Multipole Matrix Research Articles

Discrete-spectrum contribution to the Lamb-shift of the 2S and 2P hydrogenic states

A general formula due to Moses [Phys. Rev. A8, 1710 (1973)] for exact electric and magnetic multipole transition matrix elements in nonrelativistic hydrogenic atoms is reduced by carrying out additional integrations. By using the obtained reduced formula unrenormalized (finite) and renormalized Lamb-shift contributions stemming from individual discrete states as well as from the whole discrete spectrum are calculated in the case of the states 2S and 2P.

Excitations of 21+ and 31- states from (p,p') and electromagnetic measurements.

A reanalysis of (p,p') data taken on ${2}_{1}^{+}$ and ${3}_{1}^{\mathrm{\ensuremath{-}}}$ states of 37 even-even A\ensuremath{\ge}40 nuclei at energies of E50 MeV has been performed with a consistent procedure involving coupled-channels calculations and Becchetti-Greenlees optical model parameters. The matrix elements extracted from this analysis have been compared to matrix elements deduced from electromagnetic measurements of the same excitations. It is concluded that the differences between (p,p') and electromagnetic results which are often observed do not result from variations in the method of analysis of the (p,p') data. Instead, the differences are caused by experimental errors and nuclear structure effects. The nuclear structure effects are examined by extracting the ratios ${\mathit{M}}_{\mathit{n}}$/${\mathit{M}}_{\mathit{p}}$ of the neutron and proton multipole matrix elements for both the ${2}_{1}^{+}$ and ${3}_{1}^{\mathrm{\ensuremath{-}}}$ states from the proton and electromagnetic data.

Algebraic approach to matrix elements: Recurrence relations and closed formulas for hydrogenlike wave functions.

An algebraic approach to obtain ladder operators for hydrogenlike wave functions from the algebraic representation of the ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ Laguerre associated polynomials directly involved is presented. The method leads to diverse ladder operators depending on the manner as the operational form of ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ is used. As an example, this work deals with linear creation and annihilation operators of two kinds: some that influence the n energy quantum number and others acting on the l orbital number according to usual ladder properties. In addition, as a useful application of the raising and lowering operators thus obtained, generalized recurrence relations and closed formulas for the calculation of multipole matrix elements are algebraically deduced by means of a procedure that takes advantage of the relationship between the ladder operators associated with the bra and those related to the ket.

Determination of neutron and proton multipole matrix elements in 208Pb from pi - and pi + scattering at 180 MeV.

New data for elastic and inelastic ${\mathrm{\ensuremath{\pi}}}^{+}$ and ${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$ scattering by $^{208}\mathrm{Pb}$ at 180 MeV are presented. The elastic scattering data were analyzed with a standard Kisslinger potential in a coordinate space (r space) calculation using the free \ensuremath{\pi}-N amplitudes to calculate the potential coefficients. Momentum-space (p-space) calculations were also performed using the free \ensuremath{\pi}-N amplitudes to calculate on-shell matrix elements and the Landau-Tabakin model for the off-shell. The same ground-state proton and neutron densities were used in these two spaces and the fits are good. The distorted-wave impulse approximation calculations in both spaces, with the same collective model transition densities, reproduce the inelastic differential cross sections quite well. The proton and neutron transition multipole matrix elements and their ratios were calculated from the deformation parameters extracted from fits to inelastic scattering data. It is found that the ratios calculated in these two spaces are quite close to each other and close to those found in a comparison of (e,e') and (p,p') experiments, while the absolute value of matrix elements found in the p-space calculations are \ensuremath{\sim}15--20 % smaller than those from the r-space calculations. The r-space proton matrix elements are generally in good agreement with those obtained from electron scattering. The effects of the pion-nucleon interactions and form factors used in these calculations were investigated.

