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

International Tables for Crystallography (2006). Vol. C, ch. 6.1, pp. 554-595
https://doi.org/10.1107/97809553602060000600

Chapter 6.1. Intensity of diffracted intensities

P. J. Brown,a A. G. Fox,b E. N. Maslen,e M. A. O'Keefec and B. T. M. Willisd

aInstitut Laue–Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France,bCenter for Materials Science and Engineering, Naval Postgraduate School, Monterey, CA 93943-5000, USA,cNational Center for Electron Microscopy, Lawrence Berkeley National Laboratory MS-72, University of California, Berkeley, CA 94720, USA,dChemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England, and eCrystallography Centre, The University of Western Australia, Nedlands, Western Australia 6009, Australia

Section 6.1.1 covers X-ray scattering from atoms and ions. Scattering is described by the Thomson formula, including coherent (Rayleigh) and incoherent (Compton) X-ray scattering. Atomic scattering factors, calculated using relativistic Hartree–Fock or Dirac–Slater wavefunctions, give the X-ray scattering from an atom (in fact, its ensemble of electrons) in terms of that from a single electron. Free-atom scattering factors are tabulated for neutral atoms from atomic number 1 (hydrogen) to 98 (californium) over a scattering range of sin θ/λ from 0 to 6 Å−1, and for ions from H1− to Pu6+ over 0 to 2 Å−1. Analytical fits to the scattering factors are given and methods for interpolation of the tabulated factors are described. Perturbations from free-atom electron density for bound atoms are handled with generalized scattering factors expressed as spherical harmonics. Probability density functions for atom displacement due to temperature are described in terms of generalized temperature factors related to atom vibration symmetries. The final parts of Section 6.1.1 describe the role of atomic scattering factors in the computation of crystal structure factors by summation over unit-cell atoms, and the reflecting power of small crystals. Section 6.1.2 presents the basic equations governing magnetic scattering of neutrons. They are used to define the useful intermediate quantities of the magnetic interaction vector, the magnetic structure factor and the magnetic form factor, which are used in calculations of magnetic cross sections. A brief account of the way in which the magnetic scattering depends upon the neutron spin direction (neutron polarization) is included. Formulae for the scattering of neutrons by the nuclei of an atom are given in Section 6.1.3. The scattering cross sections for a single nucleus, for an element containing a mixture of isotopes, and for a single crystal are considered.

6.1.1. X-ray scattering

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E. N. Maslen,e A. G. Foxb and M. A. O'Keefec

6.1.1.1. Coherent (Rayleigh) scattering

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An electromagnetic wave incident on a tightly bound electron is scattered coherently. For an incident wave of unit amplitude with the electric vector normal to the plane of the reflection x0y containing the incident and diffracted beams (Fig. 6.1.1.1[link] ), the amplitude of the scattered wave at a distance r is [r_e/r,\eqno (6.1.1.1)]where [r_e=(\mu_0/4\pi)(e^2/m)] is the classical radius of the electron (2.818 × 10−15 m).

[Figure 6.1.1.1]

Figure 6.1.1.1 | top | pdf |

Scattering by an electron. k0 and k are the incident and scattered wavevectors, respectively.

For a wave with the electric vector parallel to the plane x0y, the amplitude of the scattered wave is [{r_e\over r}\cos2\theta.\eqno (6.1.1.2)]For unpolarized incident radiation with unit mean amplitude, the amplitude of the scattered wave is given by the Thomson formula[{r_e\over r}\bigg\{{1+\cos^22\theta\over2}\bigg\}^{1/2}.\eqno (6.1.1.3)]The corresponding intensity of scattering per unit solid angle is [I_e=I_or^2_e\bigg[{1+\cos^22\theta\over 2}\bigg]\eqno (6.1.1.4)]for an unpolarized incident beam of intensity [I_o].

6.1.1.2. Incoherent (Compton) scattering

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For scattering from a free electron, the quantum nature of the radiation must be considered. Under the impact of a photon with energy hc/λ, momentum h/λ, the recoil of an electron, initially at rest, results in a change in wavelength of [\Delta\lambda={2h\over mc}\sin^2\theta,\eqno (6.1.1.5)]a geometry similar to that in Fig. 6.1.1.1[link] being assumed. There is no fixed relationship between the phases of the incident and scattered beams – i.e. the scattering is incoherent. The intensity [I_e] predicted by the Thomson formula is modified by the correction factor [λ/(λ + Δλ)]3.

6.1.1.3. Atomic scattering factor

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For scattering by atomic electrons there are both coherent and incoherent components, with total intensity given by the Thomson formula. The phase for coherent scattering is by convention related to that of a free electron at the nucleus. There is a phase shift of π for scattering from a free electron. The scattering from an element of electron density [\rho({\bf r}_j)] has a phase difference of [i{\bf S}\cdot{\bf r}_j], where [{\bf S}=2\pi{\bf s}.\eqno (6.1.1.6)]The total amplitude for coherent scattering from the jth electron is [f_j=\textstyle\int\rho({\bf r}_j)\exp(i{\bf S}\cdot{\bf r}_j)\,{\rm d}{\bf r}_j.\eqno (6.1.1.7)]The intensity of coherent scattering is [I_{\rm coh}=I_e f^2_j.\eqno (6.1.1.8)]The intensity of Compton scattering from that electron is [I_{\rm incoh}=I_e-I_{\rm coh}=I_e(1-f^2_j).\eqno (6.1.1.9)]For an atom with atomic number Z, [I_{\rm coh}=I_e\bigg(\textstyle\sum\limits^Z_{j=1}f_j\bigg)^2\eqno (6.1.1.10)]and [I_{\rm incoh}=I_e\bigg(Z-f^2_j-\textstyle\sum\limits_{j,k}f_{jk}\bigg),\eqno (6.1.1.11)]where the correction term [f_{jk}=\int\psi^*_j\psi_k\exp(i{\bf S}\cdot{\bf r})\,{\rm d}{\bf r},\eqno (6.1.1.12)]owing to exchange, meets the requirements of the Pauli exclusion principle.

Atomic scattering factors for neutral atoms are listed in Table 6.1.1.1[link] for the range [0\lt(\sin\theta)/\lambda\lt6.0\,{\rm \AA}^{-1}]. The values for hydrogen are calculated from the analytical solution to the Schrödinger equation and are effectively zero for [(\sin\theta)/\lambda\gt1.5\,{\rm \AA}^{-1}]. Those for heavier atoms are for relativistic wavefunctions, based on the calculations of Doyle & Turner (1968[link]) using the wavefunctions of Coulthard (1967[link]) (designated RHF in Table 6.1.1.1[link]), or on those of Cromer & Waber (1968[link]) using the wavefunctions of Mann (1968a[link]) (designated *RHF). The latter are based on a more exact treatment of potential that allows for the finite size of the nucleus, but the effect on the scattering factors is small. The calculations of Cromer & Waber (1968[link]) were originally made for [0\lt(\sin\theta)/\lambda\lt2.0\,{\rm \AA}^{-1}], but these have been extended to 6 Å−1 by Fox, O'Keefe & Tabbernor (1989[link]); this has been done because there are increasing numbers of applications for high-angle scattering factors.

Table 6.1.1.1| top | pdf |
Mean atomic scattering factors in electrons for free atoms

Methods: E: exact; RHF, *RHF (see text): relativistic Hartree–Fock.

ElementHHeLiBeBCNOFNe
Z12345678910
MethodERHFRHFRHFRHFRHFRHFRHFRHFRHF
(sin [\theta])/λ (Å−1)          
0.00 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000
0.01 0.998 1.998 2.986 3.987 4.988 5.990 6.991 7.992 8.993 9.993
0.02 0.991 1.993 2.947 3.950 4.954 5.958 6.963 7.967 8.970 9.973
0.03 0.980 1.984 2.884 3.889 4.897 5.907 6.918 7.926 8.933 9.938
0.04 0.966 1.972 2.802 3.807 4.820 5.837 6.855 7.869 8.881 9.891
0.05 0.947 1.957 2.708 3.707 4.724 5.749 6.776 7.798 8.815 9.830
0.06 0.925 1.939 2.606 3.592 4.613 5.645 6.682 7.712 8.736 9.757
0.07 0.900 1.917 2.502 3.468 4.488 5.526 6.574 7.612 8.645 9.672
0.08 0.872 1.893 2.400 3.336 4.352 5.396 6.453 7.501 8.541 9.576
0.09 0.842 1.866 2.304 3.201 4.209 5.255 6.321 7.378 8.427 9.469
0.10 0.811 1.837 2.215 3.065 4.060 5.107 6.180 7.245 8.302 9.351
0.11 0.778 1.806 2.135 2.932 3.908 4.952 6.030 7.103 8.168 9.225
0.12 0.744 1.772 2.065 2.804 3.756 4.794 5.875 6.954 8.026 9.090
0.13 0.710 1.737 2.004 2.683 3.606 4.633 5.714 6.798 7.876 8.948
0.14 0.676 1.701 1.950 2.569 3.459 4.472 5.551 6.637 7.721 8.799
0.15 0.641 1.663 1.904 2.463 3.316 4.311 5.385 6.472 7.560 8.643
0.16 0.608 1.624 1.863 2.365 3.179 4.153 5.218 6.304 7.395 8.483
0.17 0.574 1.584 1.828 2.277 3.048 3.998 5.051 6.134 7.226 8.318
0.18 0.542 1.543 1.796 2.197 2.924 3.847 4.886 5.964 7.055 8.150
0.19 0.511 1.502 1.768 2.125 2.808 3.701 4.723 5.793 6.883 7.978
0.20 0.481 1.460 1.742 2.060 2.699 3.560 4.563 5.623 6.709 7.805
0.22 0.424 1.377 1.693 1.951 2.503 3.297 4.254 5.289 6.362 7.454
0.24 0.373 1.295 1.648 1.864 2.336 3.058 3.963 4.965 6.020 7.102
0.25 0.350 1.254 1.626 1.828 2.263 2.949 3.825 4.808 5.851 6.928
0.26 0.328 1.214 1.604 1.795 2.195 2.846 3.693 4.655 5.685 6.754
0.28 0.287 1.136 1.559 1.739 2.077 2.658 3.445 4.363 5.363 6.412
0.30 0.251 1.060 1.513 1.692 1.979 2.494 3.219 4.089 5.054 6.079
0.32 0.220 0.988 1.465 1.652 1.897 2.351 3.014 3.834 4.761 5.758
0.34 0.193 0.920 1.417 1.616 1.829 2.227 2.831 3.599 4.484 5.451
0.35 0.180 0.887 1.393 1.600 1.799 2.171 2.747 3.489 4.353 5.302
0.36 0.169 0.856 1.369 1.583 1.771 2.120 2.667 3.383 4.225 5.158
0.38 0.148 0.795 1.320 1.551 1.723 2.028 2.522 3.186 3.983 4.880
0.40 0.130 0.738 1.270 1.520 1.681 1.948 2.393 3.006 3.759 4.617
0.42 0.115 0.686 1.221 1.489 1.644 1.880 2.278 2.844 3.551 4.370
0.44 0.101 0.636 1.173 1.458 1.611 1.821 2.178 2.697 3.360 4.139
0.45 0.095 0.613 1.149 1.443 1.596 1.794 2.132 2.629 3.270 4.029
0.46 0.090 0.591 1.125 1.427 1.581 1.770 2.089 2.564 3.183 3.923
0.48 0.079 0.548 1.078 1.395 1.553 1.725 2.011 2.445 3.022 3.722
0.50 0.071 0.509 1.033 1.362 1.526 1.685 1.942 2.338 2.874 3.535
0.55 0.053 0.423 0.924 1.279 1.463 1.603 1.802 2.115 2.559 3.126
0.60 0.040 0.353 0.823 1.195 1.402 1.537 1.697 1.946 2.309 2.517
0.65 0.031 0.295 0.732 1.112 1.339 1.479 1.616 1.816 2.112 2.517
0.70 0.024 0.248 0.650 1.030 1.276 1.426 1.551 1.714 1.956 2.296
0.80 0.015 0.177 0.512 0.876 1.147 1.322 1.445 1.568 1.735 1.971
0.90 0.010 0.129 0.404 0.740 1.020 1.219 1.353 1.463 1.588 1.757
1.00 0.007 0.095 0.320 0.622 0.900 1.114 1.265 1.377 1.482 1.609
1.10 0.005 0.072 0.255 0.522 0.790 1.012 1.172 1.298 1.398 1.502
1.20 0.003 0.055 0.205 0.439 0.690 0.914 1.090 1.221 1.324 1.418
1.30 0.003 0.042 0.165 0.369 0.602 0.822 1.004 1.145 1.254 1.346
1.40 0.002 0.033 0.134 0.311 0.524 0.736 0.921 1.070 1.186 1.280
1.50 0.001 0.026 0.110 0.263 0.457 0.659 0.843 0.997 1.120 1.218
1.60   0.021 0.091 0.223 0.398 0.588 0.769 0.926 1.055 1.158
1.70   0.017 0.075 0.190 0.347 0.525 0.700 0.857 0.990 1.099
1.80   0.014 0.063 0.163 0.304 0.468 0.636 0.792 0.928 1.041
1.90   0.011 0.053 0.139 0.266 0.418 0.578 0.731 0.868 0.984
2.00   0.010 0.044 0.120 0.233 0.373 0.525 0.674 0.810 0.929
2.50   0.004 0.021 0.060 0.126 0.216 0.324 0.443 0.564 0.680
3.00   0.002 0.011 0.033 0.072 0.130 0.204 0.292 0.389 0.489
3.50   0.001 0.006 0.019 0.043 0.081 0.132 0.196 0.270 0.331
4.00   0.004 0.012 0.027 0.053 0.088 0.134 0.190 0.254  
5.00     0.002 0.005 0.012 0.025 0.043 0.067 0.099 0.137
6.00     0.001 0.003 0.006 0.013 0.023 0.037 0.055 0.079

