Vibrational Nuclei Research Articles

Nuclear shell model and interacting bosons

In this workshop we have seen that the interacting boson model provides us with a unified phenomenological description of vibrational, rotational and transitional nuclei.

Static Quadrupole Moments of Vibrational States in Spherical Nuclei

A simple microscopic description is presented for the static quadrupole moments of the first 2+ state in vibrational nuclei by keeping the picture that the first 2+ state is composed of a one phonon state with a small mixture of a two phonon state. The phonon mode is described by the RPA mode and the Pauli effect on the two phonon state is taken into ac­ count approximately. It is also emphasized that the inclusion of particle-particle correlation is important to obtain the reasonable magnitude of the Q-momcnt. Calculation is made systematically for the nuclei near Z=50 region. Numerical results show in general a good agreement with experiment.

Quasirotational states in vibrational nuclei

In these lectures we present some simple characteristics excitational patterns in transitional nuclei approached from the viewpoint of strongly deformed and spherical nuclei. The sign and magnitude of the product Q( j) Q(2 1) are responsible for the tendency to increase or decrease the “deformation” of a coupled system, relative to Q(2 1), and to produce strongly coupled or decoupled excitational patterns. The vibrational picture is used for classification of states in a coupled system, in terms of multiplets, and the concept of intruder states is introduced toraccount for irregularities in the yrast band. Evidence is given for states arising from coupling clusters to collective states, especially those connected with internal excitations of a cluster.

Destructive interference between direct and indirect processes in (p, t) reactions on spherical vibrational nuclei and strong resemblance to that in (p, t) reactions on well-deformed nuclei

As evidence of a phase transition of the interference between direct and inelastic multistep processes in two-neutron pickup reactions exciting the first ${2}^{+}$ state of spherical vibrational nuclei, a destructive interference has been observed in $^{A+2}\mathrm{Pd}(p, t)^{A}\mathrm{Pd}({2}_{1}^{+})$ reactions. A strong similarity between the $^{104,108,110}\mathrm{Pd}(p, t)^{102,106,108}\mathrm{Pd}({2}_{1}^{+})$ cross sections and the $^{158}\mathrm{Gd}(p, t)^{156}\mathrm{Gd}({2}_{1}^{+})$ cross section is discussed.NUCLEAR REACTIONS $^{104,108,110}\mathrm{Pd}(p, t)$, $E=52$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$, coupled channel analysis.

Excitation of particle-vibration multiplets in theA=110region by two-neutron stripping reactions

The $^{103}\mathrm{Rh}$, $^{106,108}\mathrm{Pd}$, and $^{107,109}\mathrm{Ag}(t, p)$ reactions have been studied at a bombarding energy of 17 MeV. Excitation energies and enhancement factors were obtained for a number of levels up to about 2.5 MeV excitation in the residual nuclei. The $L=0$ transitions observed indicate that these nuclei are of a transitional character, though more closely resembling pairing rotational than pairing vibrational nuclei. The low-lying negative parity levels in the Ag nuclei appear to be well described as a ($2{p}_{\frac{1}{2}}$) proton weakly coupled to the appropriate core state in the even Pd cores. $^{105}\mathrm{Rh}$ appears to display similar weak coupling characteristics. However, the higher-lying levels such as the ${3}^{\ensuremath{-}}$ based states are severely mixed with single particle levels and perhaps with other elementary modes of excitation as well.NUCLEAR REACTIONS $^{103}\mathrm{Rh}$, $^{106,108}\mathrm{Pd}$, $^{107,109}\mathrm{Ag}(t, p)$, $E(t)=17.0$ MeV, magnetic spectrograph, enriched targets; measured $\ensuremath{\sigma}(\ensuremath{\theta}, {E}_{p})$, excitation energies, DWBA analysis; $^{105}\mathrm{Rh}$, $^{108,110}\mathrm{Pd}$, $^{109,111}\mathrm{Ag}$ deduced $L$, $J$, $\ensuremath{\pi}$, enhancement factors; weak coupling model.

Description of nuclear collective motions in terms of the boson expansion technique: (II). Additional formulation and numerical calculations

A formulation of the boson expansion technique given in our previous publication has been developed further by adding the contributions of the quadrupole-pairing interaction and the coupling between the collective and noncollective modes of excitations. Based on this formalism numerical calculations were carried out to describe the properties of a number of collective nuclei found in various regions of the periodic table. In general, very good agreement with experiment was obtained both for vibrational and rotational nuclei. Comparison with earlier theoretical work is also made.

