Sums of products of Coulomb wave function over degenerate states

The sums of products of Coulomb wave function over degenerate states are expressed in terms of quadratic forms that depend on the wave function of only one state with zero orbital angular momentum l = m = 0. These sums are encountered in many fields in the physics of atoms and molecules, for example, in investigations of the perturbation of degenerate atomic energy levels of a small potential well, a delta-function potential. The sums were found in an investigation of the limit of the Coulomb Green’s function G(r, r′, E), where the energy parameter E approaches an atomic energy level: E → E _n , E _n = ?Z ²/2n ². The Green’s function found by L. Hostler and R. Pratt in 1963 was used. The result obtained is a consequence of the degeneracy of the Coulomb energy levels, which in turn is due to the four-dimensional symmetry of the Coulomb problem.     

Approximate normalized wave functions for the finite-size Coulomb problem

Using perturbation methods developed previously for the finite-size energy shift, thes-state wave functions for a bound lepton in the Coulomb field of a nucleus with an arbitrary charge distribution are developed through order (Zα)². This technique allows a determination of the finite-size contribution to the normalization of the wave function up to order (Zα)², both for small separations of the lepton and nucleus and for the lepton outside the nucleus. General features of the wave function are discussea, including the transition of the Dirac problem to the (singular) point charge limit. A practical application is developed using these results.     

Influence of 2p-2h excitations in the41Sc-41Ca Coulomb energy shift

The influence of 2p-2h core excitations on the⁴¹Sc-⁴¹Ca Coulomb energy difference is studied. A simple parametrization of the main wave function components shows that the calculated energy shift could only agree with experiment for rather unrealistic values of the amplitudes. Using an effective interaction appropriate for this region, we find that when this kind of excitations is included, the calculated Coulomb energy shift is increased by 105keV.     

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.     

AbsoluteE0 andE2 transition rates and collective states in116Sn

AbsoluteE0 andE2 transition rates in¹¹⁶Sn have been measured using several newly developed techniques. ManyE2 transitions are observed to have a collective character withB(E2) values of up to 60 W.u. The presence of deformed excited states in¹¹⁶Sn is discussed in view of the results obtained.     

Projected Hartree-Fock calculation in intermediate coupling for11B

An intermediate coupling calculation has been carried out for¹¹B using the deformed functions obtained from a Projected Hartree-Fock calculation as a basis. The resulting wave functions give an important improvement for the energy spectrum of the low-lying nonnormal parity levels. TheE2 matrix elements are enhanced with respect to the conventional shell model. Nuclear structure¹¹B; calculated levelsJ, π, B(M1),B(E2). Projected Hartree-Fock method, intermediate coupling.     

The Woods-Saxon wave functions and the beta transitions of deformed nuclei

The general expressions for the beta transition matrix elements have been obtained using the Woods-Saxon wave functions for deformed nuclei proposed byFaessler andSheline. As an application the matrix elements for the 1^?(β^?)0⁺ and 1^?(β^?)2⁺ transitions of the¹⁷⁰Tm nucleus have been calculated. The results indicate significant differences between the Nilsson and Woods-Saxon values of the nuclear parameters. Assuming a reasonableK=0 mixture in the ground state of the¹⁷⁰Tm nucleus and a value near the Ahrens-Feenberg estimate for the ∫ α¦ ∫ir ratio it is possible to obtain an important improvement of the theoretical values of the relevant observables and especially of theft values.     

Description of radiative strength functions in deformed nuclei

Radiative strength functions ofE1- andM1-transitions from ground states of doubly even deformed nuclei to states near the neutron binding energyB _n are calculated within the quasiparticle-phonon nuclear model. The wave functions of excited states include one- and two-phonon components. The calculations were made with the Pauli principle being or not included in the two-phonon components of the wave functions. It is shown that the radiativeE1- andM1-strength functions as well as the widths of giant dipole resonances in deformed nuclei are slightly influenced by the two-phonon components of the wave functions and they can be calculated in the RPA. Thek _E1- andk _M1-values are calculated for some deformed nuclei of the rare-earth and actinide region. The calculated values ofk _E1 are 1.5–2 times larger and the values ofk _M1 are somewhat less than the average values obtained in [14] from the analysis of available experimental data.     

