RIETVELD ANALYSIS OF Al INCLUDING TDS TERMS. P. Riello, P. Canton, G. Fagherazzi, Universitá di Venezia ,Italy
In the X-ray Rietveld analysis the optimized B-thermal factors are often physically unrealistic, generally smaller than the corresponding ones obtained with single crystal analysis This could be due to the inadequate analytical description of the background scattering together with the great flexibility (in the fitting procedure) of the used Bragg peak profile functions (pseudo-Voigt or Pearson VII) that may overestimate the peak areas, especially at wide 2[[theta]] Bragg angles. In order to obtain more accurate B-factors we have included in the Rietveld refinement of a powder sample of Al the main TDS correction terms starting from our procedure (P. Riello, J. Appl. Cryst. 28, 1995, 115) that is suited to describe, on physical basis, the global background scattering, Y bk(si), at the i-th s step, with s=2sin[[theta]]/[[lambda]]. Now we have expressed Ybk(si) as follows:
Ybk(si) = Kinc Yinc(si) + KTDS YTDS (si) + Yair (si),
where Kinc is the scale factor for the incoherent independent scattering, Yinc(si), and KTDS is the scale factor for the TDS terms, these two factors being related to each other as well as to the Bragg scale factor by means of specific equations obtained according to Warren's theory (B. E.Warren: X-ray Diffraction, Addison-Wesley, 1969, p. 193); Yair(si) is the air scattering contribution; YTDS(si) is the TDS contribution based on a simplified version of the theoretical approach developed by Borie (Acta Cryst., 14, 1961, 566): it contains a sum of peaked terms (centered at the hkl Bragg reflection maxima, shkl) given by: [[Sigma]]hkl j(hkl) (si / shkl) ln (gm / | si - shkl|), where j(hkl) is a multiplicity factor and gm is given by (3/[[pi]])1/3/ao, with ao the edge of the fcc unit cell of Al; these functions have been suitably truncated according to J. P. Urban (Acta Cryst. A31, 1975, 95) in order to avoid the singularity when si = shkl.
The so performed Rietveld refinement has given a BAl factor of 0.86(1)Å2, close to the BAl values reported in single crystal analyses (B. T. M. Willis & A. W. Pryor: "Thermal Vibrations in Crystallography" 1975, p.135) and the following agreement indices: Rwp=0.141, Rexp=0.071, RBragg=0.054. Using the routine polynomial background we have instead obtained the following values: BAl= 0.35(2)Å2, Rwp= 0.156, RBragg= 0.065.