Half width

The algorithm contains five minimisation procedures which are performed the same way as in the method " i.e. by minimisation of the RMS between the measured unidirectional distribution and the corresponding theoretical distribution of die z-component of the intensity of the leakage field. The aim of the first minimisation is to find initial approximations of the depth d, of the crack in the left half of its cross-section, die depth d in its right half, its half-width a, and the parameter c. The second minimisation gives approximations of d, and d and better approximations of a and c based on estimation of d,= d, and d,= d,j. Improved approximations of d] and d4 are determined by the third minimisation while fixing new estimations of d dj, dj, and dj. Computed final values dj , d/, a and c , whieh are designated by a subscript c , are provided by the fourth minimisation, based on improved estimations of d, dj, dj, and d . The fifth minimisation computes final values d, , d, dj, d while the already computed dj , d/, a and c are fixed. 

The next two temis (Lorentzians) arise from the mechanical part of the density fluctuations, the pressure fluctuations at constant entropy. These are the adiabatic sound modes (l/y)exp[-FA t ]cos[co(A) t ] with (D(k) = ck, and lead to the two spectral lines (Lorentzians) which are shifted in frequency by -ck (Stokes line) and +ck (anti-Stokes line). These are known as the Brillouin-Mandehtarn, doublet. The half-width at... 

For a given half width at half maximum in the time domain, Ar,.n =2, /, the slice width A decreases with increasing gradient strength G. ... 

Band Asymmetry. The peak asymmetry factor AF is often defined as the ratio of peak half-widths at 10% of peak height, that is, the ratio b/a, as shown in Fig. 11.2. When the asymmetry ratio lies outside the range 0.95-1.15 for a peak of k =2, the effective plate number should be calculated from the expression... 

Figure 2.5 shows, for a sample in the gas phase, a typical absorption line with a HWHM (half-width at half-maximum) of Av and a characteristic line shape. The line is not infinitely narrow even if we assume that the instmment used for observation has not imposed any broadening of its own. We shall consider three important factors that may contribute to the line width and shape. 

Figure 2.5 Typical (gaussian) absorption line showing a HWHM (half width at half maximum) of Av and a Lamb dip (dashed curve)...

Q are the absorbance and wavenumber, respectively, at the peak (center) of the band, p is the wavenumber, and y is the half width of the band at half height. Liquid band positions ate usually shifted slightly downward from vapor positions. Both band positions and widths of solute spectra are affected by solute—solvent interactions. Spectra of soHd-phase samples are similar to those of Hquids, but intermolecular interactions in soHds can be nonisotropic. In spectra of crystalline samples, vibrational bands tend to be sharper and may spHt in two, and new bands may also appear. If polarized infrared radiation is used, both crystalline samples and stressed amorphous samples (such as a stretched polymer film) show directional effects (28,29). 

Structure Chemical shifts S (p.p.m.) Reference signal Half-width (Hz) Solvent Ref... 

Plotting diagrams showing dependence of peak parameters (such as peakedness, tailing, semi-half-width of peak, peak asymmetric etc.) on peak height ... 

Kic = fracture toughness a, = size of widest microcrack (crack width for surface crack crack half-width for buried crack). 

In conclusion of this section, we write out the expressions for the density matrix of a free particle and a harmonic oscillator. In the former case p(x, x P) is a Gaussian with the half-width equal to the thermal de Broglie wavelength... 

For the simplest one-dimensional or flat-plate geometry, a simple statement of the material balance for diffusion and catalytic reactions in the pore at steady-state can be made that which diffuses in and does not come out has been converted. The depth of the pore for a flat plate is the half width L, for long, cylindrical pellets is L = dp/2 and for spherical particles L = dp/3. The varying coordinate along the pore length is x ... 

As mentioned above, the interpretation of CL cannot be unified under a simple law, and one of the fundamental difficulties involved in luminescence analysis is the lack of information on the competing nonradiative processes present in the material. In addition, the influence of defects, the surface, and various external perturbations (such as temperature, electric field, and stress) have to be taken into account in quantitative CL analysis. All these make the quantification of CL intensities difficult. Correlations between dopant concentrations and such band-shape parameters as the peak energy and the half-width of the CL emission currently are more reliable as means for the quantitative analysis of the carrier concentration. 

Fig. 3.31. Distributions (i)/(Ee) dEe of electron energy (E ) for a low-pressure HF-plasma (suffix pi, Maxwellian with temperature = 80000 K) and an electron beam (suffix eb, simplified to Gaussian shape with 40 eV half-width) (ii) rTx (Ej) ofthe Ej dependent electron impact ionization cross-section for X=Ti...

