Mean-field Basis Research Articles

β-decay of N = 126 isotones for the r-process nucleosynthesis

The β-decay properties of nuclei with neutron number N = 126 is investigated in this paper. Two different versions of the proton-neutron quasi particle random phase (pn-QRPA) model were employed to compute β-decay rates and half-lives for the N = 126 isotones. The first set of calculation solves the pn-QRPA equations using the schematic model (SM) approach. The Woods-Saxon potential was employed as a mean-field basis. A spherical shape assigned for each waiting point nuclei throughout all simulations. Both allowed Gamow-Teller (GT) and first-forbidden (FF) transitions were considered in the particle-hole (ph) channel. The second set uses the pn-QRPA model in deformed Nilsson basis to calculate β-decay rates for allowed GT and unique first-forbidden (U1F) transitions under terrestrial and stellar conditions. Our results are in agreement with shell model findings that first-forbidden transitions lead to a considerable decrement in the calculated half-lives of the isotones. Inclusion of the first-forbidden contribution led to a decent agreement of our computed terrestrial β-decay half-lives with measured ones, much better than the previous calculations. The possible implication of the waiting point nuclei on r-process nucleosynthesis is discussed briefly.

Spectral fluctuations and thermalization in the f5p-shell space

Chaos against thermalization, which was earlier investigated in the sd-shell space, is examined in the f5p-shell space. The two-body residual interaction of f5pvh with realistic single-particle energies have been utilized for carrying out f5p-shell space calculations. The present study demonstrates that the spectral fluctuations computed with realistic single-particle energies are in accordance with the Gaussian orthogonal ensemble of random matrices. Besides, the equilibrium heating is extremely related with the progress of many-body quantum chaos. Moreover, the single-particle entropy of individual states in the mean field basis is dissimilar to the thermodynamic entropy determined from the level density. The present results, which are a check of the ideas of many-body quantum chaos in a different version of the nuclear shell model applied to another part of the periodic table, are fully confirming what was formerly achieved in the sd-shell space and in this way exhibit their generality.

Embedded random matrix ensembles from nuclear structure and their recent applications

Embedded random matrix ensembles generated by random interactions (of low body rank and usually two-body) in the presence of a one-body mean field, introduced in nuclear structure physics, are now established to be indispensable in describing statistical properties of a large number of isolated finite quantum many-particle systems. Lie algebra symmetries of the interactions, as identified from nuclear shell model and the interacting boson model, led to the introduction of a variety of embedded ensembles (EEs). These ensembles with a mean field and chaos generating two-body interaction generate in three different stages, delocalization of wave functions in the Fock space of the mean-field basis states. The last stage corresponds to what one may call thermalization and complex nuclei, as seen from many shell model calculations, lie in this region. Besides briefly describing them, their recent applications to nuclear structure are presented and they are (i) nuclear level densities with interactions; (ii) orbit occupancies; (iii) neutrinoless double beta decay nuclear transition matrix elements as transition strengths. In addition, their applications are also presented briefly that go beyond nuclear structure and they are (i) fidelity, decoherence, entanglement and thermalization in isolated finite quantum systems with interactions; (ii) quantum transport in disordered networks connected by many-body interactions with centrosymmetry; (iii) semicircle to Gaussian transition in eigenvalue densities with [Formula: see text]-body random interactions and its relation to the Sachdev–Ye–Kitaev (SYK) model for majorana fermions.

Extracting the mass dependence and quantum numbers of short-range correlated pairs fromA(e,e′p)andA(e,e′pp)scattering

The nuclear mass dependence of the number of short-range correlated (SRC) proton-proton (pp) and proton-neutron (pn) pairs in nuclei is a sensitive probe of the dynamics of short-range pairs in the ground state of atomic nuclei. This work presents an analysis of electroinduced single-proton and two-proton knockout measurements off 12C, 27Al, 56Fe, and 208Pb in kinematics dominated by scattering off SRC pairs. The nuclear mass dependence of the observed A(e,e'pp)/12C(e,e'pp) cross-section ratios and the extracted number of pp- and pn-SRC pairs are much softer than the mass dependence of the total number of possible pairs. This is in agreement with a physical picture of SRC affecting predominantly nucleon-nucleon pairs in a nodeless relative-S state of the mean-field basis.

