Theory of electrodynamic levitation with a continuous sheet track — Part II

Various approximate expressions for lift and drag forces for null, normal and brake flux systems are developed from the rigorous expressions given in Part I of this work. Physical significance of the approximations used and the limits of validity are discussed. Results previously known in the literature are also discussed critically in this context.     

Moments of the intensity of a non-circular Gaussian laser speckle pattern in the diffraction field

Statistics of the intensity of a non-circular Gaussian speckle field that arises in imaging and propagation geometries was analyzed in the past but no general expressions for the moments of intensity were given. In this paper general expressions for the moments of the intensity of a non-circular speckle field are derived and their dependence on the path and beam parameters is discussed.     

The equation of state of the hard-disc fluid revisited

Some points about the search for analytical expressions for the equation of state of the hard-disc fluid are discussed in the light of the most recent advances in the field. New and accurate equations of state for this fluid are proposed.     

Exact vibration-rotation kinetic energy operators in two sets of valence coordinates for centrally connected penta-atomic molecules

The exact vibration-rotation quantum mechanical kinetic energy operator (KEO) for centrally-connected penta-atomic molecules such as methane and its isotopomers is derived for two sets of internal valence coordinates: the polyspherical coordinates and the bond-angle valence coordinates. The vibrational KEO including the pseudo-potential term is discussed for two forms of the modified Jacobian. For the rotational and vibration-rotation coupling KEO, results for two schemes of embedding the body-fixed coordinate system are presented: the bond embedding and the bisector embedding. Full expressions for the bisector embedding are too complicated to give in detail, but the working connection between the two embedding schemes is given. The future applications, including the perturbative and variational calculation of the vibrations and/or rotations for centrally connected penta-atomic molecules using the vibration-rotation KEO expressions derived in this work, are discussed.     

A study of the momentum dependence of the phase shift for finite range and Coulomb potentials and its possible applications

Closed from expressions for the partial derivative of the phase shift with respect to wave number k are obtained for a variety of potentials which can have a short range component as well as a long range Coulomb tail. Cut-off potentials, the Woods-Saxon potential, the Yukawa potentials, as well as potentials that behave like 1/r⁴ or 1/r⁶ as r tends to infinity fall into this category of short range potentials. Theoretical considerations like time delay and resonance widths are discussed. Numerical approximations based on the derived expressions for the partial derivative of the phase shift with respect to k are tested and compared with the exact value in those cases for which an analytic expressions for the partial derivative exists, such as in the case of a pure Coulomb potential or a square-well potential. This has facilitated a clear comparison of the numerical calculations. Possible applications of the formalism are discussed.     

Rotational energy,pulsars, and active nebulae

A number of different expressions for the rotational energy of a slowly rotating star have appeared in the literature. Here it is shown that these expressions are equivalent. Application to pulsars and the Crab Nebulae are discussed.Presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

Cartesian Expressions for Surface and Regular Solid Spherical Harmonics Using Binomial Coefficients and Its Use in the Evaluation of Multicenter Integrals

In this study, we present cartesian expressions for surface and regular solid spherical harmonics using binomial coefficients. The obtained results are useful for the calculation of multicenter molecular integrals over Slater type orbitals, especially for large values of quantum numbers. The main advantage of presented formulae in the evaluation of multicenter integrals is discussed.     

Third-order aberration analysis of tilted-pupil optical systems

The aberration field of an optical system with a tilted pupil is explored through expanding the vector expressions of the third-order wavefront aberrations. First, the vector forms of the wavefront aberrations are modified to obtain the aberration expressions with the pupil tilted; full field displays of coma and astigmatism in this situation are given. Then, the third-order aberration formulas with the pupil decentered and tilted simultaneously are derived and discussed. Finally, an example is taken to certify the validity of aberration distribution properties.     

Groundstate properties of the Coqblin-Schrieffer model in a crystalline field: Universality ofj=5/2 in cubic symmetry

The Bethe-ansatz equations for the Coqblin-Schrieffer model are discussed assuming an arbitary splitting of the ionic multiplet, e.g. by magnetic and axial and cubic crystal fields. Explicit expressions for the populations of thef-levels and the static spin susceptibility are obtained for the casej=5/2(Ce³⁺) in a cubic environment. The universality of this solution is discussed.     

Exact formulation of multiple scattering in a three-dimensional cylindrical geometry

Exact expressions for the source function, flux, and scattered intensity normal to the surface are developed in cylindrical coordinates for a three-dimensional, absorbing, emitting, isotropically scattering medium exposed to both diffuse and collimated radiation. Simplifications of these expressions for certain important geometries and uniform loading are presented. Also, superposition of these equations and radiative equilibrium are discussed. The generalized three-dimensional equations are shown to reduce to the familiar one-dimensional results. Also, the equations for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function are expressed in terms of the isotropic expressions.