The redistribution function for scattering by relativistic electrons

The frequency redistribution function r(ν_i, ν_f) for scattering by relativistic electrons is derived, when this scattering is isotropic in the electron rest frame and recoil effects are neglected. Unlike previous derivations, we obtain for r(ν_i, ν_f) an expression which reduces exactly to its classical value for a non-relativistic gas.     

The Hanle effect with partial frequency redistribution. Construction of a frequency-dependent polarization matrix and numerical solution by a PALI method

The polarized approximate lambda iteration (PALI) technique developed for the weak field Hanle effect relies on the decomposition of the Stokes parameters I, Q and U into six cylindrically symmetrical components. It has been applied to complete and partial frequency redistribution with redistribution matrices in which frequency redistribution is decoupled from scattering polarization. For partial frequency redistribution, the decoupling is obtained by an adequate decomposition of the frequency space into several domains. By angle-averaging frequency-dependent terms in the exact weak field Hanle redistribution matrix for a normal Zeeman triplet, we construct a redistribution matrix with coupling between frequency redistribution and polarization and no domain decomposition. The coupling is contained in generalized frequency redistribution functions that depend on the magnetic field. The redistribution matrix is expanded in the Landi Degl’Innocenti spherical tensors for polarimetry and the Stokes parameters are decomposed into cylindrically symmetrical components. A PALI method is set up for the calculation of these components. The Stokes parameters are calculated for different simple atmospheric models. The positive Q direction corresponds to the linear polarization perpendicular to the solar limb. It is shown that the frequency space decomposition may induce large errors on Stokes U in the transition region between line core and line wings but can safely be used for Stokes I and Q, the errors staying less than 1% at all the frequencies.     

Probability distribution functions for a single-particle vibrating in one dimension: experimental study and theoretical analysis

We consider the form of the rebound velocity, ν₀, particle velocity, ν, and height, h, probability density functions (PDFs) for the one-dimensional motion of a single particle on a sinusoidally oscillating base. The motion is considered in the limit of high excitation (vibration frequency ⪢ collision rate). Experimentally, we find that these PDFs are well-approximated by P_ν0(ν₀) ∞ ν₀ exp(− αν₀²), a Gaussian P_ν(ν) ∞ exp(− αν²) and a Boltzmann-type function P_h(h) ∞ exp(− 2αgh), where α is a constant and g is the acceleration due to gravity. We develop an analytical model which accurately predicts the general form for the rebound velocity PDF; the other two PDFs are then analytically shown to follow as a consequence. Scaling laws for the particle granular temperature with peak base velocity and particle-base restitution coefficient, determined from previous work, can also be predicted from the PDF. A fine scale “spiky” structure in the rebound velocity PDF is found, using numerical simulations, to be a consequence of resonance phenomena between the particle and vibrating base. Good agreement between scaling laws from the theory and simulation is found but insufficient data is obtainable to derive accuracy exponents experimentally.     

Statistical dynamics of conserved densities for the hard-sphere Lorentz gas

The statistical dynamics of a particle scattered by randomly positioned stationary hard spheres are investigated. We examine the somewhat unusual long-wavelength, low-frequency fluctuation spectra resulting from the existence of an infinite set of mutually coupled conserved densities of which the number density and energy density are two members. The analytically soluble infinite-mode correlation functions are compared with the corresponding functions obtained by truncating the set of slow modes at successively increasing orders. Furthermore, we evaluate the long-wavelenght number density autocorrelation function for a fixed speed, υ₀, in terms of a frequency-dependent diffusivity D(ω;υ₀) and obtain the fluctuation spectra of all conserved densities by an additional velocity average with the appropriate canonical weights. The effect of the frequency-dependent diffusivity D(ω;υ₀) on the density fluctuation spectra at frequencies small compared to the mean collision frequency is elucidated.     

Optical constants and dielectric function of DNA's nucleotides in UV range

We calculated frequency-dependent real n(ω) and imaginary k(ω) parts of the complex refractive index for adenine A, guanine G, cytosine C and thymine T in ultraviolet range using Kramers-Kroning analysis of their absorption spectra. The frequency-dependent real ε′(ω) and imaginary ε″(ω) parts of the dielectric function for the corresponding nucleotides were restored. We calculated also the dependence of the dielectric function on the imaginary frequency ε(iω) for the corresponding nucleotides.     

