Nonrotational States Research Articles

Distortion of high-degree solar p-mode eigenfunctions by latitudinal differential rotation

view Abstract Citations (63) References (5) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Distortion of High-Degree Solar p-Mode Eigenfunctions by Latitudinal Differential Rotation Woodard, Martin F. Abstract The eigenfunctions of nonradial oscillation of the sun depart from their customarily assumed spherical harmonic form as a result of solar rotation. This paper treats the effect on oscillation eigenfunctions of slow, axisymmetric differential rotation which is also mirror symmetric across the solar equatorial plane. A given oscillation eigenfunction, perturbed by rotation, can be expressed as a superposition of eigenfunctions of a (hypothetical) nonrotating state of the sun. The assumption of axialsymmetry implies that each of the superposed functions has the same azimuthal order, m, while mirror symmetry dictates that the values of l, the spherical harmonic degree, involved in the superposition be either all odd or all even. The treatment here is further specialized to the case of high-degree p- and f-mode oscillations, on which the effect of the Coriolis acceleration is small. It is shown that, by ignoring the Coriolis effect and making other sensible approximations, an analytic expression can be obtained for the rotationally distorted mode eigenfunctions, which is expected to be a good approximation for typical solar oscillations of degree l greater than 100. The effect of the calculated mode distortion on the measurement of solar rotation is explored. Publication: The Astrophysical Journal Pub Date: December 1989 DOI: 10.1086/168206 Bibcode: 1989ApJ...347.1176W Keywords: Eigenvectors; Solar Oscillations; Solar Rotation; Axes Of Rotation; Formalism; Spherical Harmonics; Solar Physics; SUN: OSCILLATIONS; SUN: ROTATION full text sources ADS |

111Ag utilizing the (3He, pn gamma ) reaction: A rotational nucleus with intermediate deformation.

The structure of $^{111}\mathrm{Ag}$ was studied using the $^{110}\mathrm{Pd}$${(}^{3}$He, pn\ensuremath{\gamma}) $^{111}\mathrm{Ag}$ reaction, which has populated an essentially complete set of low-energy states. The proton exit channel was isolated from competing reaction channels by operating \ensuremath{\gamma}-ray detectors in coincidence with a large-solid-angle proton detector. The experiments included \ensuremath{\gamma}-ray excitation functions, \ensuremath{\gamma}-ray angular distributions, and \ensuremath{\gamma}-\ensuremath{\gamma} coincidences. Four rotational bands have been identified. A symmetric particle-rotor model has been used to interpret these bands, and the model has identified other rotational features as well. A tentative interpretation of several nonrotational states is presented, suggesting that the mixing between rotational and nonrotational states is minimal.

Study of the shift of two-phonon poles in deformed nuclei by the method of boson expansions

The situation with regard to two-phonon collective states in even-even deformed nuclei is obscure. According to the Bohr-Mottelson model deformed nuclei must have two-phonon collective states; this was recently pointed out once more in [i]. According to the interacting boson model the low-lying states must include states with large two-boson components [2]. In the framework of the quasiparticle-phonon nuclear model [3] the study of the twophonon states showed [4] that allowance for the Pauli principle, i.e., allowance for the exact commutation relations between the phonons, leads to a shift of the two-phonon poles to higher energies. In [5], the corresponding secular equations were solved. Allowance for the Pauli principle leads to a shift of the centroids of the energy of the collective two-phonon states by 1-2 MeV. At excitation energies of 3-4 MeV the strength of the collective two-phonon states must be distributed over many nuclear levels. It was concluded on this basis in [5] that in even-even deformed nuclei there are no collective twophonon states. In the description of two-phonon collective states in deformed nuclei there is a contradiction between the quasiparticle-phonon model and the Bohr-Mottelson and interacting boson models. As is shown in [6], there is a contradiction between the quasiparticle-phonon model and the interacting boson model in the description of a number of other nonrotational states in deformed nuclei. We note that according to [7] allowance for the Pauli principle in even-even spherical nuclei does not lead to a large shift in the energies of the two-phonon states, in agreement with the experimental data on the twophonon collective states in spherical nuclei. An analysis of experimental data led to the conclusion in [8] that there are no twophonon collective states in deformed nuclei. New experimental data on 1~SEr in [9,10] showed that the levels previously treated as two-phonon levels have a dominant one-phonon component. The absence of two-phonon quadrupole states with energy less than 2 MeV in 168Er is explained in [i,Ii] by the large anharmonicity of the $ and ~ vibrations due to the shape of the nucleus in the form of a triaxial ellipsoid. The absence of two-phonon octupole states in a number of Ra, Th, and U isotopes [12] is explained in many studies by the existence of a stable octupole deformation. We note that the new experimental data on 168Er [9,13] agree well with calculations in the quasiparticle-phonon model and with the calculations in [14] of the structure of the single-phonon states and do not agree with the calculations in the interacting boson model. In view of the importance of clarifying the situation with regard to two-phonon collective states of even-even deformed nuclei, it is necessary to study the behavior of the two-phonon states by a different mathematical formalism. Since allowance for the Pauli principle in the two-phonon components of the wave functions plays a decisive role, it is expedient to make the calculations with phonons constructed from the operators of "true" bosons. It is necessary to establish whether there is a large shift of the twophonon poles in the problem when formulated in this way. The present paper is devoted to

