International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 6.3, pp. 600-608

Section 6.3.3. Absorption corrections

E. N. Maslena

aCrystallography Centre, The University of Western Australia, Nedlands, Western Australia 6009, Australia

6.3.3. Absorption corrections

| top | pdf |

The reduction in the intensity of an X-ray reflection from a uniform beam due to absorption is given by the transmission coefficient [A={1\over V}\int\exp(-\mu T)\,{\rm d}V, \eqno (6.3.3.1)]where the integration is over the volume of the crystal. The absorption correction [A^*=1/A. \eqno (6.3.3.2)]T, the path length of the X-ray beam in the crystal, is the sum of the path lengths for the incident and diffracted beams. A technique for measuring crystals for absorption measurements is described in Subsection 6.3.3.6[link].

Any least-squares analysis involving variation of the linear absorption coefficient, or equivalently an isotropic variation in crystal size, requires the weighted mean path length [\bar T=-A^{-1}{\partial A\over\partial\mu}= {1\over A^*}\;{\partial A^*\over \delta\mu}. \eqno (6.3.3.3)]This path length is also required in some analyses of extinction (Zachariasen, 1968[link]; Becker & Coppens, 1974[link]).

6.3.3.1. Special cases

| top | pdf |

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[link].

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 [\varphi] to the extended face, and the normal in the plane of the incident and diffracted beams
[A={\sin(\theta-\varphi)\over \mu\{\sin(\theta-\varphi)+\sin(\theta+\varphi)\}}](1a) [\varphi] = 0[A=1/2\mu]
(2) Reflection from a crystal slab of thickness t, with planes parallel to the extended face
[A=\{1-\exp\,(-2\mu t{\;\rm cosec}\;\theta)\}/2\mu]
(3) Transmission through a crystal slab of thickness t; the crystal planes are at [\pi/2-\varphi] to the surface, with the normal in the plane of the incident and reflected beams
[A={{\exp\{-\mu t \sec(\theta+\varphi)\}-\exp\{-\mu t \sec (\theta-\varphi) \}}\over{\displaystyle \mu\biggl [ 1-{{\sec(\theta+\varphi)}\over{ \sec(\theta-\varphi )}} \biggr] }}](3a) [\varphi=0][A=t\sec\theta\exp(-\mu t\sec\theta)]
(4) Transmission through a sphere of radius R (i.e. for a uniform X-ray beam and [\theta=0^\circ])
[A={3\over 2(\mu R)^3}[1/2-e^{-2\mu R}\{1/2+\mu R+ (\mu R)^2\}]]
(5) Reflection from a sphere of radius R (i.e. for a uniform X-ray beam, and [\theta=90^\circ])
[A={3\over 4\mu R}\left\{1/2 - {1\over16(\mu R)^2}\;[1-(1+4\mu R)\,e^{-4\mu R}]\right\}]

6.3.3.2. Cylinders and spheres

| top | pdf |

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 [\eqalignno{ A={1\over A^*} &= {1\over \pi R^2}\; \int\limits^R_0\!\int\limits^{2\pi}_0\;\exp \Big(-\mu\{[R^2-r^2\sin^2(\theta+\varphi)]^{1/2} \cr &\quad +[R^2- r^2\sin^2(\theta-\varphi)]^{1/2}\}\Big) \cr &\quad\times\cosh\,(2\mu r\sin\theta\sin\varphi)r\,{\rm d} r\,{\rm d}\varphi. & (6.3.3.4)}]Values of the absorption correction A* obtained by numerical integration by Dwiggins (1975a[link]) are listed in Table 6.3.3.2[link].

Table 6.3.3.2| top | pdf |
Values of A* for cylinders

μRθ = 0°θ = 5°θ = 10°θ = 15°θ = 20°θ = 25°θ = 30°θ = 35°θ = 40°θ = 45°θ = 50°θ = 55°θ = 60°θ = 65°θ = 70°θ = 75°θ = 80°θ = 85°θ = 90°
0.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.1 1.1843 1.1843 1.1842 1.1840 1.1838 1.1835 1.1832 1.1828 1.1823 1.1818 1.1813 1.1808 1.1802 1.1798 1.1793 1.1790 1.1787 1.1785 1.1785
0.2 1.4009 1.4007 1.4002 1.3995 1.3984 1.3970 1.3953 1.3934 1.3912 1.3889 1.3865 1.3841 1.3818 1.3796 1.3777 1.3761 1.3749 1.3741 1.3739
0.3 1.6548 1.6544 1.6531 1.6510 1.6481 1.6443 1.6398 1.6347 1.6290 1.6230 1.6169 1.6108 1.6049 1.5994 1.5946 1.5906 1.5876 1.5857 1.5851
0.4 1.9522 1.9513 1.9485 1.9439 1.9376 1.9296 1.9201 1.9094 1.8979 1.8857 1.8733 1.8611 1.8495 1.8388 1.8293 1.8215 1.8157 1.8121 1.8108
0.5 2.2996 2.2979 2.2926 2.2840 2.2721 2.2572 2.2398 2.2204 2.1996 2.1781 2.1564 2.1352 2.1152 2.0969 2.0809 2.0677 2.0579 2.0518 2.0497
0.6 2.7047 2.7017 2.6926 2.6775 2.6570 2.6317 2.6023 2.5701 2.5359 2.5010 2.4662 2.4327 2.4012 2.3728 2.3480 2.3277 2.3126 2.3033 2.3001
0.7 3.1762 3.1712 3.1561 3.1315 3.0982 3.0575 3.0111 2.9607 2.9081 2.8549 2.8028 2.7530 2.7068 2.6653 2.6295 2.6003 2.5786 2.5651 2.5606
0.8 3.7236 3.7157 3.6919 3.6532 3.6015 3.5392 3.4691 3.3941 3.3169 3.2400 3.1656 3.0953 3.0307 2.9732 2.9239 2.8839 2.8542 2.8359 2.8297
0.9 4.3578 4.3456 4.3093 4.2507 4.1733 4.0812 3.9792 3.8718 3.7629 3.6560 3.5538 3.4584 3.3717 3.2951 3.2299 3.1772 3.1383 3.1142 3.1061
1.0 5.0907 5.0724 5.0185 4.9323 4.8196 4.6877 4.5439 4.3948 4.2461 4.1022 3.9664 3.8413 3.7286 3.6298 3.5462 3.4790 3.4295 3.3990 3.3886
1.1 5.9356 5.9089 5.8305 5.7065 5.5466 5.3624 5.1649 4.9636 4.7660 4.5776 4.4022 4.2424 4.0998 3.9759 3.8717 3.7882 3.7269 3.6891 3.6763
1.2 6.907 6.869 6.757 6.582 6.360 6.109 5.8436 5.5782 5.3219 5.0811 4.8598 4.6604 4.4842 4.3322 4.2051 4.1038 4.0295 3.9838 3.9682
1.3 8.021 7.967 7.810 7.568 7.266 6.929 6.581 6.238 5.9125 5.6110 5.3376 5.0938 4.8805 4.6976 4.5456 4.4248 4.3365 4.2821 4.2636
1.4 9.294 9.219 9.003 8.674 8.268 7.826 7.376 6.942 6.536 6.166 5.8341 5.5413 5.2873 5.0711 4.8922 4.7506 4.6471 4.5835 4.5619
1.5 10.746 10.643 10.349 9.907 9.372 8.800 8.230 7.689 7.192 6.744 6.348 6.002 5.7036 5.4516 5.2441 5.0804 4.9609 4.8875 4.8625
1.6 12.397 12.257 11.862 11.276 10.581 9.852 9.141 8.477 7.877 7.344 6.877 6.473 6.128 5.8385 5.6007 5.4136 5.2773 5.1935 5.1650
1.7 14.267 14.080 13.555 12.788 11.897 10.982 10.106 9.304 8.589 7.963 7.420 6.955 6.561 6.231 5.961 5.7499 5.5960 5.5014 5.4691
1.8 16.379 16.131 15.441 14.450 13.323 12.189 11.125 10.168 9.327 8.600 7.976 7.446 7.000 6.628 6.326 6.089 5.9166 5.8107 5.7746
1.9 18.76 18.43 17.53 16.267 14.858 13.470 12.194 11.066 10.089 9.253 8.544 7.946 7.444 7.030 6.693 6.430 6.239 6.121 6.081
2.0 21.43 21.00 19.84 18.24 16.50 14.824 13.311 11.995 10.871 9.921 9.122 8.452 7.895 7.435 7.063 6.773 6.562 6.433 6.389
2.1 24.41 23.87 22.39 20.38 18.25 16.247 14.472 12.953 11.673 10.602 9.709 8.965 8.349 7.843 7.436 7.118 6.887 6.745 6.697
2.2 27.74 27.04 25.17 22.69 20.11 17.74 15.675 13.938 12.493 11.295 10.304 9.484 8.808 8.255 7.810 7.464 7.213 7.059 7.006
2.3 31.44 30.55 28.20 25.16 22.07 19.29 16.92 14.947 13.328 11.999 10.906 10.008 9.271 8.669 8.187 7.812 7.540 7.372 7.315
2.4 35.54 34.41 31.49 27.79 24.13 20.90 18.19 15.978 14.177 12.711 11.515 10.537 9.736 9.086 8.565 8.161 7.868 7.687 7.625
2.5 40.06 38.65 35.05 30.59 26.28 22.56 19.50 17.03 15.040 13.433 12.130 11.069 10.205 9.505 8.945 8.511 8.196 8.002 7.935

