中田 仁

The semi-realistic NN interactions have been applied to nuclear structure studies in the mean-field and RPA framework. It has been clarified that the tensor force plays crucial roles in M1 strengths in Pb-208 and E2 strengths to 2(1)(+) in the Sn isotopes, as well as in Z- or N-dependence of the shell structure. The semi-realistic effective NN interactions provide us with a promising tool to describe nuclear properties on microscopic basis.

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.

Shell structure in the neutron-rich Ca and Ni nuclei is investigated by the spherical Hartree-Fock calculations with semi-realisticNN interactions. Specific ingredients of the effective interaction, particularly the tensor force, often play a key role in the Z dependence of the neutron shell structure. Such examples are found in N = 32 and N = 40; N = 32 becomes magic or submagic in (52)Ca while its magicity is broken in (60)Ni, and N = 40 is submagic (though not magic) in (68)Ni but not in (60)Ca. Comments are given on the doubly magic nature of (78)Ni. We point out that the loose binding can lead to a submagic number N = 58 in (86)Ni, assisted by the weak pair coupling.

New parameter sets of Michigan three-range Yukawa-type semirealistic nucleon-nucleon interaction are obtained by taking into account the whole contribution of the interaction to the pairing. The interactions are applicable to nuclear structure studies via mean-field calculations and their extensions. Implementing self-consistent mean-field calculations for the spherical nuclei, the author confirms that the results do not change significantly from previous results, in which the parameters were determined by partly discarding the influence of the density-dependent repulsion on the pairing.

Evolutions of single-particle energies and Z = 64 sub-shell along the isotonic chain of N = 82 are investigated in the density dependent relativistic Hartree-Fock (DDRHF) theory in comparison with other commonly used mean field models such as Skyrme HF, Gogny HFB and density dependent relativistic Hartree model (DDRMF). The pairing is treated in the BCS scheme, except for Cogny HFB. It is pointed out that DDRHF reproduces well characteristic features of experimental Z-dependence of both spin-orbit and pseudo-spin-orbit splittings around the sub-shell closure Z = 64. Non-local exchange terms of the isoscalar sigma and omega couplings play dominant roles in the enhancements of the spin-orbit splitting of proton 2d states, which is the key ingredient to give the Z = 64 sub-shell closure properly. On the other hand, the pi and rho tensor contributions for the spin-orbit splitting cancel each other and the net effect becomes rather small. The enhancement of the sub-shell gaps towards Z = 64 is studied by the DDRHF, for which the local terms of the scalar and vector meson couplings are found to be important. (C) 2009 Elsevier B.V. All rights reserved.

The Gaussian expansion method (GEM) is applied to calculations of the nuclear excitations in the random-phase approximation (RPA). We adopt the mass-independent basis-set that is successful in the mean-field calculations. The RPA results obtained by the GEM are compared with those obtained by several other available methods in Ca isotopes, by using a density-dependent contact interaction along with the Woods-Saxon single-particle states. It is confirmed that energies, transition strengths and widths of their distribution are described by the GEM with good precision, for the 1(-), 2(+) and 3(-) collective states. The GEM is then applied to the self-consistent RPA calculations with the finite-range Gogny DIS interaction. The spurious center-of-mass motion is well separated from the physical states in the E1 response, and the energy-weighted sum rules for the isoscalar transitions are fulfilled reasonably well. Properties of low-energy transitions in (60)Ca are investigated in some detail. (C) 2009 Elsevier B.V. All rights reserved.

Spin-flip M1 strengths in Pb-208 have been measured in photon scattering experiments with a quasi-monochromatic, linearly polarized photon beam. The data resolve an M1 giant resonance into at least seven, possibly eight, discrete transitions at excitation energies between 7.1 and 7.4 MeV below the neutron separation energy. The M1 strengths are measured with uncertainties considerably smaller than those in a previous study, which leads to a reexamination of the total strength. Experimental results are compared with an estimation of self-consistent random phase approximation using a semirealistic interaction.

