4.31. LSABS: Absorption corrections

4.31.1. Overview

The calculation may be carried out either with an analytical method similar to the one of de Meulenaer & Tompa (1965) or by Gaussian grid integration (Busing & Levy, 1957). A major innovation in this program is the calculation of the derivatives of transmission with respect to crystal shape T/d_m, where d_m defines the distance of the m^th crystal face from an origin inside the crystal. These derivatives may be used during least-squares refinement to obtain improved estimates of the crystal shape and consequently an improved absorption correction. The program uses crystal-based azimuthal angles (Schwarzenbach & Flack, 1989, 1992) to define the diffraction geometry and is thus independent of the particular instrument (diffractometer, camera, etc.) used to collect the intensity data.

4.31.2. Methods of Absorption Correction

LSABS offers two methods of integration, the analytical method and Gaussian integration. The analytical method is more efficient than the Gaussian method in calculating the derivatives. For this reason, execution times for the two methods are comparable in the case of a coarse Gaussian grid, and in favour of the analytical method in the case of a fine Gaussian grid.

4.31.3. Analytical method

The analytical method proposed by de Meulenaer & Tompa (1965) describes the crystal shape by the set of its vertices, whereas LSABS defines it by the set of its edges. This technique facilitates an unambiguous identification of the vertices and minimizes rounding errors.

4.31.4. Gaussian grid integration

The implementation of the numerical integration of Gauss-Legendre offers a choice of grids with 6, 8, 10 ,12, 14, 16, 24, or 32 intervals for each dimension. For needle- and plate- shaped crystals the orientation of the grid may be selected to lie along one of the principal directions. For a needle, the z-axis of the grid is chosen along the needle axis identified by its zone indices [u v w]. For a plate, the z-axis of the grid is chosen along the plate normal identified by the face indices (h k l). The calculation of the derivatives T/d_m requires evaluation of surface integrals. Consequently a surface grid must be specified in addition to the 3-dimensional grid for the volume integral.

4.31.5. The Linear Absorption Coefficient µ

The linear absorption coefficient µ is calculated by LSABS using the chemical formula stored on the bdf and the X-ray cross sections stored in the program. For the moment, the available X-ray cross sections are those for CuK, MoK or AgK radiation taken from Table 2.1B of International Tables for X-ray Crystallography Vol IV (Ibers & Hamilton, 1974) for the elements from hydrogen to plutonium. For other wavelengths and elements the value of µ may be entered on the LSABS line.

4.31.6. Input and Output Data Specification

The M plane faces of a polyhedral crystal are defined by their Miller indices (h_m k_m l_m) relative to the lattice base a, b, c, and by the perpendicular distances d_m from a given point inside the crystal to each face (m = 1, ..M). For the natural faces of a crystal, (h_m k_m l_m) are integers. d_m should have been carefully measured and an estimated standard deviation of d_m should be available. The I Bragg reflections are identified by their Miller indices h_i k_i l_i and an instrument-independent crystal-based azimuthal angle _i as defined by Schwarzenbach & Flack (1989, 1992). The above data must be available on the bdf for LSABS to function correctly.

By default the nett intensities of the reflections (Schwarzenbach & Flack, 1991) are not modified by LSABS : the calculated quantities A=1/T, <T>(i.e. Tbar) and T/d_m are simply stored on the bdf.

The recommended method of placing raw diffractometer information including crystal face indices and distance data on the bdf is first to transform a diffractometer-specific file into a CIF using DIFRAC (Flack, Blanc & Schwarzenbach, 1992). This file may then be read with CIFENT (same input lines as CIFIO) to create a bdf. This process compactly loads all of the necessary raw experimental data onto a bdf in a diffractometer-independent way.

4.31.7. File Assignments

• Reads data from the input archive bdf.

• Writes data to the output archive bdf.

4.31.8. Examples

title    Create an absorption-corrected |F| bdf
CIFENT   cifin           
STARTX   upd
sgname   -C 2YC
REFCAL   neti
LSABS    gauss
shape    needle  1 2 0
grid     10 10 24
sgrid    16
REFCAL   frel
SORTRF   aver 1  frel

The above example shows the use of STARTX, REFCAL, LSABS, and SORTRF to create a bdf with |F_o| data from diffractometer data produced in CIF format by the DIFRAC program. STARTX calculates the direct and reciprocal cell metrics, loads the scattering factor tables, atomic radii, and checks the symmetry information - the sgname line is necessary in STARTX as this information is not stored with the raw diffractometer data. REFCAL calculates the reflection information. LSABS performs a Gaussian grid integration and applies the absorption correction to the net intensities in the bdf. As the crystal is needle-shaped, the Gaussian grid is oriented with the needle direction [1 2 0] parallel to the grid z axis. A finer grid is defined along the needle axis than in the other directions. A second pass through REFCAL is needed to convert the net intensities to |F_o| corrected for absorption. Finally the absorption-corrected data are sorted and averaged leaving a bdf ready for structure solution and conventional |F_o| refinement.

title    Create a net intensity bdf
CIFENT   cifin
STARTX   upd
sgname   -C 2YC
REFCAL   neti inst 1 incl
LSABS
SORTRF

A bdf with net intensity data is created. LSABS performs an analytical absorption correction and stores the resulting values in the bdf. The bdf is suitable for the refinement of the crystal shape and decay parameters.

4.31.9. References

• Blanc, E., Schwarzenbach, D. & Flack, H.D. (1991). J. Appl. Cryst. 24, 1035-1041

• Busing, W. R. & Levy, H. A. (1957). Acta Cryst. 10, 180-182.

• de Meulenaner, J. & Tompa, H. 1965. Acta Cryst. 19, 1014-1018.

• Flack, H.D., Blanc, E. & Schwarzenbach, D. (1992) J. Appl. Cryst. 25, 455-459.

• Ibers, J.A. & Hamilton, W.C. (1974) Eds. International Tables for X-rays Crystallography Vol. IV, Birmingham, England: Kynock Press

• Schwarzenbach, D. & Flack, H. D. (1989). J. Appl. Cryst. 22, 601-605.

• Schwarzenbach, D. & Flack, H. D. (1991). Acta Cryst. A47, 134-137.

• Schwarzenbach, D. & Flack, H. D. (1992). J. Appl. Cryst. 25, 69.