Termination of Rietveld refinement

Non-linear least squares technique results in finding a set of increments that are added to a set of free variables chosen to represent a certain initial approximation. Parameters, obtained in this ways, are carried over into the next refinement cycle as a more precise initial approximation. In some cases it may take a few refinement cycles to achieve the best fit, i.e. to minimize the corresponding function, while in many instances the number of required least squares steps may be quite large. Especially in Rietveld refinement, where various groups of parameters have a different and often unrelated physical origin, the ability to detect the completion of the minimization, i.e. the complete convergence of the least squares, is essential. 

Even if the refinement is stable, it should be terminated at a certain point. Because of the finite accuracy of both the data and computations, it is unrealistic to wait until increments of all least squares parameters become zero. The latter usually never happens anyway due to simplifications introduced in the non-linear least squares algorithm (see Eq. 6.9 in section 6.6). In addition to the stabilization of all figures of merit near their respective minima, another important factor that should be considered is the 

We have seen this powder diffraction pattern several times throughout this text. The histogram collected from the nearly spherical LaNi4.85Sno.15 powder, produced by high pressure gas atomization from a melt, was used to illustrate both the quality of x-ray diffraction data and as one of the examples in the ab initio crystal structure solution. To demonstrate the Rietveld refinement of this crystal structure we will begin with the profile and unit cell parameters determined from Le Bail s algorithm Table 6.3) and the model of the crystal structure determined from sequential Fourier maps as described in section 6.9 and listed in Table 6.8. 

To account for the presumably statistical distribution of Ni and Sn atoms in the 2(c) and 3(g) sites in this crystal structure, the initial distribution of atoms in the unit cell has been assumed as listed in Table 7.2. The initial profile and structural parameters are found in the input file for LHPM-Rietica on the CD, the file name is Ch7Ex01a.inp. Experimental diffraction data, collected on a Rigaku TTRAX rotating anode powder diffractometer using Cu Ka radiation in a continuous scan mode, are located on the CD in the file Ch7Ex01 CuKa.dat. 

The last column in Table 7.2 contains site occupancies by all atoms in the format required by LHPM-Rietica. The occupancy of each site ( ) is given as a product of the population parameter (g) and site multiplicity (m) divided by the multiplicity of the general site position (A )  

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