Validity of an algebraic-variational approach to the theory of collective motion for an exactly soluble shell model with R(5) symmetry

An algebraic-variational approach to the theory of collective motion previously applied in variant forms to pairing and monopole interaction models is here developed for an exactly soluble shell model Hamiltonian with R(5) symmetry. The spectrum of this class of Hamiltonian operators has previously been shown to represent a two-dimensional vibrator-rotator. The approximation scheme developed yields almost exact results up to the two-phonon level in the spherical region and goes over smoothly into a theory of the lowest states of the ground state rotational band in the deformed regime.     

The two-neutron transfer reactions (O, O) with O, C and Mg targets

Excitation functions, angular distributions and differential ranges were measured for the ²⁶Mg(¹⁸O, ¹⁶O)²⁸Mg reaction at ¹⁸O beam energies of 20–45 MeV. Excitation functions only were measured for the reactions ¹⁴C(¹⁸O, ¹⁹O)¹³C, ¹⁴C(¹⁸O, ¹⁶O)¹⁶C, ¹⁴C(¹⁸O, ²⁰O)¹²C, ¹⁴C(¹⁸O, ¹⁵N)¹⁷N and ¹⁸O(¹⁸O, ¹⁹O)¹⁷O, ¹⁸O(¹⁸O, ¹⁶O)²⁰O, ¹⁸O(¹⁸O, ¹⁵N)²¹F at ¹⁸O beam energies of 13–41 MeV. We have identified these as direct reactions in which a single neutron, a two-neutron cluster, a deuteron and a triton are transferred between projectile and target.

The cross sections for two-neutron transfer reactions were found to be relatively high and those for the ¹⁸O+¹⁸O and the ¹⁴C+¹⁸O reactions were higher than the ones of single-neutron transfers over most of the energy range.

Attempts were made to apply the theory of Buttle and Goldfarb for single-neutron transfer to the case of two-neutron transfer in the ²⁶Mg(¹⁸O, ¹⁶O)²⁸Mg reaction below the Coulomb barrier. It is shown that for those reactions for which the assumptions, implicit in the model, are valid, good agreement is obtained with experiment. We also tried to apply the diffraction model of Dar and Kozlovsky to the calculation of the angular distribution of these reactions. A good fit to the experimental results could be obtained if quite different sets of parameters were used in the calculations for the two bombarding energies.     

The reactions 18O(p,p)18O and 18O(p,p')18O1(1.98 MeV) for Ep = 3.8–6.1 MeV

Angular distributions of cross sections and analyzing powers have been measured for ¹⁸O(p, p)¹⁸O and ¹⁸O(p, p₁)¹⁸O¹ (1.98 MeV) in 25 keV intervals for proton energies between 3.8 and 6.1 MeV. A phase-shift analysis of the elastic scattering data was carried out, yielding resonance parameters for 16 levels in ¹⁹F in the excitation energy region 11.6–13.8 MeV. The results generally are in good agreement with previous work. On the basis of spin, parity, excitation energy and a comparison of reduced proton widths with reduced neutron widths of levels in ¹⁹O, an assignment of $\text{T =}\text{3}\text{2}$ could be made to at least five of the levels, including the analog of the broad $\text{3}\text{2}{}^{\mathrm{+}}$ level in ¹⁹O at 5.45 MeV. A Legendre-polynomial analysis of the inelastic scattering data suggests that the cross section for proton energies between 5.0 and 5.5 MeV is dominated by the broad $\text{3}\text{2}$⁺ resonance at E_p = 5.15 MeV.     

Proton reactions with 17O: (II). The 17O(p,d)16O reaction

Differential cross sections for the ¹⁷O(p, d₀)¹⁶O reaction have been measured at the incident proton energies 8.62, 9.56, 10.5, 11.16 and 11.44 MeV. Both the BHMM and DWBA theories have been applied to these data, and to published analysing power data for the reactions ¹⁶O(d, p₀)¹⁷O and ¹⁶O(d, p₁)¹⁷O¹. While some parameter adjustment is necessary to obtain good fits with the BHMM theory, the adjusted parameters produce predictions in good agreement with analysing power data. The DWBA theory, however, produces good fits with proton parameters deduced from measurements of proton elastic scattering from ¹⁷O, provided the deuteron parameters used give good predictions for both differential cross section and vector analysing power data for the reaction ¹⁶O(d, d₀)¹⁶O.     