ElementNaMgAlSiPSClArKCa
Z11121314151617181920
MethodRHFRHFRHFRHFRHFRHFRHFRHFRHFRHF
(sin [\theta])/λ (Å−1)          
0.00 11.000 12.000 13.000 14.000 15.000 16.000 17.000 18.000 19.000 20.000
0.01 10.980 11.978 12.976 13.976 14.977 15.979 16.980 17.981 18.963 19.959
0.02 10.922 11.914 12.903 13.904 14.909 15.915 16.919 17.924 18.854 19.838
0.03 10.830 11.811 12.786 13.787 14.798 15.809 16.820 17.830 18.683 19.645
0.04 10.709 11.674 12.629 13.628 14.646 15.665 16.683 17.700 18.462 19.392
0.05 10.568 11.507 12.439 13.434 14.458 15.484 16.511 17.536 18.204 19.091
0.06 10.412 11.319 12.222 13.209 14.237 15.271 16.306 17.340 17.924 18.758
0.07 10.249 11.116 11.987 12.961 13.990 15.030 16.073 17.116 17.630 18.405
0.08 10.084 10.903 11.739 12.695 13.721 14.764 15.814 16.865 17.332 18.045
0.09 9.920 10.687 11.485 12.417 13.435 14.478 15.533 16.591 17.032 17.685
0.10 9.760 10.472 11.230 12.134 13.138 14.177 15.234 16.298 16.733 17.331
0.11 9.605 10.262 10.978 11.849 12.834 13.865 14.921 15.988 16.436 16.987
0.12 9.455 10.059 10.733 11.567 12.527 13.546 14.597 15.665 16.138 16.655
0.13 9.309 9.864 10.498 11.292 12.223 13.224 14.266 15.331 15.841 16.334
0.14 9.166 9.678 10.273 11.025 11.922 12.902 13.932 14.991 15.543 16.024
0.15 9.027 9.502 10.059 10.769 11.629 12.583 13.597 14.647 15.243 15.723
0.16 8.888 9.334 9.857 10.525 11.345 12.270 13.263 14.301 14.941 15.430
0.17 8.751 9.175 9.667 10.293 11.072 11.964 12.934 13.957 14.638 15.142
0.18 8.613 9.023 9.487 10.074 10.811 11.668 12.611 13.615 14.334 14.859
0.19 8.475 8.876 9.318 9.868 10.563 11.382 12.297 13.279 14.031 14.580
0.20 8.335 8.735 9.158 9.673 10.327 11.109 11.991 12.949 13.728 14.304
0.22 8.052 8.465 8.862 9.319 9.894 10.598 11.413 12.315 13.130 13.760
0.24 7.764 8.205 8.592 9.004 9.510 10.138 10.881 11.721 12.550 13.225
0.25 7.618 8.078 8.465 8.859 9.335 9.927 10.633 11.441 12.268 12.961
0.26 7.471 7.951 8.341 8.722 9.170 9.727 10.398 11.172 11.994 12.701
0.28 7.176 7.698 8.103 8.467 8.869 9.363 9.964 10.671 11.468 12.194
0.30 6.881 7.446 7.873 8.231 8.600 9.039 9.576 10.216 10.977 11.705
0.32 6.588 7.194 7.648 8.011 8.357 8.752 9.231 9.807 10.521 11.240
0.34 6.298 6.943 7.426 7.800 8.134 8.494 8.923 9.441 10.103 10.800
0.35 6.156 6.817 7.316 7.698 8.029 8.376 8.782 9.272 9.908 10.590
0.36 6.015 6.691 7.205 7.597 7.928 8.262 8.649 9.113 9.722 10.388
0.38 5.739 6.442 6.985 7.398 7.733 8.051 8.403 8.820 9.375 10.004
0.40 5.471 6.194 6.766 7.202 7.547 7.856 8.181 8.558 9.061 9.650
0.42 5.214 5.951 6.548 7.008 7.367 7.673 7.979 8.322 8.778 9.324
0.44 4.967 5.712 6.330 6.815 7.190 7.501 7.794 8.110 8.522 9.025
0.45 4.848 5.595 6.222 6.719 7.103 7.417 7.706 8.011 8.403 8.885
0.46 4.731 5.480 6.115 6.622 7.017 7.335 7.621 7.917 8.290 8.752
0.48 4.506 5.253 5.902 6.431 6.845 7.174 7.459 7.739 8.080 8.502
0.50 4.293 5.034 5.692 6.240 6.674 7.017 7.305 7.575 7.889 8.275
0.55 3.811 4.520 5.186 5.769 6.250 6.633 6.941 7.207 7.474 7.788
0.60 3.398 4.059 4.713 5.312 5.829 6.254 6.595 6.875 7.125 7.392
0.65 3.048 3.652 4.277 4.878 5.418 5.877 6.254 6.560 6.814 7.057
0.70 2.754 3.297 3.883 4.470 5.020 5.505 5.915 6.252 6.523 6.762
0.80 2.305 2.729 3.221 3.750 4.284 4.790 5.245 5.639 5.961 6.228
0.90 1.997 2.317 2.712 3.164 3.649 4.138 4.607 5.036 5.406 5.717
1.00 1.784 2.022 2.330 2.702 3.122 3.570 4.023 4.460 4.859 5.209
1.10 1.634 1.812 2.049 2.346 2.698 3.092 3.509 3.931 4.337 4.710
1.20 1.524 1.660 1.841 2.076 2.364 2.699 3.070 3.462 3.855 4.233
1.30 1.438 1.546 1.687 1.872 2.104 2.384 2.704 3.056 3.423 3.791
1.40 1.367 1.459 1.571 1.717 1.903 2.133 2.405 2.713 3.045 3.391
1.50 1.304 1.387 1.481 1.598 1.747 1.935 2.162 2.427 2.722 3.039
1.60 1.247 1.326 1.408 1.505 1.626 1.779 1.967 2.192 2.450 2.733
1.70 1.191 1.270 1.346 1.430 1.530 1.655 1.811 2.000 2.221 2.470
1.80 1.137 1.219 1.292 1.367 1.453 1.557 1.686 1.844 2.033 2.250
1.90 1.084 1.169 1.243 1.313 1.389 1.477 1.585 1.717 1.876 2.063
2.00 1.032 1.120 1.195 1.264 1.333 1.411 1.502 1.614 1.748 1.908
2.50 0.791 0.892 0.979 1.056 1.122 1.182 1.240 1.301 1.367 1.444
3.00 0.591 0.691 0.783 0.867 0.942 1.009 1.069 1.123 1.174 1.225
3.50 0.438 0.527 0.615 0.699 0.777 0.849 0.915 0.974 1.028 1.078
4.00 0.325 0.401 0.478 0.566 0.632 0.705 0.773 0.836 0.895 0.949
5.00 0.183 0.234 0.290 0.349 0.411 0.474 0.536 0.597 0.657 0.715
6.00 0.107 0.141 0.179 0.222 0.268 0.316 0.367 0.419 0.472 0.524