Semirealistic nucleon-nucleon interactions applicable to the self-consistent mean-field (both Hartree-Fock and Hartree-Fock-Bogolyubov) calculations are developed by modifying the M3Y interaction. The modification is made to reproduce binding energies and rms matter radii of doubly magic nuclei, single-particle levels in Pb-208 and even-odd mass differences of the Sn isotopes. We find parameter sets with and without the tensor force. The new interactions are further checked by the saturation properties of the uniform nuclear matter, including the Landau-Migdal parameters. By the mean-field calculations, interaction dependence of the neutron drip line is investigated for the O, Ca, and Ni isotopes, and of the single-particle energies for the N=16,32,50, and 82 and Z=50 nuclei. Results of the semirealistic interactions including the tensor force are in fair agreement with available experimental data for all of these properties.

We have developed an efficient isospin projection method in the shell model Monte Carlo approach for isospin-conserving Hamiltonians. For isoscalar observables this method has the advantage of being exact sample by sample. It 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 the method to calculate 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 NZ nuclei.

We extend the shell model Monte Carlo approach to heavy deformed nuclei using a new proton-neutron formalism. The low excitation energies of such nuclei necessitate low-temperature calculations, for which a stabilization method is implemented in the canonical ensemble. We apply the method to study a well-deformed rare-earth nucleus, (162)Dy. The single-particle model space includes the 50-82 shell plus 1f(7/2) orbital for protons and the 82-126 shell plus 0h(11/2), 1g(9/2) orbitals for neutrons. We show that the spherical shell model reproduces well the rotational character of (162)Dy within this model space. We also calculate the level density of (162)Dy and find it to be in excellent agreement with the experimental level density, which we extract from several experiments.

We extensively develop a method of implementing mean-field calculations for deformed nuclei, using the Gaussian expansion method (GEM). This GEM algorithm has the following advantages: (i) it can efficiently describe the energy-dependent asymptotics of the wave functions at large r, (ii) it is applicable to various effective interactions including those with finite ranges, and (iii) the basis parameters are insensitive to nuclide, thereby many nuclei in wide mass range can be handled by a single set of bases. Superposing the spherical GEM bases with feasible truncation for the orbital angular momentum, we obtain deformed single-particle wave-functions to reasonable precision. We apply the new algorithm to the Hartree-Fock and the Hartree-Fock-Bogolyubov calculations of Mg nuclei with the Gogny interaction, by which neck structure of a deformed neutron halo is suggested for Mg-40. (C) 2008 Elsevier B.V. All rights reserved.

The La138 (T1/2=102 Gyr)-Ce138-Ce136 system is proposed to be used as a nuclear cosmochronometer for measuring the time elapsed from a supernova neutrino process. This chronometer is applied to examine a sample affected by a single nucleosynthesis episode as presolar grains in primitive meteorites. A feature of this chronometer is to evaluate the initial abundance ratio of Ce136/Ce138 using an empirical scaling law, which was found in the solar abundances. We calculate the age of the sample as a function of isotopic ratios Ce136/Ce138 and La138/Ce138 and evaluate the age uncertainty due to theoretical and observational errors. It is concluded that this chronometer can work well for a sample with the abundance ratio of La138/Ce138 ≥ 20 when the ratios of Ce136/Ce138 and La138/Ce138 are measured within the uncertainty of 20%. The availability of such samples becomes clear in recent studies of the presolar grains. We also discuss the effect of the nuclear structure to the ν process origin of La138. © 2008 The American Physical Society.

The La-138 (T-1/2=102 Gyr)-Ce-138-Ce-136 system is proposed to be used as a nuclear cosmochronometer for measuring the time elapsed from a supernova neutrino process. This chronometer is applied to examine a sample affected by a single nucleosynthesis episode as presolar grains in primitive meteorites. A feature of this chronometer is to evaluate the initial abundance ratio of Ce-136/Ce-138 using an empirical scaling law, which was found in the solar abundances. We calculate the age of the sample as a function of isotopic ratios Ce-136/Ce-138 and La-138/Ce-138 and evaluate the age uncertainty due to theoretical and observational errors. It is concluded that this chronometer can work well for a sample with the abundance ratio of La-138/Ce-138 >= 20 when the ratios of Ce-136/Ce-138 and La-138/Ce-138 are measured within the uncertainty of 20%. The availability of such samples becomes clear in recent studies of the presolar grains. We also discuss the effect of the nuclear structure to the nu process origin of La-138.

