The formula for integrated intensity from extended-face imperfect crystals – inclusion of surface-layer effects

Abstract

By reference to intensity measurements made with an asymmetrically cut extended-face lithium fluoride crystal, attention is drawn to (a) the observation that the average of measurements at asymmetrical angles + a and -a is not always equal to the value at the symmetrical position, a = 0°, as implied by W. H. Bragg [Phil. Mag. (1914), 27, 881-899] and accepted as essentially correct by later experimentalists; and (b) the associated consequence that the conclusions of Hirsch and his colleagues concerning extended-face crystals arrived at ca. 1950 [e.g. Gay & Hirsch, Brit. J. Appl. Phys. (1951). 2, 218-222] are of general rather than specific significance. It is deduced that the standard formula for the absorption factor, Ac, for such crystals [International Tables for X-ray Crystallography (1967). Vol. II], should be modified to incorporate an additional component of the form exp { -μ't [cosec (θ + α) + cosec (θ - a)]}, where μ' is the effective absorption factor for the surface layer of thickness t and α is the angle between the surface and the Bragg plane studied. The formula for the integrated intensity is then given by {{E\omega}\over{I}} = {{Q}\over{2\mu}} (1 -cot θ. tan α). exp {- μ't[cosec (θ + α) + cosec (θ - α)]}, which reduces in the symmetrical case, α = 0°, to {{E\omega}\over{I}} = {{Q}\over{2\mu}} . exp ( - 2μ't cosec θ). The factor μ't can be determined experimentally by use of an appropriately prepared asymmetrically cut crystal or by selection of a reflexion plane at an angle to the surface used for measurement. The onus should be on the experimenter to establish the magnitude of the factor and hence whether its incorporation is mandatory. Its omission in application of this technique for the determination of structure factors may have significant effects on scale factors, particularly where absolute values are sought, and on derived temperature factors and extinction parameters.