Trapping Effect on the Kinetic Critical Radius in Nucleation and Growth Processes

The critical nucleus size—above which nuclei grow, below dissolve—during diffusion controlled nucleation in binary solid-solid phase transformation process is calculated using kinetic Monte Carlo (KMC). If atomic jumps are slower in an A-rich nucleus than in the embedding B-rich matrix, the nucleus traps the A atoms approaching its surface. It doesn’t have enough time to eject A atoms before new ones arrive, even if it would be favourable thermodynamically. In this case the critical nucleus size can be even by an order of magnitude smaller than expected from equilibrium thermodynamics or without trapping. These results were published in [Z. Erdélyi et al., Acta Mater. 58 (2010) 5639]. In a recent paper M. Leitner [M. Leitner, Acta Mater. 60 (2012) 6709] has questioned our results based on the arguments that his simulations led to different results, but he could not point out the reason for the difference. In this paper we summarize our original results and on the basis of recent KMC and kinetic mean field (KMF) simulations we show that Leitner’s conclusions are not valid and we confirm again our original results.

Quantifying short-range correlations in nuclei

An approximate method to quantify the magnitude of the two-nucleon (2N) and three-nucleon (3N) short-range correlations (SRC) and their mass dependence is proposed. The proposed method relies on the concept of the "universality" or "local nuclear character" of the SRC. We quantify the SRC by computing the number of independent-particle model (IPM) nucleon pairs and triples which reveal beyond-mean-field behavior. It is argued that those can be identified by counting the number of nucleon pairs and triples in a zero relative orbital momentum state. A method to determine the quantum numbers of pairs and triples in an arbitrary mean-field basis is outlined. The mass dependence of the 2N and 3N SRC is studied. The predictions are compared to measurements. This includes the ratio of the inclusive inelastic electron scattering cross sections of nuclei to 2H and 3He at large values of the Bjorken variable. Corrections stemming from the center-of-mass motion of the pairs are estimated. We find that the relative probability per nucleon for 2N and 3N SRC has a soft dependence with mass number A and that the proton-neutron 2N SRC outnumber the proton-proton (neutron-neutron) 2N SRC. A linear relationship between the magnitude of the EMC effect and the predicted number of proton-neutron SRC pairs is observed. This provides support for the role of local nuclear dynamics on the EMC effect.

Excited states of incipient Wigner molecules

An accurate configuation-interaction method employing a mean-field basis set is used to study the excitation spectrum of localized Wigner states in the strongly interacting regime of a quasi-two-dimensional parabolic quantum dot with $N=6$ electrons. The approach achieves errors of order 1 part in 10${}^{4}$ (or better) in the energies of low-lying states for a Wigner-Seitz radius ${r}_{s}=12 {a}_{0}^{*}$--$16 {a}_{0}^{*}$, and is used to study low-lying spin, rotational, vibrational, and isomeric excitations. The vibrational excitations at ${r}_{s}=50 {a}_{0}^{*}$ are shown to correspond qualitatively with the classical normal modes of the $N=6$ electron dot, although the excitation energies agree only semiquantitatively with the classical normal-mode frequencies as a result of residual quantum fluctuations.

Isomeric and hybrid isomeric-vibrational states of Wigner molecules

An accurate configuration-interaction method employing a numerical mean-field basis set is used to study the excitation spectrum of Wigner molecules, including isomeric excitations, in small parabolic quantum dots. We find that at intermediate electron densities $({r}_{s}\ensuremath{\sim}8{a}_{0}^{\ensuremath{\ast}}--20{a}_{0}^{\ensuremath{\ast}})$, in the regime of strong interaction and partial Wigner localization, there are inversions of the usual Born-Oppenheimer ordering of isomeric and vibrational excitations, yielding low-lying isomeric excitations. Quantum-mechanical hybridization of different isomers can occur near an avoided crossing of a vibrational and an isomeric excitation. These findings suggest the possibility of observing isomers and hybrid isomeric-vibrational states experimentally.

Precise correlation energies in small parabolic quantum dots from configuration interaction

We calculate precise correlation energies of ground and low-lying excited states in circular parabolic quantum dots containing $N=2\char21{}20$ electrons by means of a configuration interaction (CI) method with a numerical, mean-field basis set. All excitations are allowed for $2\ensuremath{\le}N\ensuremath{\le}7$ (full CI), while up to hextuple excitations are included for $N=8,9$, and up to pentuple excitations for $10\ensuremath{\le}N\ensuremath{\le}20$. The energies are extrapolated to the limit of basis-set completeness and the truncation error due to restricting the number of Slater determinants is monitored. For high electron densities (Wigner-Seitz radius ${r}_{s}\ensuremath{\approx}1.7{a}_{0}^{\ensuremath{\ast}}$), the approach achieves errors of order $0.3\text{ }{\text{mHa}}^{\ensuremath{\ast}}$ for $N=3$, a few ${\text{mHa}}^{\ensuremath{\ast}}$ for $N=6\char21{}9$, rising to about $100\text{ }{\text{mHa}}^{\ensuremath{\ast}}$ for $N=20$. A comparison is made with recent quantum Monte Carlo calculations.

