The absolute measured value off(220) for Cu –the importance of extrapolation to zero extinction

Abstract

A method of deriving an extinction-free estimate of an X-ray structure-factor value is outlined. The method is based on the availability of experimental estimates of the level of extinction from the mosaic distribution of the crystal. A plot of the diffracted intensity or its corresponding structure factor (uncorrected or nominally corrected for extinction) against the percentage extinction effect yields an extinction-free value in the limit on extrapolation to zero level. Such a procedure can also reveal any inadequacy in the correction procedures by lack of internal consistency. An illustration of the potential of this procedure is given using the set of 11 experimental values off for the 220 reflexion of Cu derived by Schneider [Acta Cryst. (1977), A33, 235-243] from y-ray diffractometry. Use of the experimental estimates of the level of extinction together with the nominally corrected f data indicates the presence of systematic residual error. Linear extrapolation of these f values to zero extinction yields an estimate off(220) in the region of 16.77. A value in this region is more in accord with a Hartree-Fock value than with the band-structure value, which was favoured by Schneider's original estimate of 16.46.

The Experimental Value of f(220) for Copper

“…2 both here and in Mackenzie and Mathieson (1979)], we find that the two points are perfectly consistent with the earlier ones. This confirms the linearity of the extrapolations and strengthens our earlier conclusions (a) that the value of/(220) for eu lies in the region of 16·76 rather than 16·45 and (b) that the correction for secondary extinction made according to Darwin's transfer equations appears not to constitute complete correction.…”

“…Our earlier paper (Mackenzie and Mathieson 1979) used the published curves of Schneider (1976Schneider ( , 1977 and, by applying the same data-handling treatment as that of Schneider, arrived at essentially the same numerical results [compare Table 2 of Mackenzie and Mathieson with Fig. 5 of Schneider (1977)].…”

“…(1). The circles and the lines are from Mackenzie and Mathieson (1979); the squares refer to the results of the present paper.…”

“…So, in respect of various dynamical procedures, using either X-ray or electron diffraction, there is close accord with a value for 1(220) in the region of 16·70-16·75. Mackenzie and Mathieson (1979) used the reflectivity curves, obtained by Schneider (1976Schneider ( , 1977 for the essentially kinematic reflection of y rays, to illustrate the importance of extrapolation to zero interaction in establishing extinctionfree estimates of structure factors. Using the data for all 11 of Schneider's volume samples, they deduced a value for 1(220) of 16·76 by extrapolation of plots of estimated 1 values against experimental estimates of secondary extinction Pi to zero extinction.…”

Concerning Single Crystal Reflectivity Curves

“…When one obtains a well-defined ZWD reflectivity curve for a single reflection, one can use power-transfer relationships as a first approximation to apply point-by-point corrections (cf. Schneider 1977;Mackenzie and Mathieson 1979). With this approach, one has no absolute check as to how far the corrected curve has to be raised to attain the zero-extinction limit.…”