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"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" Of course it is proportional to the square of the number of unit cells. With more cells, more photons are scattered. The peak width also gets smaller, and its height increases, due to the interference effects. Therefore the intensity (photons/mm^2 on your high resolution detector) increases with the square of the number of unit cells. The total number of photons scattered in to the spot will be proportional to the number of unit cells. I guess it all depends on what you mean by intensity! Well James and I have already agreed that the term intensity is ill defined so I couldn't resist taking this opportunity to illustrate it (deliberately taking a different meaning from him). James of course was taking it to mean the total number of photons in a spot (sometimes called the integrated intensity). Then of course he is correct. Colin -----Original Message----- From: CCP4 bulletin board on behalf of James Holton Sent: Fri 30/01/2009 18:42 To: [log in to unmask] Subject: Re: [ccp4bb] X-ray photon correlation length 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 <DIV><FONT size="1" color="gray">This e-mail and any attachments may contain confidential, copyright and or privileged material, and are for the use of the intended addressee only. If you are not the intended addressee or an authorised recipient of the addressee please notify us of receipt by returning the e-mail and do not use, copy, retain, distribute or disclose the information in or attached to the e-mail. Any opinions expressed within this e-mail are those of the individual and not necessarily of Diamond Light Source Ltd. Diamond Light Source Ltd. cannot guarantee that this e-mail or any attachments are free from viruses and we cannot accept liability for any damage which you may sustain as a result of software viruses which may be transmitted in or with the message. 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