Section 6.3.3.2. Cylinders and spheres

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.