Restricted Model Space Research Articles

Prompt-delayed γ -ray spectroscopy of neutron-rich In119,121 isotopes

The fusion and transfer induced fission reaction $^{9}$Be($^{238}$U,~f) with 6.2 MeV/u beam energy, using a unique setup consisting of AGATA, VAMOS++ and EXOGAM detectors, was used to populate through the fission process and study the neutron-rich $^{119,121}$In isotopes. This setup enabled the prompt-delayed $\gamma$-ray spectroscopy of isotopes in the time range of $100~\rm{ns} - 200~\mu\rm{s}$. In the odd-$A$ $^{119,121}$In isotopes, indications of a short half-life $19/2^{-}$ isomeric state, in addition to the previously known $25/2^{+}$ isomeric state, were observed from the present data. Further, new prompt transitions above the $25/2^{+}$ isomer in $^{121}$In were identified along with reevaluation of its half-life. The experimental data were compared with the theoretical results obtained in the framework of large-scale shell-model calculations in a restricted model space. The $\langle \pi g_{9/2} \nu h_{11/2};I \arrowvert \hat{\mathcal{H}}\arrowvert \pi g_{9/2} \nu h_{11/2};I\rangle$ two-body matrix elements of residual interaction were modified to explain the excitation energies and the $B(E2)$ transition probabilities in the neutron-rich In isotopes. The (i) decreasing trend of $E(29/2^{+}) - E(25/2^{+})$ in odd-In (with dominant configuration $\pi g_{9/2}^{-1}\nu h_{11/2}^{-2}$ and maximum aligned spin of $29/2^{+}$) and (ii) increasing trend of $E(27/2^{+}) - E(23/2^{+})$ in odd-Sb (with dominant configuration $\pi g_{7/2}^{+1}\nu h_{11/2}^{-2}$ and maximum aligned spin of $27/2^{+}$) with increasing neutron number could be understood as a consequence of hole-hole and particle-hole interactions, respectively.

Effects of one valence proton on seniority and angular momentum of neutrons in neutron-rich Sb51122–131 isotopes

The neutron-rich $^{122-131}$Sb isotopes were produced as fission fragments in the reaction $^{9}$Be($^{238}$U,~f) with 6.2 MeV/u beam energy. An unique setup, consisting of AGATA, VAMOS++ and EXOGAM detectors, was used which enabled the prompt-delayed gamma-ray ($\gamma$) spectroscopy of fission fragments in the time range of 100 ns - 200 $\mu$s. New isomers, prompt and delayed transitions were established in the even-A $^{122-130}$Sb isotopes. In the odd-A $^{123-131}$Sb isotopes, new prompt and delayed $\gamma$-ray transitions were identified, in addition to the confirmation of the previously known isomers. The half-lives of the isomeric states and the $B(E2)$ transition probabilities of the observed transitions depopulating these isomers were extracted. The experimental data was compared with the theoretical results obtained in the framework of Large-Scale Shell-Model (LSSM) calculations in a restricted model space. Modifications of several components of the shell model interaction were introduced to obtain a consistent agreement with the excitation energies and the $B(E2)$ transition probabilities in neutron-rich Sn and Sb isotopes. The isomeric configurations in Sn and Sb were found to be relatively pure. Further, the calculations revealed that the presence of a single valence proton, mainly in the $g_{7/2}$ orbital in Sb isotopes, leads to significant mixing (due to the $\nu\pi$ interaction) of: (i) the neutron seniorities ($\upsilon_{\nu}$) and (ii) the neutron angular momentum ($I_{\nu}$). The above features have a weak impact on the excitation energies, but have an important impact on the $B(E2)$ transition probabilities. In addition, a constancy of the relative excitation energies irrespective of neutron seniority and neutron number in Sn and Sb was observed.

