Hey Dirk,
You're wrong. ;)
Well, actually, it depends on what you mean by "intensity". As Colin
pointed out the word "intensity" is practically useless unless you are
sure that you are comparing "apples to apples". In crystallography, you
get beam intensity in flux (photons/s) but the intensity of scattering
(photons/steradian/s) is proportional to the incident beam intensity (in
photons/area/s) which is also proportional to the integrated spot
intensity (photons). It is easy to get confused, so I will try to avoid
using the word "intensity" without qualifying it with some units. But I
think an unfortunate misinterpretation has arisen recently because the
"peak intensity" of a spot (which you get from certain equations in
scattering theory) is not the same as the "integrated intensity" (which
you get from mosflm).
You are right that the scattered "intensity" (photons/steradian/s) from
a mosaic block of N unit cells that is satisfying the Bragg condition is
proportional to N^2. But you have to pay attention to the units: they
are not the same as an integrated spot intensity (just photons). You
can multiply photons/steradian/s by the exposure time (if the crystal is
not moving) to get accumulated intensity (photons/steradian) but you
still need a solid angle subtended by the spot (in steradians) in order
to get a "full" intensity. You can't cheat and just use the solid angle
of a pixel because the diffracted ray from a mosaic block is much much
sharper than that. That is, the high photons/steradian only exists for
a very very small patch of solid angle, and the size of this patch is
proportional to 1/N. So, the actual integrated spot intensity (photons)
you see on your detector is still just proportional to N^2/N = N, then
number of unit cells in the mosaic block. Therefore, if you then have
"m" mosaic domains, the integrated spot intensity (photons) is
proportional to m*N. In Woolfson, this is derived in the more
traditional way of treating the crystal as always rotating, but the
result is the same.
The remarkable part of this is that the integrated spot intensity
(photons) is essentially invariant with how you divide up the unit cells
into mosaic domains. Well, okay, if N=1, then you don't really have a
crystal but an amorphous solid (seen a lot of those), so I should
qualify that so long as N > ~1000, it doesn't matter if m is 1 or 10^12,
the integrated spot intensity (photons) is still proportional to the
total number of unit cells in the crystal. This was first shown by C.
G. Darwin (1914) so I don't blame you if you can't find the original
reference. However, "Darwin's Formula" can be found in most modern
textbooks. It is Equation 9.1 in Blundell & Johnson (1976) and Equation
4.31 in Drenth (1999). You will note that the mosaic spread is not part
of this equation.
I understand it was W. L. Bragg et al. (1921) who confirmed that the
absolute scattered intensity from rock salt does indeed obey Darwin's
Formula. I confirmed it recently for lysozyme on my beamline, but never
published it as I figured I had been scooped 86 years earlier. Although
I will tell you that the trick is (once again) getting the units right.
for example: you have to enter the wavelength in meters, not Angstroms,
otherwise you're off by a factor of 10^30.
The distribution of unit cells into mosaic domains does become important
if the extent of a mosaic domain starts to become large compared with
the attenuation length of the x-ray beam, then one must invoke the
dynamical theory. Darwin derived equations for the dynamical case as
well, but these almost never apply to protein crystals. They are just
too small.
-James Holton
MAD Scientist
Dirk Kostrewa wrote:
I'm not sure where this rumor got started that the intensity
reflected from a mosaic block or otherwise perfect lattice is
proportional to the square of the number of unit cells. This is
never the case. The reason is explained in Chapter 6 of M. M.
Woolfson's excellent textbook, but the long and short of it is: yes
the instantaneous intensity (photons/steradian/s) at the
near-infinitesimal moment when a mosaic domain diffracts is
proportional to the number of unit cells squared, but this is not
useful because x-ray beams are never perfectly monochromatic nor
perfectly parallel.
Hmm, I don't have Woolfson's book at hand, so I can't read this
chapter. My current understanding is: if N unit cells scatter in
phase, the scattered total amplitude is N times the scattered
amplitude of the unit cell in that direction. Since the recorded
intensity is proportional to the square of the amplitude, the
scattered total intensity is proportional to N^2 (for simplicity, I
assume perfect sources, crystals and detectors, so I don't discuss
spot shapes). Now, if the coherence lengths is limited by the size of
the mosaic blocks, each block scatters like a tiny crystal independent
of the other mosaic blocks. Thus, if we have m < N mosaic blocks (of
equal size), each block results in a scattered intensity ~(N/m)^2, and
the m blocks add their intensities, yielding a total intensity ~N^2/m.
Dependent of the perfection of the crystal, the total scattering
intensity for the two extremes between m=N (which wouldn't make any
sense) and m=1 is proportional to between N and N^2, respectively.
Please, correct me, if I'm wrong.
