For weak scattering, treated within the Born approximation, the incoherent scattering cross section, (dσ/dΩ)inc, can be factorized as follows:
where (dσ/dΩ)0 is the cross section characterizing the interaction, in this case it is the Thomson cross section,
;
and
being the initial and final state photon polarization vectors. The dynamics of the target are contained in the incoherent scattering factor S(E1, E2, K, Z), which is usually a function of the energy transfer
, the momentum transfer K, and the atomic number Z.
The electromagnetic wave perturbs the electronic system through the vector potential A in the Hamiltonian ![[H={e\over me}{\bf p}\cdot {\bf A}+{e^2\over 2me^2}{\bf A}\cdot {\bf A}. \eqno (7.4.3.3)]](/teximages/cbch7o4/cbch7o4fd34.svg)
It produces photoelectric absorption through the
term taken in first order, Compton and Raman scattering through the
term and resonant Raman scattering through the
terms in second order.
If resonant scattering is neglected for the moment, the expression for the incoherent scattering cross section becomes
where the Born operator is summed over the j target electrons and the matrix element is summed over all final states accessible through energy conservation. In the high-energy limit of
, S(E1, E2, K, Z)
Z but as Table 7.4.3.1
shows this condition does not hold in the X-ray regime.
The evaluation of the matrix elements in equation (7.4.3.4)
was simplified by Waller & Hartree (1929
) who (i) set E2 = E1 and (ii) summed over all final states irrespective of energy conservation. Closure relationships were then invoked to reduce the incoherent scattering factor to an expression in terms of form factors
:
where
and
the latter term arising from exchange in the many-electron atom.
According to Currat, DeCicco & Weiss (1971
), equation (7.4.3.5)
can be improved by inserting the prefactor (E2/E1)2, where E2 is calculated from equation (7.4.3.1)
; the factor is an average for the factors inside the summation sign of equation (7.4.3.4)
that were neglected by Waller & Hartree. This term has been included in a few calculations of incoherent intensities [see, for example, Bloch & Mendelsohn (1974
)]. The Waller–Hartree method remains the chosen basis for the most extensive compilations of incoherent scattering factors, including those tabulated here, which were calculated by Cromer & Mann (1967
) and Cromer (1969
) from non-relativistic Hartree–Fock self-consistent-field wavefunctions. Table 7.4.3.2
is taken from the compilation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975
).
Element | (sin θ)/λ (Å−1) |
---|
0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.00 | 1.50 | 2.00 |
---|
