Transition Strength Function Research Articles

How effective is the Brink–Axel hypothesis for astrophysical weak rates?

Abstract We explore the effectiveness of the Brink–Axel hypothesis (BAH) for the computation of stellar electron capture (EC) and β-decay (BD) rates, namely that the transition strength function depends only upon the transition energy and not upon the details of the initial state. For this purpose, we calculated Gamow–Teller (GT) strength distributions for a selection of sd-shell nuclides, using two different microscopic models, namely the proton–neutron quasiparticle random phase approximation and the full configuration-interaction shell model, taking into account the first 100 states of both the initial and final nuclides. The GT transition strengths among these levels evolve with initial state energy. These transition strength functions we folded into weak-interaction mediated rates in stellar matter, specifically EC and BD rates, for a range of densities 10 g cm−3 ⩽ ρ ⩽ 1011 g cm−3 and range of temperatures 1 GK ⩽ T ⩽ 30 GK. When transitions from excited states were approximated using the BAH, augmented by so-called ‘back-resonance’ transitions, the rates were affected by up to three orders of magnitude or more at high temperatures and densities. Thus the BAH is not a reliable approximation for the calculation of stellar rates, especially in high temperature–density environments.

Complex scaling: Physics of unbound light nuclei and perspective

Abstract The complex scaling method (CSM) is one of the most powerful methods of describing the resonances with complex energy eigenstates based on non-Hermitian quantum mechanics. We present the basic application of CSM to the properties of the unbound phenomena of light nuclei. In particular, we focus on many-body resonant and non-resonant continuum states observed in unstable nuclei. We also investigate the continuum level density (CLD) in the scattering problem in terms of the Green’s function with CSM. We discuss the explicit effects of resonant and non-resonant contributions in CLD and transition strength functions.

Thouless-Valatin moment of inertia and removal of the spurious mode in the linear response theory

Symmetry breaking at the mean-field level leads to an appearance of a symmetry restoring Nambu-Goldstone (NG) mode in the linear response theory. These modes represent a special kind of collective motion of the system. However, they can interfere with the calculated intrinsic physical excitations and, hence, they are often called as spurious modes. I discuss translational and rotational NG mode and the inertia parameter associated with these modes, by using the finite amplitude method formalism. I will also discuss how to remove spurious mode from the calculated transition strength function.

Nuclear Structure of Zirconium Isotopes 90,96,106Zr Using Skyrme HF-RPA

The ground state properties of the closed-shell 90Zr and closed-subshell 96,106Zr isotopes were investigated using a self-consistent spherical Hartree-Fock HF method and Random Phase Approximation RPA with five different Skyrme type effective nucleon-nucleon interaction: KDE0v1, BsK1, SIII, SVII and SGOI. Having a large number of Skyrme-force parameterizations requires a continuous search for the best for describing the experimental data. The presented approach attempts to accurately describe the structure of the Zirconium isotopes 90, 96, 106Zr. The calculated binding energies, root-mean squares, charge distributions, single particle energies were compared with the experimental data. Moreover, the transition strength function S(E) of the isoscalar giant monopole resonance (ISGMR), Jπ = 0+, T=0, the isoscalar giant dipole resonance (ISGDR), Jπ = 1-, T=0, isoscalar giant quadrupole resonance (ISGQR), Jπ = 2+, T=0, and isoscalar giant octupole resonance (ISGOR), Jπ = 3-, T=0 were calculated for the investigated isotopes and compared with the available experimental data.

Transition sum rules in the shell model

An important characterization of electromagnetic and weak transitions in atomic nuclei are sum rules. We focus on the non-energy-weighted sum rule (NEWSR), or total strength, and the energy-weighted sum rule (EWSR); the ratio of the EWSR to the NEWSR is the centroid or average energy of transition strengths from an nuclear initial state to all allowed final states. These sum rules can be expressed as expectation values of operators, in the case of the EWSR a double commutator. While most prior applications of the double-commutator have been to special cases, we derive general formulas for matrix elements of both operators in a shell model framework (occupation space), given the input matrix elements for the nuclear Hamiltonian and for the transition operator. With these new formulas, we easily evaluate centroids of transition strength functions, with no need to calculate daughter states. We apply this simple tool to a number of nuclides, and demonstrate the sum rules follow smooth secular behavior as a function of initial energy, as well as compare the electric dipole (E1) sum rule against the famous Thomas-Reiche-Kuhn version. We also find surprising systematic behaviors for ground state electric quadrupole (E2) centroids in the $sd$-shell.

