This question by the "Mad Scientist" (here after the MS) has provoked
me to give the topic a lot of thought.  I think I can provide some
direction towards the solution, but I'm not adept enough with "The
Optical Principles of the Diffraction of X-rays" (Which people on this
BB should refer to simply as OPDXr because it is so fundamental to most
topics discussed here.) to come up with a final answer to the question
of the units of B and <u_x^2>.  My hope is that the MS, who is much
better with OPDXr than I will finish the job.

    I have been a big fan of Dimensional Analysis since high school and
have found its rigorous application to be very useful in verifying
algebraic derivations.  I learned quite early that quite a few
quantities that people usually say are "unitless" can be usefully given
meaningful units.  I think this is the root of the current issue - There
are units present in the definition of these terms that are ignored by
traditional dimensional analysis.

    As a first example, I'll consider Bragg's Law:

    2 d Sin(theta) = n lambda.

Traditionally, the units are (d - A, theta - unitless, Sin(theta) -
unitless, n - unitless, lambda - A).  While the units on each side of
the equation match (Angstrom) that's a lot of unitless quantities.
These unit assignments also create problems.  With the wavelength,
lambda, is measured in Anstrom: does that mean Anstrom/cycle,
Anstrom/radian, Anstrom/degree?  Just defining a wave length as a length
is not good enough, you have to define a length per something.  I've
created these additional rules for my personal Dimensional Analysis.

1) Angles have units.  Either radians, degrees, cycles, or (a button
    on my calculator tells me) "grd".  There are well-known conversion
    factors between these units that appear, unexplained, in popular
    equations.  For example, there are 2 Pi radians per cycle.  We see
    the term 2 Pi in many equations and usually this should be assigned
    its units.

2) Trigonometric functions have arguments that must be measured in
    radians and their results are unitless (yes, I still have unitless
    quantities).

    In Bragg's Law, I have the new unit assignments of (theta - radian,
n - cycle, and lambda - A/cycle).  Tracking these additional units
allows for tighter checking of the validity of equations.

    It is difficult to determine the units of quantities in derived
equations: you need to concentrate on the defining equations, like
Bragg's Law.  Why?  If you see Sin(theta)/lambda in some other equation,
and it comes up a lot, and you try to assign units you will say that
Sin(theta) is unitless and lambda is A/cycle so the units of
Sin(theta)/lambda is cycle/A.  Wrong!  You've forgotten that there
was an "n" in the original equation that was assumed to be 1.  It
is still there and its unit of "cycle" persists, invisibly, in
Sin(theta)/lambda.  The unit of Sin(theta)/lambda is 1/Angstrom.

    Another interesting term to analyse is 2 Pi I (hx + ky + lz).
The traditional approach is to say that fractional coordinates are
unitless, Miller indices are unitless, and the 2 Pi is just there,
don't ask.  I have additional rules:

3) The fractional coordinate x has the unit "a cell edge", y is
    "b cell edge" and z is "c cell edge".  A location that has x = 0.5
    actually means that the location is 0.5 along the a cell edge.
    This value can be converted to Angstrom with a conversion factor
    with units of Angstrom/a cell edge, and we call that conversion
    factor the A cell constant.

4) The unit of h is cycle/a cell edge.  When you think about the
    definition of Miller indices this makes sense.  When h = 5 we
    mean that there are five cycles of that set of planes along the
    a cell edge of the unit cell.

    The application of these rules shows why you never see the term
"x + y" unless the symmetry of the crystal includes an equivalence
of the a and b edges.  You can't add two numbers unless their units
match and they don't, unless the symmetry causes the units "a cell
edge" and "b cell edge" to be equivalent.  This is also true for
"h + k".

    Writing the units explicitly for our little term results in

    2 Pi I (h (cycle/a cell edge) x (a cell edge) +
            k (cycle/b cell edge) y (b cell edge) +
		   l (cycle/c cell edge) z (c cell edge))

and all the "cell edge" stuff cancels to "cycle".  Wait!  Didn't
I say that the argument of a Sin or Cos function has to be in
radian, and this term is usually such an argument?  Yes, the
factor of 2 Pi is actually 2 Pi radian/cycle and converts the
unit of the term to radian.

    If you read a lot of math books you will be confused because their
Fourier transform kernel don't include the 2 Pi that ours does.
Mathematicians are cleaver enough to define their reciprocal space
coordinates in radians from the start so they don't need to change units
later on.  Whenever you see an equation where something is actually
calculated from "h" you will see it present as "2 Pi h" because the math
wants the units to be radian/a cell edge and not cycle/a cell edge.

    Back to the original problem: what are the units of B and
<u_x^2>?  I haven't been able to work that out.  The first
wack is to say the B occurs in the term

     Exp( -B (Sin(theta)/lambda)^2)
	
and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom
and the argument of Exp, like Sin, must be radian.  This means
that the units of B must be A^2 radian.  Since B = 8 Pi^2 <u_x^2>
the units of 8 Pi^2 <u_x^2> must also be A^2 radian, but the
units of <u_x^2> are determined by the units of 8 Pi^2.  I
can't figure out the units of that without understanding the
defining equation, which is in the OPDXr somewhere.  I suspect
there are additional, hidden, units in that definition.  The
basic definition would start with the deviation of scattering
points from the Miller planes and those deviations are probably
defined in cycle or radian and later converted to Angstrom so
there are conversion factors present from the beginning.

    I'm sure that if the MS sits down with the OPDXr and follows
all these units through he will uncover the units of B, 8 Pi^2,
and <u_x^2> and the mystery will be solved.  If he doesn't do
it, I'll have to sit down with the book myself, and that will
make my head hurt.

Dale Tronrud


James Holton wrote:
> Many textbooks describe the B factor as having units of square Angstrom 
> (A^2), but then again, so does the mean square atomic displacement u^2, 
> and B = 8*pi^2*u^2.  This can become confusing if one starts to look at 
> derived units that have started to come out of the radiation damage 
> field like A^2/MGy, which relates how much the B factor of a crystal 
> changes after absorbing a given dose.  Or is it the atomic displacement 
> after a given dose?  Depends on which paper you are looking at.
> 
> It seems to me that the units of "B" and "u^2" cannot both be A^2 any 
> more than 1 radian can be equated to 1 degree.  You need a scale 
> factor.  Kind of like trying to express something in terms of "1/100 
> cm^2" without the benefit of mm^2.  Yes, mm^2 have the "dimensions" of 
> cm^2, but you can't just say 1 cm^2 when you really mean 1 mm^2! That 
> would be silly.  However, we often say B = 80 A^2", when we really mean 
> is 1 A^2 of square atomic displacements.
> The "B units", which are ~1/80th of a A^2, do not have a name.  So, I 
> think we have a "new" unit?  It is "A^2/(8pi^2)" and it is the units of 
> the "B factor" that we all know and love.  What should we call it?  I 
> nominate the "Born" after Max Born who did so much fundamental and 
> far-reaching work on the nature of disorder in crystal lattices.  The 
> unit then has the symbol "B", which will make it easy to say that the B 
> factor was "80 B".  This might be very handy indeed if, say, you had an 
> editor who insists that all reported values have units?
> 
> Anyone disagree or have a better name?
> 
> -James Holton
> MAD Scientist