The 18O(18O, 16O)20O reaction at 52 MeV

The structure of the ²⁰O nucleus was studied by the ¹⁸O(¹⁸O, ¹⁶O)²⁰O reaction at E_1ab = 52 MeV. Angular distributions for the transitions to the lowest four states in ²⁰O were obtained and analyzed with finite-range DWBA calculations. Optical potential sets were used which fit the experimental elastic scattering differential cross sections over almost the whole angular range. The two L = 0 transitions to the ground state and the 4.45 MeV state of ²⁰O populated by the ¹⁸O(¹⁸O, ¹⁶O) reaction were analyzed with exact finite-range DWBA calculations using microscopic form factors. These calculations underestimate the absolute cross sections by a factor of 11. The relative strength of the two L = 0 transitions is well reproduced in the ¹⁸O(¹⁸O, ¹⁶O) reaction. However, DWBA calculations for the ¹⁸O(t, p)²⁰O reaction overestimated the relative cross sections for the excited 0⁺ state by a factor of 6. Several model wave functions were tested for the ground-state transition. It was found that the absolute cross sections of the (¹⁸O, ¹⁶O) reaction are very sensitive to the mixing of shell-model configurations. The angular distribution shapes are also slightly dependent on the mixing.     

A spectroscopic study of 18F: (I). The 17O(p,p)17O and 17O(p, α0)14N reactions

The ¹⁷O(p, p)¹⁷O and ¹⁷O(p, α₀)¹⁴N reactions have been studied in the energy range E_p = 0.5–1.33 MeV. Excitation functions for elastic scattering measured at several angles give l_p values for six resonances and J^π; T values of 3⁺; 1 and 2⁺; 1 for the states at 6.16 and 6.28 MeV, respectively. From both reactions, J^π limitations were found for six resonant levels. The ¹⁷O(p, α₀)¹⁴N reaction also yields information on T-assignments and level formation parameters. Experimental results are discussed in terms of shell-model configurations.     

The j-dependence of the (19F, 16O) reaction

The total angular momentum transfer (j) dependence was studied for orbital angular momentum transfers l = 1, 2, and 3 in the (¹⁹F, ¹⁶O) reactions on ^{28, 30}Si and ⁶⁰Ni. In contrast to the strong j-dependence for the l = 2 transitions to ^{31, 33}P states, no distinct j-dependence was found for the l = 1 and l = 3 transitions to ⁶³Cu states. Previously reported results for the ${}^{28}\text{Si}\text{(}{}^{19}\text{F}\text{,}{}^{16}\text{O)}{}^{31}\text{P}$ reaction were re-analyzed using optical-potential sets which fit the elastic-scattering data for both the entrance and exit channels. Results of DWBA calculations without use of spin-orbit potentials were found to be out of phase with the data for all l-transfers and the j-dependence could not be reproduced. Both of these problems were alleviated by including spin-orbit forces in the optical potentials. However, a good fit to the ²⁸Si(¹⁹F, ¹⁶O)³¹P data was obtained only if the optical potentials that fit the elastic scattering data were modified for the exit channel.     

