Dear Edward, Re your em colleagues:- We are indeed happy to understand their diffraction to 5th order, by which we mean the d/5 reflection (1st order) because the two are simply different viewpoints.
Just one loose end:- The remarkable thing is that the diffraction from a crystal is largely empty. We focus on the spots, true, but the largely empty diffraction space from a crystal in a sense is a most useful aspect about the W L Bragg equation. Finally, just to mention, when I saw the laser light diffraction from a periodic ruled grating for the first time i thought:- it is magnificent. I rank it alongside the spectral lines in an atom's emission spectrum, such as the sodium D lines ie as i saw in my physics teaching lab. The red shifted hydrogen spectra of Hubble himself, available to view in the museum of the astronomical observatory in Los Angeles, are of course in a yet different, higher, league of where we are in the (expanding) universe. Yours sincerely, John Prof John R Helliwell DSc FInstP CPhys FRSC CChem F Soc Biol. Chair School of Chemistry, University of Manchester, Athena Swan Team. http://www.chemistry.manchester.ac.uk/aboutus/athena/index.html On 23 Aug 2013, at 16:34, "Edward A. Berry" <ber...@upstate.edu> wrote: > I think we are just discussing different ways of saying the same thing now. > But that can be interesting, too. If not, read no farther. > > herman.schreu...@sanofi.com wrote: >> Dear Edward, >> >> Now I am getting a little confused: If you look at a "higher order" 2n >> reflection, you will also get diffraction from the intermediate "1n" layers, >> so the structure factor you are looking at is in fact the "1n" structure >> factor. I think your original post was correct. >> > Yes- I think the original poster's question about diffraction from > the 2n planes, and whether that contributes to diffraction in the > 1n reflection, has been answered- physically they are the same thing. > > My question now is whether it is useful to consider Braggs-law "n" to > have values other than one, and whether it is useful to tie Braggs law > to the unit cell, or better to derive it for a set of equally spaced > planes (as I think it originally was derived) and later put conditions > on when those planes will diffract. > > In addition to Bragg's law one also talks about the "Bragg condition", > as somewhat related to the "diffraction condition" although maybe that > is closer to "Laue condition". > But anyway, the motivation for presenting Braggs law is to decide where > (as a function of lambda and theta) diffraction will be observed. > And in a continuos crystal (admittedly not what Braggs law was derived > for, but what the students are interested in) you don't get diffraction > without periodicity, and the spacing of the planes has to be related to > the unit cell for braggs law to help (as you say, periodicity of the planes > must match periodicity of the crystal). > > When Bragg's condition is met, points separated by d scatter in phase. > Diffraction occurs when d matches the periodicity of the material, so that > crystallographically-equivalent-by-translation points scatter in phase, > and the resultants from each unit layer (1-D unit cell) scatter in phase. > > If we are just considering equal planes separated by d with nothing between, > then the periodicity is just d, and bragg condition gives diffraction > condition. > If we are considering a crystal with continuous density, if d is equal to > a unit cell dimension and the planes are perpendicular to that axis, then > then the periodicity is d and brags law gives the (1-dimensional) diffraction > condition. > If d is some arbitrary spacing not related to periodicity of the matter, > brag condition still tells you that points separated by d along S scatter > in phase but if d has no relation to the periodicity, diffraction conditions > are not met and the different slabs thickness d will not scatter in phase. > If d is an integral submultiple of the periodicity, we get diffraction. > What is the best way to explain this? > 1. if points separated by d scatter in phase (actually out of phase by one > wavelength), > then spots separated by an integral multiple n of d will scatter in phase > (out of phase by n wavelengths). Now if n*d is the unit cell spacing, spots > separated by nd will be crystallographically equivalent, and scatter in > phase (actually out of phase by n wavelengths). > But this is more elegantly expressed by using braggs law with d' = the unit > cell spacing, nd, and n'= n. The right hand side of braggs law is calculating > the phase difference, and the left hand is saying this must be = n lambda. > That's what n is there for! > > 2. the periodicity of the set of planes must match the periodicity of the > crystal- > if d is a submultiple of the unit cell spacing, points separated by d will > scatter in > phase, but there is no relation between what exists at those points, so they > will > not interfere constructively. each slab of thickness d will have resultant > phase > (and amplitude) different from the slab above or below it. > But if d is an integral submultiple of unit cell spacing, there will be > periodicity to these slabs- the sixth slab will have the same content as > the first (or the fifth will be the same as the zero'th may be more > comfortable) > so each stack of five slabs will interfere constructively with the 5 slabs > above > it and so on throughout the crystal. > And as in the answer to original poster's question, it is the same diffraction > whether you consider it to be the first order diffraction of planes with d=c/5 > or the fifth-order diffraction from the unit cell spacing. > > I think these are equivalent in terms of the underlying physics > so this is semantics, or choosing the most intuitive explanation. > I will consider introducing Bragg's law for arbitrary planes in space > and introducing diffraction condition later with Laue condition. > And of course I should look again at some of the excellent textbooks > that are available in coming up with a plan. > > But when a colleague studying 2D crystals in cryo-EM gloats: > "I got diffraction out to the fifth order", we don't want to pour > cold water by saying, "sorry, those are first order diffraction from > from planes at 1/5 spacing!" even though it means the same thing in > terms of resolution. (OK, order of diffraction doesn't have to be > equated to "n" in braggs law, but it is convenient to do so) > > > > > > >> To summarize how I see it: >> 1) Braggs law has nothing to do with crystals or unit cells, it only >> describes diffraction from sets of planes. >> 2) However, to get constructive interference from all unit cells in the >> crystal, the periodicity of the set of planes must match the periodicity of >> the crystal, which means that only sets of planes with integer miller >> indices are allowed. >> >> So the unit cell dictates which sets of planes are able to constructively >> diffract. However, there might not be anything physically present in the >> crystal with that periodicity. In this case the corresponding reflection >> will be weak or absent. This is the kind of information we use to calculate >> our wonderful electron density maps. >> >> Best, >> Herman >> >> >> >> -----Ursprüngliche Nachricht----- >> Von: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] Im Auftrag von >> Edward A. Berry >> Gesendet: Freitag, 23. August 2013 01:01 >> An: CCP4BB@JISCMAIL.AC.UK >> Betreff: 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 >>>>> >>>>> > >>>>> >>>> >>>
