[ccp4bb] Summary : [ccp4bb] embarrassingly simple MAD phasing question
Thanks for the overwhelming response. I think I probably didn't phrase the question quite right, but I pieced together an answer to the question I wanted to ask, which hopefully is right. On Oct 13, 2010, at 1:14 PM, SHEPARD William wrote: It is very simple, the structure factor for the anomalous scatterer is FA = FN + F'A + iFA (vector addition) The vector FA is by definition always +i (90 degrees anti-clockwise) with respect to the vector FN (normal scattering), and it represents the phase lag in the scattered wave. So I guess I should have started by saying I knew f'' was imaginary, the absorption term, and always needs to be 90 degrees in phase ahead of the f' (dispersive component). So here is what I think the answer to my question is, if I understood everyone correctly: Starting with what everyone I guess thought I was asking, FA = FN + F'A + iFA (vector addition) for an absorbing atom at the origin, FN (the standard atomic scattering factor component) is purely real, and the f' dispersive term is purely real, and the f absorption term is purely imaginary (and 90 degrees ahead). Displacement from the origin rotates the resultant vector FA in the complex plane. That implies each component in the vector summation is rotated by that same phase angle, since their magnitudes aren't changed from displacement from the origin, and F must still be perpendicular to F'. Hence the absorption term F is no longer pointed in the imaginary axis direction. Put slightly differently, the fundamental requirement is that the positive 90 degree angle between f' and f must always be maintained, but their absolute orientations are only enforced for atoms at the origin. Please correct me if this is wrong. Also, since F then has a projection upon the real axis, it now has a real component (and I guess this is also an explanation for why you don't get this with centrosymmetric structures). Thanks again for everyone's help. -- Bill William G. Scott Professor Department of Chemistry and Biochemistry and The Center for the Molecular Biology of RNA 228 Sinsheimer Laboratories University of California at Santa Cruz Santa Cruz, California 95064 USA phone: +1-831-459-5367 (office) +1-831-459-5292 (lab) fax:+1-831-4593139 (fax)
Re: [ccp4bb] Summary : [ccp4bb] embarrassingly simple MAD phasing question
Dear Bill, The discussion is becoming complicated because of the mixing of notations. There is a theory or model which describes the atomic scattering factor as f = f0 + f' +if from which the structure factor is calculated. That right angle that you see in the picture you sent us with that link is a consequence of that factor i in if, therefore I would not call it a fundamental requirement. Now as you displace that particular atom from the origin, all three components receive the same phase which leaves Fa and Fa in the same relative orientation. That sounds pretty much like your explanation, but I have the impression that we have different notions of the sources of effects. This could be, though, like a discussion about whether the chicken or the egg came first, meaning that neither is more or less correct than the other. Maybe to better understand your question you could explain what you think the origin ior source of that image actually is (technically, not in terms of copyright). In a centrosymmetric structure, all phase angles are either 0degree or 180degree whyfore - as you already point out - the additional anomalous term does not affect the validity of Friedel's law. For basically the same reason you would not detect an anomalous signal from a crystal containing only one element, irrespective of the values of f' and f. The link you sent, by the way, rises the impression that in the absence of anomalous scattering phi(F) = phi(-F), but this should read phi(F) = -phi(-F). Furthermore the abbreviation -F instead of F(-h-k-l) is also misleading because F(-h-k-l) is not the same as -F(hkl). Cheers, Tim On Wed, Oct 13, 2010 at 01:58:56PM -0700, William Scott wrote: Thanks for the overwhelming response. I think I probably didn't phrase the question quite right, but I pieced together an answer to the question I wanted to ask, which hopefully is right. On Oct 13, 2010, at 1:14 PM, SHEPARD William wrote: It is very simple, the structure factor for the anomalous scatterer is FA = FN + F'A + iFA (vector addition) The vector FA is by definition always +i (90 degrees anti-clockwise) with respect to the vector FN (normal scattering), and it represents the phase lag in the scattered wave. So I guess I should have started by saying I knew f'' was imaginary, the absorption term, and always needs to be 90 degrees in phase ahead of the f' (dispersive component). So here is what I think the answer to my question is, if