On 11/12/2017 6:48 AM, Kay Diederichs wrote: > On Fri, 10 Nov 2017 14:04:26 -0800, Dale Tronrud <de...@daletronrud.com> > wrote: > ... >> >> My belief is that the fact that our spot intensities represent the >> amplitude (squared) of a series of Sin waves is the result of the hard >> work of people like Bob who give us monochromatic illumination. >> "Monochromatic" simply means it is a pure Sin wave. If Bob could get >> that shiny new ring of his to produce an electromagnetic square wave his >> users would still get diffraction patterns with spots but they would >> have to come up with programs that would perform Fourier summations of >> square waves to calculate electron density. Our instrument is an analog >> computer for calculating the Sin wave Fourier transform of the electron >> density of our crystal because we designed it to do exactly that. >> >> Dale Tronrud >> > ... > > Hi Dale, > > Well, perhaps I understand you wrongly, but I'd say if Bob would succeed in > making his synchrotron produce "square" instead of sine waves then we would > not have to change our programs too much, because a "square wave" can be > viewed as (or decomposed into) superpositions of a sine wave of a given > frequency/energy with its higher harmonics, at known amplitude ratios. > This would be similar in some way to a Laue experiment, but not using a > continuum of energies, only discrete ones. The higher harmonics would just > change the intensities a bit (e.g. the 1,2,3 reflection would get some > additional intensity from the 2,4,6 and 3,6,9 reflection), and that change > could to a large extent be accounted for computationally, like we currently > do in de-twinning with low alpha. > That would probably be done in data processing, and might not affect the > downstream steps like map calculation.
What you are describing (which is absolutely correct) sounds like a lot more programming work than writing a square-wave Fourier transform program. All I'm doing is trying to answer the very intriguing question that beginners ask, but us old-timers tend to forget - Why are the intensity of the Bragg spots the square of the amplitude of SIN waves? The answer I'm proposing is that the illumination source is a Sin wave so the diffraction spots are in reference to Sin waves. If Bob could give us square waves the spot intensity would be proportional to the square of the square wave Fourier transform of the density. If ALS could give us triangular waves their spots would tell us about the triangular wave Fourier transform. While you want to continue to live in the Sin-wave world despite having square waves in your experiment, I could be perverse and do the same from my world. Your Sin waves can be expressed as a sum of the harmonics of my square waves and I could say that the intensity of what you call the 1,2,3 reflection contains information from what I would call the 1,2,3 and 2,4,6 and 3,6,9 (and so on) reflections. The mathematics is general and not specific to Sin waves. It just happens that it is easier for Bob to provide us with Sin wave illumination and so our analysis uses Sin waves. This is quite abstract, but in the free electron laser world the pulses are getting so short that they can't make the plane-wave approximation and have to analyze their images in terms of the wave packet, with its inherent bandwidth and coherence between the individual frequencies within the packet. See, my Sin-wave bias is showing - "bandwidth" and "frequencies" both come from an insistence on reducing all problems to Sin waves. Maybe the free electron people would do better by following Ethan and thinking about wavelets... Dale Tronrud > > best, > > Kay >
