Distance least squares technique

For both structures, all final Si positions were obtained with reasonable accuracy (0.1 -0.2 A) by a 3D reconstruction of HRTEM images followed by a distance least-squares refinement. This kind of accuracy is sufficient for normal property analysis, such as catalysis, adsorption and separation, and as a starting point for structure refinement with X-ray powder diffraction data. The technique demonstrated here is general and can be applied not only to zeolites, but also to other complicated crystal structures. 

Fitting was accomplished by nonlinear least squares techniques with four adjustable parameters per component N, R, o2 and where N is the coordination number, R is the distance, a2 is the Debye-Waller factor and A/i0 is the difference between the reference functions and unknown spectra. The A/ 0 variable was iloated in the first shell only anti fixed in subsequent shells. Experimental or the-... 

Also, from equation 5, the three equations above must all be equal. Therefore if we equate them, we have six equations with six unknowns (711, Ai>7i3 A3>73i > A program using a nonlinear least-squares technique was written to solve for the unknowns. The resulting sensor parameters using the radial distances obtained from the forward solution, in which the position of the object coordinate frame is known but r, 7, and for each sensor are unknown, are given in Figure 4. As one can see, the parameter results match very closely since the percent differences are mostly negligible. 

Figure 7.17 compares the results of the two function fits to the data. In both cases all three coefficients are allowed to vary to optimize the data fit. As can be seen from the two curves and from the best fit values of the parameters, somewhat different curves and parameters are obtained depending upon whether the least squares technique minimizes the vertical distance between the theory and the data or minimizes the horizontal distance between the model and the data. This is a difference of nonlinear curve fitting as opposed to linear curve fitting. For a linear... 

The extent of gas dispersion can usually be computed from experimentally measured gas residence time distribution. The dual probe detection method followed by least square regression of data in the time domain is effective in eliminating error introduced from the usual pulse technique which could not produce an ideal Delta function input (Wu, 1988). By this method, tracer is injected at a point in the fast bed, and tracer concentration is monitored downstream of the injection point by two sampling probes spaced a given distance apart, which are connected to two individual thermal conductivity cells. The response signal produced by the first probe is taken as the input to the second probe. The difference between the concentration-versus-time curves is used to describe gas mixing. 

In the X-ray technique, the manner of normalization of the intensities can lead to significant differences in the RDFs. Habenschuss and Spedding (1979b, c, 1980) reported a detailed X-ray diffraction study of concentrated lanthanide chloride solutions (3.2 to 3.8 M) in which the data were analyzed with different constraints imposed on the resolution of the peaks in the RDF. Figure 4 shows the normalized RDFs G(r) for 10 concentrated solutions. The three peaks at ca. 2.5, 3.0 and 5.0 A correspond to Ln " -H20, H2O HjO and CI-H2O interactions, respectively. Resolution of the Ln -H20 peak from the net RDF was based on the assumption that all the peaks had Gaussian distributions which appeared acceptable for the first peak but was more questionable for the other peaks (Habenschuss and Spedding 1979c). In the least-squares fit, the CI-H2O distance was kept the same for all lanthanides while... 

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