Rayleigh Scattering Cross Section Research Articles

Contribution of the polarizability anisotropy to Rayleigh scattering

A calculation from first principles of the differential cross section for Rayleigh scattering shows that the contribution of polarizability anisotropy found in the literature is valid only in the high rotational quantum number limit of the vector coupling coefficients, that is, in the high‐temperature or classical limit. The additional term in the total Rayleigh scattering cross section arising from the polarizability anisotropy is about four times smaller than what is called the King factor in the literature. These errors, resulting from ignoring the constraints imposed by the conservation of angular momentum, can lead to significant corrections in estimating populations from stratospheric and mesospheric measurements.

Rayleigh scattering from a trapped bose condensate and the corresponding recoil atom velocity distribution

An approach has been developed that allows the Rayleigh scattering cross section to be calculated with allowance for the quantum character of motion of the center of mass of the trapped scattering particles. The shape of the line of light scattering from a Bose condensate in a parabolic trap has been studied. A shift of the scattering line center is equal to the recoil shift, while the line width depends on the chemical potential of the Bose gas and on the relaxation time of the velocity of the above-condensate recoil particles. A velocity distribution function in the beam of recoil atoms formed in the course of induced Rayleigh scattering is determined. It is shown that, under the typical experimental conditions, the characteristic width Δv/v of the recoil velocity distribution in this beam is on the order of 10−3 at a velocity v on the order of several centimeters per second.

Dissipation length scales in turbulent nonpremixed jet flames

Line imaging of Raman/Rayleigh/CO-LIF is used to investigate the energy and dissipation spectra of turbulent fluctuations in temperature and mixture fraction in several flames, including CH 4/H 2/N 2 jet flames at Reynolds numbers of 15,200 and 22,800 (DLR-A and DLR-B) and piloted CH 4/air jet flames at Reynolds numbers of 13,400, 22,400, and 33,600 (Sandia flames C, D, and E). The high signal-to-noise ratio of the 1D Rayleigh scattering images enables determination of the turbulent cutoff wavenumber from 1D dissipation spectra. The local length scale inferred from this cutoff is analogous to the Batchelor scale in nonreacting flows. The measured thermal dissipation spectra in the turbulent flames are shown to be similar to the model spectrum of Pope for turbulent kinetic energy dissipation. Furthermore, for flames with Lewis number near unity, the 1D dissipation spectra for temperature and mixture fraction are shown to follow nearly the same rolloff in the high-wavenumber range, such that the cutoff length scale for thermal dissipation is equal to or slightly smaller than the cutoff length scale for mixture fraction dissipation. Measurements from the piloted CH 4/air flames are used to demonstrate that a surrogate cutoff scale may be obtained from the dissipation spectrum of the inverse of the Rayleigh signal itself, even when the Rayleigh scattering cross section varies through the flame. This suggests that the cutoff length scale determined from Rayleigh scattering measurements may be used to define the local resolution requirements and optimal data processing procedures for accurate determination of the mean mixture fraction dissipation, based upon Raman scattering measurements or other multiscalar imaging techniques.

A combustion temperature and species standard for the calibration of laser diagnostic techniques

Optical diagnostic thermometry techniques are not generally traceable to temperature standards and due to their complexity it is often difficult to assess their absolute accuracy. This work aims to address this by calibrating a McKenna premixed flat-flame burner with a high-accuracy Rayleigh scattering thermometry system. By applying corrections for the temperature dependence of the scattering cross section, it has been possible to determine the temperature of a premixed propane/air flame with an uncertainty of less than 1%. The temperature dependence of the differential Rayleigh scattering cross section arises from changes in the mean molecular polarizability and mean molecular anisotropy. Changes in the cross section of up to 2% between room and flame temperatures have been found. The burner and associated gas mixing system are described here as a combustion temperature and species standard allowing the calibration of optical diagnostic techniques.

