中田 仁

Abstract Reducing the many-fermion problem to a set of single-particle (s.p.) equations, the Kohn-Sham (KS) theory has provided a practical tool to implement ab initio calculations of ground-state energies and densities in many-electron systems. There have been attempts to extend the KS theory so that it could describe other physical quantities, or it could be applied to other many-fermion systems. By generalizing and reformulating the KS theory in terms of the 1-body density matrix, I expose the minimal composition of the theory that enables the reduction of the many-fermion problem to the s.p. equations. Based on the reformulation, several basic issues are reconsidered. The v- and N-representabilities for the KS theory are distinguished from those for the Hohenberg-Kohn theorem. Criteria for the extendability of the KS theory are addressed.

The nuclear symmetry energy is defined by the second derivative of the energy per nucleon with respect to the proton-neutron asymmetry, and is sometimes approximated by the energy difference between the neutron matter and the symmetric matter. The accuracy of this approximation is assessed analytically and numerically within the Hartree-Fock theory using effective interactions. By decomposing the nuclear-matter energy, the relative error of each term is expressed analytically; it is constant or is a single-variable function determined by the function type. The full errors are evaluated for several effective interactions, by inserting values for the parameters. Although the errors stay within 10% up to twice the normal density irrespective of the interactions, at higher densities the accuracy of the approximation significantly depends on the interactions.

The influence of the Nambu-Goldstone (NG) mode on the energy-weighted sum (EWS) of the excitation strengths is analyzed within the random-phase approximation (RPA). When a certain symmetry is broken at the mean-field level, an NG mode emerges in the RPA, which can be represented by canonical variables forming a two-dimensional Jordan block. A general formula is rederived which separates out the NG-mode contribution to the EWS, via the projection on the subspace directed by the NG mode. As examples, the formula is applied to the E1 excitation and the rotational excitations in nuclei, reinforcing the theoretical consistency of the RPA.

The M3Y-type semirealistic interaction is applied to deformed nuclei for the first time. Constrained Hartree-Fock calculations assuming axial symmetry are implemented for the N = 20 isotones Ne-30, Mg-32, Si-34 and the N = 28 isotones Mg-40, Si-42, S-44 with the M3Y-P6 interaction. The results match the experimental data well. Effects of the realistic tensor force on the nuclear quadrupole deformation are investigated in relation to the loss of the N = 20 and 28 magic numbers. The tensor force is confirmed to favor the deformation for the N = 28 nuclei owing to the closure of the jj shell (i.e., n0f(7/)2), while favoring the sphericity for the N = 20 nuclei owing to the l(s) closure of N = 20.

Recent applications of the M3Y-type semi-realistic interaction to the nuclear mean-field approaches are presented: i) Prediction of magic numbers and ii) isotope shifts of nuclei with magic proton numbers. The results exemplify that the realistic interaction, which is derived from the bare 2N and 3N interaction, furnishes a new theoretical instrument for advancing nuclear mean-field approaches.

Properties of solutions of the random-phase approximation ( RPA) equation are reanalyzed mathematically, where it is defined as a generalized eigenvalue problem of the stability matrix S with the norm matrix N = diag(1,-1). As well as physical solutions, unphysical solutions are examined in detail, taking the possibility of Jordan blocks of the matrix NS into consideration. Two types of duality of eigenvectors and basis vectors of the Jordan blocks are pointed out and explored, which disclose many basic properties of the RPA solutions.

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 Dy-162, representing a heavy deformed nucleus, and Sm-148, 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 calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are found 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 because of 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 Sm-148 at the neutron threshold energy, which is above the pairing phase transition. This result provides support for the commonly used "back-shift" approximation, treating pairing as only affecting the excitation energy scale. When the ground state is strongly deformed, the Hartree-Fock entropy is significantly lower than the shell model Monte Carlo entropy at low temperatures because of the missing contribution of rotational degrees of freedom. However, treating the rotational bands in a simple model, we find that the entropy at moderate excitation energies is reproduced to within two units, corresponding to an error in the level density of less than an order of magnitude. We conclude with a discussion of methods that have been advocated as beyond the mean-field approximation, and their prospects to address the issues we have identified.