Temperature-dependent dispersion model of float zone crystalline silicon

In this paper, we present the temperature dependent dispersion model of float zone crystalline silicon. The theoretical background for valence electronic excitations is introduced in the theoretical part of this paper. This model is based on application of sum rules and parametrization of transition strength functions corresponding to the individual elementary phonon and electronic excitations. The parameters of the model are determined by fitting ellipsometric and spectrophotometric experimental data in the spectral range from far infrared (70cm-1) to extreme ultraviolet (40eV). The ellipsometric data were measured in the temperature range 5–700K. The excitations of the valence electrons to the conduction band are divided into the indirect and direct electronic transitions. The indirect transitions are modeled by truncated Lorentzian terms, whereas the direct transitions are modeled using Gaussian broadened piecewise smooth functions representing 3D and 2D van Hove singularities modified by excitonic effects. Since the experimental data up to high energies (40eV) are available, we are able to determine the value of the effective number of valence electrons. The Tauc–Lorentz dispersion model is used for modeling high energy electron excitations. Two slightly different values of the effective number of valence electrons are obtained for the Jellison–Modine (4.51) and Campi–Coriasso (4.37) parametrization. Our goal is to obtain the model of dielectric response of crystalline silicon which depends only on photon energy, temperature and small number of material parameters, e.g. the concentration of substituted carbon and interstitial oxygen. The model presented in this paper is accurate enough to replace tabulated values of c-Si optical constants used in the optical characterization of thin films diposited on silicon substrates. The spectral dependencies of the optical constants obtained in our work are compared to results obtained by other authors.

Monopole transition strength function of C12 in a three- α model

The energy level structure of ${}^{12}$C nucleus at a few MeV above the three-$\alpha$ threshold is still unsatisfactory known. For instance, most microscopic calculations predicted that there exist one $0^+$-state in this energy region besides the well known Hoyle state, while some experimental and theoretical studies show the existing of two $0^+$-states. In this paper, I will take a three-$\alpha$-boson (3$\alpha$) model for bound and continuum states in ${}^{12}$C, and study a transition process from the ${}^{12}$C($0_1^+$) ground state to 3$\alpha$ $0^+$ continuum states by the electric monopole ($E0$) operator. The strength distribution of the process will be calculated as a function of $3\alpha$ energy using the Faddeev three-body theory. The Hamiltonian for the $3\alpha$ system consists of two- and three-$\alpha$ potentials, and some three-$\alpha$ potentials with different range parameters will be examined. Results of the strength function show a double-peaked bump at low energy region, which can be considered as two $0^+$-states. The peak at higher energy may originate from a 3$\alpha$ resonant state. However, it is unlikely that the peak at the lower energy is related to a resonant state, which suggests that it may be due to so called "ghost anomaly". Distributions of decaying particles are also calculated.

Multipole modes in deformed nuclei within the finite amplitude method

Background: To access selected excited states of nuclei, within the framework of nuclear density functional theory, the quasiparticle random phase approximation (QRPA) is commonly used. Purpose: We present a computationally efficient, fully self-consistent framework to compute the QRPA transition strength function of an arbitrary multipole operator in axially-deformed superfluid nuclei. Methods: The method is based on the finite amplitude method (FAM) QRPA, allowing fast iterative solution of QRPA equations. A numerical implementation of the FAM-QRPA solver module has been carried out for deformed nuclei. Results: The practical feasibility of the deformed FAM module has been demonstrated. In particular, we calculate the quadrupole and octupole strength in a heavy deformed nucleus $^{240}$Pu, without any truncations in the quasiparticle space. To demonstrate the capability to calculate individual QRPA modes, we also compute low-lying negative-parity collective states in $^{154}$Sm. Conclusions: The new FAM implementation enables calculations of the QRPA strength function throughout the nuclear landscape. This will facilitate global surveys of multipole modes and beta decays, and will open new avenues for constraining the nuclear energy density functional.

Systematics of strength function sum rules

Sum rules provide useful insights into transition strength functions and are often expressed as expectation values of an operator. In this letter I demonstrate that non-energy-weighted transition sum rules have strong secular dependences on the energy of the initial state. Such non-trivial systematics have consequences: the simplification suggested by the generalized Brink–Axel hypothesis, for example, does not hold for most cases, though it weakly holds in at least some cases for electric dipole transitions. Furthermore, I show the systematics can be understood through spectral distribution theory, calculated via traces of operators and of products of operators. Seen through this lens, violation of the generalized Brink–Axel hypothesis is unsurprising: one expects sum rules to evolve with excitation energy. Furthermore, to lowest order the slope of the secular evolution can be traced to a component of the Hamiltonian being positive (repulsive) or negative (attractive).