I understood everyone correctly: Starting with what everyone I guess thought I was asking, FA = FN + F'A + iFA (vector addition) for an absorbing atom at the origin, FN (the standard atomic scattering factor component) is purely real, and the f' dispersive term is purely real, and the f absorption term is purely imaginary (and 90 degrees ahead). Displacement from the origin rotates the resultant vector FA in the complex plane. That implies each component in the vector summation is rotated by that same phase angle, since their magnitudes aren't changed from displacement from the origin, and F must still be perpendicular to F'. Hence the absorption term F is no longer pointed in the imaginary axis direction. Put slightly differently, the fundamental requirement is that the positive 90 degree angle between f' and f must always be maintained, but their absolute orientations are only enforced for atoms at the origin. Please correct me if this is wrong. Also, since F then has a projection upon the real axis, it now has a real component (and I guess this is also an explanation for why you don't get this with centrosymmetric structures). Thanks again for everyone's help. -- Bill William G. Scott Professor Department of Chemistry and Biochemistry and The Center for the Molecular Biology of RNA 228 Sinsheimer Laboratories University of California at Santa Cruz Santa Cruz, California 95064 USA phone: +1-831-459-5367 (office) +1-831-459-5292 (lab) fax:+1-831-4593139 (fax) -- -- Tim Gruene Institut fuer anorganische Chemie Tammannstr. 4 D-37077 Goettingen phone: +49 (0)551 39 22149 GPG Key ID = A46BEE1A signature.asc Description: Digital signature
Re: [ccp4bb] Summary : [ccp4bb] embarrassingly simple MAD phasing question
Bill, If I understand you correctly, the problem turns to be understanding coordinate system. The coordinate system in the plot in your original email is not a complex one but a polar coordinate system [|F| and phase (polar angle)]. In order to add the contribution of an atom with anomalous scattering, a complex coordinate is borrowed and used regionally at the tip region (why I say region not the tip) of Fp (Fp is subjected to be added by Fa and iFa), and the resultant F again is shown in the polar coordinate system again. During the adduct, Fa and Fa are perpendicular. Lijun Thanks for the overwhelming response. I think I probably didn't phrase the question quite right, but I pieced together an answer to the question I wanted to ask, which hopefully is right. On Oct 13, 2010, at 1:14 PM, SHEPARD William wrote: It is very simple, the structure factor for the anomalous scatterer is FA = FN + F'A + iFA (vector addition) The vector FA is by definition always +i (90 degrees anti- clockwise) with respect to the vector FN (normal scattering), and it represents the phase lag in the scattered wave. So I guess I should have started by saying I knew f'' was imaginary, the absorption term, and always needs to be 90 degrees in phase ahead of the f' (dispersive component). So here is what I think the answer to my question is, if I understood everyone correctly: Starting with what everyone I guess thought I was asking, FA = FN + F'A + iFA (vector addition) for an absorbing atom at the origin, FN (the standard atomic scattering factor component) is purely real, and the f' dispersive term is purely real, and the f absorption term is purely imaginary (and 90 degrees ahead). Displacement from the origin rotates the resultant vector FA in the complex plane. That implies each component in the vector summation is rotated by that same phase angle, since their magnitudes aren't changed from displacement from the origin, and F must still be perpendicular to F'. Hence the absorption term F is no longer pointed in the imaginary axis direction. Put slightly differently, the fundamental requirement is that the positive 90 degree angle between f' and f must always be maintained, but their absolute orientations are only enforced for atoms at the origin. Please correct me if this is wrong. Also, since F then has a projection upon the real axis, it now has a real component (and I guess this is also an explanation for why you don't get this with centrosymmetric structures). Thanks again for everyone's help. -- Bill William G. Scott Professor Department of Chemistry and Biochemistry and The Center for the Molecular Biology of RNA 228 Sinsheimer Laboratories University of California at Santa Cruz Santa Cruz, California 95064 USA phone: +1-831-459-5367 (office) +1-831-459-5292 (lab) fax:+1-831-4593139 (fax) Lijun Liu Cardiovascular Research Institute University of California, San Francisco 1700 4th Street, Box 2532 San Francisco, CA 94158 Phone: (415)514-2836
Re: [ccp4bb] Summary : [ccp4bb] embarrassingly simple MAD phasing question