On the Deser,Gilbert, Sudarshan representation for functions Wi(ν, q2)(i = 1, 2) describing the electroproduction process

The Deser, Gilbert, Sudarshan representation (D.G.S.R.) for the functions W_i(ν, q²) (i = 1,2) is considered as equations determining spectral functions h_i(a, α) via the values W_i(ν, q²) in the physical region of the electroproduction channel. It is shown that if W_i(ν, q²) obey the microcausality and spectrality conditions, then the equations for h_i(a, α) have solutions in the class of Schwartz temperated distributions and thereby the D.G.S.R. is proved. Formulae are obtained expressing spectral functions in the D.G.S.R. through the values of functions W_i(ν, q²) in the physical region of the electroproduction channel.     

The vibrational spectrum of thiazole between 600 and 1400 cm revisited: A combined high-resolution infrared and theoretical study

The Fourier transform infrared gas-phase spectrum of thiazole, C₃H₃NS, has been recorded in the 600-1400 cm⁻¹ wavenumber region with a resolution around 0.0030 cm⁻¹. Nine fundamental bands (ν₅(A′) to ν₁₁(A′), ν₁₅(A″), and ν₁₆(A″)) are analysed employing the Watson model. Ground-state rotational and quartic centrifugal distortion constants as well as upper state spectroscopic constants have been obtained from the fits. A detailed analysis of perturbations identified in the ν₁₁(A′) band at 866.5 cm⁻¹ enables a definitive location of the very weak ν₁₀(A′) and ν₁₄(A″) bands at 879.3 and 888.7 cm⁻¹, respectively. The three levels are analysed simultaneously by a model including Coriolis resonance using an ab initio predicted first order c-Coriolis coupling constant; second and higher order Coriolis parameters are determined. Qualitative explanations in terms of Coriolis resonances are given for a number of crossings observed in ν₅(A′), ν₆(A′), and ν₇(A′) at 1383.7, 1325.8, and 1240.5 cm⁻¹, respectively. The rotational constants, anharmonic frequencies, and vibration-rotation constants (alphas, ) calculated by quantum chemical calculations using a cc-pVTZ and TZ2P basis with B3LYP methodology, have been compared with the present experimental data. The rotation constant differences for each vibrational state, from the ground state values, are closer to experiment from the TZ2P calculations relative to those using cc-pVTZ. The values for Δ_J, Δ_JK, Δ_K, δ_J, and δ_K are close to experiment with both basis sets.     

The vapor-phase Raman spectra,Raman band contour analyses,Coriolis constants,and force constants of spherical-top molecules MX4 (M = Group IV element, X = F,Cl, Br,or I)

The Raman spectra of 21 tetrahalides of Group IV have been recorded in the vapor phase at temperatures between 20 and 420° C. The Raman shifts of the Q branches of each fundamental are reported. In agreement with the selection rules, the ν₁(a₁) fundamental of each molecule has no rotational structure. With two exceptions, the separation between the OP and RS branches of the ν₂(e) fundamental of each molecule has been measured and found to be in close agreement with that calculated from an extended form of the Placzek-Teller theory. Due to first-order Coriolis coupling, the OP, RS branch separations of the ν₃(t₂) and ν₄(t₂) fundamentals of each molecule are different from each other and from those of the ν₂(e) fundamental. It has been possible to determine Coriolis constants for most of the tetrahalides from these branch separations, using both a previously published procedure and also a new procedure in which analytical expressions for each subbranch are derived. There is close agreement between the zeta values so determined and the corresponding values determined from infrared band contour analyses in those cases where the latter are available. Force constants based on the fundamental frequencies and the zeta values are reported.     

Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the spectrum of Gaussian Schell-model (GSM) beams propagating through turbulent atmosphere is derived analytically. It is shown that, if the scaling law is valid, the normalized spectrum S(ω) of GSM beams propagating through turbulent atmosphere is the same as the normalized source spectrum S⁽⁰⁾(ω), whether GSM source is quasi-homogenous or not. On the other hand, if the scaling law fails, S(ω) of GSM is different from S⁽⁰⁾(ω). The structure constant of the refractive index, transverse coordinate of observation point and spatial correlation length of the source affect the spectrum, which is illustrated numerically.     