<p>対称エネルギーa_t_(ρ)は核子あたりエネルギーε(ρ,η_t_)を非対称度(ρ_p_-ρ_n_)/ρで展開した時の2次係数として定義される。対称エネルギーは中性子星の構造やdrip lineなどに関わる重要な量であると考えられているが、その計算は一般には易しくない。そこで、a^~^_t_=ε(ρ,1)-ε(ρ,0)による近似がなされる事がある。 今回、この近似がどれ程の精度なのかを、核物質に於いて具体的なGauss型・Yukawa型有効相互作用を用いて調べた。更にこの近似の精度を定める要因について考察した。</p>

Nuclear level densities are necessary input to the Hauser-Feshbach theory of compound nuclear reactions. However, the microscopic calculation of level densities in the presence of correlations is a challenging many-body problem. The configuration-interaction shell model provides a suitable framework for the inclusion of correlations and shell effects, but the large dimensionality of the many-particle model space has limited its application in heavy nuclei. The shell model Monte Carlo method enables calculations in spaces that are many orders of magnitude larger than spaces that can be treated by conventional diagonalization methods and has proven to be a powerful tool in the microscopic calculation of level densities. We discuss recent applications of the method in heavy nuclei.

Four quantities deducible from nuclear structure experiments have been claimed to correlate with the slope parameter L of the symmetry energy: neutron skin thickness, cross section of low-energy dipole (LED) mode, dipole polarizability alpha(D), and alpha S-D(0) (i. e., product of alpha(D) and symmetry energy S-0). By means of calculations in the Hartree-Fock plus random-phase approximation with various effective interactions, we compare the correlations between L and these four quantities. The correlation derived from different interactions and the correlation from a class of interactions that are identical in symmetric matter as well as in S-0 are simultaneously examined. These two types of correlation may behave differently, as exemplified in the correlation of aD to L. It is found that the neutron skin thickness and alpha S-D(0) correlate well to L, and therefore are suitable for narrowing down the value of L via experiments. The LED emergence and upgrowth makes the alpha S-D(0)-L correlation strong, although these correlations are disarranged when a neutron halo appears in the ground state.

It was pointed out [Phys. Rev. C 91, 021302(R) (2015)] that the isotope shifts of the Pb nuclei, the kink at N = 126 in particular, can be well described by Hartree-Fock-Bogolyubov calculations if a density-dependent LS interaction derived from the 3N interaction is incorporated. Effects of the density dependence in the LS channel on the isotope shifts are extensively investigated for the Ca, Ni, and Sn isotopes, using the semirealistic M3Y-P6 interaction and its LS modified variant M3Y-P6a, as in the Pb case. It is found that almost equal charge radii between Ca-40 and Ca-48 are reproduced, as well as the isotope shifts in a long chain of the Sn nuclei, owing to the density dependence in the LS channel. A kink is predicted at N = 82 for the isotope shifts of the Sn nuclei, in clear contrast to the interactions without the density dependence.

Differential cross sections of isoscalar and isovector spin-M1 (0(+) -&gt; 1(+)) transitions are measured using high-energy-resolution proton inelastic scattering at E-p = 295 MeV on Mg-24, Si-28, S-32, and Ar-36 at 0 degrees-14 degrees. The squared spin-M1 nuclear transition matrix elements are deduced from the measured differential cross sections by applying empirically determined unit cross sections based on the assumption of isospin symmetry. The ratios of the squared nuclear matrix elements accumulated up to E-x = 16 MeV compared to a shell-model prediction are 1.01(9) for isoscalar and 0.61(6) for isovector spin-M1 transitions, respectively. Thus, no quenching is observed for isoscalar spin-M1 transitions, while the matrix elements for isovector spin-M1 transitions are quenched by an amount comparable with the analogous GamowTeller transitions on those target nuclei.

