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Re: [ccp4bb] difference density ripples around Hg atoms |
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CCP4bb navigationCCP4bb <-- 2007 <-- August 2007 <-- 01 August 2007Subject: Re: difference density ripples around Hg atoms From: Bart Hazes bart {- dot -} hazes {- at -} UALBERTA {- dot -} CA Date: 2007-08-01 I completely agree with you and should have read your message more carefully before jumping to conclusions. I thought you suggested the ripples where not strong enough ... I'd better have my coffee now :) Anyway, I don't think I wasted your time because your expanded explanation of the convolution theorem on this particular case is very useful as a reminder of this important concept. Bart Kay Diederichs wrote: > Bart Hazes schrieb: > ... > >> W.r.t. Kay's reply I think the argument does not hold since it depends >> on how badly the data is truncated. E.g. truncated near the limit of >> diffraction will give few ripples whereas a data set truncated at >> I/SigI of 5 will have much more servious effects. >> >> Bart > > > Bart, > > if you truncate at the limit of diffraction (i.e. where there is no more > signal) you will not get any ripple at all ! > > Of course, if you truncate at a resolution where there is significant > signal (and I do agree with you in that respect: many people truncate > their datasets at too low resolution) there _will_ be Fourier ripples. > However, a ripple is never as high than the peak itself. > > To get a quantitative picture of the worst-case scenario, consider the > following: truncation means multiplication of the data with a Heaviside > function (that is 1 up to the chosen resolution limit, and 0 beyond). In > real space, this translates into a series of ripples, arising by > convolution of the true electron density with the Fourier transform of > the Heaviside function. The Fourier transform of a one-dimensional > Heaviside function is the function sin(x)/x . Convolution with sin(x)/x > has the effect of > a) broadening (or "smearing") the true electron density, resulting in a > low-resolution electron density map instead of the true one > b) adding ripples at certain distances (which can be calculated from the > resolution) around each peak. The first negative ripple has an absolute > value of less than 1/4 of the peak height, and the first positive ripple > about 1/8 of the peak height. > > So in the worst case (one-dimensional truncation of data) my estimate of > 12% was wrong - I estimated the height of the first positive ripple > whereas Klemens reported the first negative ripple! > > On the other hand, if I remember correctly, the Fourier transform of the > 3-dimensional Heaviside function (a filled sphere) is a Bessel function > that has ripples which (I think) are lower than those of the > one-dimensional Heaviside function. Surely somebody knows the function, > and its peak heights? > > best, > > Kay -- ============================================================================== Bart Hazes (Assistant Professor) Dept. of Medical Microbiology & Immunology University of Alberta 1-15 Medical Sciences Building Edmonton, Alberta Canada, T6G 2H7 phone: 1-780-492-0042 fax: 1-780-492-7521 ============================================================================== CCP4bb navigationCCP4bb <-- 2007 <-- August 2007 <-- 01 August 2007 |
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