Broadening half width

Avd Avn Avp c A AW Doppler half-width in hertz natural half-width l/47cyN pressure-broadened half-width 1 /4nyP variable of integration wavelength of radiant flux triangle function of unit height and half-width % scaled ratio (AvN -1- Avc) /ln 2/AvD... 

The complex CgHgMo(CO)3 also displays this phenomenon (379a) NMR of this in toluene solution shows a sharp singlet at 70° C, which becomes broadened (half-width 25 cps) at room temperature and freezes to a complex pattern at — 30°C. This is therefore the first known complex of cyclooctatetraene whose proton magnetic resonance shows observable change from rapid to arrested valency tautomerism. 

For the Brownian motion imbedded ELS spectrum. Ware and Flygare [26] defined the analytical resolution (not the spectral resolution discussed earlier) in terms of the ratio of the Doppler shift to the diffusion broadened half-width ... 

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. 

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. 

As time increases from —oo to 0, the half width of the wave packet y(x, t) continuously decreases and the maximum amplitude continuously increases. At t = 0 the half width attains its lowest value of flja and the maximum amplitude attains its highest value of 1 /a/2, and both values are in agreement with the wave packet in equation (1.20). As time increases from 0 to oo, the half width continuously increases and the maximum amplitude continuously decreases. Thus, as f- increases, the wave packet y(x, t) remains gaussian in shape, but broadens and flattens out in such a way that the area under the square y(x, t) of the wave packet remains constant over time at a value of (2-y/ a), in agreement with ParsevaTs theorem (1.18). 

Figure 15 gives the superposition of RR (full line) and RY (dotted plot) spectral densities at 300 K. For the RR spectral density, the anharmonic coupling parameter and the direct damping parameter were taken as unity (a0 = 1, y0 = ffioo), in order to get a broadened lineshape involving reasonable half-width (a = 1 was used systematically, for instance, in Ref. 72). For the RY spectral density, the corresponding parameters were chosen aD = 1.29, y00 = 0.85angular frequency shift (the RY model fails to obtain the low-frequency shift predicted by the RR model) and a suitable adjustment in the intensities that are irrelevant in the RR and RY models. 

The major requirement of the light source for atomic absorption is that it should emit the characteristic radiation (the spectrum) of the element to be determined at a half-width less than that of the absorption line. The natural absorption line width is about 10 4 (A), but due to broadening factors such as Doppler and collisional broadening, the real or total width for most elements at temperatures between 2000 ° and 3000 °K is typically 0.02 — 0.1 A. Hence, a high resolution monochromator is not required. 

Ideally, the emission line used should have a half-width less than that of the corresponding absorption line otherwise equation (8.4) will be invalidated. The most suitable and widely used source which fulfils this requirement is the hollow-cathode lamp, although interest has also been shown in microwave-excited electrodeless discharge tubes. Both sources produce emission lines whose halfwidths are considerably less than absorption lines observed in flames because Doppler broadening in the former is less and there is negligible collisional broadening. 

The 500 nm size is a limit value crystallites below this size tend to broaden the diffraction peaks in a spectrum, while size distributions above this value produce particularly sharp signals whose half width is a function only of the wavelength of the X-ray beam and the equipment. Signal broadening is at its maximum in materials known as X-ray amorphous substances, featuring particle size distributions below 8 nm. These afford flattened, washed-out spectra of little analytical value. 

Doppler broadening. Collisions broadening becomes less important at the lower pressures found at higher altitudes, so that the Lorentzian half-width and the Doppler half-width become comparable at altitudes of approximately 30-40 km. 

AxN, Axp, AxD, Axv line-profile half-widths in cm-1 for natural, pressure-broadened, Doppler-broadened, and combined Doppler-and pressure-broadened cases, respectively generally Ax = Av/c, where c is velocity of light... 

Av Avc Rayleigh resolution in wave-number units half-width yc/4rc observed when collision broadening dominates (given in hertz)... 

To consider gas molecules as isolated from interactions with their neighbors is often a useless approximation. When the gas has finite pressure, the molecules do in fact collide. When natural and collision broadening effects are combined, the line shape that results is also a lorentzian, but with a modified half-width at half maximum (HWHM). Twice the reciprocal of the mean time between collisions must be added to the sum of the natural lifetime reciprocals to obtain the new half-width. We may summarize by writing the probability per unit frequency of a transition at a frequency v for the combined natural and collision broadening of spectral lines for a gas under pressure ... 

When pressure broadening dominates, the situation is more complicated because the resulting Lorentzian profile contributes significant area far from the line center. A further complication in this case is that the Lorentzian half-width cannot be accurately calculated and must be measured in other experiments. If both Doppler and pressure broadening are present, however, and if the Lorentzian to Doppler half-width ratio is small, the correction necessitated by pressure broadening is small. In this situation an accurate value of the Lorentzian half-width may not be needed. Line strength in the case of combined Doppler and pressure broadening may be obtained from the equivalent width by the use of tables (Jansson and Korb, 1968). 

These emission lines have a half width of v2 X, which is consistent with the rotational broadening of the secondary component (Vrot 95 km- ). In the four... 

Very slow molecular motion (tc > 10 9 s/rad at B0 x 2.1 Tesla) leads to an increase in T, while T2 decreases (Fig. 3.20). The signals then broaden (line width at half-maximum intensity zlv1/2 1/7V). Therefore, the more sluggish macromolecules usually give poorly resolved 13C NMR spectra having a bandlike shape. 

The T2 relaxation times of 50 /rsec and 40 msec given in the preceding discussion correspond to line half-widths of 6.4 kHz and 8 Hz, respectively. Whipple et al. (265) concluded that the line widths of several hundred Hz which are obtained in practice must be due to bulk magnetic susceptibility effects. This type of line broadening is removable by MAS (273) and they were the first to obtain high resolution spectra with linewidths similar to those expected from the T2 values. 

The last two mechanisms of the broadening of atomic spectral lines are in most cases the real experimental limitations in atomic spectroscopy. The half-widths of such lines are usually of the order of 10-3 nm. 

The efficiency factor (N) is a measure of zone broadening (peak width) occurring on a column. It can be calculated for any given peak from Equation B4.2.4, where Vr is the elution volume (i.e., the total volume of eluant that has passed through the column at peak maximum) and Wh is the peak width at half the peak height. 

The half-widths of 37-39 and 78-88 Hz, respectively, for the crystalline and amorphous phases are significantly larger than 18 and 38 Hz for those of the bulk-crystallized linear polyethylene (cf. Table 1). This is caused by incorporation of minor ethyl branches. The molecular alignment in the crystalline phase is slightly disordered, and the molecular mobility in the amorphous phase will therefore be promoted. With broadening of the crystalline and amorphous resonances, the resonance of the interphase also widens in comparison to that of bulk-crystallized linear polyethylene samples. This shows that the molecular conformation is more widely distributed from partially ordered trans-rich, conformation to complete random conformation, characteristic as the transition phase from the crystalline to amorphous regions. 

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