The reduced expression for a spherical crystal of radius R is [\eqalignno{ A&={3\over 4\pi R^3}\;\int\limits^R_0\!\int\limits^1_{-1}\!\int\limits^{2\pi}_0\;\exp\Big(-\mu\{[R^2-r^2\cos^2\alpha \cr &\quad -r^2\sin^2\alpha\sin^2(\theta+\varphi)]^{1/2} \cr &\quad +[R^2-r^2\cos^2\alpha-r^2\sin^2\alpha\sin^2(\theta-\varphi)]^{1/2} \cr &\quad -2r\sin\theta\sin\alpha\sin \varphi\}\Big)r^2\,{\rm d} r\,{\rm d}(\cos\alpha)\,{\rm d}\varphi. & (6.3.3.5)}]Values of A* obtained using numerical integration by Dwiggins (1975b[link]) are listed in Table 6.3.3.3[link]. 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)[link] and (6.3.3.5)[link] may be integrated analytically, which are included in Table 6.3.3.1[link]. The comparison indicates a reliability for the tabulated values of better than 0.1%. Tables at finer intervals for cylinders and spheres for [\mu R\lt1.0] are given by Rouse, Cooper, York & Chakera (1970[link]). A tabulation up to [\mu R\lt5.0] for spheres is given by Weber (1969[link]). Interpolation for μR may be effected by the formula [A^*(\mu R) = \exp \left\{\textstyle\sum\limits^M_{m=1}\; K_m(\mu R)^m\right\}, \eqno (6.3.3.6)]where the Km are determined, for fixed θ, from the values in Tables 6.3.3.2[link] and 6.3.3.3[link].

