Analytic light-scattering solution of two merging spheres using Sh-matrices

We use the Sh-matrix formalism that contains the shape-dependent parameters of the T-matrix to derive an analytic solution for the light scattering from two merging spheres at different degrees of merging. The integral expressions for the Sh-matrix elements are simpler than those of the T-matrix elements and, for two merging spheres, these integrals can be solved analytically. Our calculations show that when the spheres merge, the primary fringes that are circular for non-merging spheres become distorted. Secondary fringes due to the interference of the waves emanating from the two spheres begin to appear when the spheres are merged approximately 50%.     

Bethe ansatz for the XXX-S chain with non-diagonal open boundaries

We consider the algebraic Bethe ansatz solution of the integrable and isotropic XXX-S Heisenberg chain with non-diagonal open boundaries. We show that the corresponding K-matrices are similar to diagonal matrices with the help of suitable transformations independent of the spectral parameter. When the boundary parameters satisfy certain constraints we are able to formulate the diagonalization of the associated double-row transfer matrix by means of the quantum inverse scattering method. This allows us to derive explicit expressions for the eigenvalues and the corresponding Bethe ansatz equations. We also present evidences that the eigenvectors can be build up in terms of multiparticle states for arbitrary S.     

Construction of a dynamical matrix for an HCP crystal using phonon data at the symmetry points of the Brillouin zone and elastic constants

General analytical expressions are obtained for the dynamical matrix D(k) and the elastic constants C _ik in an HCP crystal in terms of the Born-von Karman (BvK) parameters. An analytical method is proposed for constructing D(k) on the basis of data about the phonon frequencies ω _i (N) at the symmetry points of the Brillouin zone and the elastic constants C _ik . A number of relations between the values of ω _i (N) and C _ik are presented for conventional interaction models. It is shown that the standard method for determining BvK parameters by fitting them to experimental phonon spectra in HCP lattices is, as a rule, ambiguous, whereas the analytical method proposed allows one to find all the solutions of the problem. The methods developed are illustrated by the construction of dynamical matrices for Tb, Sc, Ti, and Co.     

Partially coherent non-paraxial vectorial Hermite-Gaussian beams and their far-field properities

The far-field theory of partially coherent vectorial non-paraxial beams is expanded and their analytical propagation expressions of the Wigner distribution function matrices and the cross-spectral-density matrices in free space are derived using the far-field approximation. Some interesting cases, in particular the vectorial non-paraxial partially coherent Hermite-Gaussian (HG) beams, are discussed and treated as special cases of our general expressions. It is shown that the f_σ and f parameters play an important role in determining vector and non-paraxiality of partially coherent HG beams. When two parameters are small enough, scalar and paraxial vectorial approximation is allowed; otherwise, non-paraxial vectorial approximation is applied. But the decided parameters additionally affect their far-field divergence angles.     

Surface-integral formulation for electromagnetic scattering in spheroidal coordinates

We derive surface-integral expressions for the Q matrices in spheroidal coordinates that allow us to compute the T matrix in spheroidal coordinates. This approach combines the advantages of the null-field method (also referred to as the extended boundary condition method) with those of the separation of variables method. For spheroidal particles we obtain explicit Q matrix expressions that display the expected symmetry properties and yield correct results in the spherical limit. Compared to surface-integral expressions for spheroids in spherical coordinates, our results are considerably simpler because the integrands do not contain radial functions.     

Vibration-rotation energies of harmonic and combination levels in tetrahedral XY4 molecules: Theory and extrapolation method

This paper presents a general method for studying overtones and combination bands of tetrahedral XY₄ molecules. Making full use of group theory, the matrix representation of the Hamiltonian restricted to any set of vibrational sublevels is easily derived from a basis of rovibrational matrices. Explicit expressions for the reduced matrix elements are developed for all the operators of any order occurring in states for which Σ_sv_s ≤ 2, including any interaction type. The parameters introduced are directly the Hamiltonian expansion coefficients and consequently have the same physical significance in any vibrational state. The treatment of overtones and combination bands for which the corresponding fundamentals are analyzed is considered in detail. In this case, the parameters of the ground and fundamental states remain fixed and only a few new parameters occurring specifically in the studied state need to be determined. In this way, the convergence of least-squares fits is greatly improved. The example of methane is used to illustrate the efficiency of this method suitable for the direct determination of potential constants.     

Partially coherent vectorial cosh-Gaussian beams beyond the paraxial approximation

The concept of partially coherent vectorial nonparaxial cosh-Gaussian (ChG) beams is introduced, and their analytical propagation expressions for the cross-spectral density matrix in free space are derived by using the generalized vectorial Rayleigh diffraction integrals. Some interesting cases, in particular, the vectorial nonparaxial Gaussian-Schell-model (GSM) beams are discussed and treated as special cases of our general expressions. It is shown that the f and f_σ parameters play a crucial role in determining the vectorial property and nonparaxiality of partially coherent ChG beams, but the decentered parameter additionally affects their behavior.     

