Oblate Limit Research Articles

Deformation dependence of breathing oscillations in Bose–Fermi mixtures at zero temperature

We study the breathing oscillations in Bose–Fermi mixtures in axially symmetric deformed traps of prolate, spherical and oblate shapes, and clarify the deformation dependence of the frequencies and the characteristics of the collective oscillations. The collective oscillations of the mixtures in the deformed traps are calculated using a scaling method. We obtain for the greatly deformed prolate and oblate limits and the spherical limit analytical expressions for the collective frequencies. The full calculation shows that the collective oscillations become consistent with the frequencies obtained analytically when the system is deformed into prolate or oblate regions. Complicated changes of the oscillation characteristics are shown to occur in the transcendental regions around the spherically deformed region. We find that these critical changes in oscillation characteristics are explained by the level crossing behaviors of the intrinsic oscillation modes. Approximate expressions are obtained for the level crossing points that determine the transcendental regions. We also compare the results from the scaling method with those from the dynamical approach.

Elliptical galaxies: Rotationally distorted, after all

On the basis of earlier investigations on homeoidally striated Mac Laurin spheroids and Jacobi ellipsoids (Caimmi and Marmo 2005, Caimmi 2006a, 2007), different sequences of configurations are defined and represented in the ellipticity-rotation plane, (O??2v). The rotation parameter, ?2v, is defined as the ratio, Erot=Eres, of kinetic energy related to the mean tangential equatorial velocity component, M(v?)2/2, to kinetic energy related to tangential equatorial component velocity dispersion, M?2 ??/2, and residual motions, M(?2 ww + ?2 33)=2. Without loss of generality (above a threshold in ellipticity values), the analysis is restricted to systems with isotropic stress tensor, which may be considered as adjoint configurations to any assigned homeoidally striated density profile with anisotropic stress tensor, different angular momentum, and equal remaining parameters. The description of configurations in the (O??2v) plane is extended in two respects, namely (a) from equilibrium to nonequilibrium figures, where the virial equations hold with additional kinetic energy, and (b) from real to imaginary rotation, where the effect is elongating instead of flattening, with respect to the rotation axis. An application is made to a subsample (N = 16) of elliptical galaxies extracted from richer samples (N = 25, N = 48) of early type galaxies investigated within the SAURON project (Cappellari et al. 2006, 2007). Sample objects are idealized as homeoidally striated MacLaurin spheroids and Jacobi ellipsoids, and their position in the (O??2v) plane is inferred from observations following a procedure outlined in an earlier paper (Caimmi 2009b). The position of related adjoint configurations with isotropic stress tensor is also determined. With a single exception (NGC 3379), slow rotators are characterized by low ellipticities (0 ? ? < 0.2), low anisotropy parameters (0 ? ? < 0.15), and low rotation parameters (0 ? ?2v < 0.15), while fast rotators show large ellipticities (0.2 ? ? < 0.65), large anisotropy parameters (0.15 ? ? < 0.35), and large rotation parameters (0.15 ? ?2v < 0.5). An alternative kinematic classification with respect to earlier attempts (Emsellem et al. 2007) requires larger samples for providing additional support to the above mentioned results. A possible interpretation of slow rotators as nonrotating at all and elongated due to negative anisotropy parameters, instead of flattened due to positive anisotropy parameters, is exploited. Finally, the elliptical side of the Hubble morphological sequence is interpreted as a sequence of equilibrium (adjoint) configurations where the ellipticity is an increasing function of the rotation parameter, slow rotators correspond to early classes (E0-E2 in the oblate limit and E2-E0 in the prolate limit) and fast rotators to late classes (E3-E6). In this view, boundaries are rotationally distorted regardless of angular momentum and stress tensor, where rotation has to be intended as due to additional kinetic energy of tangential equatorial velocity components, with respect to spherical configurations with isotropic stress tensor.