Electron scattering from vibrational nuclei

We present electron scattering form factors for the ground states and several low energy quadrupole excitations of the nuclei $^{52}\mathrm{Cr}$, $^{110}\mathrm{Pd}$, $^{114}\mathrm{Cd}$, and $^{116}\mathrm{Sn}$. For $^{116}\mathrm{Sn}$ we also present the form factor for the lowest octupole excitation. From these data we derive ground state charge distribution parameters as well as $B(\mathrm{EL})$ values. We attempt to interpret the observed ${2}^{+}$ states as the one- and two-phonon states of an anharmonic vibrational model. Predictions are made for the electromagnetic decay branching ratios and excited state electric quadrupole moments.NUCLEAR REACTIONS $e+^{52}\mathrm{Cr}$, $^{110}\mathrm{Pd}$, $^{114}\mathrm{Cd}$, $^{116}\mathrm{Sn}$; $\ensuremath{\theta}=127.5\ifmmode^\circ\else\textdegree\fi{}$, $E=40,50,60,75,92,110$ MeV; ground state charge distribution parameters; anharmonic vibrator model fits to quadrupole excitation data.

Isospin and strong coupling effects in neutron scattering from even-ASe isotopes

Differential cross sections for elastic scattering from the even-$A$ isotopes of Se have been measured at 6, 8, and 10 MeV. Inelastic scattering differential cross sections have been obtained at 6- and 8-MeV incident energies. The results have been analyzed in a coupled channel model to determine the isospin coefficients in the real and imaginary parts of the scattering potential and the deformation parameters ${\ensuremath{\beta}}_{2}$ appropriate to neutron scattering in these vibrational nuclei. Both the coupling to the first ${2}^{+}$ level and the imaginary scattering potential vary strongly between $^{76}\mathrm{Se}$ and $^{82}\mathrm{Se}$, and in such a way that their effects cancel each other for inelastic scattering. It is shown that with an adequate data set the strong coupling and isospin dependencies of neutron scattering can be separated. The normalization uncertainty of these measurements is about 7%, and the statistical precision ranges from 1 to 10%.NUCLEAR REACTIONS $^{76}\mathrm{Se}$, $^{82}\mathrm{Se}(n,n)$ and ($n,{n}^{\ensuremath{'}}$), ${E}_{n}=6\ensuremath{-}8 \mathrm{and} 10$ MeV, $^{78}\mathrm{Se}$, $^{80}\mathrm{Se}(n,n)$, and ($n,{n}^{\ensuremath{'}}$), ${E}_{n}=8$ MeV. Measured (${E}_{n},\ensuremath{\theta}$). Enriched targets. Coupled channel analysis.

A Unified Phenomenological Description of Quadrupole Excitations in Even-Even Nuclei

A phenomenological model is developed for the collective quadrupole properties of all even-even nuclei. Rotational. vibrational and transitional nuclei are included in the model on an equal footing. A Bohr-type intrinsic Hamiltonian for harmonic quadrupole vibrations about an axially deformed shape is solved exactly. States of good angular momentum are projected out of the intrinsic states, and they are made orthogonal by a Schmidt scheme. The angular-momentum and phonon-number composition of the states is analyzed at various stages; states with K = 1 are found spurious. Excitation energies for the ground, β, and γ bands are calculated as expectation values of a radically simplified nuclear Hamiltonian in our projected and orthogonalized states. With increasing deformation the calculated energies evolve smoothly from the evenly spaced phonon spectrum to the Bohr-Mottelson rotational-vibrational spectrum according to the scheme of Sheline and Sakai. Our basic model contains only two parameters (deformation d and energy scale) to fix the entire quadrupole spectrum of a nucleus. Our results are given in the form of graphs suitable for immediate application; numerical results are readily produced by our computer code. The ground bands are fitted comparably to the VMI model, while the β and γ bands are reproduced qualitatively. The nuclei 152Sm, 152Gd, and 114Cd are used as test cases. Quadrupole moments and E2 transition rates are also calculated. Intraground-band transition ratios and branching ratios from the β and γ bands are given in terms of the single parameter d. The results are applied to 152Sm with fair success. Finally we extend the model to include two more parameters (anisotropy). The improvement over the basic model is modest in view of the added parameters and computational effort.

Neutron Number Dependence of Interference in Two-Neutron Transfer Reactions and the Nuclear Deformation

