Single-particle Matrix Elements Research Articles

Relative importance of orbital and spin contributions in the study of magnetic properties in nuclei

We study the relative importance of the orbital and intrinsic spin contribution in both the diagonal magnetic single-particle matrix elements and in the matrix elements of the fermion magnetic multipole operator for many-body proton and/or neutron wave functions. The latter results shed light on the effective gyromagnetic factors for dipole ($L=1$) and octupole ($L=3$) operators, to be used in interacting boson model calculations.

Renormalized dipole moment operator in the second half of the2p−1fshell

Renormalized single-particle matrix elements of the magnetic dipole moment operator for the (${2p}_{\frac{3}{2}}$, ${1f}_{\frac{5}{2}}$, ${2p}_{\frac{1}{2}}$) space are calculated in the framework of perturbation theory involving first- and second-order excitations from the ${1f}_{\frac{7}{2}}$ core. The calculated dipole matrix elements are consistent with the empirical values deduced from the observed $M1$ transitions as well as dipole moments in several Ni and Cu isotopes.[NUCLEAR STRUCTURE Renormalized magnetic dipole moment operator; restricted valence space spanned by the ${2p}_{\frac{3}{2}}$, ${1f}_{\frac{5}{2}}$, and ${2p}_{\frac{1}{2}}$ orbits; first- and second-order diagrams; empirical values of the effective $M1$ single-particle matrix elements.]

Corrections to the free-nucleon values of the single-particle matrix elements of theM1and Gamow-Teller operators, from a comparison of shell-model predictions withsd-shell data

The magnetic dipole moments of states in mirror pairs of the $\mathrm{sd}$-shell nuclei and the strengths of the Gamow-Teller beta decays which connect them are compared with predictions based on mixed-configuration shell-model wave functions. From this analysis we extract the average effective values of the single-particle matrix elements of the $l$, $s$, and ${[{Y}^{(2)}\ensuremath{\bigotimes}s]}^{(1)}$ components of the $M1$ and Gamow-Teller operators acting on nucleons in the $0{d}_{\frac{5}{2}}$, $1{s}_{\frac{1}{2}}$, and $0{d}_{\frac{3}{2}}$ orbits. These results are compared with the recent calculations by Towner and Khanna of the corrections to the free-nucleon values of these matrix elements which arise from the effects of isobar currents, mesonic-exchange currents, and mixing with configurations outside the $\mathrm{sd}$ shell.RADIOACTIVITY $A=17\ensuremath{-}39$ nuclei, magnetic dipole moments and Gamow-Teller beta-decay strengths; shell-model wave functions, complete $0\ensuremath{\hbar}\ensuremath{\omega}$ $0{d}_{\frac{5}{2}}\ensuremath{-}1{s}_{\frac{1}{2}}\ensuremath{-}0{d}_{\frac{3}{2}}$ basis space, empirical Hamiltonian; extraction of empirical normalizations of the $l$, $s$, and ${[{Y}^{(2)}\ensuremath{\bigotimes}s]}^{(1)}$ single-particle operators.

Analysis of IBFA-model assumptions with the broken-pair model

Assuming a correspondence of boson-model states and fermion-pair states, we use the broken-pair model to compute quantities Γ ab , Λ ab j and A j and check whether these numbers fulfill the relations adopted in IBFA-model fits. Results are shown for 125Xe and 131Xe. The part of the IBFA hamiltonian which changes the number of d-bosons by one, parametrized by Γ ab , can very well be represented in its usual form as a single-particle matrix element of the quadrupole operator. The part parametrized by A j seems to be also well justified. For the part parametrized by Λ ab j , a similarity is found between our computed numbers and IBFA-model fit prescriptions. A small class of terms is missing in IBFA-model fits however.

Single-particle model of capture to continuum states

A formalism is developed for treating the single-particle description of continuum to continuum electromagnetic transitions. Convenient formulae are given for the measured observables and for the single-particle matrix elements of the transition operators. Target recoil is taken explicitly into account. The long-wavelength approximation is not made. Convergence is demonstrated both for the radial integrals and with respect to partial waves and transition multipoles. The model is compared with available experimental data. Many of the observed features are reproduced.

Level structure and lifetimes of low-excitation states in 63Cu

The lifetimes of 9 states in 63Cu up to an excitation energy of 2092 keV have been determined by means of the Doppler-shift attenuation method using the 60 Ni(α, pγ) reaction. A search was made for a suitable set of M1 and E2 single-particle matrix elements for use with shell-model eigenstates throughout the Ni, Cu and Zn isotopes. The theoretical predictions for the level spectroscopy and electromagnetic transitions and static moments in 63Cu are in satisfactory agreement with the experimental findings.

Empirical Renormalization of the One-Body Gamow-Tellerβ-Decay Matrix Elements in the1s−0dShell

..cap alpha..-decay data from nuclei with 17 < or = A < or = 39 are analyzed with full sd-space shell-model wave functions to determine the effective values of the single-particle matrix elements of the Gamow-Teller operator between all combinations of sd-shell orbitals.

