Linewidth calculation

Results of calculation of second moment line for analyzed systems were obtained by summation over 106 cells in lattice, they are presented in Table 2. The Lorentzian linewidth calculated from these M2 values... 

The atoms are stored in the bulb for about 1 s and consequently the linewidth, calculated from the uncertainty... 

Once the basic work has been done, the observed spectrum can be calculated in several different ways. If the problem is solved in tlie time domain, then the solution provides a list of transitions. Each transition is defined by four quantities the mtegrated intensity, the frequency at which it appears, the linewidth (or decay rate in the time domain) and the phase. From this list of parameters, either a spectrum or a time-domain FID can be calculated easily. The spectrum has the advantage that it can be directly compared to the experimental result. An FID can be subjected to some sort of apodization before Fourier transfomiation to the spectrum this allows additional line broadening to be added to the spectrum independent of the sumilation. 

Table I reports the observed NMR linewidths for the H/3 protons of the coordinating cysteines in a series of iron-sulfur proteins with increasing nuclearity of the cluster, and in different oxidation states. We have attempted to rationalize the linewidths on the basis of the equations describing the Solomon and Curie contributions to the nuclear transverse relaxation rate [Eqs. (1) and (2)]. When dealing with polymetallic systems, the S value of the ground state has been used in the equations. When the ground state had S = 0, reference was made to the S of the first excited state and the results were scaled for the partial population of the state. In addition, in polymetallic systems it is also important to account for the fact that the orbitals of each iron atom contribute differently to the populated levels. For each level, the enhancement of nuclear relaxation induced by each iron is proportional to the square of the contribution of its orbitals (54). In practice, one has to calculate the following coefficient for each iron atom ...

Fig. 12. Effect of a strong exchange interaction on the shape of the EPR spectrum displayed by a pair of centers A and B having identical g vEilues, = 1.89, g, = 1.96, g = 2.07, and rotated magnetic axes according to xjly, yglx, zJIzy,. (a) 9 GHz spectrum calculated with J = 0 (b) and (c) spectra calculated with J = 25 X 10 cm at 9 and 35 GHz, respectively. The spectra were calculated as described in Ref. 192) without including any dipolar terms, with the linewidths ui = cr, = oi = 0.01.

Optimum values for the probabilities may not be obtained in the case that experimental llnewidths in the spectrum are very different since only a single linewidth is used for the simulated spectra. The calculated probabilities may be stored in the database and hard copy reports may be printed-... 

Once creation of the PV A database is complete, optimized probabilities may be calculated for the experimental spectrum at hand. Since the iterative procedure is restricted to a 2048 data point region, zoom cursors are displayed and set by the user until this condition is satisfied. In this case, the methylene region was selected and an initial guess for the Bernoullian probability (Pr=0.5) and linewidth (13.0Hz) were given. Optimized values for the probability and linewidth were Pr=0.52 and 12.8Hz, respectively. 

Figure 4 shows zoomed regions of the experimental and simulated spectra. The methine region was simulated separately using the same optimized probability but with a linewidth of 8.0 Hz. At this point the user may wish to use the spectral manipulation options (overlay, subtraction, etc.), repeat the calculation, or do further simulations. 

The effect of a pressure of 80 and 150 MPa on the spin-state transition has been also studied [169], a series of spectra obtained at 150 MPa being shown in Fig. 32. The speetra show relaxation effects as line broadening and linewidth asymmetry. Calculated spectra were obtained in the same way as at ambient pressure. Rate constants for a number of temperatures are listed in Table 12, the parameter values resulting from an Arrhenius plot of the rate constants being listed in Table 13. In Fig. 33, the quantity 5g of Eq. (36) has been plotted as a... 

Here, H is the magnetic field variable and AH is the magnetic field separation between the maximum and the minimum in the derivative curve. This linewidth may be a function of orientation also however, in most calculations it is assumed to be constant. 

Fig. 6 mCd NMR chemical shifts (diamonds) and linewidths (triangles) of Cd, YZ.nYTc as a function of x. The dashed-dotted line is the most probable number of NNN Zn atoms, and the solid line is the width calculated from probability distributions. Reprinted with permission from [157]. Copyright 1987 by the American Physical Society... 

As for the width of NMR resonance lines, it is inversely proportional to mobility of resonating nuclei. The width calculated from the experimental NMR lines of the three samples is of the highest value for the sample CT ODA 5 and there is only a small difference between the linewidths of CT and CT ODA 2. 

Fig. 13. Observed and calculated pH dependence of 54.227 MHz 170 signal for the W(IV) complex (a) linewidth (oxo fast exchange ). Dashed lines 1 and 2 illustrate how the limiting value of kih was determined (6). (b) Chemical shift (oxo aqua/...

The line-broadening data as a function of pH, typically shown for the W(IV) in Figs. 13 and 14, incorporating the known pKa values (Table II), were fitted in 5 X 5 Kubo-Sack matrices describing the exchange based on the above schemes (6, 57). The experimentally determined chemical shift and linewidth data in the absence of exchange for the aqua oxo, hydroxo oxo, and dioxo species and the pH-dependent species distribution as calculated from the acid dissociation constants for the four systems were all introduced in the different matrices and the spectra were computer simulated. For each set of chosen rate con-... 

Table 5.4 A comparison of specific surface area ratio calculated from quadrupole splitting (Aq), spin-lattice relaxation rate (Rf, half-height linewidth (Avj/2) and isotherm data for an unbleached linerboard pulp beaten to various degrees.

Several general observations can be made from the spectra and the calculated ESR parameters. First, the Na+- smectites possess narrower resonance line widths than the Ca +-smectites, with the exception of the saponite. Since Ca +-smectites, unlike Na+ -smectites, do not disperse into individual platelets in aqueous suspension, the Na+-smectite films formed by drying suspensions onto a smooth flat surface have the silicate surfaces more perfectly oriented in the plane of the film. As a result, less angular variation of the z-axis of Cu + relative to the plane of the film would narrow the spectra. One can see evidence of hyperfine splitting in the gj component of the Na+-smectite spectra, but not in the Ca +-smectite spectra. Saponite, unlike the other smectites, has very similar spectral linewidths for the Na+ and Ca + form (Figure 16). Since this Na+-saponite sample does not disperse completely in water (Table II), the alignment of Na+-saponite platelets in the clay film may be no better than that of the Ca +-saponite. 

The computation of far-field radiation from a collection of incoherently radiating dipoles is in general quite a complicated problem. To calculate the angular dependence of the far-field intensity, the volume distribution of excited states must first be obtained, which, as we have seen, depends on the volume distribution of the absorbers and the electromagnetic field which stimulates them. The fields in turn depend on the frequency and linewidth of the exciting light source. Then the emission problem for the excited-state distribution (both spatial and frequency) must be solved including reorientation and depolarization effects. 

Prior knowledge allows to include fixed relations between some of the four parameters (amplitude, phase, frequency position, peak width) describing a symmetrical well-shaped resonance. Signal ratios, chemical shift difierences, linewidth relations and zero-order phase relations can be included. The reduction of the number of unknown parameters leads to a reduced calculation time, better convergence behaviour and improved results. However, the assumptions made to include the prior knowledge must be validated for each experiment. Differences between the parameter values set by the prior knowledge and the actual parameters could lead to systematic errors. 

The inverse Fourier transform of G again yields a Gaussian function g(t) = linewidth Tg of which can be numerically calculated from... 

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