ElementScTiVCrMnFeCoNiCuZn
Z21222324252627282930
MethodRHFRHFRHFRHFRHFRHFRHFRHFRHFRHF
(sin [\theta])/λ (Å−1)          
0.00 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000
0.01 20.962 21.964 22.966 23.971 24.969 25.970 26.972 27.973 28.977 29.975
0.02 20.848 21.856 22.864 23.885 24.876 25.882 26.887 27.892 28.908 29.900
0.03 20.665 21.682 22.698 23.746 24.726 25.738 26.749 27.759 28.794 29.777
0.04 20.422 21.451 22.477 23.558 24.523 25.543 26.562 27.579 28.640 29.609
0.05 20.131 21.171 22.208 23.329 24.274 25.304 26.331 27.356 28.448 29.401
0.06 19.805 20.854 21.902 23.065 23.988 25.026 26.063 27.096 28.223 29.157
0.07 19.455 20.511 21.567 22.772 23.671 24.719 25.764 26.806 27.971 28.883
0.08 19.091 20.150 21.212 22.459 23.331 24.387 25.440 26.490 27.694 28.583
0.09 18.723 19.781 20.846 22.129 22.976 24.038 25.098 26.156 27.397 28.263
0.10 18.356 19.410 20.474 21.789 22.611 23.678 24.744 25.807 27.084 27.927
0.11 17.995 19.041 20.102 21.441 22.240 23.310 24.380 25.448 26.758 27.579
0.12 17.643 18.678 19.733 21.089 21.868 22.939 24.011 25.083 26.422 27.222
0.13 17.301 18.322 19.369 20.734 21.497 22.568 23.641 24.714 26.077 26.859
0.14 16.968 17.974 19.011 20.378 21.128 22.197 23.270 24.344 25.726 26.492
0.15 16.645 17.635 18.661 20.022 20.764 21.829 22.900 23.973 25.370 26.124
0.16 16.330 17.304 18.317 19.667 20.404 21.465 22.533 23.604 25.009 25.754
0.17 16.023 16.980 17.980 19.312 20.049 21.104 22.168 23.237 24.645 25.385
0.18 15.722 16.663 17.649 18.960 19.699 20.748 21.806 22.872 24.278 25.017
0.19 15.426 16.351 17.323 18.609 19.354 20.395 21.448 22.510 23.910 24.649
0.20 15.135 16.044 17.003 18.260 19.012 20.046 21.093 22.150 23.540 24.283
0.22 14.564 15.444 16.376 17.570 18.342 19.359 20.393 21.438 22.798 23.556
0.24 14.006 14.859 15.765 16.893 17.686 18.685 19.704 20.737 22.057 22.836
0.25 13.732 14.572 15.465 16.561 17.364 18.354 19.364 20.390 21.687 22.478
0.26 13.462 14.289 15.169 16.232 17.045 18.025 19.027 20.046 21.319 22.122
0.28 12.933 13.735 14.589 15.588 16.417 17.378 18.361 19.365 20.589 21.417
0.30 12.423 13.198 14.026 14.965 15.806 16.744 17.709 18.696 19.869 20.720
0.32 11.934 12.682 13.482 14.365 15.211 16.127 17.072 18.040 19.162 20.034
0.34 11.467 12.187 12.959 13.790 14.634 15.527 16.450 17.398 18.472 19.359
0.35 11.244 11.949 12.705 13.513 14.353 15.233 16.145 17.084 18.133 19.027
0.36 11.027 11.717 12.458 13.242 14.078 14.945 15.845 16.773 17.799 18.698
0.38 10.613 11.271 11.982 12.720 13.543 14.384 15.260 16.165 17.145 18.051
0.40 10.226 10.852 11.530 12.227 13.031 13.845 14.695 15.576 16.514 17.421
0.42 9.866 10.459 11.105 11.762 12.543 13.328 14.151 15.008 15.904 16.809
0.44 9.534 10.093 10.705 11.326 12.080 12.835 13.630 14.461 15.318 16.216
0.45 9.377 9.920 10.515 11.118 11.858 12.598 13.379 14.196 15.034 15.926
0.46 9.227 9.753 10.332 10.917 11.642 12.367 13.133 13.937 14.757 15.642
0.48 8.946 9.438 9.984 10.536 11.228 11.922 12.659 13.435 14.219 15.090
0.50 8.687 9.148 9.660 10.180 10.840 11.502 12.209 12.956 13.707 14.559
0.55 8.132 8.518 8.952 9.400 9.973 10.557 11.188 11.862 12.533 13.328
0.60 7.682 8.007 8.373 8.756 9.245 9.753 10.309 10.909 11.507 12.235
0.65 7.312 7.588 7.898 8.227 8.639 9.077 9.561 10.090 10.621 11.276
0.70 6.996 7.240 7.506 7.791 8.137 8.512 8.930 9.392 9.861 10.442
0.80 6.460 6.676 6.892 7.118 7.368 7.645 7.955 8.301 8.663 9.108
0.90 5.975 6.200 6.406 6.606 6.808 7.023 7.259 7.519 7.799 8.132
1.00 5.501 5.752 5.972 6.172 6.359 6.545 6.738 6.944 7.166 7.417
1.10 5.030 5.310 5.553 5.768 5.962 6.143 6.318 6.495 6.681 6.879
1.20 4.570 4.872 5.139 5.372 5.586 5.775 5.950 6.118 6.285 6.453
1.30 4.131 4.445 4.730 4.982 5.215 5.420 5.601 5.776 5.939 6.096
1.40 3.722 4.038 4.333 4.597 4.849 5.070 5.270 5.451 5.617 5.775
1.50 3.352 3.660 3.956 4.226 4.490 4.725 4.939 5.133 5.308 5.473
1.60 3.023 3.316 3.604 3.874 4.144 4.388 4.611 4.819 5.005 5.180
1.70 2.733 3.006 3.281 3.545 3.814 4.062 4.295 4.511 4.705 4.892
1.80 2.485 2.734 2.992 3.244 3.506 3.753 3.989 4.211 4.413 4.610
1.90 2.271 2.496 2.733 2.971 3.221 3.463 3.697 3.922 4.128 4.332
2.00 2.090 2.290 2.506 2.727 2.963 3.195 3.424 3.647 3.855 4.063
2.50 1.533 1.637 1.756 1.888 2.037 2.197 2.366 2.543 2.721 2.908
3.00 1.279 1.338 1.404 1.479 1.563 1.658 1.763 1.878 2.001 2.135
3.50 1.125 1.171 1.217 1.266 1.319 1.377 1.441 1.512 1.590 1.677
4.00 0.998 1.044 1.087 1.129 1.171 1.213 1.258 1.306 1.358 1.414
5.00 0.770 0.821 0.869 0.914 0.956 0.995 1.033 1.069 1.105 1.140
6.00 0.577 0.627 0.677 0.724 0.769 0.813 0.853 0.892 0.929 0.964

ElementGaGeAsSeBrKrRbSrYZr
Z31323334353637383940
MethodRHFRHFRHFRHFRHFRHFRHFRHF*RHF*RHF
(sin [\theta])/λ (Å−1)          
0.00 31.000 32.000 33.000 34.000 35.000 36.000 37.000 38.000 39.000 40.000
0.01 30.971 31.970 32.970 33.970 34.971 35.972 36.952 37.946 38.947 39.949
0.02 30.883 31.878 32.879 33.881 34.883 35.886 36.809 37.786 38.792 39.800
0.03 30.740 31.729 32.730 33.734 34.739 35.744 36.583 37.532 38.543 39.559
0.04 30.546 31.526 32.527 33.532 34.540 35.549 36.291 37.197 38.212 39.237
0.05 30.308 31.276 32.274 33.280 34.291 35.304 35.948 36.802 37.816 38.847
0.06 30.031 30.984 31.977 32.982 33.995 35.011 35.571 36.363 37.369 38.403
0.07 29.724 30.657 31.642 32.645 33.658 34.677 35.171 35.897 36.889 37.921
0.08 29.391 30.302 31.276 32.273 33.284 34.305 34.758 35.418 36.387 37.412
0.09 29.040 29.926 30.884 31.872 32.880 33.899 34.336 34.937 35.876 36.887
0.10 28.675 29.534 30.473 31.449 32.450 33.467 33.907 34.458 35.364 36.356
0.11 28.302 29.133 30.049 31.009 32.000 33.011 33.473 33.986 34.855 35.824
0.12 27.924 28.725 29.616 30.557 31.535 32.537 33.034 33.522 34.354 35.296
0.13 27.543 28.316 29.179 30.099 31.060 32.051 32.588 33.066 33.861 34.775
0.14 27.162 27.908 28.742 29.637 30.578 31.555 32.137 32.616 33.378 34.262
0.15 26.783 27.504 28.307 29.175 30.095 31.055 31.681 32.171 32.904 33.758
0.16 26.406 27.104 27.877 28.718 29.613 30.553 31.220 31.730 32.437 33.263
0.17 26.033 26.709 27.454 28.266 29.136 30.053 30.757 31.292 31.977 32.776
0.18 25.663 26.322 27.039 27.822 28.664 29.558 30.293 30.856 31.523 32.298
0.19 25.297 25.941 26.633 27.387 28.202 29.070 29.830 30.421 31.075 31.827
0.20 24.935 25.567 26.235 26.962 27.749 28.590 29.368 29.988 30.631 31.363
0.22 24.121 24.839 25.469 26.145 26.876 27.663 28.459 29.128 29.758 30.454
0.24 23.520 24.135 24.739 25.372 26.052 26.784 27.576 28.280 28.904 29.572
0.25 23.174 23.791 24.386 25.001 25.658 26.364 27.148 27.863 28.485 29.141
0.26 22.830 23.452 24.041 24.641 25.276 25.957 26.729 27.452 28.071 28.716
0.28 22.151 22.787 23.370 23.947 24.545 25.181 25.922 26.648 27.263 27.889
0.30 21.481 22.136 22.724 23.288 23.857 24.453 25.158 25.875 26.483 27.092
0.32 20.820 21.498 22.097 22.656 23.206 23.771 24.437 25.135 25.734 26.327
0.34 20.169 20.870 21.486 22.048 22.587 23.128 23.758 24.430 25.018 25.596
0.35 19.847 20.560 21.185 21.751 22.288 22.820 23.432 24.090 24.673 25.243
0.36 19.527 20.253 20.888 21.459 21.995 22.520 23.116 23.760 24.336 24.899
0.38 18.897 19.645 20.301 20.887 21.425 21.941 22.510 23.125 23.687 24.236
0.40 18.278 19.047 19.725 20.328 20.874 21.388 21.934 22.522 23.071 23.606
0.42 17.673 18.459 19.159 19.780 20.338 20.855 21.386 21.950 22.485 23.008
0.44 17.083 17.882 18.602 19.242 19.816 20.339 20.860 21.404 21.928 22.439
0.45 16.794 17.598 18.326 18.977 19.558 20.087 20.605 21.141 21.660 22.166
0.46 16.508 17.317 18.054 18.713 19.304 19.838 20.354 20.883 21.398 21.899
0.48 15.950 16.765 17.516 18.193 18.801 19.349 19.866 20.383 20.890 21.384
0.50 15.410 16.227 16.989 17.682 18.307 18.870 19.391 19.902 20.404 20.892
0.55 14.142 14.947 15.721 16.444 17.107 17.709 18.252 18.764 19.263 19.745
0.60 12.996 13.770 14.535 15.269 15.958 16.594 17.167 17.696 18.204 18.693
0.65 11.974 12.702 13.440 14.166 14.865 15.524 16.125 16.678 17.203 17.706
0.70 11.073 11.745 12.442 13.145 13.837 14.504 15.126 15.702 16.246 16.767
0.80 9.604 10.151 10.741 11.362 12.001 12.645 13.272 13.872 14.443 14.996
0.90 8.510 8.937 9.411 9.928 10.480 11.057 11.645 12.230 12.798 13.361
1.00 7.702 8.028 8.396 8.809 9.262 9.752 10.270 10.806 11.339 11.883
1.10 7.099 7.348 7.631 7.952 8.312 8.711 9.147 9.612 10.088 10.588
1.20 6.633 6.830 7.050 7.299 7.580 7.898 8.252 8.640 9.046 9.486
1.30 6.254 6.419 6.597 6.795 7.016 7.266 7.548 7.863 8.200 8.574
1.40 5.926 6.076 6.231 6.395 6.574 6.773 6.996 7.249 7.523 7.833
1.50 5.627 5.774 5.917 6.063 6.216 6.380 6.562 6.764 6.985 7.238
1.60 5.342 5.493 5.636 5.775 5.913 6.056 6.210 6.376 6.554 6.760
1.70 5.065 5.224 5.372 5.511 5.645 5.778 5.913 6.055 6.205 6.375
1.80 4.792 4.961 5.117 5.262 5.398 5.528 5.656 5.785 5.914 6.059
1.90 4.523 4.702 4.867 5.020 5.162 5.295 5.420 5.544 5.662 5.790
2.00 4.260 4.447 4.621 4.782 4.932 5.071 5.200 5.323 5.440 5.558
2.50 3.097 3.287 3.475 3.658 3.836 4.007 4.168 4.320 4.460 4.590
3.00 2.277 2.428 2.584 2.745 2.909 3.074 3.239 3.401 3.560 3.720
3.50 1.772 1.876 1.988 2.108 2.235 2.369 2.507 2.649 2.780 2.920
4.00 1.477 1.545 1.621 1.703 1.793 1.890 1.993 2.103 2.215 2.335
5.00 1.176 1.213 1.251 1.292 1.337 1.384 1.436 1.493 1.550 1.620
6.00 0.998 1.030 1.061 1.092 1.123 1.154 1.186 1.219 1.250 1.285