We introduce spin projection methods in the shell model Monte Carlo approach and apply them to calculate the spin distribution of level densities for iron-region nuclei using the complete (pf + g(9/2)) shell. We compare the calculated distributions with the spin-cutoff model and extract an energy-dependent moment of inertia. For even-even nuclei and at low excitation energies, we observe a significant suppression of the moment of inertia and odd-even staggering in the spin dependence of level densities.

We formulate a new Bardeen-Cooper-Schrieffer (BCS)-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. With its broad applicability, this theory can be a useful tool for investigating the pairing phase transition in finite systems with the particle-number conservation. This theory provides effects of the symmetry-restoring fluctuation (SRF) for the pairing phenomena in finite fermionic systems, distinctively from those of additional quantum fluctuations. It is shown by numerical calculations that the phase transition is compatible with the conservation in this theory, and that the SRF shifts up the critical temperature (T(cr)). This shift of T(cr) occurs due to reduction of degrees-of-freedom in canonical ensembles, and decreases only slowly as the particle-number increases (or as the level spacing narrows), in contrast to the conventional BCS theory.

To investigate the pairing phase transition in finite systems with the particle-number conservation, we formulate a new BCS-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. This theory enables us to distinguish the symmetry-restoring fluctuation (SRF) from additional quantum fluctuations. By numerical calculations, it is found that the phase transition is compatible with the conservation law in this theory, and that the SRF shifts up the critical temperature. Having correct zero-temperature limit, this theory also gives new interpretation on the Pauli blocking effect in systems with odd particle number.

We suggest a La - Ce - Ce system as a new nuclear cosmic clock for measuring the time elapsed from a single neutrino process episode to the present. This clock is applied to meteoritic samples produced mainly by a single nucleosynthesis episode. The nuclear structure of La is crucial for the performance of this clock and its astrophysical origin. The lowest 1+ state in La has not been established experimentally. If this state is a β unstable isomer, La should be destroyed in the n -process and the clock cannot work well. However, this system can be used as a new cosmic thermometer. To study the nuclear structure of La, we perform a shell model calculation. Our shell model calculation shows the 1+ state may be stable against the b decay and our proposed clock can perform. © Copyright owned by the author(s). 138 138 136 138 138 138 138

We extensively develop ail algorithm of implementing the Hartree-Fock-Bogolyubov calculations, in which the Gaussian expansion method is employed. This algorithm is advantageous in describing the energy-dependent exponential and oscillatory asymptotics of the quasiparticle wave functions at large r, and in handling various effective interactions including those with finite ranges. We apply the present method to the oxygen isotopes with the Gogny interaction, keeping the spherical symmetry. In respect to the new magic numbers, effects of the pair correlation on the N = 16 and 32 nuclei are investigated. (c) 2005 Elsevier B.V. All rights reserved.

By applying the particle-number projection to the finite-temperature Bardeen-Cooper-Schrieffer (BCS) theory, the S-shaped heat capacity, which has recently been claimed to be a fingerprint of the superfluid-to-normal phase transition in nuclei, is reexamined. It is found that the particle-number (or number-parity) projection gives S shapes in the heat capacity of nuclei that look qualitatively similar to the observed ones. These S shapes are accounted for as effects of the particle-number conservation on the quasiparticle excitations and occur even when we keep the superfluidity at all temperatures by assuming a constant gap in the BCS theory. The present study illustrates significance of the conservation laws in studying phase transitions of finite systems.

Applying thermofield dynamics, we reformulate the exact quantum number projection in the finite-temperature Hartree-Fock-Bogoliubov theory. Explicit formulas are derived for the simultaneous projection of particle number and angular momentum, in parallel to the zero-temperature case. We also propose a practical method for the variation-after-projection calculation, by approximating entropy consistently with the Peierls inequality. Using quantum number projection in finite-temperature mean-field theory will be useful for studying the effects of quantum fluctuations associated with the conservation laws on thermal properties of nuclei.

