Bragg residual

The Bragg residual, Rb (this figure of merit is quite important in Rietveld refinement but has little to no use during full pattern decomposition because only observed Bragg intensities are meaningful in both Pawley and Le Bail methods) ... 

The initial model of the crystal structure results in acceptable residuals without refinement of coordinates and displacement parameters of individual atoms. When the coordinates of all atoms and the overall displacement parameter were included into the least squares, the residuals further improve (row four in Table 7.14). The biggest improvement is observed in the Bragg residual, Rb, which is expected because this figure of merit is mostly affected by the adequacy of the structural model and it is least affected by the inaccuracies in profile parameters. 

Lattice parameters and residuals for die carbide YbCoC and the silicides YbTSi (r=Mg, Fe, Cu, Pd, Ag, Pt, Au). The R-value marked with an asterisk results from a Rietveld refinement (Bragg residual)... 

Lattice parameters and residuals for the orthorhombic stannides YbTSn (T=Mg, Ni, Cu, Zn, Rh, Pd, Ag, Cd, Ir, Pt, Au, Hg). Tile R-values marked with an asterisk result fiom Rietveld refinements (Bragg residuals)... 

Both ultrasonic and radiographic techniques have shown appHcations which ate useful in determining residual stresses (27,28,33,34). Ultrasonic techniques use the acoustoelastic effect where the ultrasonic wave velocity changes with stress. The x-ray diffraction (xrd) method uses Bragg s law of diffraction of crystallographic planes to experimentally determine the strain in a material. The result is used to calculate the stress. As of this writing, whereas xrd equipment has been developed to where the technique may be conveniently appHed in the field, convenient ultrasonic stress measurement equipment has not. This latter technique has shown an abiHty to differentiate between stress reHeved and nonstress reHeved welds in laboratory experiments. 

The average conformation of an a-helix-forming polypeptide was formulated first by Zimm and Bragg (4) and then by several authors (5-9). A comprehensive survey of these theories can be found in a book by Poland and Scheraga 10) or in our companion review article (//). In this section, we outline the formulation of Nagai (5). For convenience of presentation, a peptide residue (-CO-HC R-NH-) is called helix unit when distorted to the a-helical conformation, while it is called random-coil unit when allowed to rotate about the bonds C C and C -N. These units are designated h and c. Thus a particular conformation of an a-helix-forming polypeptide chain is represented by a sequence of h and c. 

The helical fraction has here been defined as the number of helix units present in the chain under consideration relative to the total number of residues in the same chain that can assume a-helical conformation, i.e. N — 2. It should be noted that this way of defining fN differs from that of Zimm and Bragg (4), who adopted the number of hydrogen bonds formed in the chain. The difference, however, becomes important only for short chains. 

Here k is the Boltzmann constant, T is the absolute temperature, C is the number of peptide residues per cubic centimeter of solution, / and s are the helical fraction and the Zimm-Bragg parameter for the polypeptide molecule in the absence of external field, and ft is a parameter which represents the average correlation between the helix unit at the end of a helical sequence and the random-coil unit next to it. Though a detailed account of this parameter is left for Ref. (124), we note here that / = 0 corresponds to the complete absence of correlation between these two units, while ft — 1 corresponds to the case in which the dipole moment of a random coil unit points in the same direction as the axis of the preceding helix unit. 

Since the Zimm-Bragg parameters o and s of the naturally occurring amino acids (In water) cannot be obtained from studies of the helix-coil transition in homopolymers, because of experimental difficulties, a technique Is developed to circumvent these problems. It involves the study of the thermally induced transition curves for random copolymers of "guest amino acid residues in a water-soluble host" po y(amino acid). The data may be interpreted with the aid of suitable theories for the helix-coil transition in random copolymers to obtain a and s for the "guest" residues. It is shown in this paper that, for the usual ranges of parameters found for polylamino acids), one of the two lowest order approximations (corresponding to earlier treatments by Lifson and Allegra) is completely adequate. In essence, the low-order approximations hoid if o and s for the two constituents of the copolymer do not differ appreciably from each other. 

This model differs from Schwarz s2 who uses the single residue as kinetic unit in analogy to the Zimm-Bragg equilibrium model. His rate parameters satisfy, of course, Eq. (4), but in his case x y, x y. The... 

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