The original reference is Debye, P. (1914) Ann. d. Physik 43, 49, which
is in German. Waller's paper came later and the forgotten paper wich did
the math rigorously was Ott, H. (1935) Ann. d. Physik 23, 169. The best
description I have seen in English was in chapter 1 section 3 of:
James R. W. (1962) “The Optical Principles of the Diffraction of X-rays:
The Crystalline State”, Vol II. Bell & Hyman Ltd., London.
James is a big book with a lot of math in it, but it is remarkably easy
to read. Particularly chapter 1. I highly recommend it.
The long and short of the "B-factor phenomenon" is that the primary
effect of corrupting a perfect lattice by moving the "atoms" away from
their ideal positions is a drop in the Bragg intensities and a
corresponding increase in background (the elastically-scattered photons
that don't go into the spots have to go somewhere). R. W. James shows
the math to prove that the falloff of the Bragg intensities with
resolution is the Fourier transform of the histogram of atomic
displacements. It actually doesn't matter if the displacements in
adjacent unit cells are correlated or not. It is only the histogram of
displacements from the ideal lattice that is important.
If the distribution (histogram) of atomic displacements is Gaussian,
then its Fourier transform is also a Gaussian and therefore has the form
exp(-B*s^2). Where B = 8*pi*<u>^2 and u is the displacement of an atom
along the scattering vector "s" (halfway between the incident and
diffracted beams). It is interesting that movement of atoms
perpendicular to s has absolutely no effect on the Bragg intensity! In
this way, anisotropic displacement distributions lead to anisotropic
diffraction as the crystal rotates.
Waller showed that thermal vibrations (phonons) in simple crystals do
indeed produce a Gaussian distribution of atomic displacements, but it
is also interesting to note that non-Gaussian atomic displacement
distributions cannot be fully described by a B factor. For example, if
the atomic displacements have a Lorentzian shape, then the intensity
fall-off will be exponential: exp(-A*s) (the Fourier transform of a
Lorentzian is an exponential, and vice versa). I THINK this may be the
origin of using the letter B, as it is the second term in the Taylor
expansion of an arbitrary displacement distribution:
exp( ln(K) - A*s -B*s^2 - C*s^3 ...)
For Gaussian atomic displacements, all terms except B will be zero. But,
I have to admit that Debye's paper doesn't have an equation that looks
like this. In fact, even R. W. James doesn't call it "B", he calls it
"M". So, I could be wrong about the origin of "B", but I think someone
else must have written down the above equation before I did.
-James Holton
MAD Scientist
Jacob Keller wrote:
Hello Crystallographers,
does anybody have a good reference dealing with interpretations of
what B-factors (anisotropic or otherwise) really signify? In other
words, a systematic addressing of all of the possible underlying
molecular/crystal/data-collection phenomena which the B-factor
mathematically models?
Thanks in advance,
Jacob
*******************************************
Jacob Pearson Keller
Northwestern University
Medical Scientist Training Program
Dallos Laboratory
F. Searle 1-240
2240 Campus Drive
Evanston IL 60208
lab: 847.491.2438
cel: 773.608.9185
email: [EMAIL PROTECTED] <mailto:[EMAIL PROTECTED]>
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