Line broadening, equation

The crystallite dimension is calculated from the degree of broadening of the (002) reflection using the Scherrer line broadening equation namely ... 

Equation (2.27) illustrates what is called the natural line broadening. Since each atom or molecule behaves identically in this respect it is an example of homogeneous line broadening, which results in a characteristic lorentzian line shape. 

These equations do not necessarily show the actual charges the important point is that all three are single-electron events. The asterisks can be thought of as an isotopic label, but need not be anything that concrete, since certain line-broadening techniques (Section 11.5) provide EE rate constants without them. The Marcus cross relation is an expression for kA% as a function of kAA, bb> and A, the equilibrium constant for Eq. (10-67). It reads,... 

Here, ojr is the rate of spinner rotation. I is the proton spin number, 8 is the chemical shift anisotropy (CSA) and q is the asymmetric parameter of the CSA tensor. Thus, the line broadening occurs when an incoherent fluctuation frequency is very close to the coherent amplitude of proton decoupling monotonously decreased values without such interference in Figure 1. 

A very similar application of the modified Bloch equations was based in the work of Adams and Connelly.4 ESR spectra (Figure 5.8) of [Mo P(0 Me)3 2(MeC = CMc)Cp] show the expected triplet (two equivalent 31P nuclei) at 280 K, but only a doublet at 160 K. At intermediate temperatures, the lines broaden. The interpretation is that the alkyne undergoes a pendulum oscillation, which in the extrema diverts spin density from one or the other phosphite. Interestingly, the diamagnetic cation undergoes a similar motion on the NMR time scale, but then the alkyne undergoes a complete rotation. Thus, analysis of the effect leads to a measure of the rate of the oscillation. The... 

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. 

There are a number of reasons why the XRD peaks of LDH samples are often rather broad. The relatively small domain size, particularly in the (00/) direction, leads to line broadening. The Scherrer equation may be used to estimate the domain size in the a and c directions from the width of the (110) and (00/) reflections, respectively [119,120], although the inherent approximations in this method should always be borne in mind [121,122]. 

At this point it has been established that there are at least two basic mechanisms which contribute to the broad lines that are observed for the crystalline polymers. The residual zero frequency line broadening component can be analyzed in more detail. Specific attention can be given to factors which are a consequence of the chain-like character of the molecules. The local field at a given nucleus is the sum of the individual fields contributed by the neighboring magnetic nuclei. Segmental motions will induce a time dependence to the variables so that the individual contributions can be described by the equation (46)... 

The exact nature of the epimerization is still not known. NMR studies are hampered by line broadening and multiplication due to frozen rotations in the Af,A-diisopropylcar-bamoyl group at the required low temperatures. Some indications were gained by a H NMR investigation of the temperature dependence of the kinetics in the epimerization of the (—)-sparteine and (—)-a-isosparteine complexes derived from 1-lithioinden-l-yl carbamates 295 and 296, carried out in cooperation with G. Fraenkel (equation 72, Table 7). These complexes are chemically stable above 0°C, and interpretable spectra were obtained. 

A second steady-state method involves the analysis of the broadening of the nuclear magnetic resonance spectra of phospholipids in bilayers containing low concentrations of spin-labeled phospholipids. A theoretical analysis of the relation between this line broadening and diffusion rates has been given by Brulet and McConnell.3 [In this paper (6) is not correct the subsequent equations are nonetheless correct. For an alternative derivation, see Brulet.2] In this paper it is shown that a number of measurements of nuclear relaxation rates T71 of nuclei in phospholipids are consistent with lateral diffusion constants in the range 10 7 to 10 R cm2/s. 

For biased equilibria, Anet10 5 developed a method of estimating k and the population of the least stable conformer from line-broadening measurements. This method uses the expression developed by Gutowsky and Holm106 from the Bloch equations... 

Example. Kubo constructed the following model to illustrate line broadening and narrowing due to random perturbations.510 In equation (1.2) suppose that co = (o0 + a (t), where a>0 and a are constants and (r) is the dichotomic Markov process (IV.2.3). As only takes two values we may abbreviate... 

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