A new graphical version of STROTAB: The analysis and fitting of singlet–triplet spectra of asymmetric top molecules in the prolate or oblate limits

The original version of STROTAB has been modified to run under Microsoft Windows using the C++ programming language. The new version takes full advantage of the Microsoft Foundation Classes available within the Microsoft Visual C++ Version 6 development environment. Specifically, windows can be created that edit the input file, summarize the results of the least-squares fit, display the calculated and observed spectra, display whole or partial sections of the calculated spectra as a stick or Gaussian de-convoluted spectrum. A listing of the rotational quantum numbers in the cases (a) and (b) limits for each of the displayed lines is provided. A branch annotating routine provides a quick visual guide to the assignment of the spectrum. A new eigenvalue sorting method has been added as an option that complements the existing method based on the eigenvector coefficients. The new sorting method has eliminated some difficulties that may arise using the existing “Least Ambiguous Method”. The program has been extended to handle near-oblate asymmetric tops using a type III r representation. New version summary Title of program: STROTAB Version number: 2 Catalogue identifier:ADCA_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADCA_v2_0 Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Reference in CPC to previous version: 93 (1996) 241–264 Catalog identifier of previous version: ADCA Authors of previous version: R.H. Judge, E.D. Womeldorf, R.A. Morris, D.E. Shimp, D.J. Clouthier, D.L. Joo, D.C. Moule Does the new version supersede the original program: Yes Computers for which the program is designed and others on which it has been tested: Pentium Xenon, Pentium Pro and Later Operating systems or monitors under which program has been tested: Windows 98, Windows 2000, Windows XP Programming language used in the new version: ANSI C, C++, Microsoft Foundation Class (MFC) No. of lines in distributed program, including test data, etc.:11 913 No. of bytes in distributed program, including test data, etc.: 2 816 652 Memory required to execute with typical data: 7 Meg No of bits in a word: 16 No of processors used: 1 Has the code been vectorized or parallelized?: No No. of bytes in distributed program, including test data, etc.: ∼3.2 MB (compressed) Distribution format: zip file Additional keywords:near oblate top, bootstrap eigenvalue sorting, graphical environment, band contour Nature of physical problem: The least-squares/band contour fitting of the singlet–triplet spectra of asymmetric tops of orthorhombic symmetry using a basis set appropriate to the symmetric top limit (prolate or oblate) of the molecule in either Hund's case (a) or case (b) coupling situations. Method of solution: The calculation of the eigenvectors and eigenvalues remains unchanged from the earlier version. An option to sort the eigenvalues of the current J by fitting them to regular progressions formed from earlier J values (bootstrap method) can be used as an option in place of the existing method based on eigenvector coefficients. Reasons for the new version: The earlier version can only handle oblate tops by diagonalizing using the prolate limit. This has turned out to be unacceptable. An improved method of sorting eigenvalues under certain conditions is also needed. A graphical interface has been added to ease the use of the program. Summary of revisions: The Hamiltonian can now be constructed in a limit appropriate the representation for of the molecule. Sorting by an alternate method is now offered. Numerous graphical features have been added. Restrictions on complexity of the problem: The rotational quantum number restrictions are J ⩽ 255 and K (or P) ⩽127. The allowed transition frequency minus the band origin frequency must be in the range of ±10 000 cm −1. Up to five decimal places may be reported. The number of observed lines is limited by the dynamic memory and the amount of disk space available. Only molecules of symmetry D 2h, D 2 and C 2v can be accommodated in this version. Only constants of the excited triplet state may be varied.

Energy levels of the hydrogen atom in a cylindrical cavity

AbstractElectronic states of the cylindrically confined hydrogen atom are studied numerically by the finite difference (FD) approach. Energies of the low‐lying states of the system are estimated vs. the radius R and length Z of the cylinder, as well as the position of the nucleus at the axis. Under the isotropic stretch of the cylinder with the constant ratio R = Z/2, the energy level picture reminds us that in the spherically confined atom: the energies of all states regularly grow up when decreasing the size of the cavity, and the states with the same principal n, but higher values of the angular l quantum number correspond to lower energies almost everywhere. Anisotropic stretch of the cylinder from prolate to the oblate one with constant volume causes numerous level intersections and avoided crossings, with the levels ordered as 1σ, 2π, 3δ, 2σ,…, in the prolate limit case, and as 1σ, 2π, 2σ, 3δ, 3π, 3σ,…, in the oblate limit case. The results of numerical calculations are in excellent agreement with qualitative consideration of the one‐electron system in the impenetrable cavity by using both variational and perturbative approaches. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Femtosecond degenerate four‐wave mixing of pyridine and its biologically relevant derivatives

AbstractTime‐resolved degenerate four‐wave mixing experiments in the femtosecond time domain were employed to record the rotational ground‐state dynamics of the N‐heteroaromatic molecules pyridine, pyrimidine and pyrazine in the gas phase. Pyrimidine is an important molecular building block of several nucleic acid bases. The method records the time‐dependent change in third‐order polarization of a sample after coherent excitation. All three molecules are asymmetric tops in the oblate limit. From the analysis of the transients we obtained highly accurate averages of the two larger ground‐state rotational constants A and B. Copyright © 2002 John Wiley & Sons, Ltd.