Improved algorithm for calculations of Rayleigh-scattering optical depth in standard atmospheres

Precise calculations of the total Rayleigh-scattering optical depth have been performed at 88 wavelengths ranging from 0.20 to 4.00 microm for the six well-known standard atmosphere models by integrating the volume Rayleigh-scattering coefficient along the vertical atmospheric path from sea level to a 120-km height. The coefficient was determined by use of an improved algorithm based on the Ciddor algorithm [Appl. Opt. 35, 1566 (1996)], extended by us over the 0.20-0.23-microm wavelength range to evaluate the moist air refractive index as a function of wavelength, air pressure, temperature, water-vapor partial pressure, and CO2 volume concentration. The King depolarization factor was also defined taking into account the moisture conditions of air. The results indicate that the influence of water vapor on Rayleigh scattering cannot be neglected at tropospheric altitudes: for standard atmospheric conditions represented in terms of the U.S. Standard Atmosphere (1976) model, the relative variations produced by water vapor in the Rayleigh scattering parameters at a 0.50-microm wavelength turn out to be equal to -0.10% in the moist air refractivity at sea level (where the water-vapor partial pressure is equal to approximately 7.8 hPa), -0.04% in the sea-level King factor, -0.24% in the sea-level Rayleigh-scattering cross section, and -0.06% in the Rayleigh-scattering optical depth.

Rayleigh scattering cross sections of combustion species at 266, 355, and 532 nm for thermometry applications

Rayleigh scattering cross sections are measured for nine combustion species (Ar, N2, O2, CO2, CO, H2, H2O, CH4, and C3H8) at wavelengths of 266, 355, and 532 nm and at temperatures ranging from 295 to 1525 K. Experimental results show that, as laser wavelengths become shorter, polarization effects become important and the depolarization ratio of the combustion species must be accounted for in the calculation of the Rayleigh scattering cross section. Temperature effects on the scattering cross section are also measured. Only a small temperature dependence is measured for cross sections at 355 nm, resulting in a 2-8% increase in cross section at temperatures of 1500 K. This temperature dependence increases slightly for measurements at 266 nm, resulting in a 5-11% increase in cross sections at temperatures of 1450 K.

Direct measurement of the Rayleigh scattering cross section in various gases

Using the laser-based technique of cavity ring-down spectroscopy extinction measurements have been performed in various gases straightforwardly resulting in cross sections for Rayleigh scattering. For Ar and N 2 measurements are performed in the range 470–490 nm, while for CO 2 cross sections are determined in the wider range 470–570 nm. In addition to these gases also for N 2 O, CH 4 , CO, and SF 6 the scattering cross section is determined at 532 nm, a wavelength of importance for lidar applications and combustion laser diagnostics. In O 2 the cross section at 532 nm is found to depend on pressure due to collision-induced light absorption. The obtained cross sections validate the cross sections for Rayleigh scattering as derived from refractive indices and depolarization ratios through Rayleigh's theory at the few %-level, although somewhat larger discrepancies are found for CO, N 2 O and CH 4 .

Calculation of the transport properties of carbon dioxide. III. Volume viscosity, depolarized Rayleigh scattering, and nuclear spin relaxation.

Transport properties of pure carbon dioxide have been calculated from the intermolecular potential using the classical trajectory method. Results are reported in the dilute-gas limit for volume viscosity, depolarized Rayleigh scattering, and nuclear spin relaxation for temperatures ranging from 200 to 1000 K. Three recent carbon dioxide potential energy hypersurfaces have been investigated. Calculated values for the rotational collision number for all three intermolecular surfaces are consistent with the measurements and indicate that the temperature dependence of the Brau-Jonkman correlation is not applicable for carbon dioxide. The results for the depolarized Rayleigh scattering cross section and the nuclear spin relaxation cross section show that calculated values for the generally more successful potentials differ from the observations by 9% at about 290 K, although agreement is obtained for nuclear spin relaxation at about 400 K.

Reliable anisotropic dipole properties, and dispersion energy coefficients, for O2 evaluated using constrained dipole oscillator strength techniques

Constrained anisotropic dipole oscillator strength techniques are used to obtain reliable values for a wide range of anisotropic and isotropic dipole properties of O2, including the dipole–dipole dispersion energy coefficients for the interaction of O2 with O2, H2, N2, CO, He, Ne, Ar, Kr, and Xe. Some of the anisotropic constraints required for our calculations are obtained via dipole sum rules from ab initio, multireference configuration interaction (CI) wave functions for the ground state of O2. The individual dipole properties of O2 considered include the dipole oscillator strength sums Sk, k=2,1,0(−1/2)−2,−3,−4,..., the logarithmic dipole sums Lk and mean excitation energies Ik, k=2(−1)−2, and, as a function of wavelength, the dynamic polarizability and its anisotropy, the total depolarization ratio, the Rayleigh scattering cross section, and the Verdet constant. Our constrained dipole oscillator strength results are often the only reliable, and sometimes the only available, ones for many of the properties and dispersion energies considered.