Table 6.3.3.3| top | pdf |
Values of A* for spheres

μRθ = 0°θ = 5°θ = 10°θ = 15°θ = 20°θ = 25°θ = 30°θ = 35°θ = 40°θ = 45°θ = 50°θ = 55°θ = 60°θ = 65°θ = 70°θ = 75°θ = 80°θ = 85°θ = 90°
0.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.1 1.1609 1.1609 1.1609 1.1607 1.1606 1.1603 1.1600 1.1597 1.1593 1.1589 1.1586 1.1582 1.1579 1.1575 1.1572 1.1570 1.1568 1.1567 1.1567
0.2 1.3457 1.3456 1.3452 1.3447 1.3439 1.3428 1.3415 1.3400 1.3383 1.3366 1.3348 1.3331 1.3313 1.3297 1.3282 1.3271 1.3262 1.3256 1.3254
0.3 1.5574 1.5571 1.5561 1.5546 1.5525 1.5497 1.5463 1.5426 1.5383 1.5339 1.5293 1.5248 1.5204 1.5162 1.5126 1.5096 1.5074 1.5059 1.5055
0.4 1.7994 1.7988 1.7968 1.7935 1.7891 1.7833 1.7765 1.7689 1.7604 1.7515 1.7425 1.7335 1.7249 1.7169 1.7099 1.7041 1.6997 1.6970 1.6961
0.5 2.0755 2.0743 2.0706 2.0647 2.0565 2.0462 2.0340 2.0204 2.0056 1.9901 1.9745 1.9592 1.9445 1.9311 1.9194 1.9097 1.9024 1.8979 1.8964
0.6 2.3897 2.3877 2.3816 2.3715 2.3578 2.3406 2.3206 2.2984 2.2746 2.2500 2.2255 2.2015 2.1789 2.1583 2.1403 2.1257 2.1145 2.1076 2.1063
0.7 2.7467 2.7434 2.7336 2.7177 2.6959 2.6691 2.6382 2.6042 2.5683 2.5316 2.4952 2.4602 2.4274 2.3977 2.3719 2.3508 2.3351 2.3253 2.3220
0.8 3.1511 3.1461 3.1312 3.1069 3.0740 3.0339 2.9882 2.9386 2.8869 2.8347 2.7835 2.7346 2.6892 2.6484 2.6133 2.5845 2.5632 2.5499 2.5454
0.9 3.6082 3.6009 3.5789 3.5431 3.4952 3.4374 3.3723 3.3026 3.2308 3.1592 3.0898 3.0241 2.9637 2.9098 2.8634 2.8258 2.7979 2.7805 2.7747
1.0 4.1237 4.1131 4.0815 4.0304 3.9625 3.8816 3.7917 3.6966 3.6001 3.5048 3.4135 3.3280 3.2499 3.1807 3.1216 3.0738 3.0383 3.0163 3.0090
1.1 4.7035 4.6886 4.6442 4.5729 4.4790 4.3686 4.2474 4.1211 3.9945 3.8710 3.7540 3.6455 3.5470 3.4605 3.3870 3.3276 3.2838 3.2566 3.2474
1.2 5.3542 5.3335 5.2722 5.1747 5.0476 4.9001 4.7404 4.5761 4.4137 4.2571 4.1104 3.9756 3.8542 3.7483 3.6586 3.5866 3.5334 3.5005 3.4894
1.3 6.082 6.054 5.9710 5.8399 5.6710 5.4776 5.2711 5.0617 4.8573 4.6625 4.4819 4.3175 4.1706 4.0432 3.9360 3.8500 3.7868 3.7477 3.7344
1.4 6.895 6.857 6.746 6.573 6.352 6.102 5.8400 5.5774 5.3244 5.0862 4.8676 4.6703 4.4955 4.3447 4.2183 4.1174 4.0432 3.9974 3.9819
1.5 7.801 7.750 7.604 7.377 7.092 6.775 6.447 6.123 5.8143 5.5273 5.2666 5.0333 4.8281 4.6520 4.5052 4.3883 4.3024 4.2495 4.2315
1.6 8.806 8.740 8.549 8.256 7.894 7.497 7.092 6.697 6.326 5.9849 5.6780 5.4057 5.1678 4.9647 4.7961 4.6622 4.5641 4.5036 4.4830
1.7 9.920 9.834 9.587 9.214 8.759 8.268 7.774 7.299 6.859 6.458 6.101 5.7867 5.5140 5.2823 5.0907 4.9390 4.8279 4.7595 4.7361
1.8 11.151 11.040 10.725 10.254 9.689 9.088 8.492 7.928 7.411 6.946 6.535 6.176 5.8662 5.6045 5.3888 5.2184 5.0936 5.0170 4.9908
1.9 12.507 12.366 11.967 11.380 10.685 9.957 9.246 8.583 7.982 7.447 6.978 6.572 6.224 5.9308 5.6900 5.5001 5.3613 5.2760 5.2468
2.0 13.998 13.819 13.320 12.593 11.746 10.873 10.034 9.262 8.570 7.961 7.431 6.975 6.587 6.261 5.9942 5.7842 5.6307 5.5365 5.5041
2.1 15.632 15.408 14.788 13.895 12.874 11.837 10.855 9.964 9.175 8.486 7.893 7.385 6.955 6.595 6.301 6.070 5.9017 5.7982 5.7627
2.2 17.419 17.141 16.376 15.290 14.067 12.847 11.708 10.688 9.795 9.023 8.362 7.800 7.327 6.932 6.610 6.358 6.174 6.061 6.022
2.3 19.369 19.025 18.089 16.778 15.327 13.902 12.592 11.433 10.429 9.569 8.839 8.220 7.702 7.272 6.922 6.648 6.448 6.325 6.282
2.4 21.489 21.069 19.931 18.361 16.652 15.000 13.504 12.198 11.077 10.125 9.322 8.645 8.081 7.614 7.235 6.938 6.722 6.589 6.543
2.5 23.791 23.280 21.907 20.040 18.041 16.142 14.445 12.982 11.738 10.690 9.810 9.074 8.462 7.957 7.548 7.229 6.996 6.853 6.803

Subsequent interpolation as a function of θ may be effected by the interpolation formula [A^*\{\theta\}=\textstyle\sum\limits^N_{n=1}\,L_n\sin^{2n}(\theta). \eqno (6.3.3.7)]Interpolation is accurate to 0.1% with N = M = 3.

For cylinders and spheres, [\bar T] may be obtained by means of the expression [\bar T={1\over A^*}\;{{\rm d} A^*\over {\rm d}\mu} = R\left[{1\over A^*}\;{{\rm d} A^*\over{\rm d}(\mu R)}\right] \eqno (6.3.3.8)]using the values listed in Tables 6.3.3.2[link] and 6.3.3.3[link].

Values of (1/A*)[dA*/d(μR)] obtained by numerical integration by Flack & Vincent (1978[link]) for spheres with [\mu R\lt2.5] are listed in Table 6.3.3.4[link]. An equivalent table of μ(R/A*)/[dA*/d(μR)] for [\mu R\lt4.0] is given by Rigoult & Guidi-Morosini (1980[link]).