Restricted model domain time Radon transforms

The coefficients that synthesize seismic data via the hyperbolic Radon transform (HRT) are estimated by solving a linear-inverse problem. In the classical HRT, the computational cost of the inverse problem is proportional to the size of the data and the number of Radon coefficients. We have developed a strategy that significantly speeds up the implementation of time-domain HRTs. For this purpose, we have defined a restricted model space of coefficients applying hard thresholding to an initial low-resolution Radon gather. Then, an iterative solver that operated on the restricted model space was used to estimate the group of coefficients that synthesized the data. The method is illustrated with synthetic data and tested with a marine data example.

Description of pairing in finite like-particle systems in a formalism of quartets

We provide a description of the ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions in terms of four-body correlated structures (quartets). These are real superpositions of products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. We show some realistic applications of the quartet formalism in the case of Sn isotopes.

Treatment of the like-particle pairing of quartets

The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of the products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In the case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J = 0. Realistic applications of the quartet formalism are carried out for Sn isotopes with valence neutrons in the 50–82 neutron shell. The exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected BCS approach.

Multipair approach to pairing in nuclei

The ground state of a general pairing Hamiltonian for a finite nuclear system is constructed as a product of collective, real, distinct pairs. These are determined sequentially via an iterative variational procedure that resorts to diagonalizations of the Hamiltonian in restricted model spaces. Different applications of the method are provided that include comparisons with exact and projected BCS results. The quantities that are examined are correlation energies, occupation numbers and pair transfer matrix elements. In a first application within the picket-fence model, the method is seen to generate the exact ground state for pairing strengths confined in a given range. Further applications of the method concern pairing in spherically symmetric mean fields and include simple exactly solvable models as well as some realistic calculations for middle-shell Sn isotopes. In the latter applications, two different ways of defining the pairs are examined: either with J=0 or with no well-defined angular momentum. The second choice reveals to be more effective leading, under some circumstances, to solutions that are basically exact.

Identification of genes and variants associated with quantitative traits using Bayesian factor screening

We propose a factor-screening method based on a Bayesian model selection framework and apply it to Genetic Analysis Workshop 17 simulated data with unrelated individuals to identify genes and SNP variants associated with the quantitative trait Q1. A Metropolis-Hasting algorithm is implemented to generate a posterior distribution in a restricted model space and thus the marginal posterior distribution of each variant. Our framework provides flexibility to make inferences on either individual variants or genes. We obtained results for 10 simulated data sets. Our methods are able to identify FTP1 and KDR, two genes that are associated with Q1 in a majority of replicates.

A frequentist assessment of Bayesian inclusion probabilities for screening predictors

Bayesian inclusion probabilities have become a popular tool for variable assessment. From a frequentist perspective, it is often difficult to evaluate these probabilities as typically no Type I error rates are considered, neither are any explorations of power of the methods given. This paper considers how a frequentist may evaluate Bayesian inclusion probabilities for screening predictors. This evaluation looks at both unrestricted and restricted model spaces and develops a framework which a frequentist can utilize inclusion probabilities that preserve Type I error rates. Furthermore, this framework is applied to an analysis of the Arabidopsis thaliana with respect to determining quantitative trait loci associated with cotelydon opening angle.

Marine electromagnetic inverse solution appraisal and uncertainty using model‐derived basis functions and sparse geometric sampling

