Open geoffwoollard opened 5 months ago
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).
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"
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
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
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].
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