The (d,t) and (d,t) reactions on 18O and the 1p spin-orbit splitting

The reactions ¹⁸O(d, t)¹⁷O and ¹⁸O(d, τ)¹⁷N have been investigated at ifE_{d = 52 MeV}. Energy spectra of tritons and τ particles have been measured up to excitation energies of 25 MeV in ¹⁷O and 12 MeV ¹⁷N, respectively, and spectroscopic factors have been obtained by a DWBA analysis of the measured angular distributions. From a comparison of the t-and τ-spectra the distribution of $\text{T =}\text{1}\text{2}\text{and}\text{3}\text{2}$ spectroscopic strengths in ¹⁷O could be deduced and analog relations between $\text{T =}\text{3}\text{2}$ states in ¹⁷N and ¹⁷O could be established. Nearly the total $\text{T =}\text{3}\text{2}$ strengths of the $\text{1}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{and}\text{1}\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ shells and nearly the complete $\text{T =}\text{1}\text{2}$ strength of the $\text{1}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ shell have been found, whereas only one third of the $\text{T =}\text{1}\text{2}$ strength of the $\text{1}\text{p}\text{3}\text{2}$. Shell could be clearly identified. The observed centroid energies are understood from the different $\text{1}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$$\text{1}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}^{\mathrm{?}}\text{1)}$ and $\text{1}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$ effective residual interactions. This supports a strong isospin dependence of the 1p spin-orbit splitting.     

A study of the nucleus 107Ag in the core-excitation model with the (p,t) and (τ, d) reactions

The ¹⁰⁷Ag residual nucleus was studied in the core-excitation model using the (p, t) and (τ, d) reactions. The L, J, π of levels between 0.0 and 2.25 MeV was deduced from the combined reactions. The octupole state observed at 2.19 MeV in other experiments was resolved in (p, t) into a triplet of states at 2.182, 2.203 and 2.229 MeV; octupole strength was observed in (p, t) over a range from 1.144 to 2.229 MeV. Core-excitation wave functions for the quadrupole 2⁺ and 2'⁺ vibration doublets of ¹⁰⁷Ag were constructed using electromagnetic data. These wave functions, combined with data from the ¹⁰⁸Pd(p, t) core reaction, effectively reproduced the ¹⁰⁹Ag(p, t) differential cross sections to these states. The ground-state L = 0 transfer in (p, t) to ¹⁰⁷Ag was only 0.752±0.113 as strong as the corresponding transfer to ¹⁰⁶pd. this is an unexpectedly large blocking effect for an unpaired proton to exert upon a neutron-transfer reaction. An apparent dependence of the (p, t) angular distributions to states of ¹⁰⁷Ag built upon the same core excitation was observed, depending upon the J of the final state.     

Study of the 17O(α, d)19F reaction

At a bombarding energy of 47.5 MeV, the ¹⁷O(α, d)¹⁹F reaction is found to populate strongly only those positive-parity states that are known to consist predominantly of (sd)³ configurations. Distorted-wave calculations based on transfer amplitudes computed from shell-model wave functions provide good agreement to the measured angular distributions — in both shape and magnitude.     

The effective hamiltonian and spectra of the Op shell

The effective Hamiltonian for the Op shell has been calculated up to second order using modified Sussex matrix elements. The resulting spectra and binding energies are compared with experiment and the role of the effective three-body interaction is discussed.     

High resolution study of the reactions 54, 56, 58Fe(16O, 12C)58, 60, 62Ni and a comparison with (6Li,d) α-transfer spectroscopy

Spectra and angular distributions for the reactions ^{54, 56, 58}Fe(¹⁶O, ¹²C)^{58, 60, 62}Ni (E_x = 0.0–4.5 MeV) have been measured at 50 MeV with an energy resolution of 45–80 keV using a Q3D spectrograph. The selectivity of the (¹⁶O, ¹²C) reaction is found to be very similar to the (⁶Li, d) reaction. The close correspondence recently noted between the (⁶Li, d) spectra and levels strongly excited in (t, p) and (³He, n) two-nucleon transfer reactions is also observed to be present for the (¹⁶O, ¹²C) reaction. Relative α spectroscopic factors for (¹⁶O, ¹²C) and (⁶Li, d) obtained in a DWBA analysis assuming direct one-step α-cluster transfer are in very good quantitative agreement. Unnatural parity states, whose excitation is forbidden in the DWBA α-cluster approximation, are observed to be very weakly populated. These results, together with previous work on s-d shell and Ni targets, strongly suggest that the spectroscopic information provided by the (¹⁶O, ¹²C) and (⁶Li, d) reactions is essentially the same and that this information may be reliably extracted by DWBA analysis.     

