Sum Rule Strength Research Articles

Shell-model calculations of Gamow-Teller strength in 51V, 54Fe, and 59Co.

The Gamow-Teller (GT) strength functions of $^{51}\mathrm{V}$, $^{54}\mathrm{Fe}$, and $^{59}\mathrm{Co}$ measured by recent charge-exchange experiments have been studied using shell-model calculations employing two model spaces. The shell model is fairly successful at reproducing GT strength in the (p,n) direction, but the agreement is not as good in the (n,p) direction. It is found that both directions can be fit consistently by shifting single particle energies in a systematic manner. It is found that roughly 75% of the sum-rule strength can be seen in the experiments for both $^{51}\mathrm{V}$ and $^{54}\mathrm{Fe}$. It is also found that roughly 35% of the GT strength predicted by the shell model is seen in the (n,p) direction for all three nuclei, but that the quenching in the (p,n) direction varies with nucleus. The consequences of these findings for improving the calculation of weak interaction rates are discussed.

Beta decay rates of fp shell nuclei in advanced evolutionary stages of presupernova stars

Presupernova evolutionary calculations require reliable rates of beta decay and electron captures on fp shell nuclei with A > 60, as their inclusion could have appreciable effects on stellar structure and the subsequent evolution towards supernova explosion. Here we calculate the beta decay rates of a few of these nuclei by using Gamov-Teller sum rule strength that employ nucleon occupancies obtained by spectral averaging methods. The method is well suited for the finite temperatures typical of presupernova cores and includes relevant excited states of the mother nucleus.

Use of the (e,e'n) reaction to study the giant multipole resonances in 116Sn.

The giant multipole resonances in $^{116}\mathrm{Sn}$ have been studied using the (e,e'n) reaction. Data were taken at effective momentum transfers of 0.37, 0.45, and 0.55 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$ and a multipole analysis of the data was performed. The inferred multipole strength functions identify the E2 and E0 resonances as distinct peaks at 12.2 and 17.9 MeV, respectively. The energy-weighted sum-rule strengths for the E2 and E0 resonances, obtained using a Lorentzian fit to the data, are 34\ifmmode\pm\else\textpm\fi{}13% and 93\ifmmode\pm\else\textpm\fi{}37%. When compared with results from alpha scattering and pion scattering the sum-rule strengths exhibit approximate agreement, but the E0 strength identified in this measurement lies at higher excitation energy, consistent with the trend observed in heavier nuclei. The (e,e'n) data are compared with a continuum random phase approximation (RPA) calculation of the E2 and E0 strengths, and with an open-shell RPA calculation of the E2 strength. Both calculations disagree with the data in the region of the E2 resonance.

Low-lying collective quadrupole and octupole strengths in even-even nuclei.

The B(E2)\ensuremath{\uparrow} values for the first ${2}^{+}$ state of even-even nuclei in the Z\ensuremath{\ge}50 region are compared with the predictions of several theoretical models. Comparative estimates of the overall agreement with the data are provided. Gaps and discrepancies in the data and examples that show interesting features such as shape changes are discussed. The B(E2)\ensuremath{\uparrow} values are examined critically to search for the dynamical Pauli effects predicted by the fermion dynamic symmetry model. The empirical B(E2)\ensuremath{\uparrow} and B(E3)\ensuremath{\uparrow} systematics are employed to obtain a measure of the harmonicity of the quadrupole and octupole vibrations. The fraction of the energy-weighted sum-rule strength exhausted by the sum of all known low-lying ${2}^{+}$ states below 2.3 MeV is found to be surprisingly constant in the 60A250 region except near closed shells.

Giant resonances in90Zr and116Sn

The giant resonance region in90Zr and116Sn excited by 270 MeV helions has been measured up to about 35 MeV excitation energy. The low and the high energy octupole resonances are seen prominently in addition to the quadrupole and the monopole resonances. The angular distribution data for the various multipoles are satisfactorily explained by the collective model calculations. The percentatge energy weighted sum rule strengths have been determined for all the prominent resonances.

Isoscalar monopole and quadrupole strength of 16O in an alpha +12C cluster and symplectic mixed basis.

The isoscalar monopole (E0) and quadrupole (E2) transition strength of $^{16}\mathrm{O}$ is used to study how well the \ensuremath{\alpha}${+}^{12}$C cluster model works for explaining the strength distributions observed up to the isoscalar giant resonance region. The cluster model reproduces 54% of the E0 and 65% of the E2 energy weighted sum-rule strength. A symplectic basis, which naturally reproduces the sum rule, is mixed with the cluster basis to see how much of the missing strength is accounted for. A mixed basis calculation shows that the symplectic basis does not ruin the successful agreement of the cluster model with experiment obtained in the low-energy region and increases the strength by 30% for E0 and 20% for E2 below ${E}_{x}$\ensuremath{\le}40 MeV. It is predicted that both of the E0 and E2 strengths are split into two peaks in the giant resonance region with a greater spread over the excitation energy than expected.

