell model Monte Carlo (SMMC) method is a powerful technique for calculating the statistical and collective properties of nuclei in the presence of correlations in model spaces that are many orders of magnitude larger than those that can be treated by conventional diagonalization methods. We review recent advances in the development and application of SMMC to mid-mass and heavy nuclei.

Mapping the two-component atomic Fermi gas to the nuclear shell-model

The physics of a two-component cold fermi gas is now frequently addressed in laboratories. Usually this is done for large samples of tens to hundreds of thousands of particles. However, it is now possible to produce few-body systems (1-100 particles) in very tight traps where the shell structure of the external potential becomes important. A system of two-species fermionic cold atoms with an attractive zero-range interaction is analogous to a simple model of nucleus in which neutrons and protons interact only through a residual pairing interaction. In this article, we discuss how the problem of a two-component atomic fermi gas in a tight external trap can be mapped to the nuclear shell model so that readily available many-body techniques in nuclear physics, such as the Shell Model Monte Carlo (SMMC) method, can be directly applied to the study of these systems. We demonstrate an application of the SMMC method by estimating the pairing correlations in a small two-component Fermi system with moderate-to-strong short-range two-body interactions in a three-dimensional harmonic external trapping potential.

Collectivity in Heavy Nuclei in the Shell Model Monte Carlo Approach

The microscopic description of collectivity in heavy nuclei in the framework of the configuration-interaction shell model has been a major challenge. The size of the model space required for the description of heavy nuclei prohibits the use of conventional diagonalization methods. We have overcome this difficulty by using the shell model Monte Carlo (SMMC) method, which can treat model spaces that are many orders of magnitude larger than those that can be treated by conventional methods. We identify a thermal observable that can distinguish between vibrational and rotational collectivity and use it to describe the crossover from vibrational to rotational collectivity in families of even-even rare-earth isotopes. We calculate the state densities in these nuclei and find them to be in close agreement with experimental data. We also calculate the collective enhancement factors of the corresponding level densities and find that their decay with excitation energy is correlated with the pairing and shape phase transitions.

Recent Advances in the Microscopic Calculations of Level Densities by the Shell Model Monte Carlo Method

The shell model Monte Carlo (SMMC) method enables calculations in model spaces that are many orders of magnitude larger than those that can be treated by conventional methods, and is particularly suitable for the calculation of level densities in the presence of correlations. We review recent advances and applications of SMMC for the microscopic calculation of level densities. Recent developments include (i) a method to calculate accurately the ground-state energy of an odd-mass nucleus, circumventing a sign problem that originates in the projection on an odd number of particles, and (ii) a method to calculate directly level densities, which, unlike state densities, do not include the spin degeneracy of the levels. We calculated the level densities of a family of nickel isotopes $^{59-64}$Ni and of a heavy deformed rare-earth nucleus $^{162}$Dy and found them to be in close agreement with various experimental data sets.

Electron capture of nuclides 56,57,59,60Co in supernova explosive surrounding

Based on the shell model and random phase approximation theory, and using a shell-model Monte Carlo (SMMC) method, we have investigated the electron capture (EC) for nuclides 56,57,59,60Co, and the rate of change of electron fraction (RCEF) in supernova explosive surroundings. We compared our results, which were obtained by using SMMC method, with those analyzed by using Aufderheide's method. The results show that the EC rates increase greatly, even more than 6 orders of magnitude (e. g. for 57,59,60Co at ρ7=0.43, Ye=0.48). On the other hand, with the increase in temperature and density, the RCEF decreases greatly, even by 5 orders of magnitude (e. g. for 59Co at ρ7=5.86, Ye=0.47). The discussions of error factor show that in a lower density and temperature surrounding, the error is relatively great. But it may be small in the higher density and temperature surroundings.

Recent developments in the shell model Monte Carlo approach to nuclei

The shell model Monte Carlo (SMMC) approach provides a powerful method for the microscopic calculation of statistical and collective nuclear properties in model spaces that are many orders of magnitude larger than those that can be treated by conventional methods. We discuss recent applications of the method to describe the emergence of collectivity in the framework of the configuration-interaction shell model and the crossover from vibrational to rotational collectivity in families of rare-earth nuclei. We have calculated state densities of these rare-earth nuclei and find their collective enhancement factors to be correlated with the pairing and shape phase transitions. We also discuss an accurate method to calculate the ground-state energy of odd-even and odd-odd nuclei, circumventing the sign problem that originates in the projection on an odd number of particles. We have applied this method to calculate pairing gaps in families of isotopes in the iron region.