ElementNbMoTcRuRhPdAgCdInSn
Z41424344454647484950
Method*RHFRHF*RHF*RHF*RHF*RHFRHFRHFRHFRHF
(sin [\theta])/λ (Å−1)          
0.00 41.000 42.000 43.000 44.000 45.000 46.000 47.000 48.000 49.000 50.000
0.01 40.956 41.958 42.955 43.960 44.961 45.968 46.964 47.962 48.957 49.955
0.02 40.824 41.831 42.821 43.842 44.847 45.874 46.857 47.848 48.828 49.821
0.03 40.610 41.625 42.603 43.649 44.660 45.718 46.681 47.660 48.618 49.601
0.04 40.323 41.346 42.308 43.386 44.405 45.503 46.440 47.404 48.332 49.303
0.05 39.970 41.003 41.945 43.061 44.088 45.232 46.139 47.085 47.980 48.934
0.06 39.565 40.606 41.526 42.681 43.717 44.908 45.786 46.710 47.570 48.504
0.07 39.116 40.164 41.059 42.254 43.299 44.535 45.385 46.287 47.112 48.022
0.08 38.634 39.686 40.557 41.789 42.842 44.119 44.944 45.822 46.614 47.498
0.09 38.128 39.181 40.028 41.292 42.351 43.663 44.469 45.324 46.086 46.942
0.10 37.606 38.656 39.480 40.770 41.834 43.172 43.964 44.797 45.534 46.361
0.11 37.073 38.117 38.921 40.229 41.296 42.651 43.435 44.248 44.964 45.764
0.12 36.535 37.569 38.355 39.674 40.741 42.105 42.886 43.683 44.383 45.155
0.13 35.994 37.016 37.787 39.108 40.173 41.538 42.322 43.104 43.793 44.541
0.14 35.454 36.461 37.221 38.536 39.597 40.954 41.744 42.517 43.199 43.924
0.15 34.916 35.907 36.658 37.959 39.015 40.357 41.157 41.923 42.603 43.309
0.16 34.382 35.355 36.100 37.381 38.429 39.750 40.563 41.325 42.006 42.696
0.17 33.854 34.806 35.548 36.803 37.841 39.137 39.964 40.726 41.410 42.088
0.18 33.331 34.263 35.003 36.228 37.254 38.520 39.361 40.126 40.817 41.486
0.19 32.814 33.725 34.466 35.655 36.668 37.902 38.758 39.527 40.226 40.891
0.20 32.305 33.195 33.936 35.088 36.086 37.286 38.154 38.930 39.639 40.302
0.22 31.310 32.157 32.900 33.971 34.937 36.064 36.955 37.746 38.478 39.145
0.24 30.348 31.153 31.897 32.886 33.815 34.868 35.774 36.581 37.337 38.016
0.25 29.881 30.665 31.409 32.356 33.267 34.283 35.192 36.007 36.774 37.462
0.26 29.424 30.188 30.930 31.837 32.728 33.708 34.619 35.440 36.218 36.915
0.28 28.538 29.263 29.998 30.829 31.680 32.592 33.498 34.329 35.125 35.841
0.30 27.692 28.382 29.104 29.866 30.675 31.523 32.416 33.251 34.059 34.794
0.32 26.888 27.543 28.250 28.949 29.717 30.505 31.378 32.210 33.025 33.775
0.34 26.126 26.749 27.435 28.079 28.807 29.540 30.387 31.210 32.025 32.786
0.35 25.760 26.368 27.042 27.662 28.370 29.077 29.910 30.725 31.538 32.303
0.36 25.404 25.998 26.660 27.257 27.944 28.628 29.444 30.252 31.060 31.828
0.38 24.721 25.289 25.925 26.480 27.130 27.769 28.551 29.338 30.134 30.902
0.40 24.077 24.620 25.229 25.749 26.363 26.961 27.707 28.468 29.247 30.011
0.42 23.468 23.989 24.571 25.062 25.642 26.202 26.911 27.644 28.401 29.154
0.44 22.892 23.394 23.949 24.415 24.964 25.491 26.163 26.865 27.596 28.334
0.45 22.615 23.109 23.651 24.106 24.640 25.153 25.805 26.492 27.209 27.938
0.46 22.346 22.832 23.361 23.807 24.327 24.825 25.459 26.129 26.832 27.551
0.48 21.829 22.300 22.806 23.235 23.729 24.201 24.800 25.436 26.108 26.805
0.50 21.336 21.796 22.280 22.696 23.167 23.617 24.181 24.784 25.425 26.096
0.55 20.195 20.638 21.080 21.476 21.900 22.307 22.795 23.320 23.881 24.482
0.60 19.156 19.595 20.012 20.403 20.798 21.177 21.607 22.063 22.552 23.081
0.65 18.187 18.635 19.042 19.438 19.820 20.186 20.575 20.978 21.405 21.868
0.70 17.268 17.732 18.142 18.551 18.932 19.296 19.661 20.027 20.408 20.815
0.80 15.533 16.036 16.477 16.922 17.326 17.711 18.069 18.405 18.736 19.073
0.90 13.915 14.448 14.925 15.405 15.845 16.266 16.651 17.000 17.329 17.646
1.00 12.427 12.968 13.466 13.968 14.440 14.893 15.316 15.698 16.053 16.384
1.10 11.098 11.621 12.116 12.620 13.107 13.580 14.035 14.451 14.840 15.201
1.20 9.945 10.430 10.900 11.385 11.866 12.342 12.813 13.253 13.670 14.062
1.30 8.972 9.404 9.833 10.282 10.740 11.200 11.669 12.116 12.548 12.962
1.40 8.169 8.542 8.919 9.323 9.743 10.173 10.623 11.060 11.492 11.913
1.50 7.516 7.831 8.154 8.506 8.880 9.270 9.687 10.101 10.518 10.933
1.60 6.969 7.251 7.521 7.823 8.148 8.492 8.869 9.249 9.639 10.034
1.70 6.564 6.780 7.004 7.258 7.535 7.833 8.165 8.505 8.860 9.227
1.80 6.216 6.397 6.582 6.794 7.028 7.282 7.569 7.867 8.184 8.516
1.90 5.927 6.080 6.234 6.412 6.608 6.824 7.069 7.326 7.603 7.897
2.00 5.680 5.813 5.946 6.097 6.262 6.443 6.651 6.871 7.110 7.367
2.50 4.710 4.827 4.930 5.040 5.140 5.240 5.351 5.461 5.577 5.702
3.00 3.860 3.988 4.110 4.230 4.350 4.460 4.566 4.665 4.761 4.853
3.50 3.065 3.217 3.350 3.485 3.620 3.740 3.862 3.977 4.087 4.192
4.00 2.405 2.581 2.690 2.820 2.940 3.080 3.207 3.330 3.449 3.565
5.00 1.690 1.766 1.840 1.925 2.012 2.100 2.206 2.304 2.406 2.509
6.00 1.327 1.373 1.420 1.470 1.520 1.575 1.635 1.698 1.746 1.835

ElementSbTeIXeCsBaLaCePrNd
Z51525354555657585960
MethodRHF*RHFRHFRHFRHFRHF*RHF*RHF*RHF*RHF
(sin [\theta])/λ (Å−1)          
0.00 51.000 52.000 53.000 54.000 55.000 56.000 57.000 58.000 59.000 60.000
0.01 50.955 51.954 52.955 53.956 54.932 55.925 56.926 57.928 58.929 59.931
0.02 50.819 51.818 52.820 53.821 54.732 55.703 56.708 57.715 58.722 59.728
0.03 50.596 51.594 52.597 53.601 54.417 55.350 56.360 57.375 58.392 59.404
0.04 50.293 51.288 52.292 53.297 54.008 54.888 55.900 56.924 57.956 58.977
0.05 49.915 50.906 51.911 52.917 53.527 54.345 55.351 56.385 57.439 58.468
0.06 49.474 50.458 51.460 52.467 52.996 53.743 54.736 55.779 56.861 57.899
0.07 48.977 49.951 50.950 51.954 52.430 53.106 54.076 55.127 56.242 57.288
0.08 48.434 49.395 50.387 51.388 51.839 52.450 53.388 54.446 55.599 56.651
0.09 47.856 48.800 49.781 50.775 51.229 51.786 52.687 53.750 54.943 56.000
0.10 47.250 48.174 49.142 50.125 50.603 51.122 51.982 53.047 54.281 55.342
0.11 46.625 47.526 48.476 49.447 49.963 50.460 51.278 52.345 53.617 54.680
0.12 45.988 46.863 47.793 48.747 49.309 49.802 50.580 51.646 52.952 54.017
0.13 45.344 46.193 47.099 48.033 48.645 49.146 49.888 50.952 52.288 53.354
0.14 44.699 45.519 46.400 47.311 47.971 48.492 49.202 50.263 51.623 52.689
0.15 44.056 44.848 45.702 46.588 47.291 47.839 48.523 49.579 50.957 52.022
0.16 43.419 44.182 45.008 45.868 46.606 47.186 47.849 48.901 50.289 51.353
0.17 42.789 43.526 44.323 45.155 45.921 46.533 47.182 48.227 49.620 50.682
0.18 42.168 42.879 43.648 44.453 45.237 45.882 46.519 47.557 48.950 50.009
0.19 41.556 42.245 42.987 43.763 44.559 45.232 45.862 46.892 48.280 49.334
0.20 40.955 41.623 42.340 43.088 43.888 44.586 45.212 46.233 47.610 48.660
0.22 39.783 40.419 41.091 41.788 42.578 43.309 43.932 44.933 46.278 47.317
0.24 38.652 39.267 39.904 40.557 41.320 42.064 42.686 43.663 44.967 45.989
0.25 38.100 38.709 39.333 39.967 40.713 41.456 42.078 43.042 44.323 45.336
0.26 37.556 38.163 38.776 39.393 40.121 40.859 41.481 42.432 43.688 44.690
0.28 36.495 37.102 37.702 38.294 38.982 39.702 40.321 41.244 42.448 43.428
0.30 35.465 36.079 36.675 37.251 37.904 38.598 39.212 40.104 41.256 42.210
0.32 34.464 35.090 35.690 36.259 36.881 37.546 38.153 39.014 40.113 41.040
0.34 33.491 34.131 34.741 35.310 35.909 36.545 37.145 37.975 39.022 39.920
0.35 33.016 33.663 34.279 34.850 35.440 36.063 36.659 37.474 38.496 39.379
0.36 32.547 33.202 33.824 34.399 34.981 35.593 36.185 36.985 37.982 38.851
0.38 31.631 32.299 32.936 33.520 34.094 34.685 35.270 36.040 36.989 37.830
0.40 30.745 31.424 32.075 32.671 33.241 33.818 34.397 35.139 36.042 36.854
0.42 29.888 30.575 31.238 31.847 32.419 32.986 33.562 34.277 35.137 35.922
0.44 29.063 29.753 30.427 31.047 31.624 32.187 32.760 33.451 34.269 35.029
0.45 28.663 29.352 30.030 30.656 31.236 31.798 32.370 33.051 33.849 34.596
0.46 28.270 28.959 29.640 30.271 30.854 31.415 31.988 32.658 33.437 34.171
0.48 27.511 28.194 28.877 29.517 30.107 30.670 31.243 31.893 32.635 33.347
0.50 26.784 27.458 28.141 28.785 29.382 29.948 30.523 31.154 31.862 32.553
0.55 25.113 25.748 26.412 27.054 27.661 28.238 28.817 29.409 30.040 30.683
0.60 23.646 24.226 24.851 25.470 26.072 26.652 27.231 27.791 28.358 28.960
0.65 22.366 22.885 23.459 24.038 24.619 25.189 25.759 26.289 26.803 27.367
0.70 21.253 21.711 22.228 22.758 23.303 23.851 24.401 24.901 25.370 25.899
0.80 19.424 19.783 20.193 20.618 21.072 21.547 22.031 22.469 22.867 23.325
0.90 17.958 18.262 18.599 18.943 19.310 19.701 20.106 20.481 20.824 21.214
1.00 16.696 16.986 17.293 17.591 17.900 18.224 18.561 18.881 19.182 19.513
1.10 15.537 15.841 16.150 16.438 16.722 17.008 17.300 17.583 17.854 18.139
1.20 14.429 14.759 15.090 15.390 15.676 15.953 16.227 16.491 16.745 17.003
1.30 13.355 13.712 14.072 14.396 14.700 14.988 15.265 15.526 15.776 16.024
1.40 12.321 12.698 13.082 13.432 13.759 14.067 14.362 14.633 14.888 15.138
1.50 11.341 11.726 12.125 12.494 12.845 13.175 13.489 13.776 14.042 14.303
1.60 10.431 10.811 11.214 11.592 11.956 12.305 12.636 12.939 13.218 13.493
1.70 9.602 9.966 10.360 10.736 11.104 11.461 11.807 12.123 12.414 12.704
1.80 8.861 9.201 9.576 9.940 10.303 10.661 11.009 11.333 11.631 11.932
1.90 8.208 8.518 8.868 9.212 9.558 9.907 10.253 10.576 10.878 11.185
2.00 7.642 7.921 8.239 8.556 8.881 9.213 9.550 9.868 10.166 10.473
2.50 5.836 5.980 6.142 6.315 6.502 6.704 6.917 7.117 7.333 7.567
3.00 4.945 5.040 5.132 5.229 5.332 5.440 5.550 5.663 5.800 5.930
3.50 4.295 4.390 4.478 4.566 4.651 4.735 4.820 4.910 5.000 5.090
4.00 3.678 3.780 3.891 3.991 4.087 4.178 4.270 4.360 4.445 4.525
5.00 2.615 2.722 2.828 2.935 3.041 3.146 3.240 3.340 3.435 3.530
6.00 1.909 1.990 2.067 2.150 2.237 2.325 2.410 2.490 2.580 2.670