1 H | 0.343 | 0.769 | 0.937 | 0.983 | 0.995 | 0.998 | 0.994 | 0.999 | 1.000 | 1.000 | 1.000 | 1.000 | 2 He | 0.296 | 0.881 | 1.362 | 1.657 | 1.817 | 1.902 | 1.947 | 1.970 | 1.983 | 1.990 | 1.999 | 2.000 | 3 Li | 1.033 | 1.418 | 1.795 | 2.143 | 2.417 | 2.613 | 2.746 | 2.834 | 2.891 | 2.928 | 2.989 | 2.998 | 4 Be | 1.170 | 2.121 | 2.471 | 2.744 | 3.005 | 3.237 | 3.429 | 3.579 | 3.693 | 3.777 | 3.954 | 3.989 | 5 B | 1.147 | 2.531 | 3.190 | 3.499 | 3.732 | 3.948 | 4.146 | 4.320 | 4.469 | 4.590 | 4.895 | 4.973 | 6 C | 1.039 | 2.604 | 3.643 | 4.184 | 4.478 | 4.690 | 4.878 | 5.051 | 5.208 | 5.348 | 5.781 | 5.930 | 7 N | 1.08 | 2.858 | 4.097 | 4.792 | 5.182 | 5.437 | 5.635 | 5.809 | 5.968 | 6.113 | 6.630 | 6.860 | 8 O | 0.977 | 2.799 | 4.293 | 5.257 | 5.828 | 6.175 | 6.411 | 6.596 | 6.755 | 6.901 | 7.462 | 7.764 | 9 F | 0.880 | 2.691 | 4.347 | 5.552 | 6.339 | 6.832 | 7.151 | 7.376 | 7.552 | 7.703 | 8.288 | 8.648 | 10 Ne | 0.812 | 2.547 | 4.269 | 5.644 | 6.640 | 7.320 | 7.774 | 8.085 | 8.312 | 8.490 | 9.113 | 9.517 | 11 Na | 1.503 | 2.891 | 4.431 | 5.804 | 6.903 | 7.724 | 8.313 | 8.729 | 9.028 | 9.252 | 9.939 | 10.376 | 12 Mg | 2.066 | 3.444 | 4.771 | 6.064 | 7.181 | 8.086 | 8.784 | 9.304 | 9.689 | 9.975 | 10.766 | 11.229 | 13 Al | 2.264 | 4.047 | 5.250 | 6.435 | 7.523 | 8.459 | 9.225 | 9.830 | 10.296 | 10.652 | 11.592 | 12.083 | 14 Si | 2.293 | 4.520 | 5.808 | 6.903 | 7.937 | 8.867 | 9.667 | 10.330 | 10.864 | 11.286 | 12.408 | 12.937 | 15 P | 2.206 | 4.732 | 6.312 | 7.435 | 8.419 | 9.323 | 10.131 | 10.827 | 11.411 | 11.888 | 13.209 | 13.790 | 16 S | 2.151 | 4.960 | 6.795 | 8.002 | 8.960 | 9.829 | 10.626 | 11.336 | 11.952 | 12.472 | 13.990 | 14.641 | 17 Cl | 2.065 | 5.074 | 7.182 | 8.553 | 9.539 | 10.382 | 11.158 | 11.867 | 12.499 | 13.050 | 14.750 | 15.487 | 18 Ar | 1.956 | 5.033 | 7.377 | 8.998 | 10.106 | 10.967 | 11.726 | 12.424 | 13.061 | 13.629 | 15.489 | 16.324 | 19 K | 2.500 | 5.301 | 7.652 | 9.405 | 10.650 | 11.568 | 12.329 | 13.014 | 13.645 | 14.220 | 16.212 | 17.152 | 20 Ca | 3.105 | 5.690 | 7.981 | 9.790 | 11.157 | 12.163 | 12.953 | 13.635 | 14.256 | 14.830 | 16.921 | 17.970 | 21 Sc | 3.136 | 5.801 | 8.169 | 10.071 | 11.561 | 12.648 | 13.545 | 14.256 | 14.885 | 15.460 | 17.630 | 18.782 | 22 Ti | 3.114 | 5.860 | 8.312 | 10.304 | 11.901 | 13.140 | 14.093 | 14.856 | 15.509 | 16.095 | 18.334 | 19.585 | 23 V | 3.067 | 5.858 | 8.375 | 10.454 | 12.156 | 13.514 | 14.574 | 15.413 | 16.111 | 16.721 | 19.032 | 20.379 | 24 Cr | 2.609 | 5.577 | 8.206 | 10.415 | 12.264 | 13.770 | 14.960 | 15.902 | 16.670 | 17.323 | 19.730 | 21.168 | 25 Mn | 2.949 | 5.791 | 8.380 | 10.604 | 12.486 | 14.062 | 15.346 | 16.376 | 17.211 | 17.910 | 20.411 | 21.938 | 26 Fe | 2.891 | 5.781 | 8.432 | 10.733 | 12.687 | 14.343 | 15.716 | 16.831 | 17.737 | 18.488 | 21.097 | 22.704 | 27 Co | 2.832 | 5.764 | 8.469 | 10.844 | 12.867 | 14.596 | 16.050 | 17.249 | 18.229 | 19.039 | 21.777 | 23.462 | 28 Ni | 2.772 | 5.726 | 8.461 | 10.894 | 12.980 | 14.780 | 16.317 | 17.602 | 18.664 | 19.543 | 22.445 | 24.211 | 29 Cu | 2.348 | 5.455 | 8.310 | 10.778 | 12.942 | 14.847 | 16.494 | 17.885 | 19.043 | 20.002 | 23.107 | 24.957 | 30 Zn | 2.654 | 5.631 | 8.388 | 10.901 | 13.094 | 15.020 | 16.709 | 18.163 | 19.395 | 20.427 | 23.745 | 25.683 | 31 Ga | 2.791 | 5.939 | 8.599 | 11.082 | 13.290 | 15.233 | 16.947 | 18.445 | 19.734 | 20.831 | 24.370 | 26.400 | 32 Ge | 2.839 | 6.229 | 8.912 | 11.338 | 13.536 | 15.486 | 17.215 | 18.741 | 20.074 | 21.224 | 24.983 | 27.109 | 33 As | 2.793 | 6.365 | 9.236 | 11.658 | 13.828 | 15.775 | 17.511 | 19.056 | 20.420 | 21.612 | 25.583 | 27.810 | 34 Se | 2.799 | 6.589 | 9.601 | 12.033 | 14.168 | 16.098 | 17.835 | 19.391 | 20.778 | 22.003 | 26.171 | 28.504 | 35 Br | 2.771 | 6.748 | 9.940 | 12.440 | 14.552 | 16.456 | 18.185 | 19.747 | 21.149 | 22.399 | 26.747 | 29.190 | 36 Kr | 2.703 | 6.760 | 10.157 | 12.828 | 14.969 | 16.849 | 18.562 | 20.123 | 21.535 | 22.804 | 27.313 | 29.870 | 37 Rb | 3.225 | 7.062 | 10.431 | 13.206 | 15.410 | 17.282 | 18.974 | 20.526 | 21.940 | 23.221 | 27.871 | 30.543 | 38 Sr | 3.831 | 7.464 | 10.746 | 13.576 | 15.860 | 17.745 | 19.420 | 20.956 | 22.367 | 23.654 | 28.423 | 31.210 | 39 Y | 3.999 | 7.700 | 11.010 | 13.899 | 16.279 | 18.215 | 19.891 | 21.416 | 22.820 | 24.110 | 28.970 | 31.870 | 40 Zr | 4.064 | 7.879 | 11.236 | 14.176 | 16.658 | 18.672 | 20.373 | 21.895 | 23.294 | 24.583 | 29.517 | 32.522 | 41 Nb | 3.672 | 7.684 | 11.213 | 14.317 | 16.949 | 19.081 | 20.844 | 22.386 | 23.787 | 25.077 | 30.067 | 33.167 | 42 Mo | 3.625 | 7.690 | 11.260 | 14.444 | 17.196 | 19.455 | 21.300 | 22.877 | 24.288 | 25.581 | 30.620 | 33.808 | 43 Tc | 3.987 | 7.984 | 11.512 | 14.653 | 17.456 | 19.816 | 21.748 | 23.370 | 24.797 | 26.093 | 31.173 | 34.447 | 44 Ru | 3.559 | 7.857 | 11.531 | 14.782 | 17.685 | 20.150 | 22.172 | 23.855 | 25.312 | 26.621 | 31.740 | 35.081 | 45 Rh | 3.499 | 7.863 | 11.591 | 14.883 | 17.858 | 20.428 | 22.557 | 24.318 | 25.819 | 27.148 | 32.309 | 35.715 | 46 Pd | 3.103 | 7.725 | 11.441 | 14.824 | 17.943 | 26.653 | 22.904 | 24.756 | 26.316 | 27.677 | 32.888 | 36.349 | 47 Ag | 3.362 | 7.785 | 11.598 | 14.969 | 18.082 | 20.858 | 23.212 | 25.162 | 26.792 | 28.195 | 33.465 | 36.983 | 48 Cd | 3.700 | 7.980 | 11.812 | 15.185 | 18.263 | 21.064 | 23.501 | 25.546 | 27.252 | 28.705 | 34.046 | 37.618 | 49 In | 3.852 | 8.297 | 12.083 | 15.444 | 18.489 | 21.288 | 23.779 | 25.906 | 27.691 | 29.203 | 34.634 | 38.255 | 50 Sn | 3.917 | 8.615 | 12.415 | 15.746 | 18.760 | 21.541 | 24.059 | 26.252 | 28.113 | 29.687 | 35.226 | 38.894 | 51 Sb | 3.871 | 8.811 | 12.777 | 16.088 | 19.067 | 21.823 | 24.349 | 26.590 | 28.518 | 30.157 | 35.822 | 39.536 | 52 Te | 3.097 | 9.076 | 13.171 | 16.466 | 19.407 | 22.134 | 25.655 | 26.927 | 28.912 | 30.613 | 36.422 | 40.181 | 53 I | 3.903 | 9.287 | 13.564 | 16.876 | 19.227 | 22.471 | 24.980 | 27.269 | 29.298 | 31.056 | 37.024 | 40.827 | 54 Xe | 3.841 | 9.340 | 13.892 | 17.307 | 20.175 | 22.833 | 25.324 | 27.619 | 29.680 | 31.488 | 37.628 | 41.477 | 55 Cs | 4.320 | 9.615 | 14.217 | 17.753 | 20.612 | 23.228 | 25.691 | 27.981 | 30.064 | 31.914 | 38.232 | 42.129 |
|