By making use of a simple model for the nuclear deformation, the number dependence of the form factors for inelastic scattering and two-neutron transfer processes are discussed for even-even deformed nuclei in the rare earth region. The form factor amplitudes are expressed in terms of the Legendre polynomials, whose argument is a func­ tion of the number. It is easy to understand, in the perturbative sense, a number dependence of the interference between the direct and two-step processes observed in (p, t) reactions leading to the 0, + and 2, + states. The interference for the 2, + state in deformed nuclei shows a striking parallelism with that for the quadrupole vibrational state in spherical nuclei. It has been shown that the two-neutron transfer reactions on deformed nuclei 1l can be well described in terms of the coupled-channel Born approximation (CCBA), 2l which includes the effect of multi-step processes via inelastic scattering. It has also been dezywnstrated that the same is ;true in the case of the two-neutron transfer reactions leading to the collective vibrational state in spherical nuclei. 3l~sl The purpose of the present paper is to explain,. in the perturbative sense, the neutron number dependence of the interference between direct and two-step processes in the (p, t) and ( t, p) reactions leading to the ground band members in even-even deformed nuclei in the rare earth region. The explanation applies equally well to other deformed nuclei. The interference observed in the (p, t) reactions, leading to the 2 1 + state in the spherical vibrational nuclei has been treated by Udagawa and coworkers. 3l· 5l· 6l They have pointed.ouel.al in the frame­ work of the quasiparticle RPA that a number dependence of the form factor amplitudes originates from the dynamical role of the ground state correlations and the gross number dependence of the occupation coefficients vv and uu, and that the number dependence of the form factor amplitudes provides .an explanation to the sign change of the interference observed in the middle of a major shell. In § 2, the transition amplitude for the A(p, t)B reaction is given in a second­ order perturbative form of the CCBA description. In this procedure, it is shown that the amplitude of the core-excitation model studied by Kozlowsky et aFl is

Elementary excitations in vibrational nuclei

We propose to describe the entire collective spectrum of vibrational nuclei in terms of few enteracting elementary excitation modes. We discuss in detail the case in which only two elementary modes are important (the quadrupole d and the octupole f-bosons). We give explicit expressions for the energy levels and transition matrix elements.

Boson symmetries in vibrational nuclei

It is proposed that nuclei (away from closed shells and from regions of large deformation) be described as a boson gas. Exploiting the underlying O(5) symmetry analytic expressions for the eigenvalues of the boson Hamiltonian and for the transition matrix elements are obtained. The computed energy levels are in surprisingly good agreement with those recently observed in the ( α, xn γ) and (heavy ions, xn γ)

A study of the giant dipole resonance of vibrational nuclei in the 103 ≦ A ≦ 133 mass region

This paper presents a set of experimental data concerning the giant dipole resonance of nuclei (GDR) in the 103 ≦ A ≦ 133 mass region. The cross sections σ(γ,n) and σ(γ, 2n) were obtained in the energy region 8–30 MeV by means of a monochromatic photon beam produced by annihilation in flight of positons. This paper attempts also to give an interpretation of the experimental behaviour of the GDR for vibrational nuclei in the 103 ≦ A ≦ 133 mass region in terms of the simple dynamic collective model. In particular it is shown that the width of the GDR increases as β increases and as E 2+ decreases and that the theoretically predicted spreading of the dipole strength is confirmed by our experimental results. As to the characteristic behaviour of the GDR above its peak value at E 0, we come to the conclusion that the actual state of the art in ( γ, xn) research does not allow one to make an unambiguous choice between isospin splitting or electric quadrupole absorption. Finally the numerical evaluations of the different sum rules are given and some empirical results concerning the average energy of the GDR as a function of A are also presented.

Coupled-channel-Born-approximation study of (p,t) reactions leading to two-phonon states of vibrational nuclei

A coupled-channel-Born-approximation (CCBA) study is made of ($p,t$) reactions leading to two-phonon states of vibrational nuclei. A detailed numerical analysis is performed, particularly for the $^{116}\mathrm{Cd}(p,t) ^{114}\mathrm{Cd}$ reaction with the incident energy ${E}_{p}=28$ MeV. It is shown that many features observed in the reactions, which cannot be explained by the distorted-wave Born approximation (DWBA), can successfully be accounted for by CCBA. It is also shown that three-step processes play an essential role in the successful explanation of the observed data, particularly the cross sections of the ${0}^{+}$ and ${4}^{+}$ two-phonon states.[NUCLEAR REACTIONS $^{116}\mathrm{Cd}(p,t)$, $E=28$ MeV; calculated $\ensuremath{\sigma}(\ensuremath{\Theta})$.]

A study of configuration mixing in vibrational nuclei

The configuration-mixed wave functions for one-phonon and two-phonon 2 + states have been obtained from the experimental energy spectra of the even-even vibrational nuclei. The reduced transition probabilities and electric quadrupole moments have been calculated.