Energie- und Intensitätsmessungen an Konversionselektronen des Zerfalls75Se→75As

Energies and intensities of conversion lines of elevenγ transitions in the decay⁷⁵Se→⁷⁵As were measured with the Heidelberg (π/2)√13β-ray spectrometer. The experimental transition energies agree with the results of crystal diffraction spectrometer measurements; in some cases more precise values were obtained. The measured relativeK electron intensities for five transitions are in agreement with previous experiments. TheL group of the 97 keV transition was resolved for the first time and the multipolarity was determined to beE2 from theL subshell ratios.     

Sub-Coulomb transfer and fission in collisions of129Xe,132Xe and136Xe with238U

Integral cross sections for fission and for one- and two-neutron transfer reactions in the system¹³²Xe+²³⁸U were measured radiochemically in the energy range 0.7≦E/E _Coul≦1. The excitation functions for fission and transfer are found to be essentially parallel below 0.85×E _Coul. Even at the lowest energies the transfer cross sections exceed the fission cross section by more than one order of magnitude. With the other projectiles¹²⁹Xe and¹³⁶Xe different transfer cross sections illustrating their sensitivity for the ground stateQ-values,Q _gg , are observed while the fission cross sections are the same as in the¹³²Xe +²³⁸U reaction. The fission data are interpreted in terms of a continuous transition between Coulomb fission and several transfer-induced fission processes.     

Dipole γ-ray transition rates in 238U

M1 and E1 transition rates from the ground to excited states and between excited states in ²³⁸U are calculated within the quasiparticle-phonon nuclear model with the wave functions consisting of one- and two-phonon terms. We show that there are relatively large M1 transitions between 2⁺ states in the low-energy region. The fragmentation of the one-phonon states strongly affects M1 and E1 strength distributions. The correlation takes place between E1 and E3 transition strengths. We show that there are fast E1 and M1 transitions between large components of the wave functions differing by an octupole or quadrupole phonon.     

The E2 transition probability and effective charge for the neutron i132 subshell in lead isotopes

In (heavy ion, xn) reactions, X-rays and low energy γ-rays emitted by the compound nuclei are investigated using a catcher system for recoiling nuclei within the 10–100 keV energy range. The nucleide ¹⁹⁸Pb is studied by means of this technique. A 90 keV E2 (or E2 + M1) transition is identified together with information related to the 12⁺ → 10⁺E2 transition energy. The B(E2; 12⁺ → 10⁺) value is deduced. The effective charge extracted from the analysis of the reduced transition probability between two members of the ($\text{vi}{}_{\mathrm{<mtext>13</mtext><mtext>2</mtext>}}\text{)}{}^{\mathrm{?2}}$ multiplet is compared to the corresponding values for other lead isotopes.     

Gamma-rays from the24Mg(p,γ)25Al reaction for proton energiesE p=800 keV toE p=1,830 keV

The yield curve and resonance strengths of resonances in the²⁴Mg(p, γ)²⁵Al reaction for the proton energy rangeE _p=800 keV toE _p=1,830 keV, have been determined and the existence of a resonance atE _p=1.184 MeV confirmed. Single energy spectra as well as decay schemes and branching ratios are given for these resonances. An upper level of 2.0×10^?15 sec for the lifetime of the 3.079 MeV level in²⁵Al results from a Doppler shift attenuation measurement carried out on the 2.624 MeV gamma line originating in the 3.070 MeV→0.455 MeV transition in²⁵Al.     

Octupole vibrations of deformed nuclei

The octupole vibrations of deformed nuclei are studied. A strong coupling between the harmonic octupole vibrations and the quadrupole ellipsoid is assumed. This interaction gives rise to a splitting of octupole bands with different projection quantum numberK. TheE1-transitions of the octupole states are explained within the frame of the Dynamic Collective Theory ofDanos andGreiner by calculating the Coulomb interaction between octupole vibrations and giant resonances. Furthermore, theE2- andE3-transitions of the octupole states are studied. The spectrum of W¹⁸² is investigated in detail. A fair agreement between theory and experiment is obtained. Predictions for the octupole spectra of Sm¹⁵² and Gd¹⁵⁶ are given.     