To derive the equation for the jet boundary resulting from the interaction of coaxial main flow and a directing jet supplied at the distance f,) from the main outlet, this interaction was presented - as the interaction of the main jet with a sink distributed along its axis (Fig. 7.55). Considering the influence of the directing jet on the main flow boundary as AY, the half width of the resulting flow can be presented as... 

Peak asymmetry or skewing is a well-documented (4,6,7) characteristic of chromatographic peaks and is measured easily by ratioing the peak half widths at 10% height as shown ... 

Notice that P f) has a maximum at f — 1/2. The presence of this maximum can be directly traced back to the length-2 loop of the STG shown above, which yields a period-2 component in the spatial configurations. The half-width of the peak (=log(l/g)) is seen to decrease as q decreases, a natural consequence of the period-2 part of the STG being visited more often as q increases. Figure 6.4 shows in empirical estimate of the power spectrum for rule R56 obtained for a size iV = 2 = 2048 system. 

In general, increasing the temperature within the stability range of a single crystal structure modification leads to a smooth change in all three parameters of vibration spectra frequency, half-width and intensity. The dependency of the frequency (wave number) on the temperature is usually related to variations in bond lengths and force constants [370] the half-width of the band represents parameters of the particles Brownian motion [371] and the intensity of the bands is related to characteristics of the chemical bonds [372]. 

It would appear that measurement of the integrated absorption coefficient should furnish an ideal method of quantitative analysis. In practice, however, the absolute measurement of the absorption coefficients of atomic spectral lines is extremely difficult. The natural line width of an atomic spectral line is about 10 5 nm, but owing to the influence of Doppler and pressure effects, the line is broadened to about 0.002 nm at flame temperatures of2000-3000 K. To measure the absorption coefficient of a line thus broadened would require a spectrometer with a resolving power of 500000. This difficulty was overcome by Walsh,41 who used a source of sharp emission lines with a much smaller half width than the absorption line, and the radiation frequency of which is centred on the absorption frequency. In this way, the absorption coefficient at the centre of the line, Kmax, may be measured. If the profile of the absorption line is assumed to be due only to Doppler broadening, then there is a relationship between Kmax and N0. Thus the only requirement of the spectrometer is that it shall be capable of isolating the required resonance line from all other lines emitted by the source. 

The analysis can be performed for several values of the relative peak height n, using the appropriate values of f(n) taken from Table I. Thus several estimates of Ed are obtained and either an average of them is calculated with its standard deviation, or provided a dependence of Ed on n is encountered, conclusions on the variability of Ed with coverage are drawn. As an alternative, only the half-widths of the peak are treated, as long as the experimental data are not distorted by some adjacent peak. Obviously, more information is obtained in such a case. The ratios of the half-widths taken at various values of n and compared with those given in Table I represent a criterion for the fit of the value of the desorption order. Since the estimates of Ed are free of contributions of fcd, the Tm relations can be used to estimate grossly the value of fcd, similarly as in Section V.C.2.b. 

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

Fig. 3.9. (a) Dependence of the experimental half-width of the isotropic Q-branch of N2 and CO on the density ( ) CO, 295 K (+) CO in CF4, 273 K (A) CO in C02> 323 K (O) N2> 295 K (A) N2 in C02> 323 K. The error in the measurements of half-width is +0.2 cm-1, (b) The same data as in (a) but in relation to measured T = I/ojqXj. Theoretical curves for strong (curve 1) and weak (curve 2) collision limits are identical to curves A and B in Fig. 3.8. The upside-down triangles near the broken line present the difference between the actual half-width of CO in CO2 (A) and curve 1. The error in all cases is approximately the same as that indicated for CO.

With formulae (3.58), (3.59) and (3.66) Q-branch contours are calculated for CARS spectra of spherical rotators at various pressures and for various magnitudes of parameter y (Fig. 3.14). For comparison with experimental data, obtained in [162], the characteristic parameters of the spectra were extracted from these contours half-widths and shifts of the maximum subject to the density. They are plotted in Fig. 3.15 and Fig. 3.16. The corresponding experimental dependences for methane were plotted by one-parameter fitting. As a result, the cross-section for rotational energy relaxation oe is found ... 

Fig. 3.15 demonstrates that in this pressure domain the half-width of the spectrum still differs essentially from its perturbation theory estimate... 

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