Abstract:
While the shell model Monte Carlo approach has been successful in the microscopic calculation of nuclear state densities, it has been difficult to calculate accurately state densities of odd-even heavy nuclei. This is because the projection on an odd number of particles in the shell model Monte Carlo method leads to a sign problem at low temperatures, making it impractical to extract the ground-state energy in direct Monte Carlo calculations. We show that the ground-state energy can be extracted to a good precision by using level counting data at low excitation energies and the neutron resonance data at the neutron threshold energy. This allows us to extend recent applications of the shell-model Monte Carlo method in even-even rare-earth nuclei to the odd-even isotopic chains of $^{149-155}$Sm and $^{143-149}$Nd. We calculate the state densities of the odd-even samarium and neodymium isotopes and find close agreement with the state densities extracted from experimental data.

Abstract:
We present novel Monte Carlo methods for treating the interacting shell model that allow exact calculations much larger than those heretofore possible. The two-body interaction is linearized by an auxiliary field; Monte Carlo evaluation of the resulting functional integral gives ground-state or thermal expectation values of few-body operators. The ``sign problem'' generic to quantum Monte Carlo calculations is absent in a number of cases. We discuss the favorable scaling of these methods with nucleon numb er and basis size and their suitability to parallel computation.

Abstract:
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.

Abstract:
Nuclear logging is one of most important logging services provided by many oil service companies. The main parameters of interest are formation porosity, bulk density, and natural radiation. Other services are also provided from using complex nuclear logging tools, such as formation lithology/mineralogy, etc. Some parameters can be measured by using neutron logging tools and some can only be measured by using a gamma ray tool. To understand the response of nuclear logging tools, the neutron transport/diffusion theory and photon diffusion theory are needed. Unfortunately, for most cases there are no analytical answers if complex tool geometry is involved. For many years, Monte Carlo numerical models have been used by nuclear scientists in the well logging industry to address these challenges. The models have been widely employed in the optimization of nuclear logging tool design, and the development of interpretation methods for nuclear logs. They have also been used to predict the response of nuclear logging systems for forward simulation problems. In this case, the system parameters including geometry, materials and nuclear sources, etc., are pre-defined and the transportation and interactions of nuclear particles (such as neutrons, photons and/or electrons) in the regions of interest are simulated according to detailed nuclear physics theory and their nuclear cross-section data (probability of interacting). Then the deposited energies of particles entering the detectors are recorded and tallied and the tool responses to such a scenario are generated. A general-purpose code named Monte Carlo N– Particle (MCNP) has been the industry-standard for some time. In this paper, we briefly introduce the fundamental principles of Monte Carlo numerical modeling and review the physics of MCNP. Some of the latest developments of Monte Carlo Models are also reviewed. A variety of examples are presented to illustrate the uses of Monte Carlo numerical models for the development of major nuclear logging tools, including compensated neutron porosity, compensated density, natural gamma ray and a nuclear geo-mechanical tool.

Abstract:
The feasibility of shell-model calculations is radically extended by the Quantum Monte Carlo Diagonalization method with various essential improvements. The major improvements are made in the sampling for the generation of shell-model basis vectors, and in the restoration of symmetries such as angular momentum and isospin. Consequently the level structure of low-lying states can be studied with realistic interactions. After testing this method on $^{24}$Mg, we present first results for energy levels and $E2$ properties of $^{64}$Ge, indicating its large and $\gamma$-soft deformation.

Abstract:
A method for making realistic estimates of the density of levels in even-even nuclei is presented making use of the Monte Carlo shell model (MCSM). The procedure follows three basic steps: (1) computation of the thermal energy with the MCSM, (2) evaluation of the partition function by integrating the thermal energy, and (3) evaluating the level density by performing the inverse Laplace transform of the partition function using Maximum Entropy reconstruction techniques. It is found that results obtained with schematic interactions, which do not have a sign problem in the MCSM, compare well with realistic shell-model interactions provided an important isospin dependence is accounted for.

Abstract:
We present in detail a formulation of the shell model as a path integral and Monte Carlo techniques for its evaluation. The formulation, which linearizes the two-body interaction by an auxiliary field, is quite general, both in the form of the effective `one-body' Hamiltonian and in the choice of ensemble. In particular, we derive formulas for the use of general (beyond monopole) pairing operators, as well as a novel extraction of the canonical (fixed-particle number) ensemble via an activity expansion. We discuss the advantages and disadvantages of the various formulations and ensembles and give several illustrative examples. We also discuss and illustrate calculation of the imaginary-time response function and the extraction, by maximum entropy methods, of the corresponding strength function. Finally, we discuss the "sign-problem" generic to fermion Monte Carlo calculations, and prove that a wide class of interactions are free of this limitation.

Abstract:
We introduce a particle-number reprojection method in the shell model Monte Carlo that enables the calculation of observables for a series of nuclei using a Monte Carlo sampling for a single nucleus. The method is used to calculate nuclear level densities in the complete $(pf+g_{9/2})$-shell using a good-sign Hamiltonian. Level densities of odd-A and odd-odd nuclei are reliably extracted despite an additional sign problem. Both the mass and the $T_z$ dependence of the experimental level densities are well described without any adjustable parameters. The single-particle level density parameter is found to vary smoothly with mass. The odd-even staggering observed in the calculated backshift parameter follows the experimental data more closely than do empirical formulae.

Abstract:
We have developed an efficient isospin projection method in the shell model Monte Carlo approach for isospin-conserving Hamiltonians. For isoscalar observables this projection method has the advantage of being exact sample by sample. The isospin projection method allows us to take into account the proper isospin dependence of the nuclear interaction, thus avoiding a sign problem that such an interaction introduces in unprojected calculations. We apply our method in the calculation of the isospin dependence of level densities in the complete $pf+g_{9/2}$ shell. We find that isospin-dependent corrections to the total level density are particularly important for $N \sim Z$ nuclei.

Abstract:
The microscopic calculation of nuclear level densities in the presence of correlations is a difficult many-body problem. The shell model Monte Carlo method provides a powerful technique to carry out such calculations using the framework of the configuration-interaction shell model in spaces that are many orders of magnitude larger than spaces that can be treated by conventional methods. We present recent applications of the method to the calculation of level densities and their collective enhancement factors in heavy nuclei. The calculated level densities are in close agreement with experimental data.