ABSTRACTWe summarize, for marine electromagnetic inverse problems, a newly developed inverse solution appraisal and non‐linear uncertainty estimation method based on parameter reduction techniques and efficient posterior model space sampling. This method uses model compression methods to decorrelate parameters in an inverse solution and represent all feasible posterior models as linear combinations of a small number of model‐derived basis vectors and corresponding coefficients. This allows us to reduce the posterior sampling problem by orders of magnitude. We further contract this reduced‐dimensional posterior space by confining all acceptable models to a set of bounds mapped from our original parameter space. As a final step to increase efficiency, we implement a geometric sampling scheme that we use to approximate our restricted posterior by generating feasible models on adaptive, optimally‐sparse grids. The sampled equi‐feasible models are accepted according to a data misfit threshold and constitute an optimally‐sparse representation of the restricted posterior model space. Although very efficient, our method imposes a bias in the posterior space by truncating the basis expansion during the model reduction step. To investigate this, we compare two types of fast and scalable bases, the discrete cosine transform and singular value decomposition. We demonstrate that while the choice of base does influence the type of models sampled and the model rejection rates, the posterior statistics are generally compatible between the methods providing confidence in the uncertainty estimations. For the marine electromagnetic problem, we show that a representative ensemble of equivalent inverse solutions can be generated for realistically‐sized inverse problems and that solution appraisal and uncertainty inference follow directly from ensemble statistics.

Converging sequences in theab initiono-core shell model

We demonstrate the existence of multiple converging sequences in the ab initio no-core shell model. By examining the underlying theory of effective operators, we expose the physical foundations for the alternative pathways to convergence. This leads us to propose a revised strategy for evaluating effective interactions for A-body calculations in restricted model spaces. We suggest that this strategy is particularly useful for applications to nuclear processes in which states of both parities are used simultaneously, such as for transition rates. We demonstrate the utility of our strategy with large-scale calculations in light nuclei.

Structure of the isomeric states inSb123,125

Excited states in $^{123,125}\mathrm{Sb}$ have been studied following the $^{122,124}\mathrm{Sn}$($^{7}\mathrm{Li}$,\ensuremath{\alpha}2n)$^{123,125}\mathrm{Sb}$ reactions at beam energies of 35 and 37 MeV, respectively. Conversion coefficients for transitions depopulating isomeric states in $^{125}\mathrm{Sb}$ have been measured using an electron spectrometer. This has allowed firm ${J}^{\ensuremath{\pi}}$ assignments to be made and enabled the structure of the isomeric states to be compared with those in ${}^{121,127\ensuremath{-}131}$Sb and in the neighboring even-$A$ tin isotopes. The half-lives of the ${J}^{\ensuremath{\pi}}={\frac{23}{2}}^{+}$ isomeric states in $^{123}\mathrm{Sb}$ and $^{125}\mathrm{Sb}$ have been measured to be 66(4) \ensuremath{\mu}s and 272(16) ns, respectively. The half-lives of the ${J}^{\ensuremath{\pi}}={\frac{19}{2}}^{+},{\frac{19}{2}}^{\ensuremath{-}}$, and ${\frac{15}{2}}^{\ensuremath{-}}$ states in $^{125}\mathrm{Sb}$ have been measured as 31(2) ns, 28.0(7) \ensuremath{\mu}s, and 4.1(2) \ensuremath{\mu}s, respectively. The transition probabilities for the transitions depopulating these states are compared with those in ${}^{121,127\ensuremath{-}131}$Sb and with the results of shell-model calculations using the Oxbash code and the SN100PN interaction in a restricted model space.

β decay ofS40,42andCl43

Results from the study of the \ensuremath{\beta} decays of $^{40,42}\mathrm{S}$ and $^{43}\mathrm{Cl}$, produced in the fragmentation of a 70-MeV/nucleon $^{48}\mathrm{Ca}$ beam, are presented. The half-lives for $^{42}\mathrm{S}$ and $^{43}\mathrm{Cl}$ have been measured to be $1.03\ifmmode\pm\else\textpm\fi{}0.03$ s and $3.13\ifmmode\pm\else\textpm\fi{}0.09$ s, respectively. On the basis of \ensuremath{\gamma}-ray singles and $\ensuremath{\gamma}\ensuremath{\gamma}$ coincidence data, decay schemes for each of these decays have been established. Only subtle changes in low-energy nuclear structure, including switching of the order of the ${2}^{\ensuremath{-}}$ and ${3}^{\ensuremath{-}}$ states in the odd-odd Cl isotopes, were observed when passing the midpoint of the $\ensuremath{\nu}{f}_{7/2}$ subshell. These results agree well with previous shell-model calculations using restricted model spaces suggesting that deformation and shell-breaking effects in the midshell region are small. However, excitations across the $N=20$ shell gap are important to explain the low-lying positive parity states.