Table 6.3.3.4| top | pdf |
Values of (1/A*)(dA*/dμR) for spheres

μRθ = 0°θ = 5°θ = 10°θ = 15°θ = 20°θ = 25°θ = 30°θ = 35°θ = 40°θ = 45°θ = 50°θ = 55°θ = 60°θ = 65°θ = 70°θ = 75°θ = 80°θ = 85°θ = 90°
0.0 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000
0.1 1.4845 1.4842 1.4829 1.4809 1.4782 1.4739 1.4690 1.4634 1.4569 1.4504 1.4439 1.4375 1.4309 1.4248 1.4191 1.4152 1.4117 1.4096 1.4089
0.2 1.4692 1.4682 1.4650 1.4611 1.4548 1.4472 1.4374 1.4268 1.4145 1.4019 1.3879 1.3748 1.3615 1.3491 1.3385 1.3292 1.3228 1.3180 1.3168
0.3 1.4527 1.4515 1.4476 1.4400 1.4309 1.4186 1.4044 1.3886 1.3708 1.3517 1.3327 1.3128 1.2947 1.2773 1.2624 1.2494 1.2397 1.2340 1.2321
0.4 1.4360 1.4341 1.4283 1.4190 1.4058 1.3898 1.3709 1.3492 1.3265 1.3018 1.2773 1.2531 1.2296 1.2089 1.1903 1.1748 1.1628 1.1560 1.1533
0.5 1.4186 1.4161 1.4090 1.3969 1.3803 1.3598 1.3360 1.3093 1.2812 1.2522 1.2231 1.1946 1.1678 1.1434 1.1218 1.1044 1.0910 1.0825 1.0797
0.6 1.4011 1.3980 1.3890 1.3742 1.3538 1.3289 1.3006 1.2693 1.2365 1.2033 1.1700 1.1382 1.1087 1.0816 1.0577 1.0383 1.0239 1.0147 1.0115
0.7 1.3830 1.3792 1.3683 1.3507 1.3264 1.2973 1.2643 1.2286 1.1918 1.1549 1.1184 1.0839 1.0516 1.0250 0.9978 0.9767 0.9615 0.9518 0.9484
0.8 1.3641 1.3600 1.3473 1.3262 1.2984 1.2650 1.2275 1.1879 1.1473 1.1071 1.0684 1.0314 0.9976 0.9674 0.9409 0.9195 0.9034 0.8931 0.8898
0.9 1.3451 1.3401 1.3253 1.3013 1.2696 1.2321 1.1908 1.1474 1.1038 1.0608 1.0198 0.9815 0.9465 0.9152 0.8880 0.8663 0.8495 0.8391 0.8359
1.0 1.3255 1.3198 1.3029 1.2758 1.2401 1.1987 1.1535 1.1070 1.0608 1.0157 0.9733 0.9340 0.8978 0.8661 0.8392 0.8167 0.8001 0.7897 0.7859
1.1 1.3058 1.2993 1.2800 1.2497 1.2103 1.1651 1.1165 1.0670 1.0185 0.9720 0.9286 0.8886 0.8522 0.8205 0.7931 0.7709 0.7542 0.7437 0.7400
1.2 1.2851 1.2780 1.2566 1.2228 1.1799 1.1312 1.0796 1.0278 0.9777 0.9299 0.8858 0.8455 0.8093 0.7776 0.7506 0.7285 0.7120 0.7017 0.6981
1.3 1.2645 1.2563 1.2324 1.1961 1.1494 1.0967 1.0430 0.9892 0.9377 0.8895 0.8451 0.8048 0.7691 0.7378 0.7113 0.6895 0.6733 0.6631 0.6596
1.4 1.2449 1.2349 1.2090 1.1684 1.1180 1.0628 1.0068 0.9517 0.8990 0.8504 0.8064 0.7666 0.7315 0.7009 0.6749 0.6539 0.6377 0.6278 0.6243
1.5 1.2231 1.2133 1.1845 1.1398 1.0867 1.0295 0.9711 0.9145 0.8615 0.8133 0.7696 0.7308 0.6964 0.6665 0.6414 0.6209 0.6055 0.5957 0.5922
1.6 1.2015 1.1908 1.1585 1.1118 1.0555 0.9957 0.9358 0.8782 0.8261 0.7778 0.7350 0.6970 0.6638 0.6349 0.6105 0.5907 0.5758 0.5663 0.5628
1.7 1.1806 1.1681 1.1339 1.0836 1.0244 0.9621 0.9005 0.8435 0.7912 0.7444 0.7027 0.6659 0.6334 0.6057 0.5822 0.5632 0.5484 0.5394 0.5361
1.8 1.1586 1.1456 1.1087 1.0558 0.9939 0.9294 0.8669 0.8101 0.7579 0.7121 0.6711 0.6359 0.6053 0.5787 0.5561 0.5376 0.5236 0.5148 0.5117
1.9 1.1370 1.1226 1.0835 1.0275 0.9625 0.8964 0.8341 0.7774 0.7262 0.6817 0.6420 0.6078 0.5791 0.5535 0.5321 0.5144 0.5010 0.4924 0.4892
2.0 1.1152 1.0996 1.0584 0.9982 0.9318 0.8646 0.8019 0.7457 0.6962 0.6527 0.6160 0.5830 0.5550 0.5305 0.5098 0.4927 0.4799 0.4717 0.4687
2.1 1.0932 1.0772 1.0327 0.9703 0.9014 0.8340 0.7712 0.7157 0.6678 0.6259 0.5899 0.5588 0.5322 0.5088 0.4886 0.4726 0.4603 0.4523 0.4494
2.2 1.0719 1.0543 1.0074 0.9427 0.8719 0.8039 0.7421 0.6874 0.6402 0.6003 0.5658 0.5353 0.5098 0.4884 0.4699 0.4548 0.4426 0.4347 0.4311
2.3 1.0498 1.0316 0.9822 0.9150 0.8434 0.7744 0.7133 0.6605 0.6147 0.5758 0.5433 0.5141 0.4896 0.4692 0.4518 0.4363 0.4252 0.4175 0.4149
2.4 1.0275 1.0118 0.9583 0.8889 0.8147 0.7482 0.6870 0.6340 0.5918 0.5507 0.5212 0.4937 0.4699 0.4500 0.4328 0.4187 0.4076 0.4003 0.3986
2.5 1.0108 0.9691 0.9297 0.8562 0.7904 0.7074 0.6554 0.6194 0.5618 0.5289 0.4980 0.4776 0.4554 0.4315 0.4142 0.4028 0.3921 0.3883 0.3783

Alternatively, one can differentiate the interpolation formula (6.3.3.6)[link], yielding [\bar T(\mu R,\theta)={1\over\mu}\sum^M_{m=1}\;mK_m(\mu R)^m. \eqno (6.3.3.9)]In this case, however, the maximum index M = 7 is required to obtain convergence for [\mu R\le2.5]. Numerical values of the coefficients Km for cylinders and spheres evaluated by Tibballs (1982[link]) are listed in Table 6.3.3.5[link].

Table 6.3.3.5| top | pdf |
Coefficients for interpolation of A* and [\bar T]

[\theta_j]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[\sqrt{3}] −24 12 −8 48 − 24[\sqrt{3}] −3
(C−1)2,j 518 −496 − 200[\sqrt {3}] 488 −268 184 −496 + 200[\sqrt{3}] 70
(C−1)3,j −1600 1920 + 560[\sqrt{3}] −2192 1536 −1136 1920 − 560[\sqrt{3}] −448
(C−1)4,j 2432 −3520 − 640[\sqrt{3}] 4032 −3328 2752 −3520 + 640[\sqrt{3}] 1152
(C−1)5,j −1792 3072 + 256[\sqrt{3}] −3328 3072 −2816 3072 − 256[\sqrt{3}] −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 [L_m=\textstyle\sum\limits^7_{j=1}\;(C^{-1})_{mj}\,A^*_j, \eqno (6.3.3.10)]where [C_{mj} = \sin^{2m}\theta_j. \eqno (6.3.3.11)]The elements [(C^{-1})_{mj}] and the [K_m(\theta_j)] for [\theta_j] at 15° intervals in the range [0\lt\theta_j\lt90^\circ] are listed in Table 6.3.3.5[link]. Differentiating (6.3.3.7)[link] yields [A^*(\mu R, \theta)\bar T(\mu R, \theta)=\textstyle\sum\limits^M_{m=0}\;P_m\sin^{2m}\theta, \eqno (6.3.3.12)]where [P_m=R{\partial L_m \over \partial(\mu R)} = \sum^7_{j=1} (C^{-1})_{mj}\,A^*_j\bar T_j. \eqno (6.3.3.13)]Equation (6.3.3.12)[link] for path lengths is the analogue of equation (6.3.3.7)[link] 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[link]), in an extension of earlier work by Coyle & Schroeder (1971[link]). The absorption correction for the case of the cylinder axis coincident with the [\varphi] axis of a Eulerian cradle, shown in Fig. 6.3.3.1[link] , reduces to the line integral [{1\over2\tau}\;\int\limits^{2\tau}_0\;\exp\{-\mu[(z)+T(z)]\}\,{\rm d} z, \eqno (6.3.3.14)]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[link] . The equation for the ellipse is [\tau=R(1-\sin^2\theta\sin^2\chi)^{-1/2}. \eqno (6.3.3.15)]The outgoing elliptical radius v satisfies [Av^4+Bv^2+C=0, \eqno (6.3.3.16)]where [\eqalign{ A&=[1-\sin^2\theta\sin^2\chi]^2 \cr B &=-2R^2[1-\sin^2\theta\sin^2 \chi] \cr&\quad -2(\tau-z){^2}[\cos\!{^2}\,\theta-\sin\!{^2}\, \theta\cos\!{^2}\,\chi]\sin\!{^2}\,2\theta\sin{^2}\chi \cr C &=R^4+2R^2(\tau-z)^2\sin^22\theta\sin^2\chi\cos2\theta \cr &\quad+(\tau-z)^4\sin^42\theta\sin^4\chi.}]

