Hi Chen, Here is what I think:
Assuming a crystal is perfect and is being shot at 0 K, then the maximum resolution one experiment can achieve is limited by the wavelength of the X-ray. It can’t be better than the half-wavelength under the normal experimental setting (minimum d=lambda/2/sin(90)=lambda/2). With shorter wavelength, we get more reflections (an Ewald sphere with larger radius encloses more lattice points), and that’s more sampling points and more information. With infinitely short wavelength, we get infinitely detailed information. On the other hand, the seemingly continuous, detailed profile we get from a single molecule diffraction is also limited (smeared) by the same X-ray wavelength. So it is only a difference between measuring many discrete points and measuring (smeared) continuity. In other words, the continuous curve we get from the single molecule diffraction experiment does not contain information with frequency higher than that of the X-ray. It only contains information with frequency up to that of the X-ray. Recalling that the minimum d from a crystal diffraction experiment is lambda/2, then for a 1-D crystal’s unit cell (with edge length a), we are sampling it with a frequency of 2a/lambda. I think the sampling theorem says that this sampling frequency is as good as the continuous curve we get from single molecule diffraction with X-ray of wavelength lambda. With real life crystals, which are neither perfect, nor at 0 Kelvin, what makes difference is, by using a single molecule instead of a crystal, we can get away from the conformational differences of molecules found in a crystal, the defects in crystal, the heterogeneity of the crystal (e.g., the mosaicity), and probably even the background generated by solvent atoms (as the single molecule might be floating in vacuum). The packing defects and heterogeneity in a crystal is probably what limits resolution of our protein crystals in most cases. So when we are freed from the situation of having to use a crystal, in theory with short enough X-ray wavelength, by shooting at a single molecule that is not moving too much during the exposure, we can get a very high resolution that would not be achievable using a crystal. Now what weakens our high angle signal, limits the high resolution, and smears our map is the real thermo motion of the atoms, not the heterogeneity of the crystal. Then for the next step, with that highly sensitive detector and our ability of sending small pack of photons to the molecule, we might be able to get very quick snap shots of the molecule, essentially reducing the motion blur. Then in that case, what ultimately limits our resolution (or confidence of the measurement?) is probably the number of photons we can send to an atom before the absorbed energy significantly affect its location – may I say, a situation similar to that faced by the cryoEM people? Zhijie From: Chen Zhao Sent: Tuesday, January 20, 2015 10:18 PM To: CCP4BB@JISCMAIL.AC.UK Subject: [ccp4bb] A basic question about Fourier Transform Dear all, I am sorry about this slightly off-topic question. I am now a graduate TA for crystallography course and one student asked me a question that I didn't ask myself before. I don't have enough knowledge to precisely answer this question, so I am seeking for help here. The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? Thank you so much for your attention and your help in advance! Best, Chen
