Spectral moments frequency

Near the line centers, the spectral functions have sometimes been approximated by a Lorentzian. The far wings, on the other hand, may be approximated by exponential functions as Fig. 3.2 might suggest. However, better model profiles exist see Chapters 5 and 6 [421, 102, 320], Model profiles have been useful for fitting experimental spectra, for an extrapolation of measured profiles to lower or higher frequencies (which is often needed for the determination of spectral moments) and for a prediction of spectra at temperatures for which no measurements exist. We note that van der Waals dimer structures (which appear at low frequencies and low pressures) modify the Lorentzian-like appearance more or less, as we will see. 

The index n is a small integer, n = 0, 1,. .. Because of the nearly exponential fall-off of typical spectral functions, these integrals do exist. An evaluation of spectral moments is possible if good measurements over a sufficiently broad frequency band exist. (We note that units of spectral moments specified elsewhere sometimes differ from those implied here, mainly because angular frequency, a)nda), is often substituted for frequency in wavenumbers, v" dv.)... 

Table 3.1 lists measured spectral moments of rare gas mixtures at various temperatures. (We note that absorption in helium-neon mixtures has been measured recently [253]. This mixture absorbs very weakly so that pressures of 1500 bar had to be used. Under these conditions, one would expect significant many-body interactions the measurement almost certainly does not represent binary spectra.) For easy reference below, we note that the precision of the data quoted in the Table is not at all uniform. Accurate values of the moments require good absorption measurements over the whole translational frequency band, from zero to the highest frequencies where radiation is absorbed. Such data are, however, difficult to obtain. Good measurements of the absorption coefficient a(v) require ratios of transmitted to incident intensities, /(v)//o, that are significantly smaller than unity and, at the same time, of the order of unity, i.e., not too small. Since in the far infrared the lengths of absorption paths are limited to a few meters and gas densities are limited to obtain purely... 

Units of spectral moments specified by other workers may differ from those implied here because frequency in wavenumbers, v, is often replaced by angular frequency, . Also, normalization by the product of densities, Q2, is sometimes suppressed, especially if many-body interactions are considered. Furthermore, factors of h/2kT or hc/2kT have been used to the right of Eq. 3.6 in the definition of yo-... 

The translational spectra of pure liquid hydrogen have been recorded with para-H2 to ortho-H2 concentration ratios of roughly 25 75, 46 54 and 100 0, Fig. 3.9 [201, 202]. For the cases of non-vanishing ortho-H2 concentrations, the spectra have at least a superficial similarity with the binary translational spectra compare with the data shown for low frequencies (< 250 cm-1) of Fig. 3.10 below. A comparison of the spectral moments of the low-density gas and the liquid shows even quantitative agreement within the experimental uncertainties which are, however, substantial. 

The question arises whether collision-induced profiles may perhaps also be modeled by these or other simple functions, perhaps under circumstances to be defined. Such model functions would be of interest for the analysis of measured spectra, for empirical predictions of spectra at temperatures other than those of the measurements, for frequency extrapolations as may be necessary for accurate determination of spectral moments, etc. 

Whereas there is little doubt that the method of moments, as the procedure is called, is basically sound, it is obvious that for reliable results high-quality experimental data over a broad range of frequencies and temperatures are desirable. As importantly, reliable models of the interaction potential must be known. Since these requirements have rarely been met, ambiguous dipole models have sometimes been reported, especially if for the determination of the spectral moments substantial extrapolations to high or to low frequencies were involved. Furthermore, since for most works of the kind only two moments have been determined, refined dipole models that attempt to combine overlap and dispersion contributions cannot be obtained, because more than two parameters need to be determined in such case. As a consequence, empirical dipole models based on moments do not attempt to specify a dispersion component, or test theoretical values of the dispersion coefficient B(7) (Hunt 1985). 