Analyzing Quantitative Trait Loci for the Arabidopsis thaliana using Markov Chain Monte Carlo Model Composition with restricted and unrestricted model spaces

Quantitative Trait Loci (QTL) mapping is a growing field in statistical genetics. However, dealing with this type of data from a statistical perspective is often perilous. In this paper we extend and apply a Markov Chain Monte Carlo Model Composition (MC 3) technique to a data set of the Arabidopsis thaliana plant for locating the QTL mapping associated with cotyledon opening. The posterior model probabilities as well as the marginal posterior probabilities of each locus belonging to the model are presented. Furthermore, we show how the MC 3 method can be used to deal with the situation where the sample size is less than the number of parameters in a model using a restricted model space approach.

Conservation of connectivity of model-space effective interactions under a class of similarity transformation.

Effective interaction operators usually act on a restricted model space and give the same energies (for Hamiltonian) and matrix elements (for transition operators, etc.) as those of the original operators between the corresponding true eigenstates. Various types of effective operators are possible. Those well defined effective operators have been shown to be related to each other by similarity transformation. Some of the effective operators have been shown to have connected-diagram expansions. It is shown in this paper that under a class of very general similarity transformations, the connectivity is conserved. The similarity transformation between Hermitian and non-Hermitian Rayleigh-Schrodinger perturbative effective operators is one of such transformations and hence the connectivity can be deducted from each other.

Non-Hermitian perturbative effective operators: Connectivity and derivation of diagrammatic representation

Effective Hamiltonians and effective operators act on a restricted model space to give the same energies and matrix elements as those of the full Hamiltonian and operators between the corresponding true eigenstates. For the effective Hamiltonian there are two “obvious” choices: the simplest non-Hermitian effective Hamiltonian and the canonical Hermitian effective Hamiltonian. In this paper, we derive a perturbative effective operator which works together with the non-Hermitian effective Hamiltonian, prove that it can be expanded with only connected diagrams, and show how to construct the connected diagrams easily from the diagrams of the effective Hamiltonian by substitution of vertices. This effective operator is much simpler than the Hermitian effective operator and therefore is expected to be more suitable for ab initio calculations.

Level structure of 99Rh at high spins

High spins states in 99Rh were populated via the 66Zn(37Cl, 2p2n)99Rh reaction at an incident beam energy of 130 MeV. Seventeen new transitions have been observed in the present study and the level scheme has now been extended up to a spin of J∼ 25ħ and an excitation energy of about E x∼ 10 MeV. The observation of a positive parity E2 cascade based on the 9/2+ isomeric level is suggestive of collective behaviour in this nucleus up to high spins. Spherical shell model (within restricted model space) and Cranked shell model calculations were performed to obtain an insight into the observed level structure. The new collective band observed up to a spin of J∼ 25ħ is suggested to be based on (πg 9/2 3) ⊗ (νg 7/2 2) quasi-particle excitations.

Level structure of94,95,96Tcat high spins and shell-model calculations

High-spin states in the ${}^{94,95,96}\mathrm{Tc}(N=51,$ 52, and 53) nuclei have been investigated using the ${}^{65}\mathrm{Cu}{+}^{36}\mathrm{S}$ reaction at a beam energy of 142 MeV. More than 60 new transitions have been identified and placed in their level schemes, which now extend up to spin $J\ensuremath{\approx}22\ensuremath{\Elzxh}$ and excitation energies ${E}_{x}\ensuremath{\approx}12$ MeV. Spherical shell-model calculations have been performed using different model spaces. A restricted model space, using ${}^{88}\mathrm{Sr}$ as the core and the $\ensuremath{\pi}{(p}_{1/2}{,g}_{9/2})$ $\ensuremath{\nu}{(d}_{5/2}{,s}_{1/2})$ valence orbitals, reproduces the experimental excitation energies up to $J\ensuremath{\approx}14\ensuremath{\Elzxh}.$ The higher-angular-momentum states are dominated by the excitation of a ${g}_{9/2}$ neutron across the $N=50$ magic core, as indicated by large-basis shell model calculations.