Contribution of electric quadrupole and dipole-quadrupole interference terms in Coulomb breakup of 15C

The effects of electric quadrupole (E2) and dipole-quadrupole interference (E1–E2) terms in the Coulomb breakup of ¹⁵C have been investigated within the framework of eikonal approximation. The sensitivity of Coulomb breakup cross section, differential in relative energy and Longitudinal Momentum Distribution (LMD) of core fragments, towards these terms have been examined. A very small (1% of E1) contribution of E2 transition has been predicted in integrated Coulomb breakup cross section. Further it is also found that the inclusion of E2 and E1–E2 terms introduces a small asymmetry in the peak of relative energy spectrum and also increases the peak height of the spectrum. The contribution of dipole-quadrupole interference terms is clearly shown in LMD, as it introduces an asymmetry in the shape of LMD and enhances the matching between the data and predictions.     

Intrastate γ-transitions: A spectroscopic tool?

In the framework of a potential model we have calculated the various bremsstrahlung cross sections into the⁵Li ground state, includingM 1 andE 2 γ-transitions leading from the high energy wing of the⁵Li ground state resonance into states belonging to the same resonance at lower energy (intrastate transition). Our calculation supports the hypothesis of Schmalbrock et al. [1] that intrastate transitions ofM 1 andE 2 multipolarity exist. While we find a maximum cross section of roughly 1.4 nb for theE 2 transition, we predict the cross sections forM 1 intrastate transition to be less than 3·10^?5nb. An experimental observation of the intrastate γ-ray emission appears to be very difficult due to the dominance of competing resonant (M 1) and direct (E 1) capture processes. Schmalbrock et al. have suggested to deduce magnetic dipole and electric quadrupole moments of resonant states from the measurements of the respectiveM 1 andE 2 intrastate transitions. We will show that theM 1 intrastate cross sections do not yield the appropriate information to determine the magnetic dipole moment. We will also discuss thatE 2 intrastate transitions do not seem to be a suited tool to find the quadrupole moment of an unstable state.     

On the coexistence of oblate and prolate deformed bands in83Zr

Deformation parameters of the positive parity yrast band and negative parity bands in⁸³Zr are deduced from lifetimes andE2/M1 mixing ratios. Lifetimes of high spin states have been determined from recoil distance Doppler shift and Doppler shift attenuation measurements using the⁵⁴Fe(³²S,2pnγ) ⁸³Zr reaction. Ten lifetimes and five lifetimes limits were determined. The positive parity band, built on theg _9/2 K=5/2 orbital has an average deformation ¦β ₂¦=0.28(2), and shows a reduction ofE2 transition strengths in the observed backbend region at I^π≈21/2⁺. In contrast, theE2 strengths in the negative parity states show a steady increase up to I^π≈=15/2^?. These states are more strongly deformed than the positive parity states (¦β ₂¦=0.33(3)). TheE2/M1 mixing ratios show that the negative parity band hasK=3/2 and is prolate, and favour oblate deformation for the positive parity yrast band. In theK=1/2^? band theE2 strength of the 7/2^?→3/2^? transition yields a deformation ¦β ₂¦=0.26(5). The band structure is compared with calculations within the Hartree-Fock Bogoliubov cranking model.     

Self-consistent calculation of the fission barrier of 240Pu

Completely self-consistent calculations using the Skyrme force have been carried out for the fission energy curve of ²⁴⁰Pu. We use a deformed oscillator basis including 13 major shells and convergence has been checked by extending the size of the basis to 15 shells. We obtain a double humped barrier with energies E_A = 9 MeV for the first barrier, E_B = 13 MeV for the second barrier and E_II = 4 MeV for the isomeric state. Corrections to our calculation, such as inclusion of non-axial and symmetric shapes and zero-point rotational motion, are likely to improve quantitative agreement with experimental data.