Ethan Merritt wrote:
> My impression is that the coherence length from synchrotron sources
> is generally larger than the x-ray path through a protein crystal.
> But I have not gone through the exercise of plugging in specific
> storage ring energies and undulator parameters to confirm this
> impression.  Perhaps James Holton will chime in again?
>   
Hmm.  I think I should point out that (contrary to popular belief) I am 
not a physicist.  I am a biologist.  Yup. BS and PhD both in biology.  
However, since I work at a synchrotron I do have a lot of physicists and 
engineers around to talk to.  Guess some of it has rubbed off.

I passed Bernhard's question along to Howard Padmore (who is definitely 
a physicist) here at ALS and he gave me a very good description of the 
longitudinal coherence length, similar to that provided by Colin's 
posted reference:

coherence_length = lambda^2/delta-lambda

This made a lot of sense to me until I started to consider what happens 
if lambda ranges from 1 to 3 A, like it does in Laue diffraction.  One 
might expect from this formula that the coherence length would be very 
small, smaller than a typical protein unit cell, and then you would 
predict that none of the scattering from any of the unit cells 
interferes with each other and that you should see the molecular 
transform in the diffraction pattern.  But the oldest observation in 
crystallography is that Laue patterns have sharp spots.  You don't see 
the molecular transform, despite how nice that would be (no more phase 
problem!).

I think the coherence length is related to how TWO different photons can 
interfere with each other, and this is a rare event indeed.  It has 
nothing to do with x-ray diffraction as we know it.  No matter how low 
your flux is, even one photon per second, you will eventually build up 
the same diffraction pattern you get at 10^13 photons/s.  Colin is right 
that photons should be considered as waves and on the length scale of 
unit cells, it is a very good approximation to consider the 
electromagnetic wave front coming from the x-ray source to be a flat 
plane, as Bragg did in his famous construction.

So, I think perhaps Bernhard asked the wrong question?  I think the 
question should have been "how far apart can two unit cells be before 
they stop interfering with each other?"  The answer to this one is: 
quite a bit.

Consider a silicon crystal (like the ones in my monochromator).  These 
things are about 10 cm across, but every atom is in perfect alignment 
with every other.  It is one single mosaic domain that you can hold in 
your hand.  And as soon as you shine an x-ray beam on a large perfect 
crystal, lots of "weird" stuff happens.  Unlike protein crystals the 
scattering of the x-rays is so strong that the scattered wave not only 
depletes the incoming beam (it penetrates less than 1 mm and is nearly 
100% reflected), but this now very strong diffracted ray can reflect 
again on its way out of the crystal (off of the same HKL index, but 
different unit cells). Then some of that secondarily-diffracted ray will 
be in the same direction as the main beam, and interfere with it 
(extinction).  Accounting for all of this is what Ewald did in his 
so-called "dynamical theory" of diffraction.  The important thing to 
remember about perfect crystals is that a SINGLE PHOTON interacting with 
my 10 cm wide silicon crystal will experience all these dynamical 
effects.  It doesn't matter what the "coherence length" is.

Now, if a perfect crystal is really really small (much smaller than the 
interaction length of scattering), then there is no opportunity for the 
re-scattering and extinction and all that "weird stuff" to happen.  In 
this limiting case, the scattered intensity is simply proportional to 
the number of unit cells in the beam and also to |F|^2.  This is the 
basic intensity formula that Ewald showed how to integrate over all the 
depleting beams and re-scattering stuff to explain a large perfect 
crystal.  As I understand it, the fact that there were large, 
macroscopic "single" crystals that were found to still obey the formula 
for a microscopic crystal came as something of a shock in the time of 
Darwin and Ewald.  They explained this observation by supposing that 
these crystals were "ideally imperfect" and actually made up of lots of 
little perfect crystals that were mis-oriented with respect to one 
another enough so that the diffracted ray from one would be very 
unlikely to re-reflect off of another "mosaic domain" before it left the 
crystal.  Protein crystals are a very good example of ideally imperfect 
crystals.

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.  This means that for all practical 
purposes the spot must always be "integrated" over some angular width 
(such as beam divergence).  That is, you have to get rid of the 
"steradians" in the units of intensity before you can get simply 
photons/s.  The intrinsic "rocking curve" of this near-infinitely-sharp 
peak from a single mosaic block is inversely proportional to the number 
of unit cells in the mosaic block.  So, the integrated intensity 
(photons/s * exposure time) is proportional to the number of unit cells 
in the beam.  It doesn't matter how perfect the crystal is.

Okay, so I know there are a lot of people out there who don't agree with 
me on this, but please have a look at Woolfson's Ch 6 before flaming 
me.  I may be nothing more than a biologist, but I did take a few math 
classes in college and think I do understand the math in that book.

So, I think the answer Bernhard was looking for is : the size of a 
mosaic domain, which can be as much as 10 cm, or as little as a few 
dozen unit cells.

-James Holton
MAD Scientist