[Figure 6.3.3.1]

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]

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 [\varphi] axis, these equations become [\eqalign{ A &=[1-\sin^2(\theta+\beta)\sin^2\chi_1]^2 \cr B &=-2R^2[1-\sin^2 (\theta+\beta)\sin^2\chi_1] \cr &\quad -2(\tau-z)^2[\cos^2(\theta+\beta) \cr &\quad -\sin^2(\theta+\beta)\cos^2\chi_1]\sin^22\theta\sin^2\chi_1 \cr C &=R^4+2R^2(\tau-z)^2\sin^22\theta\sin^2\chi_1\cos2(\theta+\beta) \cr &\quad +(\tau-z)^4\sin^42\theta\sin^4\chi_1,}]where [\tan\beta=\sin\Gamma\sin\varphi/ [\sin\Gamma\cos\chi\cos\varphi+\sin\chi\cos\Gamma].]The roots of the quadratic equation (6.3.3.16)[link] for [v^2] 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 [T^2-2T(\tau-z)\cos2\theta+(\tau-z)^2-v^2=0. \eqno (6.3.3.17)]

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 [[1-\sin^2(\theta-\beta)\sin^2\chi_1]^{-1/2}].

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 [\eqalignno{ &\exp[-\mu(\tau_2-\tau_1+v_2-v_1)] \cr &\quad=\exp\Big(-\mu(R_2-R_1)\{[1-\sin^2(\theta-\beta)\sin^2\chi_1]^{-1/2} \cr &\qquad+[1-\sin^2(\theta+\beta)\sin^2\chi_1]^{-1/2}\}\Big), & (6.3.3.18)}]where the subscripts 1 and 2 apply to the inner and outer radii, respectively.

The integral in equation (6.3.3.14)[link] may be evaluated by Gaussian quadrature, i.e. by approximation as a weighted sum of the values of the function at the N zeros [X_i] of the Legendre polynomial of degree N in the interval [−1, +1]. The weights [w_i] for the points are tabulated by Abramowitz & Stegun (1964[link]). Further details are given in Subsection 6.3.3.4[link]. The emergent path lengths [T(z_1)] and [T(z_2)] for the case of the sheath are calculated as functions of the Gaussian variable [X_i] using the linear transformation [z_i=\tau_1X_i+\tau_2, \quad i=1,2,\ldots, N. \eqno (6.3.3.19)]This transformation converts the Gaussian variable X into the beam coordinate z for each i of the N summation points.

6.3.3.3. Analytical method for crystals with regular faces

| top | pdf |

For a crystal with regular faces, (6.3.3.1)[link] 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.[link]

[Figure 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[link]. 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,

  • (1) Crystal vertex.

  • (2) An intersection of a ray through a lit vertex with an opposite face.

  • (3) An intersection of an incident ray through a lit (i) vertex with a plane of diffracted (d) rays through a lit (d) edge, and the corresponding intersection with incident and diffracted beams interchanged.

  • (4) An intersection of a plane of incident rays through a lit (i) edge with an opposite edge, and its equivalent.

  • (5) An intersection on a shaded face of planes of incident and diffracted rays through (i) and (d) edges.

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 [\mu r_j=\mu(d_j-a_j x-b_j y-c_j z)/(a_j u+b_j v+c_jw),]where u, v, w are the direction cosines for the beam direction, and [a_j x+b_j y+c_j z=d_j] 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 [\eqalignno{ R_t &=6V_t e^{-g}H(1) \cr &={6V_t \over (b+c)}e^{-g}\left\{{h(a)-h(a+b) \over b} \,-\, {h(a+b) - h(a+b+c) \over c}\right\},\cr & & (6.3.3.20)}]where [h(x) = {1-e^{-x} \over x}. \eqno (6.3.3.21)][V_t] is the volume of the tetrahedron. For a crystal with Cartesian coordinate vertices 1, 2, 3, and 4, [V_t = \textstyle{1\over 6}\left|\matrix{ x_1-x_2&x_1-x_3&x_1-x_4 \cr y_1-y_2&y_1-y_3&y_1-y_4 \cr z_1-z_2&z_1-z_3&z_1-z_4}\right | . \eqno (6.3.3.22)]

The [g_i] are optical path lengths (i.e. path lengths rescaled by the absorption coefficient) ordered so that [g_1\lt g_2\lt g_3\lt g_4]and [g=g_1, \quad a=g_2-g_1, \quad b=g_3-g_2, \quad c=g_4-g_3. \eqno (6.3.3.23)]The transmission factor for the crystal is the sum of the scattering powers for all the tetrahedra [\sum R_t] divided by the volume [\sum V_t]. The equality of the total volume to the sum of the [V_t] 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)[link]. For [\varepsilon\lt0.3\times10^{-2}], and [\matrix{ a \lt \varepsilon\hfill &h(a)=1-a/2+a^2/3! \hfill\cr\vphantom{\bigg|} & h(b +a) =h(b)+ah_1(b)+a^2h_2(b)/2; \hfill\cr \vphantom{\bigg|} b \lt \varepsilon \hfill& h(a+b)=h(a)+bh_1(a)+b^2h_2(a)/2 \hfill\cr\vphantom{\bigg|} & [h(a)-h(a+b)]/b\hfill \cr\vphantom{\bigg|}& \quad =-h_1(a)-bh_2(a)/2-b^2h_3(a)/3!;\hfill \cr \vphantom{\bigg|}c \lt \varepsilon \hfill& [h(a+b)-h(a+b+c)]/c \hfill\cr \vphantom{\bigg|}& \quad =-h_1(a+b)-ch_2(a+b)/2 \hfill\cr \vphantom{\bigg|}& \qquad-c^2h_3(a+b)/3!;\hfill \cr \vphantom{\bigg|}b,c \lt \varepsilon\hfill & H(1)=h_2(a)/2+(2b+c)h_3(a)/3! \hfill\cr \vphantom{\bigg|}& \quad\qquad +\; (3b^2+3bc+c^2)h_4(a)/4!;\hfill \cr \vphantom{\bigg|}a,c\lt \varepsilon \hfill& h(a)=1-a/2+a^2/3! \hfill \cr \vphantom{\bigg|}&[h(a+b)-h(a+b+c)]/c \hfill\cr \vphantom{\bigg|}& \quad =-h_1(a+b)-ch_2(a+b)/2 \hfill \cr\vphantom{\bigg|} & \qquad -c^2h_3(a+b)/3!; \hfill \cr \vphantom{\bigg|}a,b\lt \varepsilon\hfill& h(a+b)=1-(a+b)/2+(a+b)^2/3! \hfill \cr\vphantom{\bigg|}& [h(a)-h(a+b)]/b \hfill \cr \vphantom{\bigg|}&\quad=1/2-a/3-b/3!+a^2/8+ab/8+b^2/4!;\hfill \cr\vphantom{\bigg|} a,b,c \lt \varepsilon\qquad& H(1)=\displaystyle{1\over3!}-{a+b \over 8}+(b-c)/4!\hfill \cr \vphantom{\bigg|}& \quad\qquad +\,[(a+b+c)(4a+3b) \hfill\cr \vphantom{\bigg|}& \quad\qquad +\,2a^2+ab+c^2]/5!;\hfill & (6.3.3.24)}]where the nth derivative of h(x) is [h_n(x)=(-)^nh(x)-\{(-)^n+n h_{n-1}(x)\}/x. \eqno (6.3.3.25)]

