Section 6.3.3. Absorption corrections

For special cases, the integral can be solved analytically, and in some of these the expression reduces to closed form. These are listed in Table 6.3.3.1.

Table 6.3.3.1| top | pdf | Transmission coefficients (1) Reflection from a crystal slab with negligible transmission; the crystal planes are inclined at an angle to the extended face, and the normal in the plane of the incident and diffracted beams (1a) = 0 (2) Reflection from a crystal slab of thickness t, with planes parallel to the extended face (3) Transmission through a crystal slab of thickness t; the crystal planes are at to the surface, with the normal in the plane of the incident and reflected beams (3a) (4) Transmission through a sphere of radius R (i.e. for a uniform X-ray beam and ) (5) Reflection from a sphere of radius R (i.e. for a uniform X-ray beam, and )

For diffraction in the equatorial plane of a cylinder of radius R within the X-ray beam, the expression for the transmission coefficient reduces to Values of the absorption correction A* obtained by numerical integration by Dwiggins (1975a) are listed in Table 6.3.3.2.

The reduced expression for a spherical crystal of radius R is Values of A* obtained using numerical integration by Dwiggins (1975b) are listed in Table 6.3.3.3. An estimate of the accuracy of the numerical integration is given by comparison with the results for special values of θ at which equations (6.3.3.4) and (6.3.3.5) may be integrated analytically, which are included in Table 6.3.3.1. The comparison indicates a reliability for the tabulated values of better than 0.1%. Tables at finer intervals for cylinders and spheres for are given by Rouse, Cooper, York & Chakera (1970). A tabulation up to for spheres is given by Weber (1969). Interpolation for μR may be effected by the formula where the K_m are determined, for fixed θ, from the values in Tables 6.3.3.2 and 6.3.3.3.

Subsequent interpolation as a function of θ may be effected by the interpolation formula Interpolation is accurate to 0.1% with N = M = 3.

For cylinders and spheres, may be obtained by means of the expression using the values listed in Tables 6.3.3.2 and 6.3.3.3.

Values of (1/A*)[dA*/d(μR)] obtained by numerical integration by Flack & Vincent (1978) for spheres with are listed in Table 6.3.3.4. An equivalent table of μ(R/A*)/[dA*/d(μR)] for is given by Rigoult & Guidi-Morosini (1980).

Alternatively, one can differentiate the interpolation formula (6.3.3.6), yielding In this case, however, the maximum index M = 7 is required to obtain convergence for . Numerical values of the coefficients K_m for cylinders and spheres evaluated by Tibballs (1982) are listed in Table 6.3.3.5.

Table 6.3.3.5| top | pdf | Coefficients for interpolation of A* and 0° 15° 30° 45° 60° 75° 90° Units K1 (sphere) 3/2 3/2 3/2 3/2 3/2 3/2 3/2   K2 −7.5234 −9.4320 −15.109 −24.3812 −35.219 −44.042 −47.745 10−2 K3 −7.0935 −10.737 −18.027 −11.088 14.265 40.021 61.084 10−3 K4 −2.3096 −2.1332 −1.4693 7.4205 24.832 44.308 37.394 10−3 K5 1.8323 1.1711 4.6784 3.0970 −10.284 −27.987 −25.879 10−3 K6 −5.1259 −1.2652 −14.491 −16.740 21.910 77.007 71.458 10−4 K7 6.0265 0.7932 16.489 21.774 −22.391 −85.570 −78.812 10−5 K1 (cylinder) 16/3π 16/3π 16/3π 16/3π 16/3π 16/3π 16/3π   K2 −5.7832 −8.1900 −15.651 −27.048 −40.317 −51.497 −55.837 10−2 K3 −14.737 −19.551 −22.883 −27.345 −8.807 26.637 41.420 10−3 K4 5.2399 1.2934 −12.301 6.844 40.689 61.371 68.963 10−3 K5 −4.0958 −2.8349 9.6249 7.503 −11.295 −29.397 −36.556 10−3 K6 13.178 12.731 −19.881 −30.211 9.4468 60.356 80.965 10−4 K7 −14.500 −14.846 14.414 34.222 3.1492 −49.206 −70.573 10−5 (C−1)0,j 3 0 0 0 0 0 0 All values multiplied by 3 to eliminate fractions (C−1)1,j −73 48 + 24 −24 12 −8 48 − 24 −3 (C−1)2,j 518 −496 − 200 488 −268 184 −496 + 200 70 (C−1)3,j −1600 1920 + 560 −2192 1536 −1136 1920 − 560 −448 (C−1)4,j 2432 −3520 − 640 4032 −3328 2752 −3520 + 640 1152 (C−1)5,j −1792 3072 + 256 −3328 3072 −2816 3072 − 256 −1280 (C−1)6,j 512 −1024 1024 −1024 1024 −1024 512

Interpolation between the tabulated θ values is obtained from the θ interpolation formula, noting that where The elements and the for at 15° intervals in the range are listed in Table 6.3.3.5. Differentiating (6.3.3.7) yields where Equation (6.3.3.12) for path lengths is the analogue of equation (6.3.3.7) for the transmission factors. It provides the basis for an interpolation formula.