Electron scattering form factors of stretched transitions using Woods-Saxon wave functions.

Electron scattering form factors for stretched transitions are computed using radial wave functions from realistic nuclear potentials, including the unbound nature of final states above particle decay thresholds. The calculated form factors are compared to data for 4/sup -/ states in /sup 12/C, /sup 14/C, and /sup 16/O, 6/sup -/ states in /sup 24/Mg, /sup 26/Mg, and /sup 28/Si, 8/sup -/ states in /sup 48/Ca, /sup 54/Fe, /sup 58/Ni, and /sup 60/Ni, the 10/sup -/ state in /sup 90/Zr, and the 14/sup -/ state in /sup 208/Pb. We assess the fraction of the single-particle sum rule strengths using these realistic nuclear potentials in place of the standard results using harmonic oscillator wave functions. Appreciably greater fractions are obtained for low mass nuclei in the present work, totalling 105% of the sum strength for /sup 12/C and 81% for /sup 16/O. Much less damping of the magnetic strength is thus experimentally observed than is the case when oscillator wave functions are used for comparison. The results of including meson exchange currents in the analysis are also discussed.

Location of the low-energy isoscalar octupole resonance in 58Ni.

The levels of $^{58}\mathrm{Ni}$ have been studied by inelastic proton scattering at ${E}_{\mathrm{p}}$=65 MeV. The fine structure of the low-energy octupole resonance in $^{58}\mathrm{Ni}$ has been observed. An octupole strength of \ensuremath{\sim}18% of the energy-weighted sum rule is obtained for the fragmented ${3}^{\mathrm{\ensuremath{-}}}$ states at ${E}_{x}$12 MeV. This sum rule strength is consistent with a simple shell model estimate.

Excitation of giant resonances in89Y by inelastic scattering of deuterons,3He nuclei and alpha particles at 43 MeV/nucleon

Elastic and inelastic scattering of deuterons, 3He and alpha particles by 89Y have been measured in the angular range from 7 degrees to about 30 degrees (lab) at incident energies of 83, 135 and 172.5 MeV, respectively. The elastic data were analysed in terms of the standard optical model. The energy spectra of the different particles show strong excitation of the giant resonance region around Ex approximately 14.5 MeV. The FWHM width of the giant resonance peak is about the same, approximately 4.5 MeV for each of the three types of incident particle. The angular distributions of the giant resonance bump are described by DWBA calculations with L=2 transfer. The derived value for the depletion of the corresponding isoscalar quadrupole energy-weighted sum rule strength is about 60% for all three particle types.

Relations between Gamow-Teller and magnetic dipole transitions.

The Gamow-Teller transitions ${J}_{i}$${=0}^{+}$ ${T}_{i}$=${T}_{z}$=1/2(N-Z)\ensuremath{\rightarrow}${J}_{f}$${=1}^{+}$ ${T}_{f}$=${T}_{z}$=[1/2(N-Z)]-1are compared with analogous M1 transitions in which the final states are the same but the initial states have ${T}_{\mathrm{zi}}$=[1/2(N-Z)]-1. In the single j shell model for the even calcium isotopes, the summed strengths for the analogous M1's (in Sc) are independent of A, but the corresponding summed Gamow-Teller strength is proportional to (N-Z) and accounts for (j+1)/(3j), i.e., (3/7 of the total ``3(N-Z)'' sum rule strength. Isospin considerations are exploited as much as possible. It is noted that configuration mixing, e.g., admixtures of j'=l-1/2, lead an enhancement of the low lying strength for the two-particle system ${(}^{42}$Ca), but to a retardation of the low lying strength in the higher A system ${(}^{48}$Ca). An analytic expression is obtained in perturbation theory with a delta residual interaction, to explain this interesting behavior. A brief survey of the experimental situation is made.

Do we understand deuteron scattering at all?: The polarization potential due to mass-three channels

The contribution to the deuteron effective local optical potential arising from coupling to triton and helion channels has been evaluated by performing finite-range coupled reaction channel calculations with the pickup channels at sum rule strength. The effective local polarization potential is evaluated by inversion of the CRC S-matrix elements. The effect is very large, with indications of underlying l-dependence. The significance of these large contributions will be discussed.

Measurements on isovector giant resonances in pion charge exchange.

We measured the energies, widths, and cross sections of the isovector monopole and dipole resonances in various nuclei between $^{40}\mathrm{Ca}$ and $^{208}\mathrm{Pb}$ with the reactions (${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$,${\ensuremath{\pi}}^{0}$). Both resonances exhaust approximately the same substantial fraction of the cross section calculated in a random-phase-approximation--distorted-wave-impulse-approximation model. The excitation energies and widths of the monopole and dipole are in good agreement with random-phase-approximation calculations and for the dipole they are also in agreement with other data. No isovector quadrupole resonance was observed, and the upper limits for the cross sections for the light elements are well below the sum rule strength for the isovector monopole and giant dipole resonance.

