Notes on Cowley (2006) "Scattering factors for the diffraction of electrons by crystalline solids" and references therein

[geoffwoollard]

Notes on Cowley, J. M. (2006). Scattering factors for the diffraction of electrons by crystalline solids. In International Tables for Crystallography (Vol. C, pp. 259–429). https://www.wiley.com/en-ie/International+Tables+for+Crystallography%2C+Volume+C%3A+Mathematical%2C+Physical+and+Chemical+Tables%2C+3rd+Edition-p-9780470710296

cowley (IUCr Series. International Tables of Crystallography) E. Prince (Ed.) - International Tables for Crystallography Volume C_ Mathematical, physical and chemical tables. C-Wiley (2007) copy.pdf

References therein

• Peng, L. M., Ren, G., Dudarev, S. L., & Whelan, M. J. (1996). Robust parameterization of elastic and absorptive electron atomic scattering factors. Acta Crystallographica Section A: Foundations of Crystallography, 52(2), 257–276. http://doi.org/10.1107/S0108767395014371

[geoffwoollard]

Units of s given under Eq 4.3.1.12 on p. 260

So inverse space. However, the convention for k in Fourier space might have to do with their convention of a Fourier transform, which has 2*pi*i*h*r in the exp (equation 4.3.1.11).

[geoffwoollard]

The text clearly states that f^B(s), i.e. "atomic scattering amplitudes" are given in table 4.3.1.1 (neutral atoms) and 4.3.1.2 (ionized atoms).

The X-ray scattering factors, i.e. f_x(s), is related to f^B(s) through the Mott formula (equation 4.3.1.14/15), and are found in a reference. However the Mott equation breaks down at low s, and is not listed in the table.

However the origin (s=0) is a special case and the Mott equation is not used but instead one can find f^B(s) with a simple equation (4.3.1.29).

The argument of f^B or f_x are is s in both cases, but in the top left corner we see the units in the tables are in sin(theta)/lambda.

Hence looking up the value of 1.0 on the table means 1 = sin(theta) / lambda, and so s = 4*pi*1.

Stated the other direction for a value of s = 4 pi, you use the value of 1.0 on the table.

This should help track down the "missing value of 4 pi"

[geoffwoollard]

Confusingly, Peng seems to have a different convention of s, where there is no 4 pi

Later on in equation 17 the variable g = 4 pi s is used

[geoffwoollard]

There is a relativisitc correction for f^B(s) from the values listed in the tables 4.3.1.1 and 4.3.1.2

[geoffwoollard]

Finally, the values of a_i and b_i are listed in 4.3.2.2 (for s up to 2 A^-1) and table 4.3.2.3 (for s up to 6 A^-1).

Note that this is likely very high resolution (depending on the 4 pi factor). Because 2 A^-1 corresponds to 0.5 A, and 6 A^-1 to 1/6 = 0.17 A.

Also, they do not agree over the range [0,2]. This confuses me at first glance. Perhaps the values in table 4.3.2.2 (for s up to 2 A^-1) are "overfit" for this range... or the values in table 4.3.2.3 (for s up to 6 A^-1) are "underfit" for the range [0,2]. Or perhaps when all 5 Gaussians are combined, the curves overlap very closely in the range [0,2].