Good summary as expected from James. "Have you ever heard of photon-photon scattering?" Well yes! See for example http://2physics.blogspot.com/2006/03/photon-photon-scattering.html which says "according to Quantum Electrodynamics (QED), particles can still be created in this emptiness of vacuum through light-light interactions." There we don't need x-ray generators, synchrotrons etc. The vacuum will do it for us. Cheers Colin
________________________________ From: CCP4 bulletin board on behalf of James Holton Sent: Tue 28/08/2007 16:56 To: [email protected] Subject: Re: [ccp4bb] Questions about diffraction For a full answer to all your questions, I refer you to the classic textbook of M. M. Woolfson "an introduction to x-ray crystallography" by Cambridge University Press. This book has been quite helpful to me of late. Unlike some similar texts I find it easy to read. There are even examples! With real numbers! Woolfson begins by describing scattering as it was originally derived by J. J. Thomson from classical mechanics of a charged mass (the electron) vibrating in an electric field (the photon), and takes you, step-by-step all the way to Bragg scattering from a rotating 3-D crystal. You have received many comments so far, so I will try not to repeat those in answering your questions here: 1) yes, x-ray sources are "non-coherent", but the photons ARE coherent on a short length scale. Specifically, this is about 0.7 micron if the x-rays are 1 A in wavelength and have a spectral dispersion of 1/7000, such as with a Si(111) monochromator. That is, after traveling 0.7 micron, two photons that were once in phase will no longer be, because they have different wavelengths. This is approximately the "coherence length" in the direction of propagation. The other two directions are ... more complicated. It is actually quite difficult to make an x-ray source with a very short spatial coherence. I've heard "talk" about shaping x-ray wavefronts in next-generation accelerators, such as the ERL planned at CHESS. Thus would have the advantage of changing the phase relationship between atoms as you described and (potentially) getting phase information directly. However, I don't think anybody has actually done this yet. 2) In general, photons do not interfere with each other. Have you ever heard of photon-photon scattering? Neither have I. However, there actually is a body of work on multi-photon correlations in scattering. Apparently, two photons can have correlated wave functions, and this leads to correlations in the arrival time of photons at the detector: something called the Hanbury-Brown and Twiss effect. The math behind it it a bit beyond me. However, theoretically, such correlated scattering events could be used to get phase information if you have a fast detector and a REALLY fast source. My colleague Ken Frankel <[EMAIL PROTECTED]> can tell you more about it if you are interested. 2b) WRT the "all electrons scattering together" question. Yes, they do all scatter "together" because they are all confined in the same atom. The total scattering is explained by integrating Thomson's scattering formula (including the phase shift) over all of the electron positions. Yes, a single photon can interact with more than one electron, just as a single photon can pass through two slits in the famous experiment by Thomas Young. For any given photon, the positions of each electron will (in the classical view) be at some well-defined position, but we experimentally integrate over a LOT of photons. If no two photons ever see a consistent arrangement of electron positions (such as when you shoot x-rays at a free electron beam) then the scattering is "incoherent" and the scattered intensity distribution you see simply follows Thomson's classical scattering formula for one electron, with the intensity multiplied by the number of electrons in the x-ray beam. However, if the "electron density" is not random but instead has some kind of consistent feature from photon to photon, then the interference of the scattered waves will also be consistent as it builds up on your detector. This "binding effect" is only significant on the length scale of the electron confinement (the size of an atom in this case), so it falls off with increasing scattering angle. Remember, the phase of the scattered wave depends on the total path length traversed by the incident and scattered photons. The phase shift between scattering at any two points in the sample will always become vanishingly small at a small scattering angle because the difference in path length from one "side" of the atom to the other "side" becomes smaller and smaller at low scattering angle. For larger scattering angles there is a larger phase shift, and the interference becomes more destructive. The quantitative angular dependence of scattering from any atom is called the "atomic form factor", and it is tabulated in ${CLIBD}/atomsf.lib. The "form factor" is why high-angle spots are weaker than low-angle spots, even if all the B-factors are zero. Atomic form factors are derived from scattering measurements on gasses (which don't have B-factors), and theoretical electron-distribution calculations have been found to be consistent with these observations. The upper limit to constructive interference from a single atom is if all of the electrons in the atom are scattering in phase. This will always occur at vanishingly small