ElementPmSmEuGdTbDyHoErTmYb
Z61626364656667686970
Method*RHF*RHFRHF*RHF*RHF*RHF*RHF*RHF*RHF*RHF
(sin [\theta])/λ (Å−1)          
0.00 61.000 62.000 63.000 64.000 65.000 66.000 67.000 68.000 69.000 70.000
0.01 60.932 61.934 62.936 63.936 64.938 65.939 66.940 67.941 68.943 69.944
0.02 60.734 61.740 62.746 63.749 64.755 65.760 66.763 67.769 68.773 69.777
0.03 60.417 61.428 62.441 63.447 64.461 65.471 66.476 67.491 68.500 69.509
0.04 59.998 61.017 62.036 63.044 64.071 65.088 66.093 67.120 68.136 69.151
0.05 59.497 60.525 61.552 62.557 63.603 64.627 65.627 66.673 67.696 68.717
0.06 58.936 59.972 61.007 62.004 63.073 64.105 65.096 66.166 67.195 68.223
0.07 58.333 59.377 60.419 61.400 62.499 63.538 64.513 65.613 66.649 67.684
0.08 57.703 58.753 59.801 60.762 61.894 62.940 63.895 65.028 66.070 67.112
0.09 57.057 58.113 59.166 60.102 61.270 62.321 63.251 64.420 65.468 66.516
0.10 56.403 57.463 58.521 59.427 60.634 61.689 62.591 63.798 64.852 65.904
0.11 55.744 56.809 57.869 58.746 59.989 61.049 61.921 63.167 64.224 65.281
0.12 55.084 56.151 57.214 58.061 59.340 60.403 61.247 62.528 63.589 64.650
0.13 54.422 55.491 56.555 57.375 58.686 59.752 60.569 61.884 62.948 64.012
0.14 53.758 54.828 55.893 56.690 58.029 59.097 59.891 61.234 62.301 63.368
0.15 53.091 54.163 55.228 56.005 57.366 58.437 59.212 60.578 61.648 62.718
0.16 52.422 53.493 54.559 55.321 56.699 57.771 58.532 59.917 60.989 62.062
0.17 51.749 52.821 53.886 54.637 56.028 57.101 57.851 59.249 60.324 61.399
0.18 51.074 52.145 53.210 53.953 55.351 56.425 57.169 58.576 59.653 60.729
0.19 50.398 51.467 52.530 53.270 54.670 55.744 56.486 57.897 58.975 60.053
0.20 49.720 50.786 51.847 52.588 53.985 55.059 55.803 57.213 58.292 59.371
0.22 48.367 49.426 50.480 51.227 52.610 53.681 54.435 55.833 56.912 57.992
0.24 47.026 48.074 49.119 49.878 51.234 52.300 53.070 54.445 55.521 56.601
0.25 46.364 47.406 48.444 49.209 50.549 51.611 52.390 53.750 54.825 55.903
0.26 45.710 46.743 47.775 48.546 49.868 50.926 51.714 53.058 54.130 55.206
0.28 44.427 45.443 46.458 47.240 48.523 49.570 50.375 51.683 52.748 53.817
0.30 43.186 44.180 45.176 45.965 47.208 48.240 49.059 50.329 51.384 52.444
0.32 41.991 42.961 43.935 44.729 45.929 46.944 47.772 49.004 50.046 51.095
0.34 40.844 41.789 42.740 43.533 44.690 45.686 46.520 47.712 48.739 49.774
0.35 40.289 41.221 42.160 42.951 44.087 45.073 45.908 47.081 48.099 49.127
0.36 39.747 40.666 41.591 42.380 43.496 44.471 45.305 46.459 47.469 48.488
0.38 38.697 39.589 40.489 41.272 42.346 43.299 44.131 45.246 46.237 47.239
0.40 37.694 38.559 39.433 40.207 41.241 42.171 42.996 44.075 45.046 46.029
0.42 36.735 37.573 38.421 39.184 40.179 41.086 41.903 42.945 43.896 44.859
0.44 35.815 36.627 37.451 38.203 39.160 40.042 40.849 41.857 42.786 43.728
0.45 35.370 36.169 36.980 37.726 38.665 39.536 40.337 41.327 42.246 43.178
0.46 34.933 35.720 36.519 37.259 38.180 39.039 39.834 40.808 41.715 42.637
0.48 34.085 34.848 35.623 36.352 37.237 38.073 38.856 39.797 40.682 41.583
0.50 33.269 34.008 34.761 35.479 36.329 37.143 37.914 38.822 39.686 40.565
0.55 31.349 32.036 32.737 33.428 34.199 34.958 35.699 36.531 37.342 38.169
0.60 29.581 30.222 30.877 31.543 32.243 32.953 33.664 34.425 35.187 35.964
0.65 27.948 28.547 29.161 29.802 30.438 31.103 31.786 32.483 33.198 33.929
0.70 26.442 27.002 27.576 28.192 28.772 29.394 30.049 30.688 31.359 32.045
0.80 23.796 24.281 24.781 25.335 25.822 26.366 26.958 27.497 28.086 28.690
0.90 21.616 22.030 22.459 22.940 23.353 23.821 24.343 24.800 25.311 25.837
1.00 19.853 20.202 20.565 20.970 21.323 21.721 22.167 22.556 22.995 23.447
1.10 18.430 18.728 19.035 19.372 19.675 20.011 20.385 20.718 21.089 21.474
1.20 17.262 17.523 17.789 18.072 18.338 18.623 18.934 19.221 19.535 19.860
1.30 16.266 16.507 16.747 16.995 17.234 17.483 17.746 17.998 18.266 18.542
1.40 15.378 15.613 15.841 16.072 16.296 16.522 16.753 16.980 17.215 17.454
1.50 14.551 14.790 15.020 15.247 15.465 15.680 15.895 16.107 16.321 16.536
1.60 13.755 14.005 14.245 14.477 14.697 14.913 15.123 15.329 15.533 15.735
1.70 12.980 13.243 13.494 13.741 13.968 14.190 14.406 14.612 14.815 15.013
1.80 12.220 12.497 12.763 13.022 13.259 13.491 13.718 13.929 14.137 14.338
1.90 11.481 11.767 12.044 12.317 12.564 12.808 13.047 13.267 13.483 13.691
2.00 10.773 11.064 11.345 11.631 11.886 12.141 12.392 12.621 12.847 13.064
2.50 7.817 8.083 8.348 8.683 8.983 9.267 9.533 9.783 10.033 10.267
3.00 6.088 6.250 6.435 6.588 6.775 6.963 7.163 7.375 7.588 7.788
3.50 5.180 5.280 5.378 5.490 5.610 5.720 5.850 5.980 6.110 6.250
4.00 4.600 4.675 4.750 4.830 4.915 5.000 5.090 5.180 5.280 5.380
5.00 3.625 3.720 3.812 3.905 3.990 4.075 4.155 4.235 4.310 4.380
6.00 2.770 2.865 2.965 3.070 3.170 3.270 3.355 3.440 3.520 3.600

ElementLuHfTaWReOsIrPtAuHg
Z71727374757677787980
Method*RHF*RHF*RHF*RHF*RHF*RHF*RHF*RHFRHFRHF
(sin [\theta])/λ (Å−1)          
0.00 71.000 72.000 73.000 74.000 75.000 76.000 77.000 78.000 79.000 80.000
0.01 70.944 71.945 72.946 73.948 74.949 75.950 76.951 77.955 78.957 79.556
0.02 70.778 71.783 72.788 73.793 74.797 75.801 76.806 77.820 78.826 79.819
0.03 70.509 71.518 72.529 73.539 74.548 75.538 76.567 77.599 78.609 79.595
0.04 70.148 71.161 72.177 73.194 74.209 75.225 76.240 77.295 78.311 79.286
0.05 69.707 70.723 71.745 72.767 73.788 74.810 75.832 76.914 77.936 78.899
0.06 69.202 70.217 71.242 72.269 73.295 74.323 75.352 76.462 77.491 78.439
0.07 68.646 69.656 70.680 71.711 72.740 73.772 74.806 75.946 76.981 77.913
0.08 68.051 69.052 70.072 71.103 72.132 73.167 74.206 75.373 76.414 77.330
0.09 67.429 68.416 69.428 70.455 71.482 72.518 73.558 74.751 75.797 76.696
0.10 66.789 67.757 68.758 69.778 70.799 71.832 72.872 74.086 75.135 76.018
0.11 66.137 67.083 68.069 69.078 70.091 71.119 72.156 73.386 74.437 75.303
0.12 65.477 66.400 67.367 68.363 69.365 70.384 71.416 72.656 73.706 74.559
0.13 64.813 65.711 66.658 67.637 68.625 69.634 70.658 71.902 72.950 73.790
0.14 64.146 65.019 65.944 66.906 67.878 68.874 69.887 71.130 72.173 73.001
0.15 63.478 64.326 65.229 66.172 67.126 68.107 69.108 70.343 71.380 72.198
0.16 62.807 63.634 64.515 65.437 66.372 67.337 68.324 69.546 70.575 71.385
0.17 62.134 62.942 63.802 64.703 65.619 66.566 67.538 68.742 69.761 70.564
0.18 61.460 62.251 63.090 63.972 64.868 65.797 66.752 67.934 68.941 69.740
0.19 60.783 61.560 62.382 63.243 64.121 65.031 65.969 67.125 68.119 68.914
0.20 60.103 60.870 61.675 62.519 63.378 64.269 65.189 66.317 67.296 68.088
0.22 58.739 59.492 60.271 61.082 61.906 62.761 63.645 64.709 65.657 66.447
0.24 57.369 58.119 58.880 59.663 60.457 61.278 62.127 63.125 64.039 64.828
0.25 56.683 57.434 58.189 58.961 59.742 60.548 61.380 62.344 63.241 64.029
0.26 55.998 56.752 57.502 58.265 59.034 59.825 60.641 61.571 62.452 63.239
0.28 54.634 55.396 56.141 56.888 57.637 58.403 59.189 60.056 60.902 61.687
0.30 53.282 54.054 54.799 55.536 56.270 57.013 57.773 58.582 59.395 60.177
0.32 51.950 52.733 53.479 54.210 54.932 55.658 56.395 57.152 57.935 58.711
0.34 50.642 51.435 52.185 52.912 53.627 54.339 55.056 55.769 56.523 57.292
0.35 49.998 50.796 51.548 52.274 52.986 53.692 54.401 55.094 55.835 56.600
0.36 49.363 50.164 50.918 51.644 52.354 53.055 53.756 54.432 55.160 55.920
0.38 48.117 48.924 49.683 50.408 51.114 51.807 52.496 53.141 53.846 54.595
0.40 46.906 47.717 48.479 49.205 49.910 50.596 51.274 51.897 52.581 53.318
0.42 45.731 46.543 47.308 48.036 48.739 49.422 50.091 50.697 51.363 52.088
0.44 44.593 45.405 46.171 46.900 47.603 48.283 48.946 49.540 50.191 50.902
0.45 44.038 44.849 45.615 46.344 47.048 47.726 48.387 48.977 49.622 50.326
0.46 43.492 44.301 45.068 45.797 46.501 47.179 47.837 48.424 49.063 49.761
0.48 42.427 43.232 43.998 44.728 45.432 46.109 46.765 47.347 47.976 48.661
0.50 41.398 42.197 42.962 43.691 44.396 45.072 45.726 46.308 46.929 47.601
0.55 38.970 39.752 40.508 41.236 41.940 42.617 43.269 43.860 44.469 45.113
0.60 36.733 37.494 38.238 38.960 39.662 40.340 40.994 41.601 42.207 42.829
0.65 34.666 35.404 36.132 36.846 37.544 38.222 38.878 39.502 40.110 40.718
0.70 32.752 33.465 34.175 34.878 35.569 36.244 36.901 37.539 38.153 38.753
0.80 29.334 29.992 30.658 31.327 31.993 32.654 33.305 33.958 34.581 35.176
0.90 26.413 27.008 27.618 28.238 28.865 29.495 30.125 30.766 31.387 31.980
1.00 23.950 24.473 25.016 25.576 26.148 26.732 27.323 27.930 28.530 29.112
1.10 21.902 22.352 22.823 23.313 23.821 24.345 24.882 25.437 25.998 26.554
1.20 20.219 20.598 20.998 21.418 21.856 22.314 22.789 23.281 23.789 24.303
1.30 18.842 19.159 19.494 19.847 20.219 20.610 21.019 21.445 21.892 22.354
1.40 17.709 17.975 18.256 18.552 18.864 19.194 19.541 19.902 20.287 20.692
1.50 16.759 16.988 17.228 17.478 17.742 18.019 18.312 18.616 18.943 19.290
1.60 15.939 16.145 16.356 16.575 16.801 17.038 17.287 17.545 17.821 18.116
1.70 15.208 15.403 15.598 15.796 15.998 16.206 16.422 16.644 16.880 17.131
1.80 14.534 14.727 14.916 15.104 15.293 15.483 15.678 15.875 16.081 16.298
1.90 13.894 14.091 14.282 14.469 14.653 14.835 15.018 15.202 15.388 15.581
2.00 13.277 13.481 13.679 13.871 14.057 14.239 14.418 14.595 14.770 14.949
2.50 10.500 10.733 10.950 11.167 11.383 11.583 11.783 11.983 12.168 12.360
3.00 8.013 8.238 8.480 8.706 8.938 9.163 9.400 9.620 9.826 10.049
3.50 6.400 6.560 6.740 6.900 7.080 7.270 7.460 7.650 7.878 8.081
4.00 5.490 5.600 5.710 5.840 5.960 6.080 6.210 6.340 6.489 6.644
5.00 4.450 4.520 4.585 4.650 4.715 4.788 4.860 4.935 5.010 5.090
6.00 3.680 3.755 3.825 3.900 3.970 4.035 4.105 4.175 4.244 4.310