Cross sections for the 11.08 (3+) and 11.096 MeV (4+) states in 16O by 12C(6Li,d) and 13C(6Li,t)

Particle and particle-γ measurements were performed to determine the cross sections for population of the 8.87 (2^?), 10.35 (4⁺), 11.08 (3⁺) and 11.096 MeV (4⁺) states in ¹⁶O by the ¹²C(⁶Li, d) and ¹³C(⁶Li, t) reactions in the energy range from 20—34 MeV. In general, statistical compound nuclear calculations correctly predict the magnitude of the cross sections of the unnatural parity states and underpredict those for the natural parity states. The population of the 10.35 MeV 4⁺ state in the ¹³C(⁶Li, t) reaction is correctly predicted by these calculations. These measurements support earlier claims that the large ¹²C(⁶Li, d) cross section to the 11.096 MeV 4⁺ state is a result of multistep processes.     

Study of the reaction 16O(7Li, 3He)20F

Angular distributions for the ¹⁶O(⁷Li,³He)²⁰F reaction, at a bombarding energy of 24 MeV, have been measured for all states below 6.25 MeV excitation, using a gas target and a multi-angle spectrograph. Low-lying core-excited states are populated much less strongly than known (sd)⁴ states. Cross sections decrease rapidly with excitation energy, but states at 4.20, 4.52, 4.58 and 5.41 MeV are quite strong — suggesting they have high spin and (sd)⁴ configurations. Previously suggested high-spin states at 4.73 and 4.76 MeV are weak, implying they are probably of core-excited character.     

Isomer ratio measurements for the reaction 29Si(18O,p2n)44m, 44gSc

Isomer ratios for the reaction ²⁹Si(¹⁸O, p2n)^44gSc, ^44gSc have been deduced from activity measurements for projectile energies between 30 and 99 MeV. Statistical model calculations show that the isomer ratio dependence on projectile energy up to about 80 MeV can be adequately described by assuming a fixed ratio of quadrupole to dipole γ-ray strengths. Such a ratio of E2/E1 strengths agrees with corresponding values deduced from the literature. The values of the γ-ray strength ratios needed to fit the experimental isomer ratios are extremely sensitive to the relative amounts of quadrupole γ-ray admixture and to the presence of discrete levels other than those which conform to the yrast line.     

Instanton-anti-instanton interaction in the O(3) non-linear σ model and an exactly soluble fermion theory

We calculate the potential energy of weakly interacting instanton and anti-instanton quarks in the O(3) non-linear σ model, which leads us to a two-component massive fermion theory. The exact solution for the ground state of the theory is built.     

15N(p,po)15N and the levels of 16O for Ex = 14.8–18.6 MeV

Angular distributions of cross section and analyzing power for elastic scattering of protons from ¹⁵N have been measured for E_p = 2.7–7.0 MeV. A phase-shift analysis of the data yields spin-parity assignments and level parameters for seventeen states of ¹⁶O in the excitation energy region 14.8–18.6MeV. Three of the resonances have not previously been identified, among them being a broad J^π = 2^? level at E_p = 6.1 MeV which is almost certainly the analog of the 2^? 1p1h state with configuration ($\text{d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{,}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}$) at E_x ∽ 5.0 MeV in ¹⁶N. The broad level previously reported near E_p = 5.0 MeV has been observed and its parameters determined. A resonance analysis of the phase shifts yielded values of E_r, Γ and Γ_p for all of the levels. The J^π assignments are in agreement with previously reported values. For resonances having J = l, the data can usually be fit with a resonant phase shift corresponding to either J = l + 1 or J = l ? 1, in addition to the phase shift for J = l. Which of the two spurious-J solutions occurs seems to depend on whether the partial wave through which the resonant state is formed is $\text{J = l +}\text{1}\text{2}\text{or}\text{J = l ?}\text{1}\text{2}$.     

Algebraic description of multiple quadrupole excitations of collective states in nuclei: (I). Representation matrices for U(6)

An algorithm for the calculation of representation matrices for the totally symmetric representations [N] of the group U(6) is described. Applications to multiple quadrupole excitation processes in nuclei are given.