The structure of isomers in N similar to 82 nuclei is reinvestigated. It is pointed out that the quasi-spin structure survives in the isomers and the final states of their decays, as a partial dynamical symmetry. Using this group structure, an extended seniority reduction formula is derived and the presence of pair excitations out of the Z = 64 core is revealed. By taking into account the core excitation, an anomaly in neutron effective charges for N = 81 and N = 83 nuclei is greatly reduced.

Lifetimes of excited states in Cu-68 were measured by the gamma-gamma-t coincidence using two BaF2 detectors through the decay of Cu-68(m) produced by the (n,p) reaction. The half-life of the 2(+) state at 84 keV was obtained as 7.84(8) ns , corresponding to the B(M1; 2(+)-->1(+)) value of 0.00777(8) mu(N)(2). A shell model calculation with a minimum model space of pip(3/2)nu(1/2)(-1) gives a good prediction of this B(M1) value by using experimental g factors of neighboring nuclei. This small B(M1) value can also be explained by a shell model calculation in the f(7/2)(-r) (p(3/2)f(5/2)p(1/2))(n+r) (r=0,1) model space.

By introducing a density-dependent contact term, M3Y-type interactions applicable to the Hartree-Fock calculations are developed. In order to view basic characters of the interactions, we carry out calculations on the uniform nuclear matter as well as on several doubly magic nuclei. It is shown that a parameter set called M3Y-P2 describes various properties similarly well to the Skyrme SLy5 and/or the Gogny D1S interactions. A remarkable difference from the SLy5 and D1S interactions is found in the spin-isospin properties in the nuclear matter, to which the one-pion-exchange potential gives a significant contribution. Affecting the single-particle energies, this difference may play a certain role in the new magic numbers in unstable nuclei.

Under the assumption that isospin T is a good quantum number, isobaric analog states and various analogous transitions among these states are expected in isobars with mass number A. Strengths of analogous Gamow-Teller (GT) and M1 transitions have been compared for an A = 26 isobar triplet with T-z = + 1, 0, and - 1, where T-z is defined by T-z = (N - Z)/2. The T-z = 1 --> 0 GT transitions from the J(pi) = 0(+) ground state of Mg-26 to excited J(pi) = 1(+) states in Al-26 were studied up to the excitation energy (E-x) of 9 MeV by using a high energy-resolution (He-3,t) reaction. The distribution of observed GT strengths was well reproduced in a shell-model calculation. The isospin symmetric T-z = -1-->0 GT transitions can be studied in the Si-26 beta decay. The GT strengths from the (He-3,t) reaction were in good agreement with the beta-decay values evaluated up to E-x = 2.7-MeV states in 26Al. The GT strengths were further compared with the strengths of analogous M1 gamma transitions in Al-26, i.e., the M1 transitions from the excited 1(+) states to the isobaric analog state of the Mg-26 ground state in Al-26. Through this comparison, contributions of spin and orbital terms in these M1 transitions were studied. The GT and M1 strengths as well as spin and orbital contributions in M1 strengths are interpreted by both the shell model and the particle-rotor model assuming a correlated proton and neutron pair around a Mg-24 core. The isospin T of each excited 1(+) state in Al-26 was also studied by examining the existence or the nonexistence of the analog 1(+) state in the T-z = 1 Mg-26 nucleus.

Lifetimes of excited states in [Formula Presented] were measured by the [Formula Presented] coincidence using two [Formula Presented] detectors through the decay of [Formula Presented] produced by the [Formula Presented] reaction. The half-life of the [Formula Presented] state at [Formula Presented] was obtained as [Formula Presented], corresponding to the [Formula Presented] value of [Formula Presented]. A shell model calculation with a minimum model space of [Formula Presented] gives a good prediction of this [Formula Presented] value by using experimental [Formula Presented] factors of neighboring nuclei. This small [Formula Presented] value can also be explained by a shell model calculation in the [Formula Presented] model space. © 2003 The American Physical Society.