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[link]. 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 [N_v] vertices [x_i,y_i,z_i] has an area [D_U=1/2 \textstyle\sum\limits^{N_v}_{i=1}\;(x_i y_{i+1}-y_i x_{i+1})=E/2. \eqno (6.3.3.26)]The corresponding vertices on the lower face may be written [x_i+q\Delta x_i], [y_i+q\Delta y_i], [z_i+q\Delta z], with q = 1. The lower face has an area [D_L=1/2(E+qF+q^2G), \quad q=1, \eqno (6.3.3.27)]where [F= \textstyle\sum\limits^{N_v}_{i=1}\;\Delta x_i y_{i+1} + \Delta y_{i+1} x_i - \Delta x_{i+1} y_i - \Delta y_i x_{i+1}]and [G= \textstyle\sum\limits^{N_v}_{i=1}\; \Delta x_i\,\Delta y_{i+1} - \Delta y_i\,\Delta x_{i+1}\eqno (6.3.3.28)]so that the volume of the slice is [V_s=1/2(z_L-z_U)(E+F/2+G/3). \eqno (6.3.3.29)]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 [T=T_U+q(T_L-T_U)=T_U+q\Delta T. \eqno (6.3.3.30)]

Thus, the total diffracting power of the slice [\eqalignno{ R_s &= 1/2(z_L-z_U)\exp (-\mu T_U) \cr &\quad \times\textstyle\int\limits^1_0\; (E+qF+q^2 G)\exp (-\mu q\Delta T)\,{\rm d} q \cr &= 1/2(z_L-z_U)\exp (-\mu T_L) \left\{ {-E \over \mu \Delta T} \,-\, {F(\mu \Delta T+1)\over(\mu \Delta T)^2} \right. \cr &\left. \quad -G{(\mu \Delta T^2+2\mu\Delta T+2) \over (\mu\Delta T)^3}\right\} \cr &\quad -1/2(z_L-z_U)\exp (-\mu T_U) \left\{{-E\over \mu\Delta T} - {F\over(\mu \Delta T)^2}-{2G \over (\mu \Delta T)^3}\right\}.\cr & & (6.3.3.31)}]

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. [A=\textstyle\sum R_s\big/\textstyle\sum V_s, \eqno (6.3.3.32)]where the crystal volume is [\sum V_s].

dA/dμ, required in calculating [\bar T] for the extinction correction, can be obtained by differentiating [R_s] for each slice with respect to μ, summing the derivatives for each slice, and dividing by [\sum V_s]. 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[link]).

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.

6.3.3.4. Gaussian integration

| top | pdf |

The integral in the transmission factor in equation (6.3.3.1)[link] 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 [\int^b_a\,f(y)\,{\rm d} y] may be approximated (Stroud & Secrest, 1966[link]) by [\int\limits^b_a f(y)\,{\rm d} y = {b-a\over 2}\sum^n_{i=1}\; w_i\; f(y_i)+R_n, \eqno (6.3.3.33)]where [y_i= \left({b-a\over 2}\right) X_i+\left({b+a \over2}\right),][X_i] is the ith zero of the Legendre polynomial [P_n(X),][w_i={2\over (1-X^2_i)}[P'_n(X_i)]^2, \eqno (6.3.3.34)]and [R_n ={(b-a)^{2n+1}(n!)^4 \over(2n+1)[(2n!)]^3}\, 2^{2n+1}f^{(2n)}(\xi), \quad -1 \lt \xi \lt 1. \eqno (6.3.3.35)]When applying this to the calculation of a transmission coefficient (Coppens, 1970[link]), we commence with the a-axis grid points [x_i] selected such that [x_i=x_{\rm min}+(x_{\rm max}-x_{\rm min})X_i, \eqno (6.3.3.36)]where the [X_i] are the Gaussian constants.

For each [x_i], a line is drawn parallel to b and points are then selected such that [y_{ij}=y_{\rm min}(x_i)+[y_{\rm max}(x_i)-y_{\rm min}(x_i)]\,X_j. \eqno (6.3.3.37)]The procedure is repeated for the c direction, yielding [z_{ijk}= z_{\rm min}(x_i,y_j)+[z_{\rm max}(s_i,y_j)-z_{\rm min}(x_i,y_j)]\,X_k. \eqno (6.3.3.38)]

To calculate the absorption corrections, the incident and diffracted wavevectors are determined. For each grid point, the sum [T_{ijk}] of the path lengths for the incident and diffracted beams is evaluated. The sum that approximates the transmission coefficient is then [A=1/V \textstyle\sum\limits_{i,\, j,\, k}\; w_iw_jw_k\exp(-\mu T_{ijk}). \eqno (6.3.3.39)]Gaussian constants are tabulated by Abramowitz & Stegun (1964[link]).

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

6.3.3.5. Empirical methods

| top | pdf |

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[link]), 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[link]).

