Spots don't fall off with the inverse square law. It is a very easy
experiment to do. Just take exposures at several distances and scale
the data together, noting the correction for air absorption.
A good reference for the underlying theory is Chapter 6 of M. M.
Woolfson's book (1997). But briefly: Diffuse scattering falls off with
the inverse square law as one would naturally expect from anything that
spreads out in all directions. Spots, however, are "reflections" and
their source point is the source of the incident beam, which is usually
far away or at worst comparable to the sample-to-detector distance.
For those who do not like my "long posts" I now direct you to the
"delete" button which should have come with your email client. Everyone
else, read on.
The source of the confusion on this topic seems largely due to the fact
that people keep dropping steradians from the units of intensity. ;)
The units of diffuse scattered intensity are photons/steradian, and so
the number of photons falling onto a pixel depends on the solid angle
subtended by the pixel at the sample. The keyword to use to learn more
about this is "differential cross section". The atomic form factor is
an example of a differential cross section (after it is squared and
multiplied by the Thomson cross section).
Spot intensity also arises from scattering and therefore fundamentally
has units of photons/steradian. Since all the atoms in the crystal
scatter coherently together, the intensity in photons/steradian can be
very very large, but only in certain directions! The solid angle over
which this high photons/steradian applies is proportionately small, and
the integrated intensity is preserved. In fact, the fundamental "grain"
of a spot is the reciprocal size of a mosaic domain (or Fresnel zone,
whichever is smaller) which can be up to 10 microns across (10,000 A)
and this corresponds to a solid angle of about 2.5e-9 steradian (using
Bragg's law). This is a LOT smaller than a pixel at normal detector
distances, and this is why the spots do not appear to spread out with
distance. The angle- and area-integrated intensity (in photons/spot)
does not depend on the distance (after correcting for air absorption).
Think of your monochromator crystals. As long as the beam is
monochromatic and the crystal is not bent or otherwise messed up, the
reflected x-ray beam will not spread out any more than the incident beam
did.
In practice, of course, the spots are broadened by other things like
mosaic spread, Nave disorder (unit cell variations, see Nave 1998),
horizontal or vertical beam divergence, spectral dispersion and the
detector point-spread function (PSF). The latter, along with the
physical size of the illuminated crystal tends to be the dominant effect
for good crystals, but obviously the PSF and crystal size does not
depend on detector distance. All the others, taken individually, tend
to spread out spots by an amount that is proportional to distance, not
the distance squared. Hence there is an advantage in "photons/pixel
from Bragg scattering" vs "photons/pixel from background" with
increasing detector distance. Different factors become "important" as
soon as their contribution to spot shape becomes comparable with the
"starting" spot shape. All the effects are literally convoluted, so the
result is the "RMS" of all contributing factors. However, if your
crystal is messed up to the point of approaching fiber diffraction (say,
with significant mosaic spread and significant Nave disorder), then the
advantage of having a bigger detector farther away over a small, close
detector starts to approach a "wash" in terms of signal/background.
It is an interesting point, however, that simultaneously increasing the
detector distance and increasing the photon energy to keep the
resolution at the edge of the detector constant has absolutely NO effect
on the photons/spot from Bragg vs diffuse scattering. It took me a
while to become convinced of this, and I still owe Colin Nave a few
beers as a result. Nevertheless, the experimental evidence was given by
Gonzalez, Denny & Nave in 1994.
HTH
-James Holton
MAD Scientist
Richard Gillilan wrote:
It seems to be widely known and observed that diffuse background
scattering decreases more rapidly with increasing detector-to-sample
distance than Bragg reflections. For example, Jim Pflugrath, in his
1999 paper (Acta Cryst 1999 D55 1718-1725) says "Since the X-ray
background falls off as the square of the distance, the expectation is
that a larger crystal-to-detector distance is better for reduction of
the x-ray background. ..."
Does anyone know of a more rigorous discussion of why background
scatter fades while Bragg reflections remain collimated with distance?
Richard Gillilan
MacCHESS