We have developed an exact isospin projection method in the framework of the shell model Monte Carlo approach that enables us to take into account the proper isospin dependence of the Hamiltonian. We apply the method to calculate level densities of N similar to Z nuclei, and demonstrate in the case of Cu-58 that the isospin projection can lead to significant corrections in N similar to Z nuclei.

Under the assumption that isospin T is a good quantum number, isobaric analog states and various analogous transitions among these states are expected in isobars with mass number A. Strengths of analogous Gamow-Teller (GT) and [Formula Presented] transitions have been compared for an [Formula Presented] isobar triplet with [Formula Presented] and [Formula Presented] where [Formula Presented] is defined by [Formula Presented] The [Formula Presented] GT transitions from the [Formula Presented] ground state of [Formula Presented] to excited [Formula Presented] states in [Formula Presented] were studied up to the excitation energy [Formula Presented] of 9 MeV by using a high energy-resolution [Formula Presented] reaction. The distribution of observed GT strengths was well reproduced in a shell-model calculation. The isospin symmetric [Formula Presented] GT transitions can be studied in the [Formula Presented] decay. The GT strengths from the [Formula Presented] reaction were in good agreement with the [Formula Presented]-decay values evaluated up to [Formula Presented] states in [Formula Presented] The GT strengths were further compared with the strengths of analogous [Formula Presented] transitions in [Formula Presented] i.e., the [Formula Presented] transitions from the excited [Formula Presented] states to the isobaric analog state of the [Formula Presented] ground state in [Formula Presented] Through this comparison, contributions of spin and orbital terms in these [Formula Presented] transitions were studied. The GT and [Formula Presented] strengths as well as spin and orbital contributions in [Formula Presented] strengths are interpreted by both the shell model and the particle-rotor model assuming a correlated proton and neutron pair around a [Formula Presented] core. The isospin T of each excited [Formula Presented] state in [Formula Presented] was also studied by examining the existence or the nonexistence of the analog [Formula Presented] state in the [Formula Presented] nucleus. © 2003 The American Physical Society.

We have developed a microscopic method to calculate nuclear level densities in the interacting shell model using auxiliary-field Monte Carlo techniques. The method has successfully been applied to nuclei in the mass range A ∼ 50 — 70. Shell effects and parity-dependence of the nuclear level densities are clarified by the present microscopic approach. © 2014 Atomic Energy Society of Japan. All Rights Reserved.

The 10(+) and 27/2(-) isomers of the Z>64, N=82 nuclei are investigated in the shell-model framework. We derive an extended seniority reduction formula for the relevant E2 transition strengths. Based on the extended formula, as well as on the approximate degeneracy among the Oh(11/2), 2s(1/2), and 1d(3/2) orbits, we argue that the B(E2) data require the Gd-146 core excitation. The energy levels of both parities and the B(E2) values are simultaneously reproduced by a multi-j shell-model calculation with the modified surface-delta interaction,if the excitations from (0g(7/2)1 d(5/2)) to (Oh(11/2)2S(1/2)1d(3/2)) are taken into account.

The [Formula Presented] and [Formula Presented] isomers of the [Formula Presented] nuclei are investigated in the shell-model framework. We derive an extended seniority reduction formula for the relevant [Formula Presented] transition strengths. Based on the extended formula, as well as on the approximate degeneracy among the [Formula Presented] and [Formula Presented] orbits, we argue that the [Formula Presented] data require the [Formula Presented] core excitation. The energy levels of both parities and the [Formula Presented] values are simultaneously reproduced by a multi-[Formula Presented] shell-model calculation with the modified surface-delta interaction, if the excitations from [Formula Presented] to [Formula Presented] are taken into account. © 2000 The American Physical Society.

A particle-number reprojection method is applied in the framework of the shell model Monte Carlo approach to calculate level densities for a family of nuclei using Monte Carlo sampling for a single nucleus. In particular we can also calculate level densities of odd-even and odd-odd nuclei despite a new sign problem introduced by the projection on an odd number of particles. The method is applied to level densities in the iron region using the complete pf + g(9/2)-shell. The single-particle level density parameter a and the backshift parameter a are extracted by fitting the microscopically calculated level densities to the backshifted Bethe formula. We find good agreement with experimental level densities with no adjustable parameters in the microscopic calculations. The parameter a is found to Vary smoothly with mass and does not show odd-even effects. The calculated backshift parameter Delta displays an odd-even staggering effect versus mass and is in better agreement with the experimental data than are the empirical values.