Intensities [H_m] are measurements for a reflection S at the angular positions [\Omega_m], [2\theta], [\chi_m], [\varphi_m]. Corrected intensities [I_m] are to be derived from the measurements by means of a correction factor [A^*_m] such that [I_m=A^*_mH_m. \eqno (6.3.3.40)]It is assumed that the correction can be written in the form of a rapidly converging Fourier series [\eqalignno {A^*_m &=\textstyle\sum\limits^\infty_{i,\,j,\,k,\, l=-\infty}\; a_{ijkl}\cos(i\Omega+j2\theta+k\chi+l\varphi) \cr & \quad + b_{ijkl}\sin(i\Omega+j2\theta +k\chi+l\varphi). &(6.3.3.41)}]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 [\Omega,2\theta,\chi,\varphi]; [\Omega,-2\theta,-\chi,\pi+\varphi]; [-\Omega,-2\theta,\pi+\chi,\varphi]; [-\Omega,2\theta,\pi-\chi,\pi+\varphi]; [\pi+\Omega,-2\theta,\chi,\varphi]; [\pi+\Omega,2\theta,-\chi,\pi+\varphi]; [\pi-\Omega, 2\theta,\pi+\chi,\varphi]; [\pi -\Omega, -2\theta,\pi-\chi,\pi+\varphi] are equivalent. In shorthand notation, the series (6.3.3.41)[link] reduces to [\eqalignno{ A^*_m &=\sum a_{\rm cccc}+a_{\rm ccsc}+a_{\rm sccc}+a_{\rm scsc} + a_{\rm sssc}+a_{\rm sscc} \cr &\quad +a_{\rm cssc}+a_{\rm cscc} +b_{\rm cccs}+b_{\rm ccss} +b_{\rm sccs} + b_{\rm scss} \cr &\quad+b_{\rm ssss}+b_{\rm sscs}+b_{\rm csss}+b_{\rm cscs}. & (6.3.3.42)}]The range of indices for some terms may be restricted by noting other symmetries in the diffraction experiment. Thus, equation (6.3.3.40)[link] will define the absorption correction for measurements of the incident-beam intensity, with [\Omega=2\theta=0]. Since with this geometry the correction will be invariant to rotation about the χ axis, the coefficients for the function involving [\cos(i\Omega)\cos(\,j2\theta)] must vanish if the χ index, k, is non-zero. By similar reasoning with the [\varphi] axis along the incident beam, one may deduce that coefficients for [\sin(i\Omega)\cos(\,j2\theta)\sin(k\chi)] will vanish unless l = 0.

Because for a given reflection all measurements are made at the same Bragg angle, the [2\theta] 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 [{1\over N_{\bf S}}\;\sum_{\bf S} A^*_{\bf S}=1, \eqno (6.3.3.43)]where [N_{\bf S}] is the number of independent reflections. Equation (6.3.3.42)[link] may be expressed in the shorthand notation [A^*_{\bf S}=\textstyle\sum\limits_{p=0}C_p\;f_{p{\bf S}}, \eqno (6.3.3.44)]where [C_p] is the coefficient in a term such as [a_{\rm ccsc}] or [b_{\rm ccss}] and [f_{p{\bf S}}] is the corresponding geometrical function. Labelling the constant geometrical term with a value of unity as [f_0] and rearranging leads to [A^*_{\bf S} = 1+\sum_{p=1}C_p\left\{\, f_{p{\bf S}}- \displaystyle{1\over N_{\bf S}}\;\sum_{\bf S}\, f_{p{\bf S}}\right\}=1+\sum_{p=1}C_pg_{p{\bf S}}, \eqno (6.3.3.45)]which defines [g_{p{\bf S}}].

Equation (6.3.3.40)[link] is now expressed as [I_{m{\bf S}}=H_{m{\bf S}}+H_{m{\bf S}}\textstyle\sum\limits_{p=1}\;C_p g_{p{\bf S}}, \eqno (6.3.3.46)]in which the coefficients [C_p] are to be chosen so that the values of [I_{m{\bf S}}] 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)[link] as [\Delta_{m{\bf S}}-H_{m{\bf S}}=-I_{\bf S}+H_{m{\bf S}}\textstyle\sum\limits_{p=1}\; C_p g_{p{\bf S}}, \eqno (6.3.3.47)]in which the mean intensity [I_{\bf S}] and the [C_p] are chosen to minimize [\textstyle\sum_{{\bf S},m}w^2_{\bf S}\Delta{}^2_{m{\bf S}}], where [\Delta_{m{\bf S}}=I_{m{\bf S}}-I_{\bf S}, \eqno (6.3.3.48)]and [w_{\bf S}] is the weight for that reflection.

If the equation to be solved [-w_{\bf S} H_{m{\bf S}}\simeq -w_{\bf S} I_m+\textstyle\sum\limits_{p=1}\,C_p g_{p{\bf S}} w_{\bf S}I_{m{\bf S}} \eqno (6.3.3.49)]is written in the shorthand form [{\bi D=FC}, \eqno (6.3.3.50)]in which D corresponds to [-w_{\bf S} H_{m{\bf S}}], the [I_m] and [C_p] correspond to C, with [(-w_{\bf S})] and [w_{\bf S} g_{p{\bf S}}H_{m{\bf S}}] corresponding to F, the solution to (6.3.3.50)[link] can be determined from the normal equations [{\bi C}=({\bi F}^T{\bi F})^{-1}{\bi F}^T{\bi D}, \eqno (6.3.3.51)]where [{\bi F}^T] 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 [C_p]) is modified. The tedious inversion of the normal equations, described by (6.3.3.51)[link], may be replaced by a simple inversion via the Gram–Schmidt orthogonalizing process, i.e. by calculating a matrix W with mutually orthogonal columns [{\bi W}_j] such that [{\bi W}_1={\bi F}_1][{\bi W}_j={\bi F}_j-\textstyle\sum\limits^{j-1}_{k=1}\;({\bi F}_j\cdot{\bi W}_k){\bi W}_k/{\bi W}^2_k. \eqno (6.3.3.52)]The minimizing of (DFC)2 is replaced by minimizing (DWA)2. Differentiating with respect to [a_j] yields [a_j={\bi D}\cdot {\bi W}_j \over {{\bi W}^2_j}. \eqno (6.3.3.53)]If equation (6.3.3.52)[link] is written as [{\bi F}={\bi WB},]where the upper triangular matrix B is [\eqalignno{ b_{ij}={\bi F}_j\cdot{\bi W}_i/{\bi W}^2_i,\quad i \lt j; \quad&b_{ij}=1, i=j; \cr &b_{ij}=0, i \gt j, &(6.3.3.54)}]the vector determining the coefficients is [{\bi C}={\bi B}^{\rm -1}{\bi A}, \eqno (6.3.3.55)]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 [\textstyle\sum\limits_{\bf S}\, w^2_{\bf S}\, H_{m{\bf S}}\Big/\textstyle\sum\limits_{\bf S}\;w^2_{\bf S}=\textstyle\sum\limits_{\bf S}\;w^2_{\bf S}\, I_{m{\bf S}}\Big/\textstyle\sum\limits_{\bf S}\;w^2_{\bf S}. \eqno (6.3.3.56)]If [m=1,2,\ldots,M], this reduces to the M constraint equations [\sum_{p=1}\;C_p\left\{ \displaystyle{\sum_{\bf S}\,w^2_{\bf S}\,H_{m{\bf S}}\,g_{p{\bf S}}\over \sum_{\bf S}\,w^2_{\bf S}}\right\} = \,\sum_{p=1}\;C_p\left\{ \displaystyle{\varepsilon w_m \sum_{\bf S}\,w^2_{\bf S}\, H_{m{\bf S}}\, g_{p{\bf S}}\over \sum_{\bf S}\,w^2_{\bf S}}\right\} =0, \eqno (6.3.3.57)]where [w_m] is the square root of the weight for the weighted mean of the equivalent reflections [H_m], defined as [H_m=\textstyle\sum\limits_{\bf S} w^2_{\bf S} H_{m{\bf S}}\Big/\textstyle\sum\limits_{\bf S} w^2_{\bf S}\quad \, \hbox{for { } each { }{\bf S}}, \eqno (6.3.3.58)]and the multiplier [\varepsilon] 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)[link], becomes [\textstyle\sum\limits_{{\bf S},m}\, w^2_{\bf S}(I_{m{\bf S}}-I_m)^2+\textstyle\sum\limits_m\, \varepsilon^2w^2_m(H_m-I_m)^2. \eqno (6.3.3.59)]

