Asymmetrical Top Research Articles

Dzhanibekov effect in a physics classroom

The purpose of this publication is to present a novel approach to the demonstration of the Dzhanibekov effect. The main idea of our version is to use a lightweight spinning top of a spherical external form but distinct principal moments of inertia floating in the upward flow of air. As a result, the Dzhanibekov effect can be easily demonstrated anywhere on Earth: in any classroom, or even in the ‘home-lab’. The proposed demonstration allows one to observe the periodical flipping motion of the asymmetrical top with the clearly seen quasi-stable rotational phase. It may also become the base for various theoretical and experimental research projects for students.

Analytic formula for the geometric phase of an asymmetric top

The motion of a handle spinning in space has an odd behavior. It seems to unexpectedly flip back and forth in a periodic manner as seen in a popular YouTube video (“Plasma Ben, Dancing T-handle in zero-g, HD,” <https://www.youtube.com/watch?v=1n-HMSCDYtM>). As an asymmetrical top, its motion is completely described by the Euler equations and the equations of motion have been known for more than a century. However, recent concepts of the geometric phase have allowed a new perspective on this classical problem. Here, we explicitly use the equations of motion to find a closed form expression for the total phase and hence the geometric phase of the force-free asymmetric top and we explore some consequences of this formula with the particular example of the spinning handle for demonstration purposes. As one of the simplest dynamical systems, the asymmetric top should be a canonical example to explore the classical analog of the Berry phase.

The semiclassical density of states for the quantum asymmetric top

In the quantization of a rotating rigid body, a top, one is concerned with the Hamiltonian operator Lα = α20L2x + α21L2y + α22L2z, where α0 ⩽ α1 ⩽ α2. An explicit formula is known for the eigenvalues of Lα in the case of the spherical top (α1 = α2 = α3) and symmetrical top (α1 = α2 ≠ α3) (Landau and Lifshitz 1981 Quantum Mechanics: Non-Relativistic Theory 3rd edn (Portsmouth, NH: Butterworth-Heinemann)). However, for the asymmetrical top, no such explicit expression exists, and the study of the spectrum is much more complex. In this paper, we compute the semiclassical density of states for the eigenvalues of the family of operators Lα = α20L2x + α21L2y + α22L2z for any α0 < α1 < α2.

Anisotropic Reorientation of 9-Methylpurine and 7-Methylpurine Molecules in Methanol Solution Studied by Combining 13C and 14N Nuclear Spin Relaxation Data and Quantum Chemical Calculations

Reorientation of 9-(trideuteromethyl)purine and 7-(trideuteromethyl)purine molecules in methanol-d4 solutions has been investigated on the basis of the interpretation of the nuclear spin relaxation rates of their 14N (or 1H) and 13C nuclei. The transverse quadrupole relaxation rates of 14N nuclei have been obtained from the line shape analysis of their 14N NMR spectra. Alternatively, the information on the longitudinal 14N relaxation rates has been obtained via the scalar relaxation of the second kind of protons coupled to 14N. The longitudinal dipolar relaxation rates of the protonated 13C nuclei in the investigated molecules have been determined by measuring their overall relaxation rates and NOE enhancement factors. The molecular geometries, scalar coupling constants, and EFG tensors needed for quantitative interpretation of the above data have been calculated theoretically [DFT B3LYP/6-311++G(2d,p) or B3PW91/6-311+G(df,pd)] including the impact of the solvent by using discrete solvation and the polarizable continuum model. The reorientation of the investigated purines has been described as rotational diffusion of an asymmetrical top. It has been found that to get a fully consistent interpretation of the relaxation data, effective C-H bond lengths being 3% longer than the calculated ones had to be used in analysis to compensate for the ground-state vibrations. The obtained rotational diffusion coefficients and orientations of the principal diffusion axes show that the investigated molecules reorient anisotropically and that the mode of their solvation is remarkably different, in spite of their structural similarity.

Mapping between the dynamic and mechanical properties of the relativistic oscillator and Euler free rigid body

In this work we investigate a formal mapping between the dynamical properties of the unidimensional relativistic oscillator and the asymmetrical rigid top at a classical level. We study the relativistic oscillator within Yamaleev's interpretation of Nambu mechanics. Such interpretation is based on the factorisation of the momenta, and as a consequence of this factorisation we are led to a three dimensional phase space. Solutions of the relativistic oscillator are given in terms of the Jacobian elliptic functions and hence we establish a correspondence of these solutions in terms of well known quantities from the rigid body theory. We also study some mechanical restrictions that appear in the mathematical development of the mapping. In particular, we find a lower bound for the relativistic frequency in order to make the mapping self-consistent and physically legitimate.