Estimating the probability density of the scattering cross section from Rayleigh scattering experiments

An important parameter in the experimental study of dynamics of combustion is the probability distribution of the effective Rayleigh scattering cross section. This cross section cannot be observed directly. Instead, pairs of measurements of laser intensities and Rayleigh scattering counts are observed. Our aim is to provide estimators for the probability density function of the scattering cross section from such measurements. The probability distribution is derived first for the number of recorded photons in the Rayleigh scattering experiment. In this approach the laser intensity measurements are treated as known covariates. This departs from the usual practice of normalizing the Rayleigh scattering counts by the laser intensities. For distributions supported on finite intervals two estimators of the probability density are given: one based on expansion of the density in orthonormal polynomials and the other based on the maximum likelihood. The latter procedure is applied to Rayleigh scattering data from an open V-shaped premixed methane–air flame.

Reliable isotropic and anisotropic dipole properties, and dipolar dispersion energy coefficients, for CO evaluated using constrained dipole oscillator strength techniques

Abstract Constrained anisotropic dipole oscillator strength methods are applied to obtain reliable results for a wide selection of anisotropic and isotropic dipole properties of CO and for the dipole-dipole dispersion energy coefficients for the interaction of CO with CO, N 2 , H 2 , He, Ne, Ar, Kr and Xe. The properties of CO evaluated include the dipole oscillator strength sums S k , k = 2, 1, 0(− 1 2 ) −2, −3, −4, ⋯, the logarithmic dipole sums L k and mean excitation energies I k , k = 2(−1) −2, and, as a function of wavelength, the dynamic polarizability and its anisotropy, the total depolarization ratio, the Rayleigh scattering cross section, and the Verdet constant. Our constrained dipole oscillator strength results are often the only reliable, and sometimes the only, results available for many of the anisotropic dipole properties and dispersion energies considered in this paper.

Single-shot temperature and mixture fraction profiles by Rayleigh scattering in the development zone of a turbulent diffusion flame

Single-shot laser Rayleigh scattering profiles have been obtained in the development zone of a turbulent diffusion flame of a diluted hydrogen jet issuing into a coflowing air stream. The Rayleigh scattering cross section that depends on the local molecular composition is calculated at each laser shot as a function of mixture fraction ( Z) and the temperature profile by the use of an algorithm involving a strained flame library. This processing provides the instantaneous profiles of T and Z, so that the radial structure of the reaction zones can be statistically investigated, apart from the flame motion, in terms of the pdf and correlations for the scalars and their gradients. The main result is the large regular spreading of peak temperature distributions down to low values that are far less than the extinction limit. This is attributed to the occurrence of local and intermittent extinctions due to the action of turbulent stretch.

Constrained anisotropic dipole oscillator strength distribution techniques, and reliable results for anisotropic and isotropic dipole molecular properties, with applications to H2 and N2

Constrained anisotropic dipole oscillator strength distribution techniques are discussed and applied to obtain reliable results for a wide variety of the anisotropic and isotropic dipole properties of H2 and N2. These include the dipole oscillator strength sumsS k, k=2, 1, −1/2(−1/2) −2, −3, −4, ..., the logarithmic dipole sumsL k and mean excitation energiesI k, k=2(−1) − 2, and, as a function of wavelength, the dynamic polarizability and the associated anisotropy, the total depolarization ratio, the Rayleigh scattering cross section, and the Verdet constant. The anisotropic components of the DOSD for a molecule are obtained from a given recommended isotropic DOSD by using a constrained least squares procedure and a series of known anisotropic constraints. Assuming that sufficient input is available, the constrained DOSD approach used in this paper is the only available method for the reliable evaluation ofall the relevant anisotropic and isotropic dipole properties for a wide variety of atoms and molecules.

Measurements of Rayleigh scattering cross sections

The authors report the results of a measurement of the Rayleigh scattering cross section of methane, nitrogen and carbon dioxide using a pulsed laser beam. In a similar study Skowronek and Alayli (1979) reported a variation in the cross section with the duration of the laser pulse. In this work the authors have been unable to reproduce Skowronek and Alayli's result and conclude that there is no effect due to the pulse duration. Their measurements agree with the classical cross section within the experimental error (+or-10% to 15%).