High-order non-born-oppenheimer electric polarizabilities for excited states of H +2 via the perturbational-variational rayleigh-ritz formalism

Large-order perturbation theory in the form of the perturbational-variational Rayleigh-Ritz (PV RR) formalism has been applied to excited energy levels in the Stark effect for H + 2,yielding non-Born-Oppenheimer electric polarizabilities through 16th, 8th, 4th, and 2nd order, respectively, for the first four excited non-rotational states; these non-Born-Oppenheimer excited states correspond to what are normally known as vibrationally excited levels of the electronic ground state. The excited-state polarizability series are found to be more divergent in each corresponding order than the ground-state one, becoming increasingly so for the higher excited states. Methods for obtaining the excited-state polarizabilities to still higher order within the framework of PV RR are discussed.

Superfluid Dissipation in RotatingHe3-B

Additional dissipation of oscillatory superflow has been observed in uniformly rotating /sup 3/He-B. The damping increases linearly with angular velocity and is interpreted in terms of a vortex-induced mutual friction similar to that observed in superfluid /sup 4/He. In the nonrotating state the damping is amplitude dependent, and may contain a contribution from remanent vorticity.

PMR study of intramolecular motions in solid p-toluidine

Wide-line PMR studies on solid p-toluidine have been made in the temperature range 77–318 K at 7.5 MHz. Analysis of second moment data suggests a rigid and non-rotating state of the crystal lattice at 77 K. The molecular and crystal structure is confirmed. The second moment gradually decreases up to 240 K (a fall of 2.8 G 2) which is indicative of quantum mechanical tunnelling of methyl protons. An observed abrupt fall in p.s.m. beyond 240 K is followed by a phase transition and is attributed to methyl and amino group reorientation upto the melting point. The potential barrier to the reorientation is found to be approximately 8.88 kcal mol −1 in the solid phase.

The quasi-adiabatic analysis of nonradial modes of stellar oscillation in the presence of slow rotation

We have computed the vibrational stability of nonradial modes of oscillation in the quasi-adiabatic approximation for slowly, uniformly rotating stars, in the Cowling approximation. Only first-order effects are taken into account in the ratio of rotation angular frequency to oscillation angular frequency. We have applied the theory to iron white dwarf models and to zero-age main-sequence modesl. We find that slow rotation renders prograde traveling waves of nonradial oscillation (m<0) slightly more unstable than retrograde modes, essentially independently of the stability of the model in a given mode in the nonrotating state.

Particle emission rates from a black hole. II. Massless particles from a rotating hole

The calculations of the first paper of this series (for nonrotating black holes) are extended to the emission rates of massless or nearly massless particles from a rotating hole and the consequent evolution of the hole. The power emitted increases as a function of the angular momentum of the hole, for a given mass, by factors of up to 13.35 for neutrinos, 107.5 for photons, and 26 380 for gravitons. Angular momentum is emitted several times faster than energy, so a rapidly rotating black hole spins down to a nearly nonrotating state before most of its mass has been given up. The third law of black-hole mechanics is proved for small perturbations of an uncharged hole, showing that it is impossible to spin up a hole to the extreme Kerr configuration. If a hole is rotating fast enough, its area and entropy initially increase with time (at an infinite rate for the extreme Kerr configuration) as heat flows into the hole from particle pairs created in the ergosphere. As the rotation decreases, the thermal emission becomes dominant, drawing heat out of the hole and decreasing its area. The lifetime of a black hole of a given mass varies with the initial rotation by a factor of only 2.0 to 2.7 (depending upon which particle species are emitted). If a nonrotating primordial black hole with initial mass 5 \ifmmode\times\else\texttimes\fi{} ${10}^{14}$ g would have just decayed away within the present age of the universe, a hole created maximally rotating would have just died if its initial mass were about 7 \ifmmode\times\else\texttimes\fi{} ${10}^{14}$ g. Primordial black holes created with larger masses would still exist today, but they would have a maximum rotation rate determined uniquely by the present mass. If they are small enough today to be emitting many hadrons, they are predicted to be very nearly nonrotating.

