Authors: Nick Spadaccini and Douglas du Boulay
Contact: Syd Hall, Crystallography Centre, University of Western Australia, Nedlands 6907, Australia
This program calculates the isotropic Zachariasen extinction coefficient r* by correlating the intensity differences of symmetry equivalent reflections with the diffraction path lengths through a nonspherical crystal. The method is optimal for complete spheres of intensity data, collected from crystals of higher symmetry.
The greater accuracy required for high precision
density studies has led to the development of XTINCT. These
studies require a particularly careful treatment of the
correction factors for absorption and extinction, both of
which depend on the size and shape of the crystal.
Extinction coefficients determined from structure factor
least squares processes (such as
XTINCT determines the isotropic extinction correction factor from the measured intensity variations of symmetrically equivalent reflections. This application relies on a non-spherical crystal in which the path lengths of symmetrically equivalent reflections are different. The precision of the method depends on several factors: the measured precision of the intensity data, the crystal shape (non-spherical but well-defined faces so that accurate path lengths can be calculated), the space group symmetry (the higher the better) and the extent of the data measured (a full sphere is highly desirable).
The correction for secondary (Type I) extinction is based on the theory of Zachariasen (1967) and the application procedure of Larson (1970). The corrected measured structure factor is .
The function adapts the formalism to several special cases of extinction. This is necessary to ensure that the parameter r* is independent of scattering angle.
case 1 - Type II, primary extinction
case 2 - Type I, secondary extinction
Parameter p specifies the fraction of total intensity incident on the crystal specimen, that is polarised perpendicularly to the diffraction plane of that specimen. In the instance of neutron diffraction Q reduces simply to
where the neutron scattering length .
The secondary extinction coefficient r* is determined by minimising the statistical variation of the intensities for symmetry equivalent reflections. The minimum variance is found when the first derivative of the variance quadratic is zero. The derivative is taken with respect to the extinction parameter r*, which in the isotopic case is a scalar variable. In practice the quadratic is approximated by a Taylor expansion about r* truncated at the second order. The zero point of the first-order differential equation is found analytically using an iterative method which terminates when the shift in r* at each iteration becomes less than 0.0001 of σ(r*). The weighting of each set of equivalents is given by so that the stronger reflection intensities dominate the refinement. In accordance with the theory, the stronger reflections more accurately reflect the effect of extinction.
XTINCT works best when applied to a full sphere of measured intensities. This provides as many equivalent reflections as possible. The strong dependence of the refinement on path length variations between equivalents restricts its application to non-spherical crystals. The procedure is applied after absorption corrections and applied to clustered F squared data.Here are the calculation steps needed for the application of XTINCT. This is when extinction is applied to |F| data. See example 4 for a typical input file which parallels these steps.
In the default mode XTINCT calculates, refines and
applies the extinction coefficient r* to the
data. This data is
output to the bdf along with the value of r* (on the
absolute scale supplied from the prior
Options on the
XTINCT line and on the
limits line provide for
various controls on this calculation. The coefficient r*
may be determined within specific regions of the data using
limits control line
(though the resulting r* is then applied to all data). The
value of r* may also be determined separately for each set
of equivalents by entering the
This is the standard run which refines and applies the extinction coefficient dat to the data. 25 reflections will be printed and the scaled r* value will be stored on the output bdf.
XTINCT print eqfr limits f2rl 1000 100000
The extinction r* will be refined on all sets of equivalents for which the average lies between 1000 and 100000. All reflections will be printed, and the Friedel related intensities will be treated as equivalent reflections.
XTINCT appx 153 22 print 40
The unscaled extinction correction r* of 153 and a sigma r* of 22 will be applied to the data. The first 40 reflections on the bdf will be printed.
SORTRF aver 2 end : important - do not use a copybdf without preceding end copybdf b a : <<<<<<<<<<<< do not save output bdf ABSORB analyt scal 7604.19 irel tbar print 20 diff a a c a orient 2 0 0 29.91 27.79 0 2 0 331.86 99.84 faceml -1 1 0 .00318 faceml 1 -1 0 .00318 faceml 1 1 0 .00364 faceml -1 -1 0 .00364 faceml 0 0 1 .00635 faceml 0 0 -1 .00635 faceml -5 -8 -2 .00440 faceml 5 0 -1 .00440 SORTRF aver 2 end : important - do not use a copybdf without preceding end copybdf b a : <<<<<<<<<<<<<<< do not save output bdf SORTRF clus ADDREF nobay lpin reduce itof2 rlp4 bdfin all remove f2rl sgf2 GENEV enot XTINCT irel SORTRF aver 2 print -9999 ADDREF nobay ffac reduce itof bdfin all