Dear Dom, No attachment here in either of your messages...
Maybe you can put it up on Dropbox or Google drive and send us the URL? Thanks, Petr On 08/23/2013 04:33 AM, Dom Bellini wrote: > Hi > > Some people emailed me saying that the attachment did not get through. > > I hope this will work. > > Sorry. > > D > > ________________________________________ > From: CCP4 bulletin board [CCP4BB@JISCMAIL.AC.UK] on behalf of Edward A. > Berry [ber...@upstate.edu] > Sent: 23 August 2013 00:01 > To: ccp4bb > Subject: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law > > OK, I see my mistake. n has nothing to do with higher-order > reflections or planes at closer spacing than unit cell dimensions. > n >1 implies larger d, like the double layer mentioned by the original > poster, and those turn out to give the same structure factor as the > n=1 reflection so we only consider n=1 (for monochromatic). > The higher order reflection from closer spaced miller planes > of course do not satisfy bragg lawat the same lambda and theta. > So I hope people will disregard my confused post (but I think the > one before was somewhat in the right direction) > > The higher order diffractions come from finding planes through > the latticethat intersect a large number of points? no- planes > corresponding to 0,0,5 in an orthorhombic crystal do not all > intersect lattice points, and anyway protein crystals aren't > made of lattice points, they havecontinuous density. > > Applying Braggs law to these closer-spaced miller planes > will tell you that points in one plane will diffract in phase. > But since the protein in the five layers between the planes > will be different, in fact the layers will not diffract in > phase and diffraction condition will not be met. > > You could say OK, each of the 5 layesr will diffract > with different amplitude and out of phase, but their > vector-sum resultant will be the same as that of > every other five layers, so diffraction from points > through the whole crystal will interfere constructively. > > Or you could say that this theta and lambda satisfy the > bragg equation with d= c axis and n=5, so that points > separated by cell dimensions, which are equal due to > the periodicity of the crystal, will diffract in phase. > That would be a use for n>1 with monochromatic light. > The points separated by the small d-spacing scatter in > phase, but that is irrelevant since they are not > crystallographically equivalent. But they also scatter in phase > (actually out of phase by 5 wavelengths) with points separated > by one unit cell, because they satisfy braggs law with > d=c and n=5 (for 0,0,5 reflection still). > So then the higher-order reflections do involve n, > but it is the small d-spacing that corresponds to n=1 > and the unit cell spacing which corresponds to the higher n. > The latter results in the diffraction condition being met. > (or am I still confused?) > (and I hope I've got my line-wrapping under control now so this won't be so > hard to read) > > > > > > > > > > > > > > > > > > > > > > > Ethan Merritt wrote: >> On Thursday, August 22, 2013 02:19:11 pm Edward A. Berry wrote: >>> One thing I find confusing is the different ways in which d is used. >>> In deriving Braggs law, d is often presented as a unit cell dimension, >>> and "n" accounts for the higher order miller planes within the cell. >> It's already been pointed out above, and you sort of paraphrase it later, >> but let me give my spin on a non-confusing order of presentation. >> >> I think it is best to tightly associate n and lambda in your mind >> (and in the mind of a student). If you solve the Bragg's law equation for >> the wavelength, you don't get a unique answer because you are actually >> solving for n*lambda rather than lambda. >> >> There is no ambiguity about the d-spacing, only about the wavelength >> that d and theta jointly select for. >> >> That's why, as James Holton mentioned, when dealing with a white radiation >> source you need to do something to get rid of the harmonics of the wavelength >> you are interested in. >> >>> But then when you ask a student to use Braggs law to calculate the >>> resolution >>> of a spot at 150 mm from the beam center at given camera length and >>> wavelength, >>> without mentioning any unit cell, they ask, "do you mean the first order >>> reflection?" >> I would answer that with "Assume a true monochromatic beam, so n is >> necessarily >> equal to 1". >> >>> Yes, it would be the first order reflection from planes whose spacing is the >>> answer i am looking for, but going back to Braggs law derived with the unit >>> cell >>> it would be a high order reflection for any reasonable sized protein >>> crystal. >> For what it's worth, when I present Bragg's law I do it in three stages. >> 1) Explain the periodicity of the lattice (use a 2D lattice for clarity). >> 2) Show that a pair of indices hk defines some set of planes (lines) >> through the lattice. >> 3) Take some arbitrary set of planes and