The weird scale factor on what we now call "B" comes from Debye's 
derivation of it (Debye 1914).  This derivation is given much more 
succinctly in Bragg's textbook (since edited by R. W. James (1962) page 
22, which I attach).   The "kappa" used therein is the magnitude of the 
incident radiation's wave vector (2*pi/lambda), so there is one factor 
of 2*pi.  The scattering vector "S" is the difference between the unit 
vectors of the incident and scattered beams, so it is 2*sin(theta) long: 
another factor of two.  Then you can see in equation 1.27 that the 
second term in the expansion is divided by 2 and squared.  The 
mathematical "reason" for this is similar to the rules for taking a 
derivative, and equation 1.27 is really just a Taylor expansion in the 
exponent.  You can also think of it as Debye assuming that the atoms 
move like simple harmonic oscillators.

So, the sum total of all the weird scale factors is to multiply by 2 to 
convert from d* to sin(theta)/lambda, multiply by 2*pi because the 
wavevector is in radians, then square it and divide by two because "B" 
is the second term in a Taylor series.  This gives a final scale factor 
of (2*2*pi)^2/2, or:

  B = 8*pi^2*<u_x^2>

where "u" is the atomic displacement vector and u_x is the component of 
that vector normal to the Bragg plane.  Remarkably, movement in the 
other two directions doesn't change the spot intensity.

You may also note that R.W. James (1962) does not explicitly use "B", 
but rather M = B*(sin(theta)/lambda)^2.  The quantity exp(-2*M) is what 
is traditionally known as the "Debye factor". The "2" in exp(-2*M) puts 
it on an intensity scale, just like the polarization factor, Lorentz 
factor and absorption factor.  The splitting of "M" into "B" and 
"(sin(theta)/lambda)^2" was probably done in Waller's thesis (which I 
don't have), but I think it does make sense to use the letter "B" if you 
decide to simplify R. W. James's equation 1.27 as:

  exp(-A*s-B*s^2-C*s^3 ... ).

abbreviating "sin(theta)/lambda" as "s".

As for the question of WHY Debye did this in terms of "s" and not "d", I 
think it could simply be because he had not yet head of "Bragg's Law", 
which was published only a few months earlier (Bragg & Bragg Proc. R. 
Soc. Lon. 1913).  Papers didn't spread as quickly then as they do now.  
Why was the "sin(theta)/lambda formalism" kept?  I think because Bragg 
defined "d" as the spacing between two planes of a crystal lattice, and 
so using "d" in the "general scattering equation" is arguably 
inappropriate.  Since the B factor connects disordered crystals with 
diffuse scattering and completely amorphous substances, it probably 
actually is a good idea to keep it in terms of "r dot S", where "r" is 
the position vector and "S" is the difference between the directions of 
the incident and scattered beams.

The fundamental problem with getting rid of all the "scale factors" is 
because reciprocal space is not actually an inverse-distance space, it 
is an angle space.  That's why all those pesky factors of 2*pi keep 
popping up everywhere and mixing with factors of two.  For example, 
there are people who follow the gospel of "q", but I've never really 
understood why.

-James Holton
MAD Scientist

On 10/12/2011 6:55 AM, Phil Evans wrote:
> I've been struggling a bit to understand the definition of B-factors, particularly anisotropic Bs, and I think I've finally more-or-less got my head around the various definitions of B, U, beta etc, but one thing puzzles me.
>
> It seems to me that the natural measure of length in reciprocal space is d* = 1/d = 2 sin theta/lambda
>
> but the "conventional" term for B-factor in the structure factor expression is exp(-B s^2) where s = sin theta/lambda = d*/2 ie exp(-B (d*/2)^2)
>
> Why not exp (-B' d*^2) which would seem more sensible? (B' = B/4) Why the factor of 4?
>
> Or should we just get used to U instead?
>
> My guess is that it is a historical accident (or relic), ie that is the definition because that's the way it is
>
> Does anyone understand where this comes from?
>
> Phil