Electron capture decay of 245Bk (4.90 d) and 246Bk (1.80 d)

The electron capture decay schemes of 245Bk and 246Bk have been investigated by measuring the γ-ray and conversion-electron spectra of mass-separated 245Bk and 246Bk samples. The γ-ray spectra were measured with a 25 cm 3 Ge(Li) spectrometer and the conversion-electron spectra were measured with a cooled Si (Li) detector. Multipolarities of most of the transitions in 245Cm and 246Cm were deduced. The half-lives of 245Bk and 246Bk were determined by following the decay of the 252.85 and 798.7 keV photopeaks and were found to be 4.90 ± 0.03 d and 1.80 ± 0.02 d, respectively. The α/(α+EC) ratio for the 245Bk decay was measured to be (1.2 ± 0.1) × 10 −3. On the basis of the present investigation the following non-rotational states were identified in 245Cm: 7 2 + [624], 0; 5 2 + [622], 252.85; 1 2 + 355.95; 3 2 − vibrational, 633.65; and 1 2 + [620], 740.95 keV. The 3 2 − state at 633.65 keV is int as a K π = 2 − phonon coupled to the i 7 2 + [624] single-particle state. Our measurements of 246Bk γ-ray and conversion-electron energies and intensities confirm previous level assignments in 246Cm.

The level structure of 189Os

The level structure of 189Os has been studied by (d, p), (d, t) and (d, d') reaction spectroscopy at E d = 12.1 MeV. Assignments of a number of levels at excitation energies below ≈ 1700 keV are given. The assignments are discussed in terms of a unified model based on the Nilsson model including pairing, rotational motion and attenuated Coriolis coupling. Deviations between predicted and experimental excitation energies and wave functions are generally found to be consistent with trends observed in 187Os and in the odd W isotopes. Evidence for the existence of collective non-rotational states is found from the (d, d') reactions. Results of ( 3He, α) and high resolution γ-ray and conversion electron studies were also included at various stages of the investigation to supplement the data from the deuteron induced reactions. Comparisons between calculated and measured B(E2) values are found to indicate an intrinsic quadrupole moment of the ground-state band of ≈ 5.0 b, in agreement with values in adjacent even Os isotopes. Details of the Coriolis coupling calculations are given.

Octupole vibrations and ground state correlations

The relationship between ground-state correlations and collectiveness is investigated for the case of low-lying nonrotational states in the rare earth nuclei. Both octupole and quadrupole modes of excitation are studied and the quasiparticle virtual populations associated with each of them are discussed. The relative importance of particle-particle and particle-hole interaction matrix elements is also analyzed in connection with the shape of the correlation patterns. The fundamental role of the Nilsson + BCS scheme is emphasized and the consistency of the quasiparticle random-phase approximation is established.

A comparative study of the diffusion of C 2H 6, C 2H 4, and nC 5H 12 in erionite and in type a zeolite

Experimental kinetic and equilibrium data are presented for the sorption of C 2H 6, C 2H 4, and nC 5H 12 in synthetic 4 A and 5 A zeolites and in natural erionite. The Henry's Law constants and heats of sorption are somewhat greater for erionite than for the type A zeolites, reflecting the difference in cavity size and framework density. The concentration dependence of the diffusivity is similar for all three zeolites and is adequately correlated by Darken's equation. The activation energies for diffusion in erionite are similar although somewhat larger than for the 5 A sieve and this is consistent with the smaller minimum free diameter of the distorted octagonal windows of erionite as compared with the symmetric eight-membered windows of the 5 A sieve. The experimental values of the pre-exponential factors for the diffusion of the smaller molecules (C 2H 4 and C 2H 6) in both erionite and type A zeolites are similar to the values estimated from transition state theory, assuming a nonrotating transition state.

Particle-particle and particle-hole interactions and non-rotational collective nuclear states

The structure of the γ-vibrational band-heads in the rare-earth region is examined in relation to the different types of interaction matrix element that appear within the framework of the quasi-particle random phase approximation. The systematic trends of the force coupling constants, the B(E2) transition rates and the quasi-particle numbers in the correlated ground state of the nuclei are analysed in particular situations in which the interaction between quasiparticles contains either particle-particle or particle-hole terms, or both. The effects of schematic and non-schematic model calculations are discussed.