In the case of a cylindrical crystal much larger than the X-ray beam, the absorption correction has been determined by Coyle (1972), in an extension of earlier work by Coyle & Schroeder (1971). The absorption correction for the case of the cylinder axis coincident with the axis of a Eulerian cradle, shown in Fig. 6.3.3.1 , reduces to the line integral where z and T(z) are the path lengths for the incident and diffracted beams, respectively. τ is the radius, along the line of the incident beam, of the ellipse described by the cross section of the crystal in the plane of diffraction, shown in Fig. 6.3.3.2 . The equation for the ellipse is The outgoing elliptical radius v satisfies where

Figure 6.3.3.1| top | pdf | Geometry of the Eulerian cradle with the axis of a cylindrical specimen coincident with the φ axis.

Figure 6.3.3.2| top | pdf | Cross section of the plane of diffraction for a cylindrical specimen coincident with the φ axis.

In the case where the cylinder axis is inclined at an angle Γ to the axis, these equations become where The roots of the quadratic equation (6.3.3.16) for are real and positive for reflection from within the crystal. The convergent path length T is given by the positive root of the triangle formula

It should be noted that the volume of the specimen irradiated changes with the angular settings of the diffractometer. Normalization to constant volume requires that the absorption correction be multiplied by the volume-correction factor .

The method readily extends to the case of a cylindrical window or sheath, such as used for mounting an unstable crystal of conventional size. The correction in this case is where the subscripts 1 and 2 apply to the inner and outer radii, respectively.

The integral in equation (6.3.3.14) may be evaluated by Gaussian quadrature, i.e. by approximation as a weighted sum of the values of the function at the N zeros of the Legendre polynomial of degree N in the interval [−1, +1]. The weights for the points are tabulated by Abramowitz & Stegun (1964). Further details are given in Subsection 6.3.3.4. The emergent path lengths and for the case of the sheath are calculated as functions of the Gaussian variable using the linear transformation This transformation converts the Gaussian variable X into the beam coordinate z for each i of the N summation points.

For a crystal with regular faces, (6.3.3.1) may be integrated exactly, giving the correction in analytical form. In its simplest form, the analytical method applies to specimens with no re-entrant angles. It is efficient for crystals with a small number of faces. Its accuracy does not depend on the size of the absorption coefficient. The principles can be illustrated by reference to the two-dimensional case of a triangular crystal shown in Fig.6.3.3.3.

Figure 6.3.3.3| top | pdf | The crystal ABC divided into polygons by the dashed lines AE and CF parallel to the incident (i) and diffracted (d) beams, respectively. A locus of constant absorption is shown dotted.

The crystal is divided into polygons ADC, AFD, CDE, and BEDF as shown. The radiation incident on each polygon enters through one face of the crystal, and is either absorbed or emerges through another. Within each polygon, the loci of constant absorption are the straight lines dotted in Fig. 6.3.3.3. It is convenient to subdivide BEDF into the triangles BEF and EDF. By the derivation of an expression for the contribution of a triangular crystal to the scattering, including allowance for absorption, and with the sum taken over the component triangles ADC, AFD, CDE, BEF, and EDF, the correction for absorption can be calculated.

A three-dimensional crystal is divided into polyhedra, for each of which the radiation enters through one crystal face and leaves through another. Corners for the polyhedra are of five types, namely,

For each vertex x, y, z, the sum of the path lengths to each of the crystal faces is calculated, and multiplied by the absorption coefficient μ to give the optical path length using the equation where u, v, w are the direction cosines for the beam direction, and is the equation for the crystal face. The minimum for all j is the path length to the surface.

The analytical expression for the scattering power for each polyhedron, including the effect of absorption, can be expressed in a convenient form by subdividing the polyhedra into tetrahedra. The auxiliary points define the corners of the tetrahedra.