ElementTlPbBiPoAtRnFrRaAcTh
Z81828384858687888990
Method*RHFRHFRHF*RHF*RHFRHF*RHF*RHF*RHF*RHF
(sin [\theta])/λ (Å−1)          
0.00 81.000 82.000 83.000 84.000 85.000 86.000 87.000 88.000 89.000 90.000
0.01 80.950 81.949 82.947 83.944 84.944 85.945 86.922 87.915 88.915 89.916
0.02 80.799 81.792 82.784 83.778 84.776 85.777 86.694 87.664 88.664 89.669
0.03 80.553 81.536 82.518 83.506 84.502 85.502 86.332 87.263 88.260 89.269
0.04 80.217 81.186 82.154 83.134 84.125 85.123 85.854 86.734 87.723 88.735
0.05 79.798 80.750 81.700 82.669 83.654 84.649 85.286 86.104 87.077 88.085
0.06 79.305 80.237 81.167 82.121 83.098 84.087 84.647 85.397 86.346 87.344
0.07 78.748 79.656 80.563 81.501 82.466 83.448 83.955 84.638 85.553 86.533
0.08 78.134 79.018 79.901 80.819 81.770 82.742 83.222 83.845 84.719 85.672
0.09 77.473 78.332 79.189 80.086 81.020 81.979 82.457 83.030 83.859 84.779
0.10 76.773 77.607 78.438 79.312 80.226 81.169 81.666 82.202 82.985 83.867
0.11 76.042 76.851 77.657 78.506 79.398 80.322 80.852 81.368 82.105 82.946
0.12 75.284 76.071 76.852 77.677 78.545 79.448 80.018 80.528 81.225 82.025
0.13 74.507 75.274 76.032 76.831 77.674 78.554 79.167 79.685 80.348 81.107
0.14 73.715 74.464 75.202 75.976 76.794 77.648 78.303 78.839 79.474 80.196
0.15 72.912 73.645 74.365 75.117 75.908 76.737 77.430 77.990 78.605 79.294
0.16 72.101 72.822 73.527 74.257 75.023 75.826 76.550 77.138 77.739 78.400
0.17 71.285 71.997 72.689 73.400 74.143 74.920 75.667 76.285 76.879 77.516
0.18 70.467 71.172 71.855 72.549 73.269 74.021 74.785 75.431 76.023 76.642
0.19 69.648 70.349 71.026 71.706 72.405 73.133 73.907 74.578 75.172 75.777
0.20 68.830 69.530 70.203 70.871 71.553 72.258 73.035 73.728 74.326 74.922
0.22 67.205 67.907 68.578 69.232 69.885 70.552 71.320 72.043 72.654 73.242
0.24 65.600 66.310 66.987 67.634 68.269 68.907 69.653 70.389 71.014 71.602
0.25 64.807 65.523 66.204 66.852 67.481 68.109 68.841 69.576 70.208 70.798
0.26 64.022 64.743 65.430 66.080 66.706 67.325 68.043 68.775 69.412 70.005
0.28 62.478 63.210 63.909 64.567 65.193 65.802 66.491 67.210 67.855 68.454
0.30 60.970 61.712 62.425 63.093 63.725 64.332 64.996 65.696 66.345 66.951
0.32 59.503 60.253 60.977 61.658 62.301 62.912 63.556 64.235 64.884 65.497
0.34 58.079 58.833 59.566 60.260 60.915 61.535 62.167 62.826 63.473 64.091
0.35 57.383 58.138 58.875 59.575 60.236 60.862 61.489 62.140 62.785 63.405
0.36 56.698 57.453 58.193 58.899 59.566 60.198 60.823 61.466 62.110 62.731
0.38 55.362 56.116 56.859 57.573 58.253 58.898 59.520 60.151 60.792 61.416
0.40 54.072 54.820 55.563 56.283 56.974 57.631 58.256 58.879 59.517 60.143
0.42 52.826 53.567 54.306 55.029 55.728 56.397 57.026 57.646 58.282 58.910
0.44 51.625 52.356 53.089 53.811 54.515 55.194 55.829 56.448 57.084 57.713
0.45 51.041 51.766 52.495 53.215 53.921 54.604 55.242 55.862 56.497 57.127
0.46 50.467 51.187 51.910 52.629 53.335 54.021 54.663 55.284 55.919 56.550
0.48 49.352 50.058 50.771 51.483 52.189 52.879 53.527 54.151 54.787 55.419
0.50 48.276 48.969 49.669 50.373 51.075 51.767 52.420 53.048 53.684 54.317
0.55 45.753 46.411 47.077 47.752 48.435 49.119 49.777 50.413 51.050 51.684
0.60 43.442 44.069 44.700 45.343 45.997 46.659 47.310 47.948 48.580 49.211
0.65 41.313 41.914 42.517 43.127 43.750 44.384 45.017 45.646 46.268 46.889
0.70 39.337 39.921 40.501 41.085 41.678 42.281 42.891 43.504 44.110 44.716
0.80 35.755 36.322 36.879 37.430 37.980 38.533 39.095 39.664 40.229 40.795
0.90 32.561 33.127 33.680 34.220 34.751 35.277 35.804 36.335 36.863 37.391
1.00 29.687 30.252 30.805 31.344 31.872 32.389 32.900 33.408 33.912 34.413
1.10 27.109 27.662 28.208 28.744 29.271 29.787 30.292 30.790 31.283 31.770
1.20 24.824 25.350 25.875 26.397 26.915 27.426 27.926 28.418 28.906 29.387
1.30 22.827 23.313 23.804 24.298 24.794 25.291 25.779 26.263 26.744 27.219
1.40 21.110 21.546 21.992 22.446 22.909 23.379 23.845 24.312 24.779 25.244
1.50 19.652 20.034 20.429 20.836 21.256 21.689 22.123 22.564 23.008 23.454
1.60 18.424 18.754 19.097 19.453 19.826 20.215 20.608 21.014 21.427 21.846
1.70 17.394 17.674 17.969 18.277 18.602 18.944 19.295 19.660 20.036 20.421
1.80 16.524 16.764 17.017 17.281 17.562 17.859 18.165 18.488 18.823 19.170
1.90 15.780 15.989 16.207 16.435 16.677 16.934 17.199 17.481 17.776 18.083
2.00 15.131 15.317 15.510 15.711 15.922 16.143 16.377 16.623 16.880 17.149
2.50 12.530 12.724 12.896 13.060 13.230 13.386 13.550 13.700 13.860 14.020
3.00 10.270 10.482 10.690 10.900 11.090 11.282 11.460 11.640 11.815 11.980
3.50 8.290 8.495 8.704 8.910 9.120 9.329 9.530 9.730 9.930 10.130
4.00 6.800 6.973 7.145 7.320 7.500 7.686 7.878 8.070 8.255 8.440
5.00 5.175 5.260 5.351 5.440 5.540 5.650 5.755 5.870 5.933 6.118
6.00 4.374 4.441 4.505 4.567 4.630 4.702 4.768 4.840 4.910 4.982