A quantitative evaluation of the reliability of calculated decay properties of nuclei in the mass region A=24?28

Electromagnetic transition rates and logft values were calculated for transitions between positive-parity states in theA=24−28 mass region. The wave functions used were taken from a previous paper. In general we found satisfactory agreement with experiment. In order to have a measure of the stability of the results against changes in the Hamiltonian a method was developed for assigning errors to calculated transition properties. The renormalized single-particle matrix elements of theE2 andM 1 transition operators were determined in a phenomenological way. To this end use was made of the errors just mentioned. It was found that good agreement was obtained with bare-nucleonM 1 singleparticle matrix elements and a state independent effective isoscalar charge for theE 2 operator. Predictions for static moments are given.

Survey of single-particle states in the mass regionA&gt;228

A deformed single-particle potential mode, with residual interactions, is applied to an analysis of states in odd-mass nuclides with $Ag228$, and configuration assignments are presented for many levels. The systematics of energy level spacings throughout the actinides are studied, and the values of the nuclear deformation parameters that are appropriate to these nuclides are discussed. Tables of occupation probabilities and single-particle matrix elements are provided to facilitate the comparison of future measurements with theoretical expectations.

Causality, equal-time algebra, and light-cone commutators

Extending previous work we discuss the sum rules that follow from causality, scaling, and equal-time algebra for the nonforward (spinless) single-particle matrix element of the commutator of two conserved vector currents, using the recently introduced refined infinite-momentum technique. The complete set of sum rules is found to include those obtained from light-cone commutators. It also contains some additional ones.

Operator product expansions at large distances and Regge asymptotic behavior

The class of theories is considered in which the simplest high-energy scattering processes are described by the exchange of a Regge pole or cut and in which light-cone behavior is described by operator product expansions. The scattering processes are those in which an initial particle interacts with an initial cluster of other particles to produce a final particle and final cluster. The high energy limit is that in which the energy of the initial particle relative to the energy of any particle in the clusters becomes large with fixed masses, momentum transfers and cluster subenergies. It is shown that these limits can be completely characterized by an operator statement on the product of the interpolating fields of the initial and final particles, independently of the initial and final clusters. Given then that these limits correspond to large distance limits in space-time, operator product expansions are obtained which describe in an operator manner the behavior of a product A( x) B(0) of local fields in the large distance limit x 0 → ∞ with x 2 fixed. This provides an exact operator picture of a Regge pole being built out of an infinite set of elementary spin exchanges. The crucial ingredient in the derivation is the factorization property of Regge poles or Regge cut discontinuities. The leading light-cone contribution to the scattering amplitude will then have the same factorization properties. The operator expansion is derived first for single particle matrix elements, first using commutativity of the Regge and scaling limits and second using diagonalized Bethe-Salpeter equations for the matrix elements of the local fields in the light-cone expansion. It is then deduced for single-double particle matrix elements using the second method where now helicity sums play an important role. It is finally derived for arbitrary matrix elements. Non-leading Regge poles and Regge cuts can be similarly described by using also non-leading light-cone singularities.

Evaluation of the Nuclear Charge Form Factor with Intrinsic Hyperspherical-Coordinate Wave Functions

Evaluation of the nuclear charge form factor requires the evaluation of a many-body matrix element. In this paper it is shown how this many-body matrix element can be reduced to an integral over single-particle matrix elements when hyperspherical coordinates are used.

Position operators in relativistic quantum theory: Analysis of algebraic operator relationships

This work is the first of a series of three papers examining different aspects of position operators in relativistic quantum theory. In this paper the properties of the position operator X and the velocity operator V are derived for single particle matrix elements in the context of the Poincaré generator algebra. Both the physical meaning and the mathematical implications of each property are discussed. The algebraic structure of the extended set of relationships including the Poincaré generators, X and V is examined. It is found that this set defines an infinite algebra which is intractable mathematically. The Casimir operators of the Poincaré algebra are required to be Casimir operators for X and V, a new condition on V is formulated, and a simple solution for K is constructed. These conditions, together with familiar position operator properties, give the constraints and solutions for the extended algebra.

Shell-model calculations on the nickel isotopes

Binding and excitation energies, electromagnetic transition rates and multipole moments have been calculated for 57–66Ni from many-particle shell-model wave functions with neutrons in the 2 p 3 2 , 1 f 5 2 and 2 p 1 2 orbits outside the 56Ni core. The efiective two-body matrix elements are obtained from the modified surface delta interaction. The average absolute deviation between the calculated and experimental binding and excitation energies is 0.11 MeV (with the exclusion of 66Ni). Electric quadrupole transition rates and moments are calculated with an effective neutron charge e n = (1.70±0.08) e, obtained from a least-squares fit to experimental data. Magnetic dipole transition rates and moments follow from five fitted effective reduced singleparticle matrix elements. The average absolute deviations between theory and experiment for the strongest E2 transitions (1–15 W.u.) and for the strongest M1 transitions (0.01–0.14 W.u.) are 3.5 and 0.03 W.u., while the average measured strengths are 7.0 and 0.07 W.u., respectively.