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[link]). A similar method minimizing the difference between observed and calculated structure factors is described by Walker & Stuart (1983[link]). 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[link]), North, Phillips & Mathews (1968[link]), Flack (1974[link]), Stuart & Walker (1979[link]), Lee & Ruble (1977a[link],b[link]), Schwager, Bartels & Huber (1973[link]), and Santoro & Wlodawer (1980[link]).

6.3.3.6. Measuring crystals for absorption

| top | pdf |

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 ([\varphi]) is zero. During rotation, each face will at some stage be oriented with its normal [N({\bf S})] 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 [\varphi_N], the appearance of a typical face ABCD will be as indicated in Fig. 6.3.3.4.[link]

[Figure 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 [x\sin\varphi_N+y\cos\varphi_N+z\tan\chi=Y]or, equivalently, [(x\sin\varphi_N+y\cos\varphi_N)\cot\chi+z=Z.]

For a crystal oriented on an Eulerian cradle, it is necessary to specify the orientation of the crystal, i.e. the angles [\Omega,\chi,\varphi] 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.[link]

[Figure 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, [\varphi] 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.

References

Abramowitz, M. & Stegun, I. A. (1964). Handbook of mathematical functions, p. 916. National Bureau of Standards Publication AMS 55.
Alcock, N. W., Pawley, G. S., Rourke, C. P. & Levine, M. R. (1972). An improvement in the algorithm for absorption correction by the analytical method. Acta Cryst. A28, 440–444.
Becker, P. J. & Coppens, P. (1974). Extinction within the limit of validity of the Darwin transfer equations. I. General formalisms for primary and secondary extinction and their application to spherical crystals. Acta Cryst. A30, 129–147.
Coppens, P. (1970). The evaluation of absorption and extinction in single crystal structure analysis. Crystallographic computing, edited by F. R. Ahmed, S. R. Hall & C. P. Huber, pp. 255–270. Copenhagen: Munksgaard.
Coyle, B. A. (1972). Absorption and volume corrections for a cylindrical specimen, larger than the beam, and in general orientation. Acta Cryst. A28, 231–233.
Coyle, B. A. & Schroeder, L. W. (1971). Absorption and volume corrections for a cylindrical sample, larger than the X-ray beam, employed in Eulerian geometry. Acta Cryst. A27, 291–295.
Dwiggins, C. W. Jr (1975a). Rapid calculation of X-ray absorption correction factors for cylinders to an accuracy of 0.1%. Acta Cryst. A31, 146–148.
Dwiggins, C. W. Jr (1975b). Rapid calculation of X-ray absorption correction factors for spheres to an accuracy of 0.05%. Acta Cryst. A31, 395–396.
Flack, H. D. (1974). Automatic absorption correction using intensity measurements from azimuthal scans. Acta Cryst. A30, 569–573.
Flack, H. D. (1977). An empirical absorption–extinction correction technique. Acta Cryst. A33, 890–898
Flack, H. D. & Vincent, M. G. (1978). Absorption weighted mean path lengths for spheres. Acta Cryst. A34, 489–491.
Graaff, R. A. G. de (1973). A Monte Carlo method for the calculation of transmission factors. Acta Cryst. A29, 298–301.
Graaff, R. A. G. de (1977). On the calculation of transmission factors. Acta Cryst. A33, 859.
Katayama, C., Sakabe, N. & Sakabe, K. (1972). A statistical evaluation of absorption. Acta Cryst. A28, 293–295.
Kopfmann, G. & Huber, R. (1968). A method of absorption correction for X-ray intensity measurements. Acta Cryst. A24, 348–351.
Lee, B. & Ruble, J. R. (1977a). A semi-empirical absorption-correction technique for symmetric crystals in single-crystal X-ray crystallography. I. Acta Cryst. A33, 629–637.
Lee, B. & Ruble, J. R. (1977b). A semi-empirical absorption-correction technique for symmetric crystals in single-crystal X-ray crystallography. II. Acta Cryst. A33, 637–641.
North, A. C. T., Phillips, D. C. & Mathews, F. S. (1968). A semi-empirical method of absorption correction. Acta Cryst. A24, 351–359.
Rigoult, J. & Guidi-Morosini, C. (1980). An accurate calculation of [\bar T]μ for spherical crystals. Acta Cryst. A36, 149–151.
Rouse, K. D., Cooper, M. J., York, E. J. & Chakera, A. (1970). Absorption corrections for neutron diffraction. Acta Cryst. A26, 682–691.
Santoro, A. & Wlodawer, A. (1980). Absorption corrections for Weissenberg diffractometers. Acta Cryst. A36, 442–450.
Schwager, P., Bartels, K. & Huber, R. (1973). A simple empirical absorption-correction method for X-ray intensity data films. Acta Cryst. A29, 291–295.
Stroud, A. H. & Secrest, D. (1966). Gaussian quadrature formulas. New Jersey: Prentice-Hall.
Stuart, D. & Walker, N. (1979). An empirical method for correcting rotation-camera data for absorption and decay effects. Acta Cryst. A35, 925–933.
Tibballs, J. E. (1982). The rapid computation of mean path lengths for cylinders and spheres. Acta Cryst. A38, 161–163.
Walker, N. & Stuart, D. (1983). An empirical method for correcting diffractometer data for absorption effects. Acta Cryst. A39, 158–166.
Weber, K. (1969). Eine neue Absorptionsfactortafel für kugelförmige Proben. Acta Cryst. B25, 1174–1178.
Zachariasen, W. H. (1968). Extinction and Borrmann effect in mosaic crystals. Acta Cryst. A24, 421–424.








































to end of page
to top of page