Nuclear level densities are required for estimating statistical nuclear reaction rates. The shell model Monte Carlo method is a powerful approach for microscopic calculation of state densities in very large model spaces. However, these state densities include the spin degeneracy of each energy level, whereas experiments often measure level densities, in which each level is counted only once. To enable the direct comparison of theory with experiments, we introduce a method to calculate directly the level density in the shell model Monte Carlo approach. The method employs a projection on the minimal absolute value of the magnetic quantum number. We apply the method to nuclei in the iron region and to the strongly deformed rare-earth nucleus Dy-162. We find very good agreement with experimental data obtained by various methods, including level counting at low energies, charged particle spectra and Oslo method data at intermediate energies, neutron and proton resonance data, and Ericson's fluctuation analysis at higher excitation energies. We also extract a thermal moment of inertia from the ratio between the state density and the level density, and observe that in even-even nuclei it exhibits a signature of a phase transition to a superconducting phase below a certain excitation energy.

To study the Gamow-Teller (GT) transitions from the T-z = +1 nucleus Ca-42 to the T-z = 0 nucleus Sc-42, where T-z is the z component of isospin T, we performed a (p, n)-type (He-3, t) charge-exchange reaction at 140 MeV/nucleon and scattering angles around 0 degrees. With an energy resolution of 29 keV, states excited by GT transitions (GT states) could be studied accurately. The reduced GT transition strengths B(GT) were derived up to the excitation energy of 13 MeV, assuming the proportionality between the cross sections at 0 degrees and B(GT) values. The main part of the observed GT transition strength is concentrated in the lowest 0.611-MeV, J(pi) = 1(+) GT state. All the other states at higher energies are weakly excited. Shell-model calculations could reproduce the gross feature of the experimental B(GT) distribution, and random-phase-approximation calculations including an attractive isoscalar interaction showed that the 0.611-MeV state has a collective nature. It was found that this state has all of the properties of a "low-energy super-Gamow-Teller state." It is expected that low-lying J(pi) = 1(+) GT states have T = 0 in the T-z = 0 nucleus Sc-42. However, T = 1 states are situated in a higher energy region. Assuming an isospin-analogous structure in A = 42 isobars, analogous T = 1, 1(+) states are also expected in Ca-42. Comparing the Ca-42(He-3, t)Sc-42 and Ca-42(p, p') spectra measured at 0 degrees, candidates for T = 1 GT states could be found in the 10-12-MeV region of Sc-42. They were all weakly excited. The mass dependence of the GT strength distributions in Sc isotopes is also discussed.

The shell model Monte Carlo (SMMC) approach enables the microscopic calculation of nuclear state densities in model spaces that are many orders of magnitude larger than those that can be treated by conventional diagonalization techniques. However, it has been difficult to calculate accurate state densities of odd-mass heavy nuclei as a function of excitation energy. This is because of a sign problem that arises from the projection on an odd number of particles at low temperatures, making it difficult to calculate accurate ground-state energies of odd-mass nuclei in direct Monte Carlo calculations. Here we extract the ground-state energy from a one-parameter fit of the SMMC thermal energy to the thermal energy that is determined from experimental data. This enables us to calculate the state densities of the odd-even isotopes Sm149-155 and Nd143-149 as a function of excitation energy. We find close agreement with state densities extracted from experimental data. Our results demonstrate that the state densities of the odd-mass samarium and neodymium isotopes can be consistently reproduced using the same family of Hamiltonians that describe the neighboring even-mass isotopes within the configuration-interaction shell model approach.