Equation 5.9 is in essence the ensemble average of the total dipole moment squared. It is given in a form suitable for numerical computation [315], The computation of the spectral moment yi, on the other hand, begins with the integration of Eq. 5.1 over all frequencies,... 

Spectral moments may be computed from expressions such as Eqs. 5.15 or 5.16. Furthermore, the theory of virial expansions of the spectral moments has shown that we may consider two- and three-body systems, without regard to the actual number of atoms contained in a sample if gas densities are not too high. Near the low-density limit, if mixtures of non-polar gases well above the liquefaction point are considered, a nearly pure binary spectrum may be expected (except near zero frequencies, where the intercollisional process generates a relatively sharp absorption dip due to many-body interactions.) In this subsection, we will sketch the computations necessary for the actual evaluation of the binary moments of low order, especially Eqs. 5.19 and 5.25, along with some higher moments. 

Relationship with the intercollisional dip. The cancellation effect described by the doubly primed spectral moments y(naab>", y, abb ", is of course related to the intercollisional interference process observed near zero frequency, Fig. 3.5. The important difference is that the spectral moments are ternary quantities by design while the intercollisional dip is affected by many-body processes. 

Summarizing, it may be said that virial expansions of spectral line shapes of induced spectra exist for frequencies much greater than the reciprocal mean free time between collisions. The coefficients of the density squared and density cubed terms represent the effects of purely binary and ternary collisions, respectively. At the present time, computations of the spectral component do not exist except in the form of the spectral moments see the previous Section for details. 

For any given potential and dipole function, at a fixed temperature, the classical and quantum profiles (and their spectral moments) are uniquely defined. If a desymmetrization procedure applied to the classical profile is to be meaningful, it must result in a close approximation of the quantum profile over the required frequency band, or the procedure is a dangerous one to use. On the other hand, if a procedure can be identified which will approximate the quantum profile closely, one may be able to use classical line shapes (which are inexpensive to compute), even in the far wings of induced spectral lines a computation of quantum line shapes may then be unnecessary. 

Despite the obvious power of quantum line shape calculations for the analysis of measured collision-induced absorption spectra, a need persists for simple but accurate model line profiles, especially for extrapolating experimental spectra to both low and high frequencies for an accurate determination of the spectral moments. Reliable model profiles are also useful for line shape analyses, i.e., for representing complex spectra as a superposition of lines (where this is possible). 

Starting from the assumption that the only known information concerning the line shape is a finite number of spectral moments, a quantity called information is computed from the probability for finding a given spectral component at a given frequency this quantity is then minimized. Alternatively, one may maximize the number of configurations of the various spectral components. This process yields an expression for the spectral density as function of frequency which contains a small number of parameters which are then related to the known spectral moments. If classical (i.e., Boltzmann) statistics are employed, information theory predicts a line shape of the form... 

H2-H2 rototranslational spectra. For the significant A1A2AL induction components, Table 4.11, values of the various spectral functions have been computed at frequencies from 0 to 1800 cm-1 and for temperatures from 40 to 300 K, Fig. 6.3 [282]. As a test of these line shape computations, the zeroth, first and second spectral moments have been computed in two independent ways by integration of the spectral functions with respect to frequency, Eq. 3.4, and also from the quantum sum formulae, Eqs. 6.13, 6.16, and 6.21. Agreement of the numerical results within 0.3% is observed for the 0223, 2023 components, and 1% for the other less important components. This agreement indicates that the line shape computations are as accurate as numerical tests with varying grid widths, etc., have indicated, namely about 1% see Table 6.2 as an example (p. 293). 

The two main nuclear modes affecting electronic energies of the donor and acceptor are intramolecular vibrations of the molecular skeleton of the donor-acceptor complex and molecular motions of the solvent. If these two nuclear modes are uncoupled, one can arrive at a set of simple relations between the two spectral moments of absorption and/or emission transitions and the activation parameters of ET. The most transparent representation is achieved when the quantum intramolecular vibrations are represented by a single, effective vibrational mode with the frequency Vy (Einstein model). 