Single-particle energies and wave functions for the saxon-woods potential and the levels of odd-A nuclei in the actinide region

Abstract The Saxon-Woods single-particle energies and wave functions are given for the description of deformed nuclei in the region 234 ≦ A ≦ 250. The energies and the wave functions for the non-rotational states of odd- A nuclei are calculated taking into account the interaction of quasiparticles with phonons. The latter is found to be more important for nuclei in the actinide region than in the rare-earth one. A rather good description of the energies of low-lying non-rotational states in 235 U, 237 U, 241 Pu, 245 Cm, 247 Cm, 249 Cm, 237 Np, 241 Am and 249 Bk is obtained. The calculated Saxon-Woods single-particle energies and wave functions may serve as a basis for calculating various characteristics of nuclei in the actinide region.

Multi-step stripping on deformed nuclei with non-rotational states involved

An expression for the deuteron stripping cross section is obtained, taking into account the intermediate excitations of the rotational and non-rotational types. All the rotational states are included using the adiabatic approximation, while the other types of nuclear excitations are treated by the coupled equations. The result is illustrated using perturbation theory. The spectroscopic amplitudes of the intermediate transitions are calculated in the framework of the microscopic nuclear model. The theory is applied to the 154Gd(d, p) 155Gd stripping reaction. A detailed analysis is made.

Plasma Diffusion and Stability in Toroidal Systems

The problems of resistive diffusion and of rotation about the magnetic axis in toroidal systems are considered. They are shown to be related to the question of stability of drift-acoustic waves on which viscosity has an important influence. A nonrotating state is always unstable, but if the appropriate criterion is satisfied, there are stable rotating states for which an enhanced diffusion is calculated.

The decay scheme of 165Tm

The electron-capture decay 165 Tm → 165 Er has been investigated with the six-gap β-ray spectrometer as well as Ge(Li) and NaI(Tl) γ-ray detectors. The single γ-ray spectrum, e-γ and γ-γ coincidence spectra have been studied. Some tentative levels have been confirmed and several new established. The structure of the non-rotational states in 165Er is discussed in terms of a model in which the interaction of quasi-particles with quadrupole and octupole phonons is considered. The energy spacing between states with a predominant single-particle character is compared with the energy spacing calculated on the basis of the pairing force theory.

Surface delta interaction and non-rotational collective states in rare-earth nuclei

The surface delta interaction (SDI) has been applied to a calculation of the γ- and K = 0 octupole vibrational states in the rare-earth region. The residual interaction used here has only one parameter, which can be determined by means of the odd-even mass differences, while the pairing plus quadrupole (or octupole) model utilized for this type of calculation requires three essentially unrelated coupling strengths. The results obtained in this paper compare reasonably well with the experimental ones, being in this sense as good as those yielded by the pairing-plus-quadrupole (or octupole) model.

Structure of the ground and excited states of odd-mass deformed nuclei in the region 153 ≦ A ≦ 187

The interactions of quasi-particles with quadrupole and octupole phonons in deformed odd-mass nuclei are considered in the framework of the method of approximate second quantization. The properties of phonons are determined in investigating even-mass nuclei. In considering the interaction of quasi-particles with phonons in odd-mass nuclei, there is therefore not a single free parameter. The peculiarities of the secular equations are studied and it is shown that for many states, besides the interaction of quasi-particles with gamma-vibrational phonons, the interactions of quasi-particles with octupole and beta-vibrational phonons are very important. The energies and wave functions for the majority of odd-mass nuclei in the range 151 ≦ A ≦ 187 are found. It is shown that the interaction of quasi-particles with ohonons leads to the appearance of admixtures to the one-quasi-particle states and improves the description of these states as compared to the independent quasi-particle model. It is shown that this interaction leads to the formation of the collective non-rotational states and of the complex structure states in odd-mass nuclei. The calculated energies of the excited states, values of B(E2), the decoupling parameters a and the spectroscopic factors agree rather well with the corresponding experimental data. It is concluded that the interaction of quasi-particles with phonons gives a correct description of the structure of the ground and excited states of deformed odd-mass nuclei.

Quasi-particle and collective structure of the states of even, strongly-deformed nuclei

The structure of the non-rotational excited states of even, deformed nuclei in the ranges 150≦ A ≦ 186 and 228 ≦ A ≦ 254 is investigated. The energies of the first and second K π = 2 + and 0 + states are calculated taking into account the blocking effect in the framework of the method of approximate second quantization. These energies agree sufficiently well with the corresponding experimental data. The structure of the excited states is investigated and it is found that in most cases the lowest lying K π = 2 +, 0 + states possess clearly expressed collective properties. In some cases the energies of quadrupole excited states lie near the secular equation poles and the structure of K π = 2 + states is similar to the two-quasi-particle ones. The particularities of the K π = 0 + states are investigated. The spectroscopic factors are calculated for direct nuclear reactions and the relative probabilities are found for beta transitions to the K π = 0 + states. A satisfactory agreement is obtained between theory and experiment. The role of the F and Q forbiddennesses in the superfluid nuclear model is investigated.