We investigate the effects of the 3N interaction, which effectively adds a density-dependent term to the LS channel, on the isotopes shifts of the Pb nuclei. With the strength so as to keep the ls splitting of the single-nucleon orbits, the density dependence in the LS channel tends to shrink the wave functions of the j = l + 1/2 orbits while making the j = l - 1/2 functions distribute more broadly. Thereby the kink in the isotope shifts of the Pb nuclei at N = 126 becomes stronger, owing to the attraction from neutrons occupying 0i(11)/(2) in N &gt; 126. The density dependence in the LS channel enables us to reproduce the data of the isotope shifts by the Hartree-Fock-Bogoliubov calculations in a long chain of neutron numbers, even without degeneracy between the n1g(9/2) and n0i(11/2) levels. We exemplify it by the semirealistic M3Y-P6 interaction.

We have extensively investigated characters of the low-energy E1 strengths in N &gt; Z nuclei, by analyzing the transition densities obtained by the HF+RPA calculations with several effective interactions. Crossover behavior has been confirmed, from the skin mode at low energy to the pn mode at higher energy. Decomposing the E1 strengths into the skin-mode, pn-mode and interference fractions, we show that the ratio of the skin-mode strength to the full strength may be regarded as a generic function of the excitation energy, insensitive to nuclides and effective interactions, particularly beyond Ni.

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.

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.

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.

Magic numbers are predicted in a wide range of the nuclear chart by self-consistent mean-field calculations with the M3Y-P6 and P7 semi-realistic NN interactions. The magic numbers are identified by vanishing pair correlations in the spherical Hartree-Fock-Bogolyubov regime. We also identify submagic numbers when the energy gain due to the pairing is sufficiently small. It is found that the results with M3Y-P6 correspond well to the known data, apart from a few exceptions. For some of the magic or submagic numbers, the prediction differs from that with the Gogny-D1S or D1M interaction. The roles of the tensor force and the spin-isospin channel originating from the one-pion exchange potential are investigated in the Z- or N-dependence of the shell gap.

We study quantum phase transitions in the asymmetric variation of the three-leg Heisenberg tube for half-odd-integer spin, with a modulation of one of the rung exchange couplings J(perpendicular to)'. while the other two are kept constant J(perpendicular to). We focus on the strong rung-coupling regime J(perpendicular to) &gt;&gt; J(parallel to), where J(parallel to) is the leg coupling, and analyze the effective spin-orbital model with a transverse crystal field in detail. Applying the Abelian bosonization to the effective model, we find that the system is in the dimer phase for the general half-odd-integer-spin cases without the rung modulation; the phase transition between the dimer and Tomonaga-Luttinger-liquid phases induced by the rung modulation is of the SU(2)-symmetric Berezinskii-Kosterlitz-Thouless type. Moreover, we perform a level spectroscopy analysis for the effective model for spin-1/2 using exact diagonalization, to determine the precise transition point vertical bar J(perpendicular to)' - J(perpendicular to)|/J(parallel to) similar to 0.283 in the strong rung-coupling limit. The presence of the dimer phase in a small but finite region is also confirmed by a density-matrix renormalization group calculation on the original spin-tube model.

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.

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 Ni59-64 and of a heavy deformed rare-earth nucleus Dy-162 and found them to be in close agreement with various experimental data sets.

Electric dipole (El) reduced transition probability B(El) of Zr-90 was obtained by the inelastic proton scattering near 0 degrees using a 295 MeV proton beam and multipole decomposition analysis of the angular distribution by the distorted-wave Born approximation with the Hartree-Fock plus random-phase approximation model and inclusion of El Coulomb excitation, and the El strength of the pygmy dipole resonance was found in the vicinity of the neutron threshold in the low-energy tail of the giant dipole resonance. Using the data, we plan to determine the precise dipole polarizability alpha(D) which is defined as an inversely energy-weighted sum value of the elecrric dipole strength. The dipole polarizability is expected to constrain the symmetry energy term of the neutron matter equation of state. Thus systematical measurement of the dipole polarizability is important.