Multipole matrix elements for hydrogen atom

The orthogonality relations of the eigenfunctions in the discrete and continuum states for the hydrogen-like atom are proved. The multipole matrix elements ⟨n, l|ra|m, j⟩ are given as the analytic expressions which are represented by Appell's hypergeometric function of two variables. For the particular case, the electric dipole and electric quadrupole matrix elements ⟨n, l|r|m, l − 1⟩, ⟨n, l|r2|m, l⟩ and ⟨n, l|r2|m, l − 2⟩ are represented as the sum of the hypergeometric functions and the electric dipole matrix elements ⟨n, l|r|m, l − 1⟩ are consistent with the results obtained by Gordon.

Elastic and inelastic scattering of 280 and 489 MeV protons from 58Ni

Nonrelativistic and relativistic collective deformed potential models have been used to study the excitation of seven states in 58Ni by 280 and 489 MeV unpolarized protons. Optical model potentials were determined at each energy by fitting elastic scattering data using both of the models. Deformation lengths were also obtained using both models with no dependence on incident energy observed for either model. The average value of the hadronic deformation lengths along with the charge deformation lengths were used to calculate neutron to proton multipole matrix elements ( M n M p ). Five transitions, 2 + 1, 1.45 MeV; 2 + 3, 3.04 MeV; 2 + 4, 3.26 MeV; 4 + 1, 2.46 MeV; and the 4 + 5, 4.75 MeV state were determined to be isoscalar, i.e. M n M p = N Z . The 4 + 2, 3.63 MeV state was observed to be a proton-like transition with M n M p = 0.43 ± 0.18. At small scattering angles, the relativistic model gave consistently larger cross sections than the nonrelativistic model. This effect was much larger at 280 MeV than at 489 MeV.

Existence of zeros in the Coulomb multipole matrix element with retardation

The parity of the number of zeros of one-photon multipole transition matrix elements for an electron in a point Coulomb field is examined, generalising on some multipole transitions in the long-wavelength limit. Results are obtained for the general multipole matrix element with fixed angular momentum, Delta l not=0, retardation being included. For low multipolarities, results are compared with those obtained in the long-wavelength approximation, and cases in which results differ are indicated. The method involves determining the sign of the matrix element both at very low photon energy (for unphysical situations below the photoelectric threshold not corresponding to real bound-bound transitions) and in the limit of high photon energy.

Studies of [formula omitted] reactions and electromagnetic currents in rotational nuclei

The general formalism is presented for electron scattering studies involving excitation of a discrete state followed by the emission of any number of real photons which are detected in coincidence with the scattered electron. Special limited situations are discussed in which only a single real photon is emitted and in which one multipole dominates in the γ-decay. The formalism is applied in a treatment of low-lying collective excitations in nuclei. Observables which become accessible from measurements of angular distributions in the (e, e′γ) reaction permit the determination of transverse multipole matrix elements through sensitivity to transverse/longitudinal interferences, and so yield information on intrinsic nuclear electromagnetic current distributions where heretofore in almost all cases only charge distribution information has been available. Illustrative examples of vibrational and rotational excitations are presented to help elucidate the degreee of sensitivity to the nature of the current flow. Detailed results are given for rotational bands described within the context of the deformed Hartree-Fock method (using projected Hartree-Fock and cranking approximations, and using several effective forces) applied to two selected nuclei, a light nucleus, 20Ne and a rare-earth nucleus, 166Er.

Polarization in coincidence electron scattering from nuclei

Theoretical analysis of the uses of leptonic and nuclear polarization degrees of freedom in the extraction of information on nuclear structure and dynamics from exclusive electron-nucleus scattering is presented. The formalism is discussed in its most general form in terms of multipole matrix elements of the hardronic current with all particles polarized and is then specialized to a variety of cases of interest in intermediate-energy nuclear physics. These studies suggest that control of polarization degrees of freedom will provide an important means to extend the usefulness of lepton-scattering experiments for the measurement of the electromagnetic properties of nuclei.

Alternative forms of the Coulomb approximation for bound-bound multipole transitions

We discuss three different formulations of the Coulomb approximation that allow us to avoid numerical difficulties encountered with the standard one in the case of transitions between highly excited states. A common feature of these alternative (though not equivalent) methods is the fact that the radial multipole matrix elements are expressed in terms of standard or generalized hypergeometric functions. The recurrence relations satisfied by the latter can be used to bypass the loss of accuracy resulting from dramatic cancellations among their individual terms.