On Oct 13, 2010, at 3:09 PM, Tim Gruene wrote: Dear Bill, The discussion is becoming complicated because of the mixing of notations. There is a theory or model which describes the atomic scattering factor as f = f0 + f' +if from which the structure factor is calculated. That right angle that you see in the picture you sent us with that link is a consequence of that factor i in if, therefore I would not call it a fundamental requirement. Dear Tim: Sorry, I mean that it was a fundamental requirement (restriction) that the absorption term must remain 90 degrees out of phase with the dispersive term, regardless of the absolute phase angle of their resultant. (And it must be retarded, due to absorption and conservation of energy, as Bernhard pointed out). Now as you displace that particular atom from the origin, all three components receive the same phase which leaves Fa and Fa in the same relative orientation. That is what I was trying to say. (But on the drive home from work, I was plagued by renewed doubt, for why then would there be a relative angle between f0 and f' that was anything but 0 degrees or 180 degrees?) That sounds pretty much like your explanation, but I have the impression that we have different notions of the sources of effects. I started to reply to emails in the inverse order I received them, realized I had forgotten to go to a boring meeting that in the end never went down anyway, and mixed notation from that email and the figure (or my false memory of the figure). This could be, though, like a discussion about whether the chicken or the egg came first, meaning that neither is more or less correct than the other. Maybe to better understand your question you could explain what you think the origin ior source of that image actually is (technically, not in terms of copyright). Google image search. Someone pointed out it was Bernhard's originally. (Sorry!) My understanding of the origin of the effect illustrated in the figure is roughly as follows: 1. We treat X-ray scattering using the assumptions inherent in the First-Order Born Approximation to elastic scattering (the photon interacts with an electron described by a spherically symmetric potential that is purely real). The photon scatters once, and elastically, so the atomic scattering factor is simply the Fourier Transform of the spherically symmetric (real) potential, or, using a couple of algebra steps and Poisson's equation, it is the Fourier Transform of the (real) electron density. The reality and symmetry of the potential ultimately are manifested in Friedel symmetry. The scattering from a spherically symmetric potential (not an electron per se) at the origin has a purely real amplitude and a phase of 1 (phase angle of zero) and nonzero phase angles result from displacement of the real scattering potential from the origin, so for elastic (non-absorptive) scattering, the resultant phase is a consequence purely of spatial displacement. Hopefully I am right so far ... 2. Within the confines of the approximation we use (or, equivalently, Fraunhofer Diffraction, if you want to stick to a purely classical treatment), absorption is modeled with a complex potential. The imaginary term added to the potential could account for both emission and absorption, but conservation of energy dictates it be the latter (hence the absolute value of the orientation of f). There is a good treatment of this in James; Blundell and Johnson glosses over the some of the essential steps in the derivation. But the main point is that the F (the imaginary component) arises as a consequence of absorption, which we model as a complex potential. This is distinct from what arises from path length displacement. In a centrosymmetric structure, all phase angles are either 0degree or 180degree whyfore - as you already point out - the additional anomalous term does not affect the validity of Friedel's law. For basically the same reason you would not detect an anomalous signal from a crystal containing only one element, irrespective of the values of f' and f. The link you sent, by the way, rises the impression that in the absence of anomalous scattering phi(F) = phi(-F), but this should read phi(F) = -phi(-F). Furthermore the abbreviation -F instead of F(-h-k-l) is also misleading because F(-h-k-l) is not the same as -F(hkl). Here's the figures I made that I actually used in my lecture. I realize now the way I made the first one set me up for this confusion: http://sage.ucsc.edu/~chem200a/2009/slides/diffraction_004/diffraction_004.html_files/diffraction_004.022-001.jpg http://sage.ucsc.edu/~chem200a/2009/slides/diffraction_004/diffraction_004.html_files/diffraction_004.023-009.jpg These are actually composite images of several-step slides, and for some reason a couple of labels didn't appear when I exported from Keynote, so I looked for something else with