Polarization of multiple scattered light in planetary atmospheres from solution of the coupled linear integral equations derived for mixed rayleigh-isotropic scattering

A set of coupled integral equations describing nonconservative multiple scattering for a mixed isotropic and Rayleigh single scattering phase function in inhomogeneous, plane-parallel planetary atmospheres is derived. The equations are applicable for the frequency redistributions MRE (monochromatic radiative equilibrium), CFR (complete frequency redistribution), or PFR (partial frequency redistribution). Solution of the equations permits one to calculate the intensity and degree of polarization in an arbitrary direction outside or inside the plane-parallel scattering medium. The equations are readily adaptable to more complicated geometries. Solutions for several cases are presented to demonstrate the versatility and validity of the method. These include a calculation of MRE pure Rayleigh scattering of sunlight in an optically-thick planetary atmosphere to demonstrate agreement with the results of Coulson⁽¹⁾et al. (1960), calculations of the altitude profile of the degree of polarization of the earth's Ly-α 1216A and helium 584A dayglow, and the center to limb variation of the degree of polarization of the sunlight diffusely reflected from a distant planet such that the scattering is pure Rayleigh and conservative at the top of the planet's atmosphere, varying smoothly to conditions of nonconservative and pure isotropic scattering deep in the planet's atmosphere. Tables of functions that one might utilize (without resort to a digital computer) to obtain solutions in the escape function approximation are also given.     

Preresonance Raman intensity studies on the fundamentals of the group IV tetrahalides

The intensities of all four fundamentals of the tetrachlorides, tetrabromides, and tetraiodides of carbon, silicon, germanium, tin, and titanium (MX₄) have been determined in cyclohexane solutions relative to the intensity of the 806 cm^?1 band of cyclohexane as internal standard. The comparisons have been effected at four different wavelengths in each case (647.1, 568.2, 514.5, and 488.0 nm), the solutions of the more strongly colored halides being rotated at ca. 1600 rpm in order to obviate thermal decomposition. The fundamentals ν₁(a₁) and ν₃(t₂), but not ν₂(e) and ν₄(t₂) were found to exhibit preresonance enhancement of their intensities as the exciting frequency (ν₀) was made to approach that of the first allowed electronic transition (ν_e) of each molecule. The extent of the observed enhancement increases in the order MCl < MBr < MI and CX < SiX < GeX < SnX < TiX, consistent with the known decrease in ν_e along each series. Molecular and bond polarizability derivatives have been calculated from the Raman intensities of the four fundamentals (extrapolated to zero exciting frequency). Complete quadratic force fields for each molecule are established on the assumption of the Wolkenstein intensity theory.     

Deviations from LTE in a stellar atmosphere

Deviations from LTE are investigated in an atmosphere of hydrogen atoms with one bound level, satisfying the equations of radiative, hydrostatic, and statistical equilibrium. The departure coefficient and the kinetic temperature as functions of the frequency dependence of the radiative cross section are studied analytically and numerically. Near the outer boundary of the atmosphere, the departure coefficient b is smaller than unity when the radiative cross section α_ν grows with frequency ν faster than ν²; b exceeds unity otherwise. Far from the boundary the departure coefficient tends to exceed unity for any frequency dependence of α_ν. Overpopulation (b > 1) always implies that the kinetic temperature in the statistical equilibrium atmosphere is higher than the temperature in the corresponding LTE atmosphere. Upper and lower bounds on the kinetic temperature are given for an atmosphere with deviations from LTE only in the optically shallow layers when the emergent intensity can be described by a radiation temperature.     

Rotationally inelastic,classical scattering from an anisotropic rigid shell potential of rotation symmetry

Rotational excitation in collisions of structureless atoms and diatomic rigid rotor molecules interacting by a rigid potential shell is considered in classical mechanics. The double differential cross sectionJ(u ^*,θ) for final (over initial) relative velocityu ^*=ν′/ν and deflection byθ is analytically related to the shell form in the case of vanishing initial molecular rotation.J(u ^*,θ) exhibits the strong structure of “bulge” scattering or “orientational rainbows” which has been observed in the K?N₂ and K-CO systems and is expected to occur in rotationally inelastic collisions of many nonreactive systems under appropriate scattering conditions. The present results elucidate the nature of the sensitive and direct relation of bulge scattering to the anisotropy of the intermolecular potential.     

