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