Multipole theory of composite pulses

From the results for a pure pulse from the multipole theory of nuclear magnetic resonance, it is possible to obtain general analytical expressions for the decomposition of a single pulse into a product of a number of constituent pulses. These pulses, which are represented as Wigner rotation matrices, have the angles as functions of the off resonance frequency and the rf amplitude. By multiplying 3 × 3 matrices n times it is possible to generate the analytical expressions for n composite pulses which describe the three components of the magnetization vector. These are exact for all off resonance conditions and for any spin magnitude. Our theoretical results agree with experimental data presented here and elsewhere.     

Non-hermitian random matrix models

We introduce an extension of the diagrammatic rules in random matrix theory and apply it to non-hermitian random matrix models using the 1/N approximation. A number of one-and two-point functions are evaluated on their holomorphic and non-holomorphic supports to leading order in 1/N. The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic cotrelations. The generic form for the two-point functions is obtained, generalizing the concept of macroscopic universality to non-hermitian random matrices. We show that the holomorphic and non-holomorphic one- and two-point functions condition the behavior of pertinent partition functions to order O(1/N). We derive explicit conditions for the location and distribution of their singularities. Most of our analytical results are found to be in good agreement with numerical calculations using large ensembles of complex matrices.     

Diagonal vibrational matrix elements for an anharmonic oscillator

General expressions for the diagonal rotationless matrix elements of any power of the internuclear displacement from equilibrium are obtained for ¹Σ-state diatomic molecule using an iterative relation. These expressions, including contributions arising from terms of the ninth power in the Dunham potential, are obtained in analytical form. As an application, sixth- and seventh-order contributions to the matrix element 〈0?x¹?〉 are calculated for the first eight powers of the displacement.     

Another way to shape the comprehensive analytical approach describing electromagnetic energy distribution through four-slab-layer structures

Starting from the generalised four-slab layer in electromagnetic theory and photonics, this paper introduces a convenient method and a new proper change of variables in order to obtain the global analytical expressions of the power flows in such multilayer structures for the TE_m and TM_m optical modes. These proper changes of variables and relevant definitions of apt new parameters (Θ, W, Y and ξ) allow us to derive and shape new general analytical formulations and normalizations in terms of power flows. According to such specific parameters, it can be noted that such a comprehensive result brings in an effective criteria form of the classical results ascribed to three-slab problems. Moreover, we have verified with specific cases regarding three-slab problems the validity of our new global frame for analysing power flows. It clearly appears that classical three-slab-waveguide expressions directly stem from our formulation. Naturally, this global four-slab-waveguides approach can be used directly to the analytical calculus of corresponding ratios of power between the different layers, such as the core compared with buffer layers as upper and lower claddings.     

Characterization of the <Emphasis Type="Italic">T</Emphasis> <Subscript>1</Subscript> state of porphyrin molecules on the basis of numerical modeling of decrease and increase of fluorescence intensity kinetics

For a number of porphyrin molecules, it was shown that their excitation by light pulses of a rectangular step shape and a certain duration led to experimentally observed dynamical decrease and increase of fluorescence kinetics due to changes in the population of the lower (metastable) triplet T₁ state. On the basis of exact analytical expressions for a three-energy-level model, a simple analytical relationships between the rate constants of intramolecular processes and experimentally measured parameters of the fluorescence kinetics were obtained. The three-dimensional isotropic orientation of the molecules in the framework of the chosen model was taken into account by numerical methods and allowed to simulate adequately the experimentally observed fluorescence kinetics. The T₁ state lifetimes of the studied porphyrin molecules in polymer matrices were determined from experimental curves using numerical methods for solving inverse problems. The obtained values correlated with literature data. Features and advantages of this approach were discussed.     

Optical Mueller matrices in terms of geometric algebra

Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl_3,0 algebra and using isomorphism between Cl_3,0 and even Cl_3,1⁺ subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl_3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented.     

Random matrix ensembles with random interactions: Results for EGUE(2)-SU(4)

We introduce in this paper embedded Gaussian unitary ensemble of random matrices, for m fermions in Ω number of single particle orbits, generated by random twobody interactions that are SU(4) scalar, called EGUE(2)-SU(4). Here the SU(4) algebra corresponds to Wigner’s supermultiplet SU(4) symmetry in nuclei. Formulation based on Wigner-Racah algebra of the embedding algebra U(4Ω) ? U(Ω) ? SU(4) allows for analytical treatment of this ensemble and using this analytical formulas are derived for the covariances in energy centroids and spectral variances. It is found that these covariances increase in magnitude as we go from EGUE(2) to EGUE(2)-s to EGUE(2)-SU(4) implying that symmetries may be responsible for chaos in finite interacting quantum systems.