Mapping between the dynamic and mechanical properties of the relativistic oscillator and Euler free rigid body

In this work we investigate a formal mapping between the dynamical properties of the unidimensional relativistic oscillator and the asymmetrical rigid top at a classical level. We study the relativistic oscillator within Yamaleev’s interpretation of Nambu mechanics. Such interpretation is based on the factorisation of the momenta, and as a consequence of this factorisation we are led to a three dimensional phase space. Solutions of the relativistic oscillator are given in terms of the Jacobian elliptic functions and hence we establish a correspondence of these solutions in terms of well known quantities from the rigid body theory. We also study some mechanical restrictions that appear in the mathematical development of the mapping. In particular, we find a lower bound for the relativistic frequency in order to make the mapping self-consistent and physically legitimate.

APPLICATION OF CLASSICAL FORMULATIONS OF QUANTUM MECHANICAL TIME-DEPENDENT VARIATIONAL PRINCIPLES TO THE SHELL AND COLLECTIVE MODELS

The pairing off of two-dimensional vortices with opposite orientation and constant strength has its analog in nuclear pairing forces, where the constant vortex strength corresponds to the projection of the angular momentum on the symmetry axis. This occurs as a second-order phase transition for a critical value of the interaction strength. Interactions leading to configurational mixing are analyzed in terms of Euler's equation of an asymmetrical top in the strong coupling limit. The dynamics of pairing forces, configurational mixing, and deformation alignment, due to quadrupole forces and the coupling of the total angular momentum to the intrinsic spin of the odd nucleon, are analyzed by imposing constraints on the coefficients of a quadratic rotational Hamiltonian. Processes leading to deformation alignment give rise to precessional motion of the total angular momentum about the nuclear symmetry axis.

Magnetic shielding tensors of carbon nuclei in 3,5-dichlorophenylacetylene, a theoretical ab initio and experimental study

Ab initio GIAO-CHF and DFT(B3LYP) molecular geometry optimization and magnetic shielding tensor calculations of carbon nuclei of 3,5-dichlorophenylacetylene have been performed using 6–311G** and 6–311+G(2d,p) basis sets. The isotropic 13C chemical shifts, needed for comparison, have been measured in C6D12 solution. The principal elements of the shielding tensor of the carbon nuclei in the investigated molecule in the solid state have been determined from an intensity analysis of the spinning sidebands in 1H-13C CP/MAS NMR spectra. Shielding anisotropy parameters of the acetylenic carbons have been independently determined using the method based on proton-coupled 13C nematic phase spectra as well as from the interpretation of the 13C longitudinal relaxation rates. The latter data have been analysed assuming the molecular reorientation to be the rotational diffusion of an asymmetrical top, which has provided, apart from the diffusion coefficients, an additional check on the reliability of the theoretical calculation of the shielding tensors. In general, satisfactory agreement between the theoretical and experimental results has been achieved.

Self-consistent mean field electrodynamics of turbulent dynamos

A turbulent dynamo in a conducting fluid is accompanied by the generation of small-scale magnetic fields, which are much stronger than the mean dynamo-generated magnetic field. These small-scale fields modify the α effect in such a way as to stabilize the dynamo process, α=(α0+β0R⋅/B∇×R)/(1+R2), where α0, β0 are the standard kinematic dynamo parameters, and R is proportional to the mean magnetic field B0, R=B0/(4πρV2/Rm)1/2, ρ is the fluid density, V is the characteristic turbulent velocity, and Rm is the magnetic Reynolds number. The derivation of this formula is illustrated using a simple model—the turbulent dynamo for an asymmetrical top.

A unified four-terminal JFET static model for circuit simulation

Junction field-effect transistors (JFETs) are useful for signal mixing purposes because of the isolated top- and bottom-gate terminals in such devices. Difficulties often arise, however, when one simulates the operation of a four-terminal JFET because the conventional JFET model treats the top and bottom gates as a single terminal. In this paper, we develop a unified four-terminal JFET static model covering both linear and saturation regions and including important device physics such as subthreshold behavior and asymmetrical top- and bottom-gate depletion layer thicknesses. Experimental data measured from JFETs fabricated at Harris Semiconductor is included in support of the model.