Cross section and analyzing power measurements for the giant resonance region in 208Pb with 200-MeV protons.

The giant resonance region in $^{208}\mathrm{Pb}$ has been studied using inelastic scattering of 200-MeV polarized and unpolarized protons. Both differential cross sections \ensuremath{\sigma}(theta) and analyzing power ${A}_{y}$(theta) measurements were made. The isoscalar giant quadrupole resonance at 10.6\ifmmode\pm\else\textpm\fi{}0.5 MeV, and the combined isovector giant dipole resonance and isoscalar giant monopole resonance at 14.0\ifmmode\pm\else\textpm\fi{}0.6 MeV are clearly observed. Within uncertainties all three giant resonances are in accord with full energy-weighted sum rule depletions, based on comparisons with macroscopic distorted wave Born approximation calculations. Data for a peak at 20.9\ifmmode\pm\else\textpm\fi{}1.0 MeV are found to be consistent with an isoscalar giant octupole resonance exhausting 36\ifmmode\pm\else\textpm\fi{}12 % of the energy-weighted sum rule strength. A peak located at an excitation energy of 12.0\ifmmode\pm\else\textpm\fi{}0.7 MeV is shown to be a 2\ensuremath{\Elzxh}\ensuremath{\omega}, L=4 transition depleting 8\ifmmode\pm\else\textpm\fi{}3 % of the hexadecapole energy-weighted sum rule strength. The measured reduced transition probability for the ${3}^{\mathrm{\ensuremath{-}}}$ state at 2.614 MeV is consistent both with the accepted value and with results measured at 334 and 800 MeV. Our results for the ${3}^{\mathrm{\ensuremath{-}}}$ state at 2.614 MeV and for the giant quadrupole resonance do not support the anomalous behavior found in earlier studies at 104, 156, and 201 MeV. This shows that the macroscopic distorted wave Born approximation can be used to extract deformation lengths providing meaningful comparison with electromagnetic energy-weighted sum rules for proton inelastic scattering to at least 800 MeV incident proton energy.

The 48Ca(d→, 3He) 47K reaction at 80 MeV

Abstract A high-resolution measurement of the 48Ca(d→, 3He) 47K reaction has been carried out. Seventeen final states are found, up to an excitation of 8.35 MeV. The measured vector analysing powers allow spin determination for most of these states. The 2s 1/2 and ld 3/2 hole strengths observed are found to exhaust the respective sum rules, while 64% of the sum-rule strength for 1d 5/2 hole states is found. It is surmised that the remainder of this strength resides at higher excitation energies. The amount of proton ground-state correlations is deduced from the strength of a newly observed 7/2 − state in 47K.

Pion inelastic scattering to giant resonances and low-lying collective states in 118Sn and 40Ca.

A comparison of differential cross sections for inelastic scattering of 130 MeV positive and negative pions has been made for studying the giant resonance region in $^{118}\mathrm{Sn}$ and $^{40}\mathrm{Ca}$. In addition, several of the low-lying collective states in each target were examined. Comparison to distorted wave impulse approximation calculations allowed collective deformation lengths \ensuremath{\beta}R and fractions of the energy-weighted sum rules to be extracted. For bound collective states, the strengths for ${\ensuremath{\pi}}^{+}$ and ${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$ were nearly equal, as has been found for other scattering probes. Giant quadrupole and dipole excitations in $^{40}\mathrm{Ca}$ (T=0) had nearly equal ${\ensuremath{\pi}}^{+}$ and ${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$ sum rule strengths. Quadrupole and monopole excitations in $^{118}\mathrm{Sn}$ near 14 MeV show about twice the ${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$ transition strengths compared to those for ${\ensuremath{\pi}}^{+}$, contrary to expectations from simple isoscalar models.

Analog giant dipole Ni(n, p) at 59.6 MeV

A broad enhancement above the continuum observed in 58, 60, 62, 64Ni(n, p) at 59.6 MeV is interpreted as the parent analog of the T> component of the giant dipole resonance. A continuum parameterization is used to extract the resonance cross section. The angular distributions for the state are most consistent with the Steinwedel-Jensen model for the giant dipole. The sum-rule strength is somewhat greater than observed with photonuclear measurements. A measurement of the blocking of the cross section with neutron excess is made, and is consistent with theoretical predictions.