Half-lives of rp-process waiting point nuclei

We give results of microscopic calculations for the half-lives of various proton-rich nuclei in the mass region A = 60-90, which are involved in the astrophysical rp-process, and which are needed as input parameters of numerical simulations in Nuclear Astrophysics. The microscopic formalism consists of a deformed QRPA approach that involves a self-consistent quasiparticle deformed Skyrme Hartree-Fock basis and residual spin-isospin separable forces in both the particle-hole and particle-particle channels. The strength of the particle-hole residual interaction is chosen to be consistent with the Skyrme effective force and mean-field basis, while that of the particle-particle is globally fixed to κ = 0.07 MeV after a judicious choice from comparison to experimental half-lives. We study and discuss the sensitivity of the half-lives to deformation and residual interactions.

Metal-insulator transition in the quarter-filled frustrated checkerboard lattice

We study the electronic structure and correlations in the geometrically frustrated two dimensional checkerboard lattice. In the large U limit considered here we start from an extended Hubbard model of spinless fermions at half-filling. We investigate the model within two distinct Green's function approaches: In the first approach a single-site representation decoupling scheme is used that includes the effect of nearest neighbor charge fluctuations. In the second approach a cluster representation leading to a 'multiorbital' model is investigated which includes intra-cluster correlations exactly and those between clusters on a mean field basis. It is demonstrated that with increasing nearest-neighbor Coulomb interaction V both approaches lead to a metal-insulator transition with an associated 'Mott-Hubbard' like gap caused by V. Within the single site approach we also explore the possibility of charge order. Furthermore we investigate the evolution of the quasiparticle bands as funtion of V.

Gamow–Teller strength distributions in Fe and Ni stable isotopes

We study Gamow–Teller strength distributions in some selected nuclei of particular astrophysical interest within the iron mass region. The theoretical framework is based on a proton–neutron quasiparticle random phase approximation built on a deformed selfconsistent mean field basis obtained from two-body density-dependent Skyrme forces. We compare our results to available experimental information obtained from ( n, p) and ( p, n) charge exchange reactions.

Single-particle entropy in(1+2)-body random matrix ensembles

Random matrix ensembles defined by a mean-field one-body plus a chaos generating random two-body interaction (called embedded Gaussian orthogonal ensembles of (1+2)-body interactions[EGOE(1+2)]) predict for the entropy defined by the occupation numbers of single-particle states, in the chaotic domain, an essentially one parameter Gaussian form for their energy dependence. Numerical embedded ensemble calculations are compared with the theory. In addition, it is shown that the single-particle entropy, thermodynamic entropy defined by the state density and information entropy defined by wave functions in the mean-field basis for EGOE(1+2) describe the results known for interacting Fermi systems such as those obtained from nuclear shell model.

Entropy production and wave packet dynamics in the Fock space of closed chaotic many-body systems.

Highly excited many-particle states in quantum systems such as nuclei, atoms, quantum dots, spin systems, quantum computers, etc., can be considered as "chaotic" superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is due to a very high level density of many-body states that are easily mixed by a residual interaction between particles (quasiparticles). For such systems, we have derived simple analytical expressions for the time dependence of the energy width of wave packets, as well as for the entropy, number of principal basis components, and inverse participation ratio, and tested them in numerical experiments. It is shown that the energy width Delta(t) increases linearly and very quickly saturates. The entropy of a system increases quadratically, S(t) approximately t(2), at small times, and afterward can grow linearly, S(t) approximately t, before saturation. Correspondingly, the number of principal components determined by the entropy N(pc) approximately exp[S(t)] or by the inverse participation ratio increases exponentially fast before saturation. These results are explained in terms of a cascade model which describes the flow of excitation in the Fock space of basis components. Finally, the striking phenomenon of damped oscillations in the Fock space at the transition to equilibrium is discussed.