use it to draw the Bragg >> construction. >> >> This way the Bragg diagram refers to a particular set of planes, >> d refers to the resolution of that set of planes, and n=1 for a >> monochromatic X-ray source. The unit cell comes back into it only if you >> try to interpret the Bragg indices belonging to that set of planes. >> >> Ethan >> >> >>> Maybe the mistake is in bringing the unit cell into the derivation in the >>> first place, just define it in terms of >>> planes. But it is the periodicity of the crystal that results in the >>> diffraction condition, so we need the unit cell >>> there. The protein is not periodic at the higher d-spacing we are talking >>> about now (one of its fourier components is, >>> and that is what this reflection is probing.) >>> eab >>> >>> Gregg Crichlow wrote: >>>> I thank everybody for the interesting thread. (I'm sort of a nerd; I find >>>> this interesting.) I generally would always >>>> ignore that �n� in Bragg's Law when performing calculations on data, but >>>> its presence was always looming in the back of >>>> my head. But now that the issue arises, I find it interesting to return to >>>> the derivation of Bragg's Law that mimics >>>> reflection geometry from parallel planes. Please let me know whether this >>>> analysis is correct. >>>> >>>> To obtain constructive 'interference', the extra distance travelled by the >>>> photon from one plane relative to the other >>>> must be a multiple of the wavelength. >>>> >>>> ________\_/_________ >>>> >>>> ________\|/_________ >>>> >>>> The vertical line is the spacing "d" between planes, and theta is the >>>> angle of incidence of the photons to the planes >>>> (slanted lines for incident and diffracted photon - hard to draw in an >>>> email window). The extra distance travelled by >>>> the photon is 2*d*sin(theta), so this must be some multiple of the >>>> wavelength: 2dsin(theta)=n*lambda. >>>> >>>> But from this derivation, �d� just represents the distance between /any/ >>>> two parallel planes that meet this Bragg >>>> condition � not only consecutive planes in a set of Miller planes. >>>> However, when we mention d-spacing with regards to a >>>> data set, we usually are referring to the spacing between /consecutive/ >>>> planes. [The (200) spot represents d=a/2 >>>> although there are also planes that are spaced by a, 3a/2, 2a, etc]. So >>>> the minimum d-spacing for any spot would be the >>>> n=1 case. The n=2,3,4 etc, correspond to planes farther apart, also >>>> represented by d in the Bragg eq (based on this >>>> derivation) but really are 2d, 3d, 4d etc, by the way we define �d�. So we >>>> are really dealing with >>>> 2*n*d*sin(theta)=n*lambda, and so the �n�s� cancel out. (Of course, I�m >>>> dealing with the monochromatic case.) >>>> >>>> I never really saw it this way until I was forced to think about >>>> it by this new thread � does this makes sense? >>>> >>>> Gregg >>>> >>>> -----Original Message----- >>>> From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of >>>> Edward A. Berry >>>> Sent: Thursday, August 22, 2013 2:16 PM >>>> To: CCP4BB@JISCMAIL.AC.UK >>>> Subject: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's >>>> law >>>> >>>> herman.schreu...@sanofi.com <mailto:herman.schreu...@sanofi.com> wrote: >>>> >>>> > Dear James, >>>> >>>> > thank you very much for this answer. I had also been wondering about >>>> it. To clearify it for myself, and maybe for a >>>> few other bulletin board readers, I reworked the Bragg formula to: >>>> >>>> > >>>> >>>> > sin(theta) = n*Lamda / 2*d >>>> >>>> > >>>> >>>> > which means that if we take n=2, for the same sin(theta) d becomes >>>> twice as big as well, which means that we describe >>>> interference with a wave from a second layer of the same stack of planes, >>>> which means that we are still looking at the >>>> same structure factor. >>>> >>>> > >>>> >>>> > Best, >>>> >>>> > Herman >>>> >>>> > >>>> >>>> > >>>> >>>> This is how I see it as well- if you do a Bragg-law construct with two >>>> periods of d and consider the second order >>>> diffraction from the double layer, and compare it to the single-layer case >>>> you will see it is the same wave traveling >>>> the same path with the same phase at each point. When you integrate >>>> rho(r) dot S dr, the complex exponential will have >>>> a factor of 2 because it is second order, so the spatial frequency is the >>>> same. (I haven't actually shown this, being a >>>> math-challenged biologist, but put it on my list of things to do). >>>> >>>> So we could calculate the structure factor as either first order >>>> diffraction from the conventional d or second order >>>> diffraction from spacing of 2d and get the same result. by convention we >>>> use first order diffraction only. >>>> >>>> (same would hold for 3'd order diffraction from 3 layers etc.) >>>> >>>> > -----Urspr�ngliche Nachricht----- >>>> >>>> > Von: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] Im Auftrag von >>>> >>>> > James Holton >>>> >>>> > Gesendet: Donnerstag, 22. August 2013 08:55 >>>> >>>> > An: CCP4BB@JISCMAIL.AC.UK <mailto:CCP4BB@JISCMAIL.AC.UK> >>>> >>>> > Betreff: Re: [ccp4bb] Dependency of theta on n/d in Bragg's law >>>> >>>> > >>>> >>>> > Well, yes, but that's something of an anachronism. Technically, a >>>> >>>> > "Miller index" of h,k,l can only be a triplet of prime numbers >>>> (Miller, W. (1839). A treatise on crystallography. >>>> For J. & JJ Deighton.). This is because Miller was trying to explain >>>> crystal facets, and facets don't have >>>> "harmonics". This might be why Bragg decided to put an "n" in there. But >>>> it seems that fairly rapidly after people >>>> starting diffracting x-rays off of crystals, the "Miller Index" became >>>> generalized to h,k,l as integers, and we never >>>> looked back. >>>> >>>> > >>>> >>>> > It is a mistake, however, to think that there are contributions from >>>> different structure factors in a given spot. >>>> That does not happen. The "harmonics" you are thinking of are actually >>>> part of the Fourier transform. Once you do the >>>> FFT, each h,k,l has a unique "F" and the intensity of a spot is >>>> proportional to just one F. >>>> >>>> > >>>> >>>> > The only way you CAN get multiple Fs in the same spot is in Laue >>>> diffraction. Note that the "n" is next to lambda, >>>> not "d". And yes, in Laue you do get single spots with multiple hkl >>>> indices (and therefore multiple structure factors) >>>> coming off the crystal in exactly the same direction. Despite being at >>>> different wavelengths they land in exactly the >>>> same place on the detector. This is one of the more annoying things you >>>> have to deal with in Laue. >>>> >>>> > >>>> >>>> > A common example of this is the "harmonic contamination" problem in >>>> >>>> > beamline x-ray beams. Most beamlines use the h,k,l = 1,1,1 reflection >>>> >>>> > from a large single crystal of silicon as a diffraction grating to >>>> >>>> > select the wavelength for the experiment. This crystal is exposed to >>>> >>>> > "white" beam, so in every monochromator you are actually doing a Laue >>>> >>>> > diffraction experiment on a "small molecule" crystal. One good reason >>>> >>>> > for using Si(111) is because Si(222) is a systematic absence, so you >>>> >>>> > don't have to worry about the lambda/2 x-rays going down the pipe at >>>> >>>> > the same angle as the "lambda" you selected. However, Si(333) is not >>>> >>>> > absent, and unfortunately also corresponds to the 3rd peak in the >>>> >>>> > emission spectrum of an undulator set to have the fundamental coincide >>>> >>>> > with the Si(111)-reflected wavelength. This is probably why the >>>> >>>> > "third harmonic" is often the term used to describe the reflection >>>> >>>> > from Si(333), even for beamlines that don't have an undulator. But, >>>> >>>> > technically, Si(333) is n >>>> >>>> ot a "har >>>> >>>> monic" of Si(111). They are different reciprocal lattice points and each >>>> has its own structure factor. It is only the >>>> undulator that has "harmonics". >>>> >>>> > >>>> >>>> > However, after the monochromator you generally don't worry too much >>>> about the n=2 situation for: >>>> >>>> > n*lambda = 2*d*sin(theta) >>>> >>>> > because there just aren't any photons at that wavelength. Hope that >>>> makes sense. >>>> >>>> > >>>> >>>> > -James Holton >>>> >>>> > MAD Scientist >>>> >>>> > >>>> >>>> > >>>> >>>> > On 8/20/2013 7:36 AM, Pietro Roversi wrote: >>>> >>>> >> Dear all, >>>> >>>> >> >>>> >>>> >> I am shocked by my own ignorance, and you feel free to do the same, >>>> >>>> >> but do you agree with me that according to Bragg's Law a diffraction >>>> >>>> >> maximum at an angle theta has contributions to its intensity from >>>> >>>> >> planes at a spacing d for order 1, planes of spacing 2*d for order >>>> >>>> >> n=2, etc. etc.? >>>> >>>> >> >>>> >>>> >> In other words as the diffraction angle is a function of n/d: >>>> >>>> >> >>>> >>>> >> theta=arcsin(lambda/2 * n/d) >>>> >>>> >> >>>> >>>> >> several indices are associated with diffraction at the same angle? >>>> >>>> >> >>>> >>>> >> (I guess one could also prove the same result by a number of Ewald >>>> >>>> >> constructions using Ewald spheres of radius (1/n*lambda with n=1,2,3 >>>> >>>> >> ...) >>>> >>>> >> >>>> >>>> >> All textbooks I know on the argument neglect to mention this and in >>>> >>>> >> fact only n=1 is ever considered. >>>> >>>> >> >>>> >>>> >> Does anybody know a book where this trivial issue is discussed? >>>> >>>> >> >>>> >>>> >> Thanks! >>>> >>>> >> >>>> >>>> >> Ciao >>>> >>>> >> >>>> >>>> >> Pietro >>>> >>>> >> >>>> >>>> >> >>>> >>>> >> >>>> >>>> >> Sent from my Desktop >>>> >>>> >> >>>> >>>> >> Dr. Pietro Roversi >>>> >>>> >> Oxford University Biochemistry Department - Glycobiology Division >>>> >>>> >> South Parks Road Oxford OX1 3QU England - UK Tel. 0044 1865 275339 >>>> >>>> > >>>> > > -- Petr Leiman EPFL BSP 415 CH-1015 Lausanne Switzerand Office: +41 21 69 30 441 Mobile: +41 79 538 7647 Fax: +41 21 69 30 422
