The Cromer-Mann coefficients are in reciprocal space, and written in
terms of s = sin(theta)/lambda, where lambda is the wavelength in
Angstrom and "theta" is the Bragg angle (half the take-off angle of the
spot). To convert them into real space you need to do an inverse
Fourier transform, which is easy in this case because the function is a
sum of Gaussians and the Fourier transform of a Gaussian is another
Gaussian. For example, the reciprocal-space equation for the structure
factor of a carbon atom is:
C_sf(s)=C_a1*exp(-C_b1*s*s)+C_a2*exp(-C_b2*s*s)+C_a3*exp(-C_b3*s*s)+C_a4*exp(-C_b4*s*s)+C_c
where the C_a(i) C_b(i) and C_c coefficients are the ones you will find
in ${CLIBD}/atomsf.lib .
C_c = 0.215600;
C_a1 = 2.310000; C_a2 = 1.020000; C_a3 = 1.588600; C_a4 = 0.865000;
C_b1 = 20.843899; C_b2 = 10.207500; C_b3 = 0.568700; C_b4 = 51.651199;
The atomic B-factor (capital B) is then applied by simply multiplying
this whole thing by the exponential factor: C_sf(s)*exp(-B*s*s) . I
should note that these coefficients are NOT the ones published by Cromer
& Mann (1968), rather they appear to be a re-evaluation of the
Cromer-Mann 9-parameter fit, and the correct citation for them is Table
6.1.1.4 of International Tables vol C.
The real-space version is this:
C_ff(x,B) = \
+C_a1*(4*pi/(C_b1+B))**1.5*exp(-4*pi**2/(C_b1+B)*x*x) \
+C_a2*(4*pi/(C_b2+B))**1.5*exp(-4*pi**2/(C_b2+B)*x*x) \
+C_a3*(4*pi/(C_b3+B))**1.5*exp(-4*pi**2/(C_b3+B)*x*x) \
+C_a4*(4*pi/(C_b4+B))**1.5*exp(-4*pi**2/(C_b4+B)*x*x) \
+C_c *(4*pi/(B))**1.5*exp(-4*pi**2/(B)*x*x);
The coefficients are the same as for the reciprocal-space version. The
variations of 4*pi are to account for the change of units in the Fourier
Transform. Note that the peak height at the center gets "squished" by
the 3/2 power of the B factor. This is necessary to preserve the
integrated number of electrons over the whole atomic electron density
cloud, which must always be 6 in the case of carbon.
You can perhaps see the problem here when the atomic B factor is zero,
you get a divide-by-zero in the last term. This has perhaps led to the
misconception that atoms with zero B factor have infinite electron
density. In fact, they don't. The divide-by-zero is just a problem
with the 9-parameter representation. The "constant term" is quite
convenient in reciprocal space, but horrible in real space. Sad thing
is we don't really need it. You can get perfectly reasonable
8-parameter fits with the "c" term set to zero, but they just don't work
as well at ultra-high resolution, which was historically the reason for
the research into this. At the end of the day, all these things are are
fits to the data in ITC Vol C Table 6.1.1.1.
The reality for all but a few macromolecular structures is that we have
atomic B factors on the order or 20. With this amount of blurring we
might as well not even bother distinguishing atom types. The atomic
number is really the only difference. But it doesn't hurt that much to
use the full 9-parameter tabulation, and it is just fine for B factors
of 5 or more. For applications with B less than that you might want to
download the data from 6.1.1.1, re-do the fit, and make your own atomsf.lib.
For those who are interested in a C-like syntax of all the atomic form
factors I have tabulated them here:
https://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
and in real space:
https://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
These are ready to go into gnuplot, but other languages I'm sure will
work with a little editing.
-James Holton
MAD Scientist
On 6/28/2018 4:50 AM, Aaron Oakley wrote:
> Dear CCP4ers,
>
> The Cromer-Mann coefficients ai, bi, c (i = 1 to 4) describing the non-dispersive part of the atomic scattering factor f(s) for a neutral atom as a function of s=(sin theta / lambda) is:
>
> f(s) = sum(i=1...4) ai*exp(-b*s^2) + c
>
> Is it correct to interpret this in terms of electron density rho for said atom as a function of distance r from centre:
>
> rho(r) = sum(i=1…4) a(i) * [4*pi/ (bi + B)]^1.5 * exp[-4*pi^2*r^2 / (bi + B)]
> + c * [4*pi / B]^1.5 * exp[-4*pi^2*r^2 / B]
>
> Where ai, bi and c are the aforementioned Comer-Mann coefficients and B is the temperature factor?
>
> With thanks,
>
> a++
>
>
> Aaron Oakley
> Associate Professor
> School of Chemistry and Molecular Bioscience | Molecular Horizons | Faculty of Science, Medicine and Health
> University of Wollongong NSW 2522 Australia
> T +61 2 4221 4347 | F +61 2 4221 4287
>
>
>
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