Collective model distorted-wave Born approximation analysis of 500-MeV proton scattering from 40Ca.

A vibrational collective model has been used to analyze 15 bound states in /sup 40/Ca excited by inelastic scattering of 500 MeV polarized protons. It is shown that for those states which have surface peaked charge transition densities the collective model describes the shape and magnitude of the angular distributions quite well. The hadronic deformation lengths extracted are shown to be constant over the incident proton energy range of 25--800 MeV with the exception of data at 185 MeV. The average values for the hadronic deformation lengths are used to calculate the ratio of the neutron to proton multipole matrix elements for five states. Elastic scattering and the 3/sub 1//sup -/, 2/sub 1//sup +/, and 5/sub 1//sup -/ transitions have also been studied using the relativistic collective model.

Status of the Nuclear Shell Model

In this review we concentrate on recent results of shell model calculations and related experimental data obtained for the sd-shell nuclei. Model spaces and effective interactions, Gamow-Teller, magnetic dipole, electric quadrupole, and higher multipole matrix elements, and electron scattering form factors are discussed. (AIP)

Isospin distribution of quadrupole strength in 118Sn: Comparison with pion, nucleon, and electron scattering.

The isospin structure of the quadrupole strength in $^{118}\mathrm{Sn}$ is examined by a variety of means in an attempt to understand the surprisingly large amount of isovector strength extracted from some recent ${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$/${\ensuremath{\pi}}^{+}$ experiments in the excitation-energy region (below 2\ensuremath{\Elzxh}\ensuremath{\omega}) expected for the isoscalar giant-quadrupole resonance. The ratio of the giant-quadrupole resonance neutron and proton multipole matrix elements for $^{118}\mathrm{Sn}$ determined from ${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$/${\ensuremath{\pi}}^{+}$ data is ${M}_{\mathrm{n}/{M}_{\mathrm{p}=1.9\ifmmode\pm\else\textpm\fi{}0.4}}$ compared to calculations which range from 1.1 to about N/Z (1.36). It is demonstrated that this large ratio has unrealistic consequences in random-phase approximation mixing of the giant states into the first ${2}^{+}$ state transition (core polarization), which has been independently studied by other probes and is in rather good agreement with quasiparticle random-phase approximation calculations, which have ${M}_{\mathrm{n}/{M}_{\mathrm{p}\mathrm{\ensuremath{\le}}\mathrm{N}/\mathrm{Z}}}$ for the isoscalar giant-quadrupole resonance. Quasiparticle random-phase approximation transition densities are used to calculate pion, proton, and neutron cross sections to the isoscalar giant-quadrupole resonance and the first ${2}^{+}$ state using microscopic reaction models. A comparison of B(E2) from the same structure model is made to the various data for the first ${2}^{+}$ transition and to the $^{116}\mathrm{Sn}$(e,e') data on the isoscalar giant-quadrupole resonance. Although not completely conclusive because of the lack of reliable (e,e') data, the evidence from all these comparisons is that the ${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$/${\ensuremath{\pi}}^{+}$ results are at odds with the results of other probes and with nuclear structure theory and that the problem seems to be with the ${\ensuremath{\pi}}^{+}$ scattering results.

Van der Waals forces, sum rules, and pseudostate expansions.

We reexamine the pseudostate expansion used previously to evaluate van der Waals (vdW) coefficients and finite-mass corrections. In particular, we wish to understand the very rapid convergence of the expansion, which for the case of two hydrogenic atoms requires only ten terms to achieve an accuracy of 1\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}10}$. The analysis proceeds from the representation of the vdW coefficients as integrals of single-atom frequency-dependent polarizabilities. These in turn can be expanded in power series with coefficients of the form ${\mathcal{J}}_{n}$${M}^{2}$(nl)/(${E}_{\mathrm{nl}}$-${E}_{1s}$${)}^{k}$, where M(nl) is a multipole matrix element between the ground (1s) and excited (nl) states. Although defined as sums over an infinite set of hydrogenic states, the coefficients of each order k are reproduced exactly by a finite sum over suitable pseudostates. By analytic continuation, the frequency-dependent polarizabilities computed using pseudostates are expected to be very close to the exact values, and hence the rapid convergence of the vdW coefficients is understood.