Total and parity-projected level densities of iron-region nuclei are calculated microscopically by using Monte Carlo methods for the nuclear shell model in the complete (pf + 0g(9/2))-shell. The calculated total level density is found to be in good agreement with the experimental level density. The Monte Carlo calculations offer a significant improvement over the thermal Hartree-Fock approximation. Contrary to the Fermi gas model, it is found that the level density has a significant parity-dependence in the neutron resonance region. The systematics of the level density parameters (including shell effects) in the iron region is presented.

We use the auxiliary fields Monte Carlo method for the shell model in the complete (pf + Og(9/2)) shell to calculate level densities. We introduce parity projection techniques which enable us to calculate the parity dependence of the level density. Results are presented for Fe-56, where the calculated total level density is found to be in good agreement with the experimental level density. The parity-projected densities are well described by a backshifted Bethe formula, but with significant dependence of the single-particle level-density and backshift parameters on parity. We compare our exact results with those of the thermal Hartree-Fock approximation.

In-beam gamma-ray spectroscopy studies of high-spin states of (62,64)2n have been performed via the fusion-evaporation reaction Ca-40 + Si-28 and a charged-particle filter. Levels up to an excitation energy of about 14 MeV have been observed. The results are compared with a shell-model calculation in the k less than or equal to 3 model space with (0f(5/2) 1p(3/2) 1p(1/2))(A-56-k) (0g(9/2))(k). The parity changes in yrast sequences can be well understood in terms of crossing of level sequences with different k configurations.

The distribution of E2 excitations from the ground state is studied for sd-shell even-even nuclei by a realistic shell-model. It is shown that the Q-phonon scheme for the 2(1)(+) state is quite reasonable in this region. The origin of the deviations from the pure Q-phonon scheme can be explained in terms of a Hamiltonian with broken SU(3) symmetry. The symmetry breaking is generated by non-degenerate single particle energies. (C) 1997 Published by Elsevier Science B.V.

We present a new method by which wave functions with simple structure an renormalized so as to contain more complicated structure. This method, called H-n-cooling method, is applied to the study of the quadrupole collective motion of Fe-56, Cr-54, Fe-58, and Cr-56. Th, shell-model wave functions of lowest-lying states of these nuclei are well treated by this method. By using the wave functions obtained via the H-n-cooling method, interacting boson model-2 parameters are derived from a realistic shell-model Hamiltonian and transition operators. The Majorana interaction becomes sizably repulsive, primarily as an effect of the renormalization. The bosonic E2 effective charges are enhanced due to the renormalization, while a quenching occurs in the M1 and M3 parameters for proton bosons. It is shown that the chi parameters take similar values in the Hamiltonian and in the E2 operator.

Gamow-Teller (GT) transitions in N = 28-30 nuclei are studied in terms of a large-scale realistic shell-model calculation, by using Towner's microscopic parameters. B(GT) values to low-lying final states are reproduced with a reasonable accuracy. Several gross properties with respect to the GT transitions are investigated with this set of wavefunctions and operator. While the calculated total GT(-) strengths show no apparent disagreement with the measured ones, the calculated total GT(+) strengths are somewhat larger than those obtained from charge-exchange experiments. With respect to the Ikeda sum rule, the proportionality of S-GT to (N - Z) persists to an excellent approximation, with a quenching factor of 0.68. For the relative GT(-) strengths among possible isospin components, the lowest isospin component gathers a greater proportion than expected by the squared CG coefficients of the isospin coupling. It turns out that these relative strengths are insensitive to the size of the model space. The systematics of the summed B(GT) values are discussed for each isospin component.

A new boson mapping procedure is proposed by using the NCQP approximation, which was introduced by Li. This method is applicable to systems with non-degenerate orbits, while it agrees with the OAI mapping in the degenerate limit.