The total diffracted intensity is proportional to the sum of contributions, one from each tetrahedron, of the form where is the volume of the tetrahedron. For a crystal with Cartesian coordinate vertices 1, 2, 3, and 4,

The are optical path lengths (i.e. path lengths rescaled by the absorption coefficient) ordered so that and The transmission factor for the crystal is the sum of the scattering powers for all the tetrahedra divided by the volume . The equality of the total volume to the sum of the values for the component tetrahedra provides a useful check on the accuracy of the calculations, since the total volume is independent of the beam directions, and must be the same for all reflections.

When any of a, b, and c are small, asymptotic forms are required for the expressions in (6.3.3.20). For , and where the nth derivative of h(x) is

An alternative method of calculating the scattering power of each Howells polyhedron is based on a subdivision into slices. Within each polyhedron, the loci of constant absorption are planes, equivalent to the dotted lines for the two-dimensional example in Fig. 6.3.3.3. The loci may be determined from the path lengths of rays diffracted at each vertex of the polyhedron. The sum of the path lengths in the incident and diffracted directions is found for each vertex, and the loci determined by interpolation. The slices into which each polyhedron is divided are bounded at the upper and lower faces by planes parallel to the loci of constant absorption, such that at least one vertex of the polyhedron lies on those planes.

The volume of the slice is determined from the coordinates of the vertices on each of the opposite faces. Dummy vertices are inserted if necessary to make the number of vertices on the top and bottom faces identical. For simplicity, an axis (z) is chosen perpendicular to the upper face. This locus of constant absorption with vertices has an area The corresponding vertices on the lower face may be written , , , with q = 1. The lower face has an area where and so that the volume of the slice is The diffracting power of an element of the slice, allowing for absorption, is D(q)exp(−μT) dz, where T is the total path length of the rays diffracted from this plane. Because of the definition of the Howells polyhedron, the path length

Thus, the total diffracting power of the slice

The transmission factor for the Howells polyhedron is obtained by summing over the slices, and that for the whole crystal is obtained by summing over the polyhedra, i.e. where the crystal volume is .

dA/dμ, required in calculating for the extinction correction, can be obtained by differentiating for each slice with respect to μ, summing the derivatives for each slice, and dividing by . To reduce rounding errors in calculation, it may be desirable to rescale the crystal dimensions so that the path lengths are of the order of unity, multiplying the absorption coefficient by the inverse of the scale factor. Further details are given by Alcock, Pawley, Rourke & Levine (1972).

The number of component tetrahedra or slices, which determines the time and precision required for calculation, is a rapidly increasing function of the number of crystal faces. The method may be computationally prohibitive for crystals with complex shapes.

The integral in the transmission factor in equation (6.3.3.1) may be approximated by a sum over grid points spaced at intervals through the crystal volume. It is usually convenient to orient the grid parallel to the crystallographic axes. The grid is non-isometric, the points being chosen weighted by Gaussian constants to minimize the difference between the weighted sum at those points and the exact value of the integral.

Thus, an integral such as may be approximated (Stroud & Secrest, 1966) by where is the ith zero of the Legendre polynomial and When applying this to the calculation of a transmission coefficient (Coppens, 1970), we commence with the a-axis grid points selected such that where the are the Gaussian constants.

For each , a line is drawn parallel to b and points are then selected such that The procedure is repeated for the c direction, yielding

To calculate the absorption corrections, the incident and diffracted wavevectors are determined. For each grid point, the sum of the path lengths for the incident and diffracted beams is evaluated. The sum that approximates the transmission coefficient is then Gaussian constants are tabulated by Abramowitz & Stegun (1964).

Alternative schemes based on Monte Carlo and three-dimensional parabolic integration are described by de Graaff (1973, 1977).

Some crystals do not have regular faces, or cannot be measured because these are obscured by the crystal mounting. If corrections based on measurements of the crystal shape are not feasible, absorption measurements may be estimated, either from the intensities of the same reflection at different azimuthal angles ψ (see Subsection 6.3.3.6), or from measurements of equivalent reflections, by empirical methods.

There are variants of the method related to differences in experimental technique. The principles may be illustrated by reference to the procedure for a four-circle diffractometer (Flack, 1977).

Intensities are measurements for a reflection S at the angular positions , , , . Corrected intensities are to be derived from the measurements by means of a correction factor such that It is assumed that the correction can be written in the form of a rapidly converging Fourier series The form of the geometrical terms may be simplified by taking advantage of the symmetry of the four-circle diffractometer. If it is assumed that diffraction is invariant to reversal of the incident and diffracted beams, the settings ; ; ; ; ; ; ; are equivalent. In shorthand notation, the series (6.3.3.41) reduces to The range of indices for some terms may be restricted by noting other symmetries in the diffraction experiment. Thus, equation (6.3.3.40) will define the absorption correction for measurements of the incident-beam intensity, with . Since with this geometry the correction will be invariant to rotation about the χ axis, the coefficients for the function involving must vanish if the χ index, k, is non-zero. By similar reasoning with the axis along the incident beam, one may deduce that coefficients for will vanish unless l = 0.