Applying the semirealistic NN interactions that include a realistic tensor force to the Hartree-Fock calculations, we investigate tensor-force effects on the single-particle levels in Ca isotopes. The semirealistic interactions successfully describe the experimental difference between É(p1s 1/2) and É(p0d3/2) (denoted by ΔÉ 13) both at 40Ca and 48Ca, confirming the importance of the tensor force. The tensor force plays a role in the N dependence of ΔÉ 13 also in neutron-rich Ca nuclei. While the p1s1/2- p0d3/2 inversion is predicted in heavier Ca nuclei as in 48Ca, it takes place only for N≥46, delayed by the tensor force. We further investigate the possibility of proton bubble structure in Ar, which is suggested by the p1s1/2-p0d3/2 inversion in 48Ca and more neutron rich Ca nuclei, by using spherical Hartree-Fock-Bogolyubov calculations. Even with the inversion at 48Ca the pair correlation prohibits prominent bubble distribution in 46Ar. Bubble structure in Ar is unlikely also near the neutron drip line because of either unboundness or deformation. However, 34Si remains a candidate for proton bubble structure, owing to the large shell gap between p1s1/2 and p0d5/2. © 2013 American Physical Society.

Applying the semirealistic NN interactions that include a realistic tensor force to the Hartree-Fock calculations, we investigate tensor-force effects on the single-particle levels in Ca isotopes. The semirealistic interactions successfully describe the experimental difference between epsilon(p1s(1/2)) and epsilon(p0d(3/2)) (denoted by Delta epsilon(13)) both at Ca-40 and Ca-48, confirming the importance of the tensor force. The tensor force plays a role in the N dependence of Delta epsilon(13) also in neutron-rich Ca nuclei. While the p1s(1/2)-p0d(3/2) inversion is predicted in heavier Ca nuclei as in Ca-48, it takes place only for N &gt;= 46, delayed by the tensor force. We further investigate the possibility of proton bubble structure in Ar, which is suggested by the p1s(1/2)-p0d(3/2) inversion in Ca-48 and more neutron rich Ca nuclei, by using spherical Hartree-Fock-Bogolyubov calculations. Even with the inversion at Ca-48 the pair correlation prohibits prominent bubble distribution in Ar-46. Bubble structure in Ar is unlikely also near the neutron drip line because of either unboundness or deformation. However, Si-34 remains a candidate for proton bubble structure, owing to the large shell gap between p1s(1/2) and p0d(5/2).

The character of the low-energy E1 excitations is investigated by analyzing transition densities obtained from the RPA calculations in the doubly magic nuclei. We propose a decomposition method of the E1 excitations into the pn mode (i.e., oscillation between protons and neutrons) and the skin mode (i.e., oscillation of the neutron skin against the inner core) via the transition densities, by which their mixing is handled in a straightforward manner. Crossover behavior of the E1 excitations is found, from the skin mode at low energy to the pn mode at higher energy. The ratio of the skin-mode strength to the full strength turns out to be insensitive to the nuclides and to the effective interactions in the energy region of the crossover. Depending on the excitation energy, the observed low-energy E1 excitations are not necessarily dominated by the skin mode, as exemplified for 90Zr. © 2013 American Physical Society.

Electromagnetic dipole transitions in 56Fe were measured in photon-scattering experiments with a linearly polarized photon beam. The parity quantum numbers of the excited dipole states were determined by the intensity asymmetry of resonantly scattered γ rays with respect to the polarization plane of the incident photon beam. While the summed magnetic dipole (M1) strength was determined as ΣB(M1)↑=3.52(17) μN2 at excitation energies between 7 and 10 MeV, the summed electric dipole (E1) strength below 10 MeV was obtained as ΣB(E1)↑=78.0(15)×10-3 e2 fm2. The observed M1 strength was compared with shell-model predictions in the pf shell using the GXPF1J and KB3G effective interactions. In addition, the E1 strength was compared with random-phase approximation calculations with the Skyrme interaction. © 2013 American Physical Society.