Shell-model calculations for masses 27, 28 and 29: electromagnetic transition rates and multipole moments

Electromagnetic transition probabilities, multipole moments and log ƒt values have been calculated from many-particle shell-model wave functions in a truncated 1 d 5 2 2 s 1 2 1 d 3 2 config space, with a maximum of four holes in the 1 d 5 2 subshell. The electric quadrupole transition strengths and moments are reproduced very well in a least-squares fit to 74 experimental data with one parameter, for the isoscalar effective charge, yielding the values e p = 1.6 e and e n = 0.6 e. The results for magnetic dipole transition strengths and moments follow from adjusting two effective reduced single-particle matrix elements in separate least-squares fits to 17 experimental data in A = 27 nuclei and 21 data in A = 29 nuclei. The average absolute deviations between theory and experiment for E2 and M1 transition strengths are 3.0 and 0.05 W.u., while the average measured strengths are 7.7 and 0.08 W.u., respectively. Transitions from excited states above E x = 4.8 MeV in 27A1 and from some low-lying states in 27Al and 28Si are poorly reproduced by the present model. Calculated strengths of transitions from analogue states are given. Previous conclusions about the single-particle character of M1 transitions and the collective behaviour of E2 transitions are confirmed. The experimental data of seven A = 27–29 nuclei are well reproduced in one general treatment with an appreciably lower number of free parameters than are required to obtain comparable results in collective model calculations.

Critical Remarks on the Asymptotic Symmetry Scheme

The asymptotic symmetry scheme proposed by Oneda and Matsuda is analyzed rather critically on two counts. First of all, we discuss the underlying reasons given for the scheme's basic assumption, that single-particle matrix elements of the SU(3) raising operator ${V}^{K}$ are effectively not renormalized, even when the symmetry is broken, and we find that some of the arguments are not completely convincing. Second, it is shown that most of the results of the scheme can be obtained without using this basic assumption, and that it is only most of the results derived from equal-time commutators of the form $[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}^{K},{A}^{i}]=0$ which are necessary consequences of this assumption. These latter results include all of their intermultiplet mass sum rules together with a few other predictions such as the degeneracy of the ${\ensuremath{\Sigma}}^{0}$ and ${\ensuremath{\Lambda}}^{0}$, and, in general, these are not as well satisfied experimentally as the other results.

Aspects of the Scalar-Dominance Hypothesis

The scalar-dominance hypothesis, incorporating $\ensuremath{\epsilon}(700)\ensuremath{-}{\ensuremath{\epsilon}}^{\ensuremath{'}}(1060)$ mixing, is applied to a large class of elastic, single-particle matrix elements of the energy-momentum trace operator.

Many-particle shell model calculation of electromagnetic transition rates and multipole moments in A = 30–34 nuclei

Many-particle shell model wavefunctions have been used to calculate magnetic dipole and electric quadrupole transition probabilities and moments as well as log ft values for even-parity states in the A = 30–34 region. The calculations were performed: (i) for bare-nucleon g factors and charges; (ii) for effective g factors and charges obtained from a least-squares fit to experimental data; (iii) for effective single-particle matrix elements obtained from an analogous fit. For the magnetic dipole transitions and moments it appeared necessary to perform separate fits for the A = 30–32 and 33–34 regions. The final best fits for strong transitions are excellent. The average deviation between theory and experiment with three effective single-particle matrix elements for M1 transitions and effective charges for E2 transitions is about 30 %, only slightly larger than the average experimental error. Show cases are 31Si and 33S where the calculated lifetimes, branching ratios and mixing ratios (including the sign) all agree with the measured values within the experimental error. Poor fits, however, are obtained for transitions involving 1 + states and, more generally, states for which the experimental and theoretical spectroscopic factors are in disagreement. Strong M1 transitions generally have a pronounced singe-particle character (with one contribution in the matrix element predominating), whereas all E2 transitions have a collective character (many small contributions).

Octet behaviour of single-particle matrix elements 〈B′| Hw|B〉 and 〈M′| Hw|M〉 using a weak current-current quark Hamiltonian

The SU(3) transformation behaviour of single-particle matrix elements 〈B′| Hw(0)|B〉 and 〈M′| Hw(0)|M〉 is investigated under the assumption that the transitions are induced by a weak current-current type quark Hamiltonian. The baryons B and mesons M are taken to be members of the 56 and 35 multiplets, respectively. For Fermi quarks the baryon matrix elements vanish identically and the meson matrix elements transform as octets. For para quarks and in the three-triplet model the baryon matrix elements transform as octets, whereas the meson matrix elements have a strong 27 component. A slightly modified Hamiltonian is introduced which induces octet transitions for both mesons and baryons.

Shell-model calculations on deformed nuclei

A slightly modified deformed nuclear single-particle potential is investigated in detail in the rare earth and actinide regions. Its parameters are determined so as to reproduce optimally the observed single-particle level order in deformed nuclei. The most important single-particle matrix elements connected with decoupling factors and magnetic moments are calculated, and in these cases, a Y 4 0 term in the potential is included. Some collective properties such as quadrupole moments are calculated.