Because for a given reflection all measurements are made at the same Bragg angle, the dependence of the correction cannot be determined by empirical methods. This factor in A is obtained from the absorption correction for a spherical crystal of equivalent radius.

Since an empirical absorption correction is defined only to within a scale factor, the scale must be specified by applying a constraint such that where is the number of independent reflections. Equation (6.3.3.42) may be expressed in the shorthand notation where is the coefficient in a term such as or and is the corresponding geometrical function. Labelling the constant geometrical term with a value of unity as and rearranging leads to which defines .

Equation (6.3.3.40) is now expressed as in which the coefficients are to be chosen so that the values of for each S are as near equal as possible. Since the values within each set will not be exactly equal, we rewrite (6.3.3.46) as in which the mean intensity and the are chosen to minimize , where and is the weight for that reflection.

If the equation to be solved is written in the shorthand form in which D corresponds to , the and correspond to C, with and corresponding to F, the solution to (6.3.3.50) can be determined from the normal equations where is the transpose of F. This procedure suffers from the disadvantages of requiring a matrix inversion whenever the set of trial functions (i.e. those multiplied by the coefficients ) is modified. The tedious inversion of the normal equations, described by (6.3.3.51), may be replaced by a simple inversion via the Gram–Schmidt orthogonalizing process, i.e. by calculating a matrix W with mutually orthogonal columns such that The minimizing of (D − FC)² is replaced by minimizing (D − WA)². Differentiating with respect to yields If equation (6.3.3.52) is written as where the upper triangular matrix B is the vector determining the coefficients is in which the inversion of B is straightforward.

In difficult cases, with data affected by errors in addition to absorption, the method described may give physically unreasonable absorption corrections for some reflections. In such cases, it may help to impose the approximate constraints If , this reduces to the M constraint equations where is the square root of the weight for the weighted mean of the equivalent reflections , defined as and the multiplier controls the strength with which the additional constraints are enforced. With the additional constraint equations, the sum of squares to be minimized, corresponding to (6.3.3.48), becomes

A closely related procedure expressing the absorption corrections as Fourier series in polar angles for the incident and diffracted beams is described by Katayama, Sakabe & Sakabe (1972). A similar method minimizing the difference between observed and calculated structure factors is described by Walker & Stuart (1983). Other experimental techniques for measuring data for empirical absorption corrections that could be analysed by the Fourier-series method are described by Kopfmann & Huber (1968), North, Phillips & Mathews (1968), Flack (1974), Stuart & Walker (1979), Lee & Ruble (1977a,b), Schwager, Bartels & Huber (1973), and Santoro & Wlodawer (1980).

In general, A depends both on the shape of the crystal and on its orientation with respect to the incident and diffracted beams. To measure the shape of the crystal, a measuring microscope is mounted in the xy plane, and the crystal rotated about the z axis at right angles to that plane. A rotation about the z axis changes the orientation of the crystal x and y coordinates with respect to those (X and Y) for the measuring device. The x axis is directed from crystal to microscope when the angle of rotation about the z axis () is zero. During rotation, each face will at some stage be oriented with its normal perpendicular to the line of view, i.e. in the XY plane for instrument coordinates. If the angle of rotation at that orientation is denoted , the appearance of a typical face ABCD will be as indicated in Fig. 6.3.3.4.

Figure 6.3.3.4| top | pdf | Crystal oriented with the normal N(S) to the face ABCD in the plane of view.

The equation for the plane is or, equivalently,

For a crystal oriented on an Eulerian cradle, it is necessary to specify the orientation of the crystal, i.e. the angles in which the measurements of the diffraction intensities are made. In a reflecting position, the reciprocal-lattice vector S, which is normal to the Bragg planes, bisects the angle between the incident and diffracted beams, as shown in Fig. 6.3.3.5.

Figure 6.3.3.5| top | pdf | Geometry of the Eulerian cradle in the bisecting position.

If the crystal is rotated about the reciprocal-lattice vector S, varying the angle ψ, the crystal remains in a reflecting position. That is, there is a degree of freedom in the scattering experiment that enables the same reflection to be observed at different sets of Ω, χ, values. The path length varies with ψ, except for spherical crystals. In order to calculate an absorption correction, the value of ψ and its origin must be specified. For a crystal mounted on an Eulerian cradle, the bisecting position, with Ω = θ, is usually chosen as the origin for ψ.