New parameter sets of the semi-realistic nucleon-nucleon interaction are developed by modifying the M3Y interaction but maintaining the tensor channels and the longest-range central channels. The modification is made so as to reproduce microscopic results of neutron-matter energies, in addition to the measured binding energies of doubly magic nuclei including 100Sn and the even-odd mass differences of the Z=50 and N=82 nuclei in the self-consistent mean-field calculations. Separation energies of the proton- or neutron-magic nuclei are shown to be in fair agreement with the experimental data. With the new parameter-sets M3Y-P6 and P7, the isotropic spin-saturated symmetric nuclear matter remains stable in the density range as wide as 260, while keeping desirable results of the previous parameter set on finite nuclei. Isotope shifts of the Pb nuclei and tensor-force effects on shell structure are discussed. © 2013 American Physical Society.

Heavy nuclei exhibit a crossover from vibrational to rotational collectivity as the number of neutrons or protons increases from shell closure towards midshell, but the microscopic description of this crossover has been a major challenge. We apply the shell model Monte Carlo approach to families of even-even samarium and neodymium isotopes and identify a microscopic signature of the crossover from vibrational to rotational collectivity in the low-temperature behavior of &lt; J(2)&gt;(T), where J is the total spin and T is the temperature. This signature agrees well with its values extracted from experimental data. We also calculate the state densities of these nuclei and find them to be in very good agreement with experimental data. Finally, we define a collective enhancement factor from the ratio of the total state density to the intrinsic state density as calculated in the finite-temperature Hartree-Fock-Bogoliubov approximation. The decay of this enhancement factor with excitation energy is found to correlate with the pairing and shape phase transitions in these nuclei. DOI: 10.1103/PhysRevLett.110.042502

A high-resolution measurement of inelastic proton scattering off Zr-90 near 0 degrees was performed at 295 MeV with a focus on a pronounced strength previously reported in the low-energy tail of giant dipole resonance. A forest of fine structure was observed in the excitation energy region 7-12 MeV. A multipole decomposition analysis of the angular distribution for the forest was carried out using the ECIS95 distorted-wave Born approximation code with the Hartree-Fock plus random-phase approximation model of E1 and M1 transition densities and inclusion of E1 Coulomb excitation. The analysis separated pygmy dipole and M1 resonances in the forest at E-PDR 9.15 +/- 0.18 MeV with Gamma(PDR) = 2.91 +/- 0.64 MeV and at E-M1 = 9.53 +/- 0.06 MeV with Gamma(M1) = 2.70 +/- 0.17 MeV in the Lorentzian function, respectively. The B(E1) up arrow value for pygmy dipole resonance over 7-11 MeV is 0.75 +/- 0.08 e(2) fm(2), which corresponds to 2.1 +/- 0: 2% of the Thomas-Reiche-Kuhn sum rule.

We extend the shell-model Monte Carlo applications to the rare-earth region to include the odd-even nucleus Dy-161. The projection on an odd number of particles leads to a sign problem at low temperatures making it impractical to extract the ground-state energy in direct calculations. We use level counting data at low energies and neutron resonance data to extract the shell model ground-state energy to good precision. We then calculate the level density of Dy-161 and find it in very good agreement with the level density extracted from experimental data.

The mean-field approximation predicts pairing and shape phase transitions in nuclei as a function of temperature or excitation energy. However, in the finite nucleus the singularities of these phase transitions are smoothed out by quantal and thermal fluctuations. An interesting question is whether signatures of these transitions survive despite the large fluctuations. The shell model Monte Carlo (SMMC) approach enables us to calculate the statistical properties of nuclei beyond the mean-field approximation in model spaces that are many orders of magnitude larger than spaces that can be treated by conventional diagonalization methods. We have extended the SMMC method to heavy nuclei and used it to study the transition from vibrational (spherical) to rotational (deformed) nuclei in families of rare-earth isotopes. We have calculated collective enhancement factors of level densities as a function of excitation energy and found that the decay of the vibrational and rotational enhancements is well correlated with the pairing and shape phase transitions, respectively.

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.