Level structures of96,97,98Ruat high angular momentum

The high-spin level structures of {sup 96,97,98}Ru (Z=44) have been investigated using the {sup 65}Cu({sup 36}S,pxn){sup 96,97,98}Ru (x=4,3,2) reactions. About 130 new transitions have been observed and unambiguously placed in the decay schemes of these nuclei. The level schemes have been extended up to spin J{approx}22{endash} 34{h_bar}, and excitation energies E{sub x}{approx}20{minus}24 MeV. Spherical shell model calculations have been performed and theoretical level energies compared with experimental values. Calculations using {sup 88}Sr as the core give a reasonable agreement for the observed energy levels up to J{le}16{h_bar} (the maximum angular momentum possible within this restricted model space). The higher angular momentum states are dominated by the {nu}(g{sub 9/2}){sup {minus}1} configuration, associated with breaking of the N=50 core. The experimental indications for this `core breaking` are (i) the observation of {gamma} rays with energies E{sub {gamma}}{approx} 2 MeV at intermediate spins and (ii) the fragmentation of {gamma}-ray intensity into a number of branches. A cascade of E2 transitions of nearly equal transition energies has been observed in {sup 98}Ru after the core breaking (J{ge}16{h_bar}), possibly manifesting vibrational behavior. {copyright} {ital 1998} {ital The American Physical Society}

Fully microscopic model of 200 MeV proton-12C elastic and inelastic scattering.

An effective two nucleon (NN) interaction in the nuclear medium is defined from an accurate mapping of the NN g matrices obtained by solving the Brueckner-Bethe-Goldstone equations for infinite nuclear matter. That effective interaction is used in a fully microscopic calculation of the nonlocal effective proton${\mathrm{\ensuremath{-}}}^{12}$C interaction from which we obtain predictions of the differential cross section and analyzing power for 200 MeV elastic scattering. The relative motion wave functions so found are used as the distorted waves in a distorted wave approximation (DWA) study of select inelastic scattering events. The effective NN interaction is used as the transition operator in those calculations. The relevant nuclear spectroscopy for the elastic and DWA (p,p\ensuremath{'}) calculations is found from a full (0+2)\ensuremath{\Elzxh}\ensuremath{\omega} shell model evaluation of the positive parity states while a restricted shell model space (1+3)\ensuremath{\Elzxh}\ensuremath{\omega} has been used to give the negative parity states. Results are compared with those of the 0p-shell model of Cohen and Kurath or with those based upon axially symmetric, projected Hartree-Fock calculations. The diverse structure model wave functions are assessed by using them in calculations to compare with measured longitudinal, transverse electric, and transverse magnetic form factors from electron scattering to many of the excited states of $^{12}\mathrm{C}$. Using those models of the structure of $^{12}\mathrm{C}$ in our completely microscopic model of the elastic and inelastic scattering of 200 MeV protons, good fits have been found to the cross section and analyzing power data.

Effective Operators in Time-Independent Approach

An effective operator Θeff is defined so that it operates in a restricted model space and gives the same matrix element as that of the original operator Θ between the corresponding true eigenstates. The Θeff is determined dependently on the model-space eigenstates, and therefore various types of Θeff are possible. General solutions for Θeff are derived in the time-independent and algebraic approach. These Θeff contain the usual non-Hermitian and Hermitian effective operators as special cases. The explicit expansion form for Θeff is given by extending the \hatQ-box formalism of Kuo et al. developed in the derivation of the effective interaction to the problem of constructing the effective operator.