Spin-isospin excitations and nuclear models

Gamow-Teller (GT) transitions are discussed in the context of the nuclear shell model. It is pointed out that the GT strength function contains information both about the character of the nuclear constituents and about the effective residual nucleon-nucleon force. A major part of the strength function is inaccessible to beta decay, but the (p,n) reaction can be used to probe GT transitions without the energetic limitations of beta decay. The experimental techniques are reviewed. The (p,n) data are surveyed. Only about 55% of the sum rule strength is seen. The possibility that the missing strength signifies the presence of delta or quark degrees of freedom in the constituent particles of the nuclear model is discussed.

Isoscalar giant resonances inPb208—in particular2ℏωL=0, 2, 4, and 6excitations—studied in small angleαscattering

Inelastic excitation of giant resonances has been studied in 172 MeV $\ensuremath{\alpha}$ scattering from $^{208}\mathrm{Pb}$ at scattering angles of 1.5\ifmmode^\circ\else\textdegree\fi{}-8\ifmmode^\circ\else\textdegree\fi{}. Fine structures in the region of the giant monopole and quadrupole resonances indicate multipolarities also different from $L=0 \mathrm{and} 2$. The isoscalar $L=0 \mathrm{and} 2$ strengths in the giant resonance region yield (90\ifmmode\pm\else\textpm\fi{}20)% and (70\ifmmode\pm\else\textpm\fi{}20)% of the corresponding energy weighted sum rule limit. This is consistent with other hadron scattering results, with higher energy electron scattering, and with microscopic random phase approximation results. However, the strength for $L=2$ excitations is significantly larger than extracted from lower energy electron scattering. Fine structures observed are compared with high resolution (p,p\ensuremath{'}) and (e,e\ensuremath{'}) results. At scattering angles 5\ifmmode^\circ\else\textdegree\fi{}-8\ifmmode^\circ\else\textdegree\fi{} new structures have been observed at ${E}_{x}=12.5 \mathrm{and} 16.0$ MeV consistent with multipolarities $L=4 \mathrm{and} 6$. Together with the contribution of higher multipolarity extracted at lower excitation energies from the angular distributions of giant monopole and quadrupole excitations, this indicates rather broad distributions of $2\ensuremath{\hbar}\ensuremath{\omega}$ $L=4 \mathrm{and} 6$ excitations with centroid energies of 12.2 and 13.7 MeV and widths of 4.5 and 7.8 MeV, respectively. The extracted sum rule strengths for $L=4 \mathrm{and} 6$ are in the order of 30% in each case. In the region of the giant monopole resonance evidence for additional $L=2$ excitation was found (\ensuremath{\le}14% energy weighted sum rule strength). The fact that strong contributions of $L>0$ are obtained solves the problem of a former too large monopole cross section in $\ensuremath{\alpha}$ scattering which was inconsistent with microscopic descriptions. A recently suggested "antiscaling" mode for the giant monopole resonance, in which the nuclear surface vibrates opposite in phase to the "scaling" mode, can be ruled out by our experimental data. The existence of two odd parity giant resonances at ${E}_{x}=18.7$ MeV ($L=3$) MeV ($L=1$) with widths of 5.0 and 5.9 MeV, respectively, are confirmed by our small angle scattering experiments. Our description of the dipole compressional mode is in excellent agreement with 200 MeV proton scattering data.NUCLEAR REACTIONS $^{208}\mathrm{Pb}(\ensuremath{\alpha}, {\ensuremath{\alpha}}^{\ensuremath{'}})$; ${E}_{\ensuremath{\alpha}}=172$ MeV. Small angle scattering, $\ensuremath{\theta}=1.5\ifmmode^\circ\else\textdegree\fi{}\ensuremath{-}8\ifmmode^\circ\else\textdegree\fi{}$; measured $\ensuremath{\sigma}(\ensuremath{\theta})$. Deduced excitation strength distributions for giant resonances of multipolarities $L=0\ensuremath{-}6$.

The characteristics of electric dipole strength built on highly-excited continuum states

High-energy γ-ray spectra from heavy-ion fusion reactions have been measured with low background and high resolution over a broad mass range. The data have been fitted with statistical-evaporation calculations in which the γ-ray strength associated with each level follows a lorentzian. The extracted resonance widths are generally much larger than the width of the giant dipole resonance built on the ground state of the compound system (gs-GDR). The centroids of these excited-state resonances are shifted from those of the gs-GDRs, but no simple A −1 3 dependence is evident. The most surprising result is that the extracted sum rule strengths of these resonances track those of the gs-GDRs observed in photoabsorption.

M1 strengths and sum rules in the Wigner supermultiplet scheme

The non-energy-weighted (NEW) sum-rule strength for the M1 operator and the B(M1) strengths in the ground-state region are evaluated in the SU(4) basis. Here, one combines the spectral distribution method of calculating the intensities of the different irreducible representations with the method of evaluation of M1 strengths between two SU(4) ST states through SU(4) algebra. Comparison with experimental and shell-model values in the lower half of the ds shell is made.