Absolute Line Intensities in Acetylene: The 1.5-μm Region

We measured 305 absolute line intensities in the ν₁+ν₃(Σ⁺_u)-0(Σ⁺_g) band of ¹²C₂H₂ and ¹³C¹²CH₂ and the ν₁+ν₂+(ν¹₄+ν⁻¹₅)⁰(Σ⁺_u)-0(Σ⁺_g), ν₁+ν₃+ν¹₄(Π_g)-ν¹₄(Π_g), and ν₁+ν₃+ν₅¹(Π_u)-ν₅¹(Π_u) bands of the main isotopomer, all observed near 1.5 μm. The absolute intensity of these bands are respectively 6.4882 (34), 0.12337 (10), 0.083746 (71), 0.58771 (28), and 0.32126 (11) cm⁻² atm⁻¹ at 296 K. In addition, we also determined Herman-Wallis factors for the first time in this spectral region.     

Analysis of the hot bands in the 2ν9 band region of propyne-d3

The rovibrational spectrum of 2ν₉ band of CD₃CCH is overlapped by two prominent hot bands identified as (2ν₉⁰+ν₁₀^±1)(E)←ν₁₀^±1(E) and 3ν₉^±1(E)←ν₉^±1(E), where ν₁₀ and ν₉ are the degenerate CCC and CCH bending fundamental vibrations, respectively. Assignment of lines to the transitions of these hot bands were carried out with the help of the high-resolution spectra recorded at ∼195 K and at room temperature. Molecular parameters for these hot bands have been obtained from the rotational analysis of the partially resolved K-structure lines. Only Q-head of the third hot band , originating from the lower 2ν₁₀ state could be identified.     

The scattering effects of the redistribution function RIII(x′, n′; x, n) on spectral line formation

The effects of frequency and angular redistribution on line formation are examined in static isothermal atmospheres for scattering described by the redistribution function R_III(x′, n′; x, n). For an optically thin atmosphere, it is found that the emergent line intensities obtained using R_III are essentially the same as R_I and R_II. In the case of a semi-infinite atmosphere, the emergent line intensities obtained using R_III, whether angle-averaged or angle-dependent, did not differ substantially from the complete redistribution results.     

Open and Closed World Models in Kaluza-Klein-Theory with Variables G and Λ

The field equation of higher dimensions theory, have been applied in the area of cosmology. The resulting differential equations are solved for open and closed. We derive a relation between the Einstein constant G(t) and the cosmological constant Λ(t) from the conservation law T _μ ^ν _;ν =0. We give a specific form of Λ(t) to solve the non-linear differential equations. Some cosmological parameters are calculated and some relevant cosmological problems are discussed.     

Optical and magneto-optical properties of room-temperature ferromagnetic In(Ga)MnAs layers, deposited by pulse laser ablation

Spectral dependences of refractive and absorption indices n(hν), k(hν) (hν=1.2-4.4 eV) and the transversal Kerr effect δ(hν) (hν=0.5-4.4 eV) in In(Ga)MnAs layers fabricated by laser deposition have been investigated. Spectra of the diagonal and off-diagonal components of the dielectric permittivity tensor of these layers have been calculated. Comparison of the spectral dependences δ(hν), ε′(hν) and ε′₂×(hν)² of the In(Ga)MnAs layers with similar spectra for MnAs has been carried out. Particular features in the spectra of the In(Ga)MnAs layers have been explained by a competition of contributions of the In_1−x(Ga_1−x)Mn_xAs host and MnAs inclusions.     

Infrared and laser Raman gas spectra of SnH4

In the present work the ν₃(f₂) Raman band and the 2ν₃ infrared band of SnH₄ are reported. Analyses have been performed for the ν₃(f₂) Raman band and the ν₂(e), ν₃(f₂), ν₄(f₂) and 2ν₃ infrared bands. The interatomic distance in the ground state has been found to be 1.7108 ± 0.0010 Å.     

Microwave spectrum of Trioxane in four excited degenerate vibrational states,especially in the 2ν20 (E) state and the combination state ν7 (A1) + ν20 (E)

The microwave spectra of the Trioxane molecule in the doubly excited degenerate vibrational state 2ν₂₀(E) and in the combination state ν₇(A₁) + ν₂₀(E) are reported for several J values. Molecular constants of these states have been determined. The excited states ν₂₀(E) = 1 and ν₁₉(E) = 1 previously studied by a perturbation formula have been analyzed by a direct diagonalization method of the energy matrix. Parameters which give the variation of the rotational constant B_V as a function of the quantum number v have been obtained.