ElementPaUNpPuAmCmBkCf
Z9192939495969798
Method*RHFRHF*RHF*RHF*RHF*RHF*RHF*RHF
(sin [\theta])/λ (Å−1)        
0.00 91.000 92.000 93.000 94.000 95.000 96.000 97.000 98.000
0.01 90.919 91.922 92.922 93.924 94.926 95.926 96.928 97.929
0.02 90.678 91.687 92.691 93.701 94.706 95.708 96.713 97.718
0.03 90.290 91.307 92.318 93.340 94.352 95.354 96.365 97.375
0.04 89.772 90.798 91.817 92.857 93.877 94.877 95.895 96.912
0.05 89.144 90.180 91.208 92.271 93.299 94.294 95.320 96.344
0.06 88.427 89.474 90.510 91.601 92.638 93.623 94.656 95.688
0.07 87.644 88.699 89.742 90.866 91.910 92.879 93.920 94.961
0.08 86.813 87.874 88.923 90.082 91.131 92.081 93.129 94.176
0.09 85.950 87.014 88.067 89.261 90.315 91.241 92.294 93.347
0.10 85.066 86.130 87.186 88.413 89.470 90.371 91.429 92.486
0.11 84.170 85.232 86.288 87.547 88.605 89.479 90.540 91.601
0.12 83.269 84.326 85.380 86.665 87.723 88.573 89.635 90.699
0.13 82.366 83.417 84.467 85.772 86.829 87.656 88.718 89.783
0.14 81.463 82.505 83.550 84.870 85.924 86.731 87.793 88.858
0.15 80.563 81.595 82.632 83.961 85.011 85.802 86.862 87.926
0.16 79.665 80.685 81.715 83.044 84.090 84.869 85.926 86.989
0.17 78.771 79.779 80.799 82.123 83.163 83.934 84.988 86.048
0.18 77.881 78.875 79.885 81.198 82.231 82.998 84.047 85.103
0.19 76.995 77.975 78.973 80.271 81.296 82.062 83.105 84.157
0.20 76.115 77.080 78.066 79.343 80.360 81.126 82.163 83.210
0.22 74.375 75.308 76.267 77.493 78.490 79.263 80.285 81.318
0.24 72.668 73.568 74.496 75.663 76.636 77.419 78.421 79.437
0.25 71.829 72.712 73.624 74.759 75.719 76.507 77.498 78.504
0.26 71.001 71.866 72.763 73.865 74.811 75.603 76.582 77.577
0.28 69.380 70.211 71.074 72.110 73.027 73.824 74.777 75.749
0.30 67.810 68.607 69.436 70.408 71.293 72.091 73.016 73.960
0.32 66.294 67.058 67.853 68.763 69.615 70.409 71.303 72.219
0.34 64.832 65.564 66.326 67.178 67.997 68.783 69.645 70.531
0.35 64.121 64.838 65.584 66.409 67.212 67.991 68.838 69.707
0.36 63.423 64.126 64.857 65.655 66.441 67.214 68.045 68.898
0.38 62.066 62.742 63.443 64.193 64.947 65.705 66.503 67.325
0.40 60.758 61.409 62.083 62.789 63.513 64.254 65.020 65.810
0.42 59.495 60.125 60.775 61.442 62.137 62.859 63.595 64.354
0.44 58.274 58.886 59.514 60.147 60.816 61.519 62.226 62.954
0.45 57.679 58.283 58.901 59.518 60.175 60.869 61.562 62.276
0.46 57.093 57.689 58.298 58.901 59.546 60.231 60.910 61.610
0.48 55.948 56.531 57.124 57.702 58.325 58.992 59.646 60.319
0.50 54.836 55.410 55.989 56.544 57.148 57.798 58.430 59.078
0.55 52.191 52.748 53.303 53.819 54.385 54.998 55.581 56.176
0.60 49.719 50.268 50.808 51.302 51.842 52.427 52.974 53.528
0.65 47.405 47.950 48.483 48.967 49.490 50.052 50.574 51.098
0.70 45.241 45.784 46.312 46.794 47.307 47.850 48.354 48.858
0.80 41.333 41.869 42.390 42.879 43.380 43.894 44.380 44.859
0.90 37.930 38.454 38.966 39.465 39.958 40.449 40.926 41.395
1.00 34.946 35.458 35.961 36.465 36.952 37.426 37.898 38.361
1.10 32.292 32.794 33.289 33.793 34.276 34.740 35.209 35.671
1.20 29.897 30.391 30.879 31.379 31.858 32.318 32.786 33.247
1.30 27.714 28.199 28.680 29.172 29.648 30.106 30.572 31.033
1.40 25.720 26.192 26.662 27.142 27.611 28.068 28.530 28.989
1.50 23.905 24.360 24.813 25.275 25.733 26.184 26.639 27.093
1.60 22.266 22.699 23.128 23.566 24.006 24.446 24.889 25.332
1.70 20.807 21.207 21.609 22.019 22.435 22.857 23.281 23.708
1.80 19.518 19.886 20.253 20.630 21.018 21.415 21.815 22.221
1.90 18.394 18.723 19.055 19.398 19.754 20.121 20.496 20.872
2.00 17.423 17.713 18.012 18.319 18.640 18.975 19.315 19.665
2.50 14.180 14.341 14.503 14.664 14.826 14.988 15.150 15.311
3.00 12.150 12.294 12.475 12.656 12.838 13.019 13.200 13.381
3.50 10.320 10.495 10.695 10.895 11.095 11.295 11.495 11.695
4.00 8.630 8.823 9.008 9.193 9.378 9.563 9.748 9.933
5.00 6.250 6.378 6.489 6.602 6.713 6.825 6.937 7.049
6.00 5.055 5.136 5.206 5.275 5.345 5.414 5.484 5.553

For a detailed study of the effect of changes in the electron density due to chemical bonding and lattice formation, a more general procedure is necessary, as described in Subsection 6.1.1.4[link]. The changes due to chemical bonding are small in absolute terms, and are relatively small except in the case of hydrogen.

A more approximate treatment is adequate for many purposes. An isotropic approximation to the scattering factor for bonded hydrogen, based on an analysis of the hydrogen molecule by Stewart, Davidson & Simpson (1965[link]), is listed in Table 6.1.1.2[link].

Table 6.1.1.2| top | pdf |
Spherical bonded hydrogen-atom scattering factors from Stewart, Davidson & Simpson (1965[link])

(sin [\theta])/λ (Å−1)f
0.0000 1.0000
0.0215 0.9924
0.0429 0.9704
0.0644 0.9352
0.0859 0.8892
0.1073 0.8350
0.1288 0.7752
0.1503 0.7125
0.1718 0.6492
0.1932 0.5871
0.2147 0.5277
0.2576 0.4201
0.3006 0.3301
0.3435 0.2573
0.3864 0.1998
0.4294 0.1552
0.4723 0.1208
0.5153 0.0945
0.5582 0.0744
0.6011 0.0592
0.6441 0.0474
0.6870 0.0383
0.7300 0.0311
0.7729 0.0254
0.8158 0.0208
0.8588 0.0171
0.9017 0.0140
0.9447 0.0116
0.9876 0.0096
1.0305 0.0080
1.0735 0.0066
1.1164 0.0056
1.1593 0.0047
1.2023 0.0040
1.2452 0.0035
1.2882 0.0031
1.3311 0.0027
1.3740 0.0025
1.4170 0.0022
1.4599 0.0020
1.5029 0.0018
1.5458 0.0016
1.5887 0.0015
1.6317 0.0013
1.6746 0.0011
1.7176 0.0010

Scattering for ionic models of solids may be related to the scattering factors for the corresponding free ions. Values for some of the more chemically significant ions are listed in Table 6.1.1.3[link]. For H, Li+ and Be2+ these are based on the correlated electron calculations of Thakkar & Smith (1992[link]). For other ions lighter than rubidium, values are based on the Hartree–Fock calculations of Cromer & Mann (1968[link]), using the wavefunctions of Mann (1968b[link]). For the heavier ions, the calculations are by Cromer & Waber (1968[link]), based on relativistic Dirac–Slater wavefunctions, which are a good approximation to the corresponding relativistic Hartree–Fock wavefunctions. If ionic scattering factors are required for values of [(\sin\theta)/\lambda] greater than those shown in Table 6.1.1.3[link], the free-atom scattering factors of Table 6.1.1.1[link] can be used because high-angle scattering is dominated by core electrons and is therefore very little affected by ionicity.

Table 6.1.1.3| top | pdf |
Mean atomic scattering factors in electrons for chemically significant ions

Methods: C: correlated; HF: non-relativistic Hartree–Fock; RHF: relativistic Hartree–Fock; *DS: modified Dirac–Slater.

ElementH1−Li1+Be2+CvalO1−F1−Na1+Mg2+Al3+Sival
Z13468911121314
MethodCCCHFHFHFRHFRHFHFHF
(sin [\theta])/λ (Å−1)          
0.00 2.000 2.000 2.000 6.000 9.000 10.000 10.000 10.000 10.000 14.000
0.01 1.983 1.999 1.999 5.989 8.986 9.988 9.995 9.997 9.997 13.973
0.02 1.933 1.997 1.999 5.956 8.945 9.953 9.981 9.986 9.989 13.894
0.03 1.857 1.994 1.997 5.903 8.878 9.895 9.958 9.969 9.976 13.766
0.04 1.763 1.990 1.995 5.829 8.785 9.816 9.925 9.945 9.957 13.593
0.05 1.659 1.984 1.992 5.738 8.670 9.716 9.883 9.914 9.933 13.381
0.06 1.550 1.977 1.988 5.629 8.534 9.597 9.833 9.876 9.904 13.138
0.07 1.442 1.968 1.983 5.507 8.381 9.461 9.773 9.832 9.870 12.870
0.08 1.338 1.959 1.978 5.372 8.211 9.309 9.705 9.782 9.831 12.586
0.09 1.238 1.948 1.973 5.227 8.029 9.144 9.630 9.725 9.787 12.293
0.10 1.145 1.936 1.966 5.074 7.836 8.967 9.546 9.662 9.738 11.995
0.11 1.058 1.923 1.959 4.916 7.635 8.781 9.455 9.594 9.684 11.700
0.12 0.978 1.909 1.952 4.754 7.429 8.586 9.357 9.519 9.625 11.410
0.13 0.904 1.894 1.944 4.591 7.218 8.386 9.253 9.440 9.563 11.130
0.14 0.836 1.877 1.935 4.428 7.005 8.181 9.142 9.355 9.495 10.862
0.15 0.773 1.860 1.925 4.267 6.792 7.973 9.026 9.265 9.424 10.608
0.16 0.715 1.842 1.915 4.109 6.579 7.762 8.904 9.171 9.349 10.368
0.17 0.661 1.823 1.905 3.954 6.368 7.551 8.777 9.072 9.270 10.143
0.18 0.612 1.804 1.894 3.805 6.160 7.341 8.647 8.969 9.187 9.933
0.19 0.567 1.783 1.882 3.661 5.956 7.131 8.512 8.862 9.101 9.737
0.20 0.526 1.762 1.870 3.523 5.756 6.924 8.374 8.751 9.011 9.553
0.22 0.452 1.718 1.845 3.266 5.371 6.517 8.089 8.521 8.823 9.222
0.24 0.390 1.671 1.817 3.035 5.008 6.126 7.795 8.280 8.623 8.931
0.25 0.362 1.647 1.803 2.930 4.836 5.937 7.646 8.156 8.520 8.798
0.26 0.337 1.623 1.788 2.831 4.670 5.753 7.496 8.030 8.414 8.671
0.28 0.291 1.573 1.758 2.651 4.357 5.399 7.195 7.774 8.198 8.435
0.30 0.253 1.523 1.726 2.495 4.068 5.067 6.894 7.513 7.975 8.214
0.32 0.220 1.471 1.692 2.358 3.804 4.756 6.597 7.251 7.747 8.005
0.34 0.192 1.419 1.658 2.241 3.564 4.467 6.304 6.987 7.515 7.803
0.35 0.179 1.394 1.641 2.188 3.452 4.330 6.160 6.856 7.399 7.704
0.36 0.168 1.368 1.623 2.139 3.345 4.199 6.018 6.725 7.282 7.606
0.38 0.147 1.316 1.587 2.050 3.147 3.951 5.739 6.465 7.047 7.410
0.40 0.129 1.265 1.551 1.974 2.969 3.724 5.471 6.210 6.813 7.215
0.42 0.113 1.215 1.514 1.907 2.808 3.514 5.212 5.959 6.581 7.021
0.44 0.100 1.165 1.476 1.849 2.663 3.322 4.964 5.715 6.350 6.826
0.45 0.094 1.141 1.458 1.822 2.597 3.233 4.845 5.595 6.237 6.729
0.46 0.089 1.117 1.439 1.798 2.533 3.147 4.728 5.477 6.124 6.632
0.48 0.079 1.069 1.401 1.752 2.417 2.987 4.503 5.247 5.901 6.437
0.50 0.070 1.023 1.364 1.711 2.313 2.841 4.290 5.025 5.683 6.244
0.55 0.0526 0.914 1.270 1.624 2.097 2.531 3.808 4.508 5.162 5.766
0.60 0.0401 0.814 1.179 1.552 1.934 2.288 3.395 4.046 4.681 5.303
0.65 0.0311 0.724 1.091 1.488 1.808 2.096 3.046 3.641 4.243 4.865
0.70 0.0243 0.643 1.007 1.428 1.710 1.945 2.753 3.288 3.851 4.455
0.80 0.0155 0.507 0.852 1.315 1.567 1.729 2.305 2.724 3.195 3.734
0.90 0.0102 0.400 0.717 1.204 1.463 1.585 1.997 2.315 2.693 3.150
1.00 0.0070 0.317 0.602 1.096 1.376 1.481 1.785 2.023 2.319 2.691
1.10 0.0049 0.253 0.505 0.992 1.296 1.397 1.635 1.813 2.041 2.338
1.20 0.0036 0.203 0.424 0.894 1.219 1.322 1.524 1.662 1.837 2.069
1.30 0.0026 0.164 0.357 0.802 1.143 1.252 1.438 1.548 1.685 1.867
1.40 0.0020 0.133 0.301 0.718 1.067 1.184 1.367 1.460 1.570 1.713
1.50 0.0015 0.109 0.255 0.642 0.994 1.117 1.304 1.388 1.479 1.595
1.60 0.0012 0.090 0.216       1.246 1.326    
1.70 0.0009 0.075 0.184       1.191 1.270    
1.80 0.0008 0.062 0.157       1.137 1.218    
1.90 0.0006 0.053 0.135       1.084 1.168    
2.00 0.0005 0.044 0.116       1.032 1.119    