Boson description of collective states: (II). Description of odd vibrational nuclei

By addition of the so-called ideal quasiparticle to the boson space one can represent the odd fermion states in that product space. In such a way one finds various representations of the fermion operators in terms of the boson operators and ideal quasiparticles. From these boson expansions of the fermion operators a finite one is selected by considering non-unitary transformations. Thus, the direct generalization, of the Dyson representation for even systems is given for the case of odd systems. The Hamiltonian can be divided into three parts: the boson term which describes the vibrational motion of the even core, the unperturbed motion of the quasiparticle, and the interaction between the quasiparticle and the bosons. This interaction consists of two terms, one of which agrees with the term used by Kisslinger and Sorensen 2), which is usually called the dynamical interaction, and the additional term is due to the antisymmetrization between the extra particle and the even core. The latter term can be identified as kinematical interaction which is responsible for the anomalous coupling states. For example, it is demonstrated that this term produces qualitatively the same splitting of the one-phonon multiplet as was obtained by Kuriyama et al. 3) for the j-shell. Furthermore, it is shown for the more complicated case of 117Sn that the effect of this additional interaction between phonons and quasiparticle is important when many shells to the states in the odd nucleus are taken into account.

The 110Pd, 116Cd(p, p'γ) reactions

The possibility of extending our knowledge about higher-energy collective-type states among the vibrational nuclei by observing γ-rays from states excited by inelastically scattered protons has been examined. Gamma rays from 110Pd and 116Cd were detected in coincidence with inelastically scattered protons; the incident protons had energies of 12 and 13 MeV. Gamma rays, most of them previously unreported, were observed from 17 levels in 110Pd and 14 levels in 116Cd below ≈ 2.5 MeV. The results are compared to predictions of a phonon- model interpretation of the levels. Evidence is shown for the existence of the three-quadrupole phonon quintet in 110Pd.

Xe136(p,p′)Xe136andXe136(p,d)Xe135Reactions

Low-lying states of $^{136}\mathrm{Xe}$ with excitation energies up to 3.3 MeV are investigated via inelastic scattering of 13.982-MeV protons. Spin, parity, and deformation parameters are extracted for 13 excited states through distorted-wave Born-approximation (DWBA) analysis using a collective-model form factor. The results obtained indicate that $^{136}\mathrm{Xe}$ is not a typical vibrational nucleus. A microscopic calculation for the level spectrum and the inelastic form factor, based on the quasiparticle random-phase approximation, is performed. DWBA angular distributions are calculated using a microscopic form factor. The level spectrum and the absolute magnitudes of the cross sections for most of the ${4}^{+}$, ${6}^{+}$, and ${3}^{\ensuremath{-}}$ observed states are reasonably well reproduced through these calculations. For the ${2}^{+}$ states, however, the predicted level density and the distribution of the inelastic strengths disagree with the observed data. In addition, three deuteron groups leading to states of $^{135}\mathrm{Xe}$ are observed. The orbital-angular-momentum-transfer values and the spectroscopic factors for these states are obtained through DWBA analysis of the data.

Electron scattering from one- and two-phonon vibrational states

Admixtures of one- and two-phonon states are required to fit (e, e′) form factors for the lowest two 2 + states in several even-even vibrational nuclei. These anharmonic wave functions correctly reproduce radiative branching ratios and excited state quadrupole moments.

Mössbauer studies and nuclear quadrupole moments of186, 188, 189, 190 Os

Mossbauer transmission experiments with the 137, 155, 69 and 187 keV gamma rays of186, 188, 189, 190Os, respectively, yielded the electric quadrupole splitting of these gamma resonance lines in OsO2 and OsP2. From the results the following ratios of quadrupole moments were derived:Q2+(186Os, 137 keV):Q2+(188Os, 155keV):Q2+(190Os, 187 keV):Q3/2−(189Os, g.s.)=(+1.100±0.020): 1.0:(+0.863±0.051): (−0.586±0.011) andQ5/2−(189Os, 69 keV)/Q3/2−(189Os, g.s.)=−0.735 ±0.012. The ratios for180, 188, 190Os are, within their limits of error, in agreement with the expectation of the rotational model, indicating that the pairing-plusquadrupole model calculations of Kumar and Baranger predict too rapid a transition form rotational to vibrational nuclei. Applying this argument to186Os in particular and using the measured ratios, one obtains a set of values for the quadrupole moments themselves, namelyQ2+ (186Os)=−(1.50 ± 0.10)b,Q2+(188Os)=−(1.36± 0.09) b,Q2+ (190Os)=−(1.18 ± 0.08) b,Q3/2−(189Os)=+ (0.80 ± 0.06) b, andQ5/2− (189Os)=− (0.59 ± 0.05) b. For the electric field gradient at Os nuclei in Re metal a value ofVzz=−(3.3 ± 0.6) · 1017 V/cm2 was found. A measurement with a magnetically split source yielded δ=+ 0.685 ± 0.025 for theE2/M1 mixing parameter of the 69 keV transition of189Os,g5/2−/g3/2−=0.895 ± 0.006 for the ratio of theg-factors of the 69 keV state and the groundstate, andHi=−(1135 ± 20) kOe for the hyperfine field at Os nuclei in an iron matrix.