Positions of zeros in the bound-bound multipole matrix elements of the point Coulomb field.

Minima in cross sections (Cooper minima) are known to occur for all types of transitions in atoms. These minima are caused by zeros in the transition matrix elements which are the result of a zero value for the radial integral of a particular energy. These minima can occur in both the bound-bound and bound-free transition matrix elements. In this paper the approximate position (energy) of the zeros in the bound-bound quadrupole and octupole matrix elements of the point Coulomb field are presented. The positions are found to change in the same manner as those found in the continuum, even to the extent that a zero first occurring in a bound-bound matrix element will move into the corresponding bound-free matrix element.

Validity of collective model DWBA analysis for intermediate energy proton scattering to low-lying states of 208Pb

New measurements of inelastic proton scattering to low-lying collective states of 208Pb at 200 and 400 MeV are reported. Deformation lengths ( δ H = βR) extracted from angular distributions for the 3 − (2.614), 5 − 1 (3.198 MeV), 5 − 2 (3.709 MeV), 2 + (4.086 MeV) and 4 + (4.324 MeV) states are in good accord with values extracted at other incident proton energies. The fact that the deformation lengths for these levels are independent of incident proton energy within experimental uncertainty provides further support of the validity of the collective DWBA for medium energy proton scattering to strongly excited states. Advantage is taken of this to extract more precise values for the ratio of neutron to proton multipole matrix elements for both the low-lying states and the giant quadrupole resonance at 10.6 MeV.

Positions of zeros in the multipole matrix elements of the point Coulomb field.

Minima in the photoionization cross sections (Cooper minima) are known to occur for all types of transitions in atoms, even for dipole transitions from the ground state. These minima are caused by zeros in the transition matrix elements which are, in turn, the result of a zero value for the radial integral of a particular energy. For transitions involving the point Coulomb field there are no zeros in the dipole matrix elements, but there are zeros in higher multipole transitions. The zeros in the quadrupole matrix elements involve transitions between low-angular-momentum bound states and free states with two more units of angular momentum. Zeros in octupole matrix elements fall into two categories. The first involves transitions between low-angular-momentum bound states and free states with three more units of angular momentum and the second involves transitions between all s and p states. In this paper the position (energy) of the zeros in the multipole matrix elements of the point Coulomb field are examined. Why the dipole matrix elements of hydrogen have no zeros and what causes the zeros in the higher-multipole matrix elements are then discussed.

Recurrence relations for coulomb excitation electric multipole radial matrix elements

Recurrence relations for coulomb excitation electric multipole radial matrix elements

The reversal effect for neutron and proton multipole matrix elements in higher 2 + transitions

Recent experimental evidence is cited showing that there commonly exists a higher prominent 2 + state which has a reversal of the inequality of multiple matrix elements M n and M p from that of the first 2 + state of even nuclei. An explanation in terms of the spin-orbit intruder is presented for nuclei beyond the s-d shell, and examples are given.

Direct neutron scattering from 182W and 184W

Differential elastic and inelastic neutron scattering cross sections for 182W and 184W have been measured at incident energies 4.87 and 6.00 MeV. Cross sections for the first (0 +, 2 +, 4 +, 6 +), second (0 +, 2 +), and some higher excitations are presented. Angular distributions exhibit direct reaction characteristics, suggesting that compound cross sections for these states are small. This is supported by statistical-model calculations. Coupled-channel calculations of cross sections are made using a phenomenological deformed optical potential. Quadrupole and hexadecapole deformations have been searched to optimize fits. The necessity of introducing a β 6 deformation is investigated. Electric multipole transition matrix elements, used in the coupled-channel analysis, are obtained from the rotation-vibration model and the dynamic-deformation theory.