An analysis of the rotational stability of the layout back somersault

A numerical solution of Euler’s equations for a freely rotating asymmetrical top is applied to an analysis of the relative stability of rotation of a gymnast performing a layout back somersault with and without a full twist. The results suggest that successful performance of the stunt without a full twist has less tolerance for error in the initial rotational velocity than does the stunt with a full twist. This analysis provides an interesting illustration that can be used in advanced undergraduate mechanics courses. The analysis also demonstrates the potential for increased utilization of computers and numerical techniques in the undergraduate curriculum.

Time-convolutionless projection operator formalism for elimination of fast variables. Applications to Brownian motion

The relation between the time-convolutionless projection operator formalism and the methods developed by Kubo and van Kampen in the context of linear stochastic differential equations is discussed in detail. It is shown how this formalism may be used to develop a systematic procedure for the elimination of “fast variables”. This procedure is applied to the case of a free Brownian particle and to that of a heavily damped Brownian particle in the presence of an external force to obtain a reduced description in terms of the position variable: the results are found to be in complete agreement with the previous works on this problem. The rotational Brownian motion of an asymmetrical top in the presence of an external torque is also considered and an equation for the reduced probability distribution is derived which contains all the previous works on this problem as special cases.

Quantization in riemannian space

Expressions for the operators of momentum and kinetic energy in the Riemannian space are obtained from general quantummechanical requirements. The results make it possible to simplify the derivation of the Schrodinger equation in generalized coordinates from the classical Lagrangian function. The asymmetrical top is considered as an example. (auth)

A new basis for the representations of the rotation group. Lamé and Heun polynomials

The representation theory of the rotation group O(3) is developed in a new basis, consisting of eigenfunctions of the operator E = −4(L12 + rL22), where 0 &amp;lt; r &amp;lt; 1 and Li are generators. This basis |Jλ〉 is shown to be a unique nonequivalent alternative to the canonical basis (eigenfunctions of L3). The functions |Jλ〉 are constructed as linear combinations of canonical basis functions and are shown to fall into four symmetry classes, distinguished by their behavior under reflections of the inidividual space axes. Algebraic equations for the eigenvalues λ of E are derived. When realized in terms of functions on an O(3) sphere, the basis |Jλ〉 consists of products of two Lamé polynomials, obtained by separating variables in the corresponding Laplace equation in elliptic coordinates. When realized in a space of functions of one complex variable, |Jλ〉 are Heun polynomials. Applications of the new basis in elementary particle, nuclear, and molecular physics are pointed out, due in particular to the symmetric form of |Jλ〉 as functions on a sphere and to the fact that they are the wavefunctions of an asymmetrical top.

Fine structure of some infrared bands of CH 2F 2 and CH 2ClF

Some infrared bands of CH 2F 2 and CH 2ClF have been observed in the near infrared (PbS region). For all observed bands of both molecules, the R Q and P Q branches have been resolved and the molecular constants [ A − 1 2 (B + C) ] determined. For one type- C band of CH 2F 2, the P P and R R branches were resolved, and for this molecule, the 1 2 (B + C) molecular constant was determined. From these parameters the molecular dimensions were calculated. The 2.219μ band of CH 2F 2 presents an asymmetric shape in the strong, unresolved Q branch, and its is shown that this asymmetry is due to the fact that the CH 2F 2 molecule is a slightly asymmetrical top.

Energy Levels of Triatomic Molecules

The mathematical theory of the asymmetrical top and of triatomic molecules has been considerably simplified by removing two of the Eulerian angles from the wave equation by a special artifice. Asymptotic expressions for the eigenvalues of the former and the centrifugal and Coriolis corrections to be applied to the energy levels of the latter have been obtained in closed forms for values of J exceeding 9 without solving any secular equation.