ElementSi4+Cl1−K1+Ca2+Sc3+Ti2+Ti3+Ti4+V2+V3+
Z14171920212222222323
MethodHFRHFRHFRHFHFHFHFHFRHFHF
(sin [\theta])/λ (Å−1)          
0.00 10.000 18.000 18.000 18.000 18.000 20.000 19.000 18.000 21.000 20.000
0.01 9.998 17.972 17.986 17.989 17.991 19.988 18.990 17.992 20.988 19.990
0.02 9.991 17.888 17.943 17.955 17.963 19.951 18.962 17.969 20.952 19.961
0.03 9.981 17.751 17.872 17.899 17.917 19.891 18.914 17.930 20.892 19.913
0.04 9.966 17.563 17.774 17.821 17.853 19.807 18.848 17.877 20.808 19.846
0.05 9.947 17.330 17.649 17.721 17.771 19.701 18.764 17.808 20.702 19.760
0.06 9.924 17.057 17.499 17.601 17.672 19.572 18.662 17.725 20.573 19.657
0.07 9.896 16.750 17.325 17.462 17.556 19.423 18.543 17.628 20.424 19.536
0.08 9.865 16.415 17.129 17.303 17.424 19.253 18.407 17.516 20.255 19.398
0.09 9.829 16.058 16.912 17.127 17.278 19.065 18.255 17.392 20.066 19.244
0.10 9.790 15.685 16.677 16.935 17.116 18.860 18.089 17.255 19.861 19.075
0.11 9.747 15.301 16.426 16.727 16.941 18.639 17.909 17.106 19.639 18.892
0.12 9.700 14.911 16.160 16.506 16.754 18.404 17.716 16.946 19.402 18.695
0.13 9.649 14.519 15.882 16.272 16.555 18.156 17.510 16.775 19.152 18.485
0.14 9.595 14.130 15.594 16.028 16.345 17.896 17.294 16.593 18.890 18.265
0.15 9.537 13.747 15.297 15.774 16.126 17.626 17.067 16.403 18.618 18.033
0.16 9.476 13.371 14.994 15.512 15.898 17.348 16.832 16.205 18.336 17.793
0.17 9.411 13.006 14.688 15.244 15.662 17.062 16.589 15.998 18.047 17.544
0.18 9.343 12.653 14.378 14.970 15.421 16.771 16.339 15.785 17.751 17.287
0.19 9.272 12.313 14.069 14.692 15.173 16.475 16.083 15.566 17.450 17.025
0.20 9.199 11.987 13.760 14.412 14.922 16.176 15.822 15.342 17.146 16.757
0.22 9.043 11.379 13.150 13.850 14.410 15.574 15.291 14.881 16.529 16.210
0.24 8.877 10.832 12.560 13.292 13.893 14.972 14.752 14.408 15.910 15.653
0.25 8.790 10.580 12.275 13.017 13.634 14.673 14.482 14.170 15.602 15.373
0.26 8.701 10.343 11.997 12.745 13.377 14.377 14.213 13.930 15.296 15.093
0.28 8.518 9.908 11.467 12.217 12.869 13.797 13.680 13.452 14.694 14.537
0.30 8.327 9.524 10.972 11.713 12.374 13.236 13.157 12.979 14.107 13.989
0.32 8.131 9.184 10.515 11.235 11.896 12.697 12.650 12.515 13.541 13.455
0.34 7.929 8.884 10.097 10.787 11.438 12.184 12.162 12.064 12.998 12.938
0.35 7.827 8.746 9.901 10.575 11.218 11.938 11.926 11.844 12.736 12.687
0.36 7.724 8.616 9.715 10.370 11.004 11.698 11.696 11.628 12.481 12.441
0.38 7.516 8.377 9.369 9.984 10.595 11.242 11.254 11.211 11.991 11.967
0.40 7.306 8.162 9.056 9.629 10.212 10.815 10.837 10.815 11.530 11.517
0.42 7.095 7.965 8.773 9.303 9.855 10.417 10.446 10.439 11.096 11.092
0.44 6.884 7.785 8.518 9.006 9.524 10.047 10.080 10.086 10.692 10.692
0.45 6.779 7.699 8.399 8.867 9.368 9.873 9.907 9.917 10.500 10.502
0.46 6.674 7.616 8.287 8.734 9.218 9.706 9.740 9.754 10.315 10.318
0.48 6.465 7.457 8.077 8.487 8.937 9.391 9.426 9.445 9.965 9.969
0.50 6.259 7.305 7.886 8.262 8.678 9.102 9.135 9.158 9.641 9.645
0.55 5.755 6.945 7.474 7.781 8.121 8.477 8.503 8.529 8.935 8.936
0.60 5.277 6.600 7.125 7.389 7.670 7.972 7.990 8.012 8.359 8.354
0.65 4.830 6.259 6.814 7.058 7.298 7.560 7.571 7.588 7.889 7.878
0.70 4.418 5.920 6.523 6.764 6.982 7.216 7.222 7.234 7.501 7.485
0.80 3.701 5.248 5.962 6.231 6.445 6.656 6.658 6.664 6.892 6.870
0.90 3.124 4.608 5.406 5.719 5.961 6.179 6.182 6.189 6.407 6.384
1.00 2.673 4.024 4.859 5.209 5.488 5.728 5.734 5.745 5.973 5.950
1.10 2.326 3.509 4.336 4.710 5.017 5.282 5.291 5.306 5.553 5.531
1.20 2.063 3.070 3.854 4.232 4.556 4.840 4.852 4.870 5.137 5.116
1.30 1.864 2.705 3.423 3.790 4.115 4.411 4.425 4.443 4.727 4.705
1.40 1.712 2.405 3.045 3.390 3.706 4.004 4.017 4.035 4.330 4.307
1.50 1.595 2.162 2.722 3.038 3.335 3.626 3.638 3.655 3.952 3.929
1.60   1.968 2.449 2.732         3.600  
1.70   1.811 2.221 2.470         3.278  
1.80   1.686 2.033 2.250         2.989  
1.90   1.585 1.877 2.064         2.731  
2.00   1.502 1.749 1.909         2.505  

ElementV5+Cr2+Cr3+Mn2+Mn3+Mn4+Fe2+Fe3+Co2+Co3+
Z23242425252526262727
MethodHFHFHFRHFHFHFRHFRHFRHFHF
(sin [\theta])/λ (Å−1)          
0.00 18.000 22.000 21.000 23.000 22.000 21.000 24.000 23.000 25.000 24.000
0.01 17.993 21.988 20.990 22.988 21.990 20.992 23.989 22.991 24.989 23.990
0.02 17.974 21.952 20.961 22.953 21.961 20.968 23.954 22.962 24.954 23.962
0.03 17.941 21.892 20.913 22.894 21.913 20.927 23.895 22.914 24.897 23.914
0.04 17.895 21.808 20.845 22.812 21.846 20.871 23.814 22.848 24.818 23.848
0.05 17.837 21.702 20.759 22.707 21.760 20.799 23.711 22.763 24.716 23.764
0.06 17.766 21.574 20.655 22.581 21.656 20.712 23.587 22.660 24.593 23.661
0.07 17.682 21.425 20.534 22.433 21.534 20.610 23.441 22.539 24.450 23.541
0.08 17.587 21.256 20.395 22.266 21.395 20.493 23.276 22.401 24.287 23.404
0.09 17.480 21.067 20.240 22.080 21.240 20.363 23.091 22.247 24.104 23.250
0.10 17.362 20.861 20.069 21.875 21.070 20.218 22.889 22.078 23.904 23.081
0.11 17.234 20.638 19.884 21.654 20.884 20.061 22.669 21.893 23.687 22.896
0.12 17.095 20.400 19.685 21.418 20.684 19.891 22.435 21.695 23.455 22.698
0.13 16.946 20.148 19.474 21.167 20.472 19.710 22.185 21.483 23.207 22.486
0.14 16.789 19.884 19.250 20.904 20.247 19.517 21.923 21.258 22.946 22.261
0.15 16.622 19.609 19.016 20.629 20.011 19.315 21.648 21.023 22.673 22.024
0.16 16.448 19.324 18.772 20.344 19.765 19.102 21.363 20.776 22.389 21.777
0.17 16.266 19.030 18.519 20.050 19.509 18.881 21.068 20.521 22.095 21.520
0.18 16.078 18.729 18.258 19.748 19.246 18.652 20.765 20.256 21.791 21.253
0.19 15.883 18.423 17.991 19.440 18.975 18.415 20.455 19.984 21.481 20.978
0.20 15.683 18.112 17.718 19.126 18.697 18.172 20.140 19.705 21.164 20.696
0.22 15.268 17.481 17.157 18.488 18.127 17.669 19.494 19.130 20.514 20.114
0.24 14.839 16.845 16.585 17.841 17.543 17.149 18.838 18.538 19.850 19.513
0.25 14.620 16.527 16.297 17.517 17.247 16.884 18.508 18.238 19.516 19.207
0.26 14.399 16.210 16.008 17.193 16.951 16.617 18.178 17.937 19.180 18.899
0.28 13.955 15.584 15.431 16.551 16.357 16.079 17.520 17.331 18.510 18.280
0.30 13.509 14.972 14.862 15.920 15.768 15.540 16.871 16.727 17.845 17.659
0.32 13.067 14.378 14.303 15.304 15.187 15.005 16.234 16.130 17.191 17.043
0.34 12.631 13.805 13.759 14.707 14.619 14.477 15.614 15.543 16.550 16.435
0.35 12.417 13.528 13.494 14.417 14.341 14.217 15.312 15.254 16.236 16.135
0.36 12.205 13.257 13.234 14.132 14.068 13.961 15.014 14.970 15.927 15.838
0.38 11.792 12.734 12.730 13.581 13.536 13.458 14.436 14.414 15.324 15.258
0.40 11.392 12.238 12.248 13.055 13.024 12.972 13.881 13.877 14.743 14.694
0.42 11.010 11.770 11.790 12.556 12.536 12.504 13.352 13.361 14.186 14.151
0.44 10.644 11.330 11.357 12.083 12.072 12.057 12.848 12.868 13.653 13.629
0.45 10.469 11.121 11.150 11.857 11.848 11.841 12.606 12.630 13.396 13.376
0.46 10.298 10.918 10.950 11.638 11.632 11.630 12.370 12.398 13.146 13.129
0.48 9.970 10.533 10.567 11.219 11.216 11.225 11.919 11.953 12.664 12.652
0.50 9.662 10.174 10.210 10.827 10.826 10.843 11.494 11.531 12.207 12.200
0.55 8.973 9.386 9.419 9.954 9.956 9.982 10.542 10.581 11.176 11.171
0.60 8.396 8.737 8.764 9.229 9.223 9.252 9.737 9.772 10.293 10.286
0.65 7.915 8.205 8.224 8.626 8.615 8.641 9.063 9.092 9.546 9.534
0.70 7.515 7.766 7.779 8.128 8.111 8.132 8.501 8.523 8.917 8.900
0.80 6.888 7.091 7.095 7.365 7.341 7.352 7.640 7.651 7.948 7.921
0.90 6.399 6.578 6.580 6.808 6.779 6.785 7.023 7.026 7.257 7.224
1.00 5.968 6.143 6.145 6.360 6.330 6.334 6.546 6.548 6.739 6.703
1.10 5.556 5.738 5.742 5.963 5.933 5.938 6.144 6.145 6.320 6.283
1.20 5.147 5.341 5.348 5.585 5.555 5.562 5.775 5.778 5.951 5.913
1.30 4.741 4.949 4.958 5.213 5.183 5.193 5.419 5.423 5.605 5.566
1.40 4.344 4.564 4.573 4.846 4.815 4.826 5.068 5.074 5.268 5.228
1.50 3.965 4.191 4.202 4.487 4.454 4.467 4.722 4.729 4.936 4.895
1.60       4.140     4.384 4.392 4.609  
1.70       3.810     4.058 4.066 4.291  
1.80       3.502     3.749 3.757 3.985  
1.90       3.218     3.459 3.467 3.694  
2.00