Broadening of dielectric response and sum rule conservation

Different types of broadening of the dielectric response are studied with respect to the preservation of the Thomas–Reiche–Kuhn sum rule. It is found that only the broadening of the dielectric function and transition strength function conserve this sum rule, whereas the broadening of the transition probability function (joint density of states) increases or decreases the sum. The effect of different kinds of broadening is demonstrated for interband and intraband direct electronic transitions using simplified rectangular models. It is shown that the broadening of the dielectric function is more suitable for interband transitions while broadening of the transition strength function is more suitable for intraband transitions.

Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models

The classical f-sum rule is generalized using quantum mechanical Thomas–Reiche–Kuhn sum rule to include nucleonic contribution, i.e. lattice vibrations. The sum rule is formulated for the transition strength function defined as a continuous condensed-matter equivalent of the oscillator strength known for discrete transitions in atomic spectra. The application of such formulated sum rule allows construction of dispersion models containing a fitting parameter directly related to the atomic density of material. The dielectric response expressed using the transition strength function is split into individual contributions such as direct and indirect electronic interband transitions including excitonic effect, excitations of electrons to the high-energy states existing above the conduction band, core-electron excitations and phonon absorption. The presented models reflect understanding of structure of disordered and crystalline materials on the basis of quantum theory of solids. The usual term ‘joint density of states’, that should be used only for electronic transitions in the one-electron approximation, is replaced by the more general term ‘transition density’.

Fine structure of strength functions for beta decays of atomic nuclei

The β transition strength function S β(E) is the nuclear excitation energy distribution of moduli squared of the β-decay-type matrix elements. The function S β(E) determines the characteristics of β decay, the spectra of accompanying radiation, and the probabilities of delayed processes following the β decay. Until recently, tools widely used for experimental investigation of the S β(E) structure have been total absorption gamma spectrometers and total absorption gamma-ray spectroscopy (TAGS) which could not provide high energy resolution. Development of experimental techniques allows nuclear spectroscopy methods with high energy resolution to be used for studying the fine structure of S β(E). A thorough investigation of this kind has been carried out for a number of nuclei produced at the YASNAPP-2 facility in Dubna. In this review, studies involving works on measuring the fine structure of S β(E) in spherical and deformed nuclei are analyzed. Modern nuclear spectroscopy methods made it possible to identify the splitting of peaks from nuclear deformation in S β(E) for Gamow-Teller (GT) transitions. The resonance nature of S β(E) for first-forbidden (FF) transitions in both spherical and deformed nuclei is experimentally proven. It is shown that for some nuclear excitation energies, FF transitions can be comparable in intensity with GT transitions. Criteria for verifying the completeness of nuclear decay schemes are considered. The S β(E) functions obtained by TAGS and by the high-resolution spectroscopy are compared.

β-decay of [formula omitted]Ca

The β-decay of the Tz=−52 nucleus 35Ca was studied at the LISE3 spectrometer at GANIL. The 35Ca decay scheme was deduced from its β-delayed proton emission into the ground and excited states of 34Ar and from its β-delayed two-proton emission. The 35Ca half-life was determined to be 25.7±0.2 ms. The measured transition strength function B(GT) is compared to results obtained from large-scale sd-shell model calculations.

Structure and reactions of three-body Borromean halos

A three-body model for two-neutron halos is applied on 6He and 11Li to describe ground and continuum properties, S-matrix poles, transition strength functions, invariant mass spectra and momentum distributions, including diffraction, in break-up reactions. We relate these properties to those of 5He and 10Li. Essentially all measured results are reproduced, except the p-resonance observed at about 0.5 MeV in 10Li.

High resolution Gamow-teller strength measurement via β-decay of 65Ge

The β + decay of 65Ge was studied by β-delayed proton and γ-ray emission. The He-jet technique was used. Altogether 28 proton groups (1.0 MeV< E p<2.1 MeV) were observed with proton energy resolution of about 10 keV. The proton spectrum apparently consists of discrete proton groups due to a selective population of excited states of 65Ga by Gamow-Teller β + decay. The total proton branching ratio was measured to be (1.1±0.3)×10 −4. Upper limit of log ft values for beta transitions associated with the observed delayed proton groups and γ-rays were calculated and an experimental beta transition strength function was obtained. The single level structure of 65Ga was studied by the 64Zn(p, γ) 65Ga resonance reaction in the proton energy range 1.1 MeV< E p<2.1 MeV with 1.0 keV overall energy resolution and about 170 resonance levels were observed. Angular distribution and resonance decay measurements were done to study the nature of selected single levels and decay properties of low energy levels. The proton binding energy of 65Ga was measured to be B p = 3943.0±1.0 keV.

New Resonances in Photon Transition Strength Functions

Received 19 December 1962DOI:https://doi.org/10.1103/PhysRevLett.10.299©1963 American Physical Society