Vibrational Analysis of the Absorption System of Sulphur Dioxide atλ3400−2600

Photographs of the bands of sulphur dioxide in the region of $\ensuremath{\lambda}3400\ensuremath{-}2600$ have been taken under low, medium and high dispersion, both at room temperature and at 200\ifmmode^\circ\else\textdegree\fi{}K; the pressure of the absorbing gas was varied from 0.3 mm to 480 mm. Thirty bands can be represented by the formula $\ensuremath{\nu}=29622+770{{v}^{\ensuremath{'}}}_{1}+320{{v}^{\ensuremath{'}}}_{2}+813{{v}^{\ensuremath{'}}}_{3}\ensuremath{-}6{{v}^{\ensuremath{'}2}}_{1}\ensuremath{-}2.5{{v}^{\ensuremath{'}2}}_{2}\ensuremath{-}20{{v}^{\ensuremath{'}}}_{1}{{v}^{\ensuremath{'}}}_{2}\ensuremath{-}25{{v}^{\ensuremath{'}}}_{2}{{v}^{\ensuremath{'}}}_{3}\ensuremath{-}15{{v}^{\ensuremath{'}}}_{1}{{v}^{\ensuremath{'}}}_{3},$ where ${{v}^{\ensuremath{'}}}_{1}$, ${{v}^{\ensuremath{'}}}_{2}$, ${{v}^{\ensuremath{'}}}_{3}$, are the quantum numbers of the symmetrical valence, the deformation and the antisymmetrical vibrations respectively. The three fundamental frequencies for infinitesimal vibrations in the upper electronic state are ${{v}^{\ensuremath{'}}}_{1}=794$, ${{v}^{\ensuremath{'}}}_{2}=345$ and ${{v}^{\ensuremath{'}}}_{3}=833$ ${\mathrm{cm}}^{\ensuremath{-}1}$. In addition, twelve bands have been identified that correspond to transitions from excited vibrational levels in the normal state. The relatively long ${{v}^{\ensuremath{'}}}_{1}$ and ${{v}^{\ensuremath{'}}}_{2}$ progressions indicate that both the bond distance and the angle have changed considerably in the transition to the excited electronic state. The vibrationless transition at 29622 ${\mathrm{cm}}^{\ensuremath{-}1}$ is weak, as one would have expected from considerations of the Franck-Condon principle. Substituting the three fundamental frequencies in the formula based on a valence force model, one obtains a value of 100\ifmmode^\circ\else\textdegree\fi{} for the apex angle in the excited state, as compared with 120\ifmmode^\circ\else\textdegree\fi{} in the normal state. The absence of any regularity in the rotational structure supports the resulting conclusion that the molecule in its upper state has become a more asymmetrical top.

The Rotational Energy of Polyatomic Molecules

It is shown that the quantized energy of a polyatomic molecule is approximately separable into the internal electronic and vibrational energy plus the rotational energy with the latter determined by solving the conventional problem of the rigid asymmetrical top. Because of the large oscillating terms in the Hamiltonian function due to interaction between vibration and rotation, this conclusion is not as obvious as it sounds, and seemed, if anything, to be contradicted by Eckart's recent investigation on the choice of a reference frame. The discrepancy, however, disappears when a second order perturbation calculation is made with Eckart's coordinates.

On the Asymmetrical Top in Quantum Mechanics

An elaboration and a more complete analysis of Witmer's work on the asymmetrical top treated as a perturbation of the symmetrical one show that one can deduce the rigorous solution of the problem from those of the algebraic equations of degree $2j+1$ or less. Without actually solving such equations, we find the terms divisible into even and odd groups just as in the case of the sigma-type doubling in diatomic molecules treated by Kronig, Van Vleck and others. For the case where the asymmetry is slight, an explicit expression for the separation of such similar doublets is obtained. The selection rules, which are rigorous for any degree of asymmetry, consist of the following: (a) Kronig's rule; (b) $\ensuremath{\Delta}j=0, \ifmmode\pm\else\textpm\fi{}1$; $\ensuremath{\Delta}m=0, \ifmmode\pm\else\textpm\fi{}1$; and (c) rule for the quantum number sigma, $\ensuremath{\Delta}\ensuremath{\sigma}=\mathrm{even}\mathrm{for}\mathrm{electric}\mathrm{moment}$ in $z$ direction and $\ensuremath{\Delta}\ensuremath{\sigma}=\mathrm{odd}\mathrm{for}\mathrm{moment}$ in $x\ensuremath{-}y$ plane. The effect of the electronic motions on the rotation of a polyatomic molecule as a whole is also briefly discussed.