The electron capture of nuclides 55Co and 56Ni in the process of stellar core collapse

Using Shell-Model Monte Carlo (SMMC) method and Random Phase Approximation (RPA) theory, the electron capture (EC) and the electron capture cross section (ECCS) of nuclei 55Co and 56Ni are investigated. We also discuss the rates of the change of electron fraction (RCEF) and the error factor C, which is compared our results with those of AFUD, which calculated by the method of Aufderheide. The results show that the ECCS and the EC rates for 55Co and 56Ni increased about four orders of magnitude at relative high temperature (such as T9=5,7,9). On the other hand, the RCEF for two nuclides decreased greatly, and even exceed four orders of magnitude. The error factor shows ours is agreed reasonably well with AUFD under the higher density surroundings (e.g. ρ7=106, Ye=0.43; ρ7=506, Ye=0.42; ρ7=4010, Ye=0.41). But under the lower density surroundings (e.g. ρ7=3.36, Ye=0.48) the maximum error is ∼14.5 % for 55Co but is ∼14.0 % for 56Ni. The error is ∼9.5 % and ∼9.0 % for 55Co, 56Ni at ρ7=5.86, Ye=0.47 respectively.

Shell model Monte Carlo approach: the heavy nuclei

The auxiliary-field Monte Carlo (AFMC) method, also known as the shell model Monte Carlo (SMMC), enables us to calculate microscopically statistical and collective properties of nuclei in the presence of strong correlations. These calculations can be carried out in shell model spaces that are many orders of magnitude larger than spaces that can be treated with conventional diagonalization methods. A recent major development has been the extension of the AFMC approach to heavy nuclei. Such applications to heavy nuclei have been a major challenge. On the conceptual level, a crucial question is whether a truncated spherical shell model Hamiltonian can describe the proper collectivity observed in heavy nuclei and, in particular, the rotational character of strongly deformed nuclei. On the technical level, the low excitation energies make it necessary to perform calculations down to much lower temperatures. At such low temperatures the propagator becomes ill-conditioned and requires the introduction of stabilization methods.

Extended pool of electron-capture rates for core-collapse supernovae simulations

The mechanism behind core-collapse supernova explosions is not yet fully understood. As computers get faster, more complex processes can be included in supernova models and the modelling can be performed in greater detail. Improved nuclear physics data play a role in improving the understanding of processes taking place during the core-collapse phase of this phenomenon. Large-scale shell model (LSSM) and shell model Monte Carlo (SMMC) calculations combined with the random phase approximation (RPA) have yielded two complementary pools of electron-capture and neutrino-nucleus reaction rates for iron group nuclei and beyond. Recently, the electron-capture SMMC+RPA pool was extended by including more neutron-rich nuclei to improve its predictions. At present the average rate is built from data on more than 250 nuclei. This presentation is focused on the features of this new, enlarged pool of electron-capture rates. The accuracy of predictions made with this extended pool is estimated.

Shell model Monte Carlo studies of pairing correlations and level densities in medium-mass nuclei

We have performed shell model Monte Carlo (SMMC) calculations of even–even, odd- A and odd–odd nuclei in the mass A ∼ 70 region as function of temperature. The calculations have been performed in the complete ( p f − g d s ) shell using a pairing + quadrupole residual interaction. We calculate the level densities for these nuclei and investigate the influence of pairing correlations.

Nuclear level statistics: Extending shell model theory to higher temperatures

The Shell Model Monte Carlo (SMMC) approach has been applied to calculate level densities and partition functions to temperatures up to ~ 1.5 - 2 MeV, with the maximal temperature limited by the size of the configuration space. Here we develop an extension of the theory that can be used to higher temperatures, taking into account the large configuration space that is needed. We first examine the configuration space limitation using an independent-particle model that includes both bound states and the continuum. The larger configuration space is then combined with the SMMC under the assumption that the effects on the partition function are factorizable. The method is demonstrated for nuclei in the iron region, extending the calculated partition functions and level densities up to T ~ 4 MeV. We find that the back-shifted Bethe formula has a much larger range of validity than was suspected from previous theory. The present theory also shows more clearly the effects of the pairing phase transition on the heat capacity.

Temperature dependence of the symmetry energy

We perform large-scale shell model Monte Carlo (SMMC) calculations for many nuclei in the mass range A=56-65 in the complete pfg_{9/2}d_{5/2} model space using an effective quadrupole-quadrupole+pairing residual interaction. Our calculations are performed at finite temperatures between T=0.33-2 MeV. Our main focus is the temperature dependence of the symmetry energy which we determine from the energy differences between various isobaric pairs with the same pairing structure and at different temperatures. Our SMMC studies are consistent with an increase of the symmetry energy with temperature. We also investigate possible consequences for core-collapse supernovae events.