Homoclinic billiard orbitsinside symmetrically perturbed ellipsoids

The billiard motion inside an ellipsoid ofℝ3 is completely integrable. Ifthe ellipsoid is not of revolution, thereare many orbits bi-asymptotic to its majoraxis. The set of bi-asymptotic orbits isdescribed from a geometrical, dynamicaland topological point of view. It containseight surfaces, called separatrices.The splitting of the separatrices under symmetric perturbations of theellipsoid is studied using a symplectic discrete version of thePoincaré-Melnikov method, with special emphasis on the followingsituations:close to the flat limit(when the minor axis of the ellipsoid is small enough), close to the oblatelimit(when the ellipsoid is close to an ellipsoid of revolution around its minoraxis)and close to the prolate limit(when the ellipsoid is close to an ellipsoid of revolution around its majoraxis).It is proved that any non-quadratic entire symmetric perturbation breaksthe integrability and splits the separatrices, although (at least) 16symmetric homoclinic orbits persist. Close to the flat limit, these orbitsbecome transverse under very general polynomial perturbations of theellipsoid.Finally, a particular quartic symmetric perturbation is analysed in greatdetail. Close to the flat and to the oblate limits, the 16 symmetrichomoclinic orbits are the unique primary homoclinic orbits. Close to theprolate limit, the number of primary homoclinic orbits undergoesinfinitely many bifurcations. The first bifurcation curves are computednumerically.The planar and high-dimensional cases are also discussed.

The mm-Wave Rotational Spectrum of CBrClF2(Halon BCF): Observation of a NewR-Type Band and Generalization of Conditions for Oblate-Type Band Formation

The rotational spectrum of the best known halon molecule, CBrClF2, was measured in the region 160–320 GHz. The data set was extended toJ″ = 110 and the weakbQandaRrotational transitions were also assigned and measured. Accurate spectroscopic constants are reported for the ground state in the four most abundant isotopic species of the molecule, and for the lowest vibrational satellite in the79Br35Cl and81Br35Cl isotopic species. Broadband rotational spectra showed an unusual diffuseR-type band characterized by quantum number changes ΔJ= −1, ΔK−1= 2, and ΔK+1= −3 between successive lines down the band. It is shown that this band is a variant of the so-called type II bands, which will form under the most general oblate limit condition (A + B) = (l/m)2C, wheremandlare integers. Such bands are expected to be common in high-Jrotational spectra and their key properties are given.

Geometrical percolation threshold of overlapping ellipsoids.

A recurrent problem in materials science is the prediction of the percolation threshold of suspensions and composites containing complex-shaped constituents. We consider an idealized material built up from freely overlapping objects randomly placed in a matrix, and numerically compute the geometrical percolation threshold ${\mathit{p}}_{\mathit{c}}$ where the objects first form a continuous phase. Ellipsoids of revolution, ranging from the extreme oblate limit of platelike particles to the extreme prolate limit of needlelike particles, are used to study the influence of object shape on the value of ${\mathit{p}}_{\mathit{c}}$. The reciprocal threshold 1/${\mathit{p}}_{\mathit{c}}$ (${\mathit{p}}_{\mathit{c}}$ equals the critical volume fraction occupied by the overlapping ellipsoids) is found to scale linearly with the ratio of the larger ellipsoid dimension to the smaller dimension in both the needle and plate limits. Ratios of the estimates of ${\mathit{p}}_{\mathit{c}}$ are taken with other important functionals of object shape (surface area, mean radius of curvature, radius of gyration, electrostatic capacity, excluded volume, and intrinsic conductivity) in an attempt to obtain a universal description of ${\mathit{p}}_{\mathit{c}}$. Unfortunately, none of the possibilities considered proves to be invariant over the entire shape range, so that ${\mathit{p}}_{\mathit{c}}$ appears to be a rather unique functional of object shape. It is conjectured, based on the numerical evidence, that 1/${\mathit{p}}_{\mathit{c}}$ is minimal for a sphere of all objects having a finite volume.