scattering angles (forward scattering). Since there is no phase shift the amplitudes add, and you square the amplitude to get intensity. Experimentally, the forward scattering intensity (that is: low-angle scattering extrapolated to zero angle) from any particle is equal to Z^2 multiplied by Thomson's formula (where Z is the number of electrons in the particle). This is true for everything from He gas to protein molecules in solution (as observed by SAXS). The reason why you multiply the intensity by just Z (and not Z^2) when the electrons are not confined to atoms is because out-of-phase amplitudes add "in quadrature": Ftotal=sqrt(sum(F^2)) and I~Ftotal^2. In-phase amplitudes just add: Ftotal=sum(F), and I~Ftotal^2. In multi-atom particles (such as proteins) you have two (or more) atoms that are constrained to be "near" one another. By "near" I mean separated by a distance corresponding to a scattering angle that will clear the beamstop. In this case, the scattering from one atom can interfere with that from the other (just as the scattering from electrons within the same atom interfere with each other). Taking the vector sum of the form factors of the two atoms together explains the total scattering. I have personally confirmed this with the scattering from N2 gas from the cryo-stream at my beamline! N2 has 14 electrons worth of forward scattering, not 7. If the distance between the two atoms is not infinitely precise, then the Gaussian distribution of atom-atom vectors in real space becomes a Gaussian in reciprocal space that you multiply by the "perfect" two-atom form factor. There is some debate over who first called this a "B factor", but the name has certainly stuck. 3) As Dale pointed out, the energy goes into other reflections. Don't forget F(0,0,0). That is a REAL reflection, and it is always on the Ewald sphere. If you manage to orient a crystal so that no visible Bragg peaks intersect the Ewald sphere, then all the elastically scattered photons will go into F(0,0,0). Incidentally, for visible light the interference of F(0,0,0) with the main beam gives rise to the "index of refraction" effect. There is an index of refraction for x-rays, but it is much smaller. These and other effects of conservation of energy are accounted for in the "dynamical theory" of diffraction, and this is what is used for the so-called "three beam" phasing technique. The interaction of scattering and absorption gives rise to anomalous dispersion, and this is also explained by the dynamical theory. Bragg's Law and other familiar equations come to us from the "kinematic approximation" to the dynamical theory. This approximation involves ignoring small "violations" to conservation of energy and conservation of momentum. For example, an elastically scattered photon has changed direction (momentum) without changing energy (wavelength). This approximation does ignore conservation of momentum, but the small amount of momentum transferred from the crystal to the photon is distributed evenly over all the atoms in the crystal, so the "recoil" motion of the crystal is very small (and safe to ignore in crystallography). Other things ignored by the kinematic approximation are secondary scattering and depletion of the primary (and diffracted) beam intensity as it looses photons to scattering. For protein crystals, the kinematic approximation is very good since there are many spots and they are all very weak when compared to the incident beam. I bought a different book (not Woolfson) on the dynamical theory so that I could understand all this more quantitatively. This book has served well putting me to sleep every night... -James Holton MAD Scientist Michel Fodje wrote: > Dear Crystallographers, > Here are a few paradoxes about diffraction I would like to get some > answers about: > > 1. In every description of Braggs' law I've seen, the in-coming waves > have to be in phase. Why is that? Given that the sources used for > diffraction studies are mostly non-coherent. > > 2. Trying to derive the diffraction condition for a pair of non-coherent > waves with a phase difference of 'y' where 0 < y < 2pi, I obtain the > following diffraction condition > y * (lambda/2pi) = 2d sin (theta) > i.e. the phase difference y = 4pi * sin(theta) * d / lambda > This seems to imply that diffraction will occur if the incident waves > are not in phase but the phase difference still satisfies the above > condition. One may be able to envision a case where for a given distance > d, the diffracting condition will be met for various angles depending on > the phase shift of the waves diffracting. Does this make sense? Has > anyone looked at the significance of this relationship before? Any > pointers will be welcome. > > 3. What happens to the photon energy when waves destructively interfere > as mentioned in the text books. Doesn't 'destructive interference' > appear to violate the first and second laws of thermodynamics? Besides, > since the sources are non-coherent, how come the photon 'waves' don't > annihilate each other before reaching the sample? If they were coherent, > would we just end up with a single wave any how? With what will it > interfere to cause diffraction? > > I'm sure some of these may have some really obvious answers I may be > missing. > > Thanks, > > Michel > <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. 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