Energy and isospin dependence of nuclear chaos

Energy levels and wave functions, obtained in realistic shell-model calculations for Ca and Sc isotopes up to $A=52$ and ${}^{46}\mathrm{Ti},$ are analyzed using standard statistics such as the nearest level spacing distribution, the Dyson-Mehta ${\ensuremath{\Delta}}_{3},$ and the mean localization length. These statistics are calculated for different energy regions of the spectrum. For all the Ca isotopes, in the ground state region the energy levels show strong deviations from Gaussian orthogonal ensemble predictions. It is shown that a transition to a more chaotic regime takes place as excitation energy increases. However, even when the full spectrum is taken into account, the ${\ensuremath{\Delta}}_{3}$ and the degree of localization of the eigenfunctions in the mean-field basis prove that Ca isotopes are less chaotic than Sc isotopes. A comparison with ${}^{46}\mathrm{Ti}$ shows that this nucleus is still more chaotic. Thus we find a clear isospin dependence in the degree of nuclear chaoticity.

Mean Field Theory Interpretation of Solar Polarity Reversal

A mechanism of the polarity reversal of the solar magnetic field is explored on the basis of the mean field or turbulent dynamo theory. In the low-latitude region of the convective zone, the toroidal magnetic field, which is the origin of sunspots, is generated by the rotational motion of fluids, with the turbulent cross helicity as the intermediary. This field generates the poloidal field of dipole type through the alpha or turbulent helicity effect. The latter, in turn, contributes to the annihilation of the turbulent cross helicity, resulting in the decay of the toroidal magnetic field. This process indicates less room for the occurrence of the fully developed poloidal field in the low-latitude region and paves the way for the polarity reversal through the change of the sign of the turbulent cross helicity. A simple model mimicking the periodic polarity reversal is presented, and the relationship of the reversal period to the ratio of the poloidal to toroidal fields is given. The meridional-flow velocity at the solar surface is estimated, giving a result consistent with observations.

Time Dynamics in Chaotic Many-body Systems: Can Chaos Destroy a Quantum Computer?

Highly excited many-particle states in quantum systems (nuclei, atoms, quantum dots, spin systems, quantum computers) can be ‘chaotic’ superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is a result of the very high energy level density of many-body states which can be easily mixed by a residual interaction between particles. We consider the time dynamics of wave functions and increase of entropy in such chaotic systems. As an example, we present the time evolution in a closed quantum computer. A time scale for the entropy S(t) increase is t c ~τ 0 /(n log 2 n), where τ 0 is the qubit ‘lifetime’, n is the number of qubits, S(0) = 0 and S(t c )=1. At t _ t c the entropy is small: S ~nt 2 J 2 log 2 (1/t 2 J2 ), where J is the inter-qubit interaction strength. At t > t c the number of ‘wrong’ states increases exponentially as 2 S(t) . Therefore, t c may be interpreted as a maximal time for operation of a quantum computer. At t >>t c the system entropy approaches that for chaotic eigenstates.

Invariant correlational entropy and complexity of quantum states

We define correlational (von Neumann) entropy for an individual quantum state of a system whose time-independent hamiltonian contains random parameters and is treated as a member of a statistical ensemble. This entropy is representation independent and can be calculated as a trace functional of the density matrix which describes the system in its interaction with the noise source. We analyze perturbation theory in order to show the evolution from the pure state to the mixed one. Exactly solvable examples illustrate the use of correlational entropy as a measure of the degree of complexity in comparison with other available suggestions such as basis-dependent information entropy. It is shown in particular that a harmonic oscillator in a uniform field of random strength comes to a quasithermal equilibrium; we discuss the relation between effective temperature and canonical equilibrium temperature. The notion of correlational entropy is applied to a realistic numerical caculation in the framework of the nuclear shell model. In this system, which reveals generic signatures of quantum chaos, correlational entropy and information entropy calculated in the mean field basis display similar qualitative behavior.

QUANTUM CHAOS AND COMPLEXITY IN NUCLEI

▪ Abstract A mesoscopic system with strong interaction between constituents, the atomic nucleus is a good candidate for studying the manifestations of quantum chaos. Recent experimental and theoretical developments give clear evidence of the dominant role chaotic dynamics plays not only in local level statistics, but also in the damping of collective motion, isospin and parity nonconservation, thermalization, and decay into continuum. In this review, those and other phenomena are discussed from the general viewpoint of quantum chaos and complexity. The complexity of eigenstates measured by information entropy in the mean-field basis is more informative than the standard signatures of chaos in the level statistics. It evolves regularly as a function of excitation energy and provides an alternative temperature scale. This sheds new light on the problems of foundations of statistical mechanics and quantum decoherence.

Chaos vs thermalization in the nuclear shell model.

Generic signatures of quantum chaos found in realistic shell model calculations are compared with thermal statistical equilibrium. We show the similarity of the informational entropy of individual eigenfunctions in the mean-field basis to the thermodynamical entropy found from the level density. Mean occupation numbers of single-particle orbitals agree with the Fermi-Dirac distribution despite the strong nucleon interaction.