Rayleigh scattering for density measurements in premixed flames

Rayleigh scattering measurements for molecular number density in turbulent, premixed CH4-air flames are discussed, and data for both flamelet passage time distributions and power spectral density functions are reported and compared to the recent predictions of Bray, Libby and Moss (1984). Measurement problems associated with variations in mixture-averaged Rayleigh scattering cross section, index of refraction fluctuations, finite spatial and temporal resolution and with scattering from particles are discussed. It is concluded that these effects are relatively minor in the reported experiments. Correction procedures are suggested for the effects of cross section variation and of finite resolution.

An analytic solution to the Rayleigh scattering cross section integral

The total Rayleigh scattering cross section integral has been integrated analytically using atomic form factors described as a sum of exponentials. This form of the scattering integral eliminates the need for numerical integration using interpolated values of form factors. The result of the integration is expressed in a simple analytical form that can be numerically integrated over the spectra from an X-ray tube.

On the molecular scattering in the terrestrial atmosphere : An empirical formula for its calculation in the homosphere

A recent determination by D. R. Bates of the Rayleigh scattering cross section (σ RS ) for air from 0.2 to 1 μm leads to a simple empirical formula (λ in μm) σ RS = 4.02 × 10 −28 λ 4+x cm 2 where x = 0.389λ + 0.09426 λ − 0.3228 for the spectral region 0.2 μm < λ < 0.55 μm ; the accuracy is within ±0.5%. From the visible at 0.55 μm to the infrared (i.r.) at 1 μm, the same accuracy can be obtained using a constant value, x = 0.04. The formula accounts for the degree of depolarization which varies with the wavelength according to the latest determination by Bates.

Rayleigh scattering by air

The Rayleigh scattering cross section for air is calculated from published experimental and theoretical results account being taken of the dispersion of the King correction factor. The spectral region 0.2 mμ ⩽ λ ⩽ 1.0 mμ is covered.

Effect of the momentum transfer on the Rayleigh scattering cross section

The Rayleigh scattering cross section is measured with a short-duration pulse of laser light in different gases. It depends on the pulse duration, involving a time constant $\ensuremath{\tau}$, following the expression $\frac{{S}_{\mathrm{exp}}}{{S}_{\mathrm{theor}}}=(\frac{{\ensuremath{\tau}}_{1}}{\ensuremath{\tau}})(1\ensuremath{-}{e}^{\ensuremath{-}\frac{\ensuremath{\Delta}t}{\ensuremath{\tau}}})$, where ${S}_{\mathrm{exp}}$ and ${S}_{\mathrm{theor}}$ are the experimental and the theoretical cross sections and ${\ensuremath{\tau}}_{1}$ is the classical time constant corresponding to the dipole damping. The time constant $\ensuremath{\tau}$ was determined with a ruby laser whose pulse half-width was varied between 6 and 200 nsec. Using a tunable dye laser and its nitrogen laser pump, the authors have found that $\ensuremath{\tau}$ is proportional to ${\ensuremath{\lambda}}^{2}$ (square of the wavelength). The variation of $\ensuremath{\tau}$ for gases of different molecular sizes shows that $\ensuremath{\tau}$ is proportional to their diameter $a$. The angular distribution of the scattered light has been determined and found to be favored in the forward direction. The constant $\ensuremath{\tau}$ is proportional to $sin(\frac{\ensuremath{\theta}}{2})$ (where $\ensuremath{\theta}$ is the scattering angle), i.e., to the momentum transfer. By analogy with a corpuscular collision, dimensional considerations lead to the formula $\ensuremath{\tau}=0.85{(\frac{\ensuremath{\lambda}}{{\ensuremath{\lambda}}_{c}})}^{2}(\frac{a}{c})sin(\frac{\ensuremath{\theta}}{2})$ (${\ensuremath{\lambda}}_{C}$ is the Compton wavelength), which well describes the experimental results.

Refractivity and dispersion of hydrogen in the visible and near infrared

The refractivity of natural hydrogen gas has been measured at eighteen wavelengths from 0.4047 to 1.6945 μm by means of a corner-reflector interferometer. The resulting refractivity at 0.5462252 μm, and at standard conditions, is 139.304 × 10−6. Sixteen points of our data are well fitted by the dispersion formula 106(n − 1) = 21.113 + 12723.2/(111 − σ2), where σ is wave number in reciprocal micrometers. The combination of our data with recent measurements in the ultraviolet by Smith, Huber, and Parkinson is fitted by the formula 106(n − 1) = [14895.6/(180.7 − σ2)] + [4903.7/(92 − σ2)]. Comparisons are made with the literature in respect to refractivity, dispersion, dielectric constant, molecular polarizability, Rayleigh scattering cross section, and Verdet constant.