Usually this method is used on an H-depleted molecular graph, truncated expansions being obtained considering only fragments up to a user-defined size. Some methods for -> log P estimations are based on cluster expansion. Moreover, a new method for the calculation of embedding frequencies for acyclic trees based on spectral moments of iterated line graph sequence was proposed recently [Estrada, 1999]. 

Typical input data for a moment tensor inversion consist of the network geometry (coordinates of the sensors) knowledge of sensor polarity (i.e. whether an upwards deflection at the sensor indicates a compression or dilatation) sensor orientation source coordinates P and/or S-wave displacement amplitudes recorded at each sensor (time-domain inversion) or P and/or S-wave spectral amplitudes (frequency-domain inversion) and the polarities of the wave phases. 

The other modes are coupled only by bilinear terms resulting from the transformation of the unperturbed Hamiltonian Hq, and they become decoupled in the limiting case of equal frequencies. The spectral moments are correctly reproduced up to third order by this approach and the bandshapes of multi-mode JT systems are accurately reproduced in this way for a large range of system parameters. ... 

A remark applies to the experimental values of deduced from experimental bandshapes. Since we deal with total (multimolecular) correlation functions, their moments might also contain contributions arising from the intercorrelations of different molecules - or multimolecular effects - since cross terms in the statistical averages do not necessarily vanish. On this account it is necessary to estimate the extent of high-frequency collective effects if any, whenever torques are to be computed from the spectral moments. 

In the analysis of stochastic processes, an important role is played by the so-called spectral moments (SM), introduced by Vanmarcke (1972), important for the definition of some characteristics of stochastic processes and in reliability analysis (Michaelov et al. (1999)). These quantities are defined as the moments of the one-sided PSD with respect to the frequency origin. Let Sy (co) be the PSD of the earthquake acceleration with 5y(ffl) = 5y(—co). Let Gy(co) be the one-sided PSD defined... 

In this entry only two problems have been presented representation of the power spectral density (PSD) and correlation by means of complex fractional moments and filter equations. Regarding the first problem, it has been shown that both PSD and correlation may be represented by the so-called fractional spectral moments (FSMs). The latter are the extension of the classical spectral moments introduced by Vanmarcke (1972) to complex order. The appealing in using these FSM is that they are able to reconstruct both PSD and correlation. As a result, it may be stated that FSM functiOTi is another equivalent representation of PSD and correlation. Moreover it has been shown that with a limited number of informations (few FSM), the whole PSD and correlation may be restored, including the trend at infinity. Extension to multivariate seismic process has been also provided in both frequency... 

The approximate solution given in Eq. 30 provides accurate results for low and intermediate frequencies. Therefore the author considered the solution proposed by Vanmarcke and Gasparini (1977) given by Eq. 10 with the adjusted spread factor as further improvement of Eq. 19. As the peak factor evaluation depends on the whole power-spectral density function, which is intrinsically related to the spectral characteristics of the unknown response process (see Eqs. 9, 13, 14, and 20), the author developed an algorithm which enables to update the spectral characteristics of the response process every frequency step of definition of the frequency domain. The iterative scheme mainly differs from the procedure proposed by Sundararajan (1980) as not only the zeroth-order spectral moment is updated at each step but also the peak factor. To this aim the... 

Evolutionary frequency response function Evolutionary power spectral density function Gaussian zero-mean random models of seismic accelerations Non-geometric spectral moments Stochastic analysis... 

The main steps of the described approach are (i) the use of modal analysis to decouple the equation of motion (ii) the determination, in state variable, of the evolutionary frequency response vector functions and of the evolutionary power spectral density function matrix of the structural response and (iii) the evaluatiOTi of the nongeometric spectral moments as weU as the spectral characteristics of the stochastic response of linear systems subjected to stationary... 

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