Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
On Sun, 2013-09-01 at 22:28 +0200, James Holton wrote: ... but Bragg's genius was in simplifying all this to a little rule which tells you how much to turn the crystal to see a given spot. We sort of take this for granted now that we have automated diffractometers that do all the math for us, but in 1914 realizing that the rules or ordinary optics could be applied to x-rays and crystals was a pretty important step forward. The history of all this can be read in chapter 5 of Fifty Years of X-Ray Diffraction, edited by Ewald, that can be found here: http://www.iucr.org/publ/50yearsofxraydiffraction As noted on page 58, recognising that the principles of optics could be applied to X-rays and crystals meant that W.H. Bragg had to abandon his cherished view of X-rays as having a purely particle-like nature. An object lesson in open-mindedness, I think. Regards, Peter. -- Peter Keller Tel.: +44 (0)1223 353033 Global Phasing Ltd., Fax.: +44 (0)1223 366889 Sheraton House, Castle Park, Cambridge CB3 0AX United Kingdom
Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
Perhaps some of the confusion here arises because Bragg's Law is not a Fourier transform. Remember, in the standard diagram of Bragg's Law, there are only two atoms that are d apart. The full diffraction pattern from just two atoms actually looks like this: http://bl831.als.lbl.gov/~jamesh/nearBragg/intimage_twoatom.img This is an ADSC format image, so you can look at it in your favorite diffraction image viewer, such as ADXV, imosflm, HKL2000, XDSviewer, ipdisp, fit2d, whatever you like. Or, you can substitute png for img in the filename and look at it in your web browser. Notice how there are 9 bands for only 2 atoms? If you look at the *.img file you can see that the d spacing of the middle of each line is indeed 10 A, 5A, 3.33A, and 2.5A. Just as Bragg's Law predicts for n=1,2,3,4 because the two atoms were 10 A apart (d = 10 A) and the wavelength was 1 A. But what about the corners? The 2.5 A band reads a d-spacing of 1.65 A at the corners of the detector! Also, if you look at the central band, it passes through the direct beam (d=infinity), but at the edge of the detector it reads 2.14 A! Does this mean that Bragg's Law is wrong!? Of course not, it just means that Bragg's Law is one dimensional. Strictly speaking, it is about planes of atoms, not individual atoms themselves. The Fourier transform of two dots is indeed a series of bands (an interference pattern), but the Fourier transform of two planes (edge-on to the beam) is this: http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/20A_disks.img What? A caterpillar? How does that happen? Well, it helps to look at the diffraction pattern of a single plane: http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/20A_disk.img I should point out here that I'm not modelling an infinite plane, but rather a disk 20A in radius. This is why the edge of the caterpillar has a d-spacing of 40 A. If it were an infinite plane, its Fourier transform would be an infinitely thin line, visible at only one point: the origin. Which is not all that interesting. The halo around the main line is because the plane has a hard edge, and so its Fourier transform has fringes (its a sinc function). The reason why it does not run from the top of the image to the bottom is because the Ewald sphere (a geometric representation of Bragg's law) is curved, but the Fourier transform of a disk is a straight line in reciprocal space. By giving the plane a finite size you can more easily see that the diffraction pattern of a stack of two planes is nothing more than the diffraction pattern of one plane, multiplied by that of two points. This is a fundamental property of Fourier transforms: convolution becomes a product in reciprocal space. Where convolution is nothing more than copying an object to different places in space, and in this case these places are the two points in the Bragg diagram. But, still, why the caterpillar? It is because the Ewald sphere is curved, so the reciprocal-space line only brushes against it for a few orders. We can, however, get more orders by tilting the planes by some angle theta, such as the 11.53 degrees that satisfies n*lambda = 2*d*sin(theta) for n = 4. That is this image: http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_20A_disks.png Yes, you can still see the caterpillar, but clearly the 4th spot up is brighter than all but the 0th-order one. The only reason why it is not identical in intensity is because of the inverse square law: the pixels on the detector for the 4th-order reflection are a little further away from the sample than the zeroeth-order ones. As the planes get wider: http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_20A_disks.png http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_40A_disks.png http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_80A_disks.png http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_160A_disks.png the caterpillar gets thinner you see less and less of the n=1,2,3 orders. For an infinite pair of planes, there will be only two intersection points: the origin and the n=4 spot. This is not because the intermediate orders are not there, they are just not satisfying the Bragg condition, and neither are their fringes. Of course, with only two planes, even the infinite-plane spot will be much fatter in the vertical. Formally, about half as fat as the distance between the spots. This is because the interference pattern for only two points is still there. But if you have three, four or five planes, you get these: http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_20A_disk.png http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_20A_disks.png http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_20A_3disks.png http://bl831.als.lbl.gov/~jamesh/nearBragg/BraggsLaw/tilted_20A_4disks.png
Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
On 22 August 2013 07:54, James Holton jmhol...@lbl.gov wrote: 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. Yes but I think it would be a pity if we lost IMO the important distinction in meaning between Miller indices as defined above as co-prime integers and (for want of a better term) reflection indices as found in an MTZ file. For example, Stout Jensen makes a careful distinction between them (as I recall they call reflection indices something like general indices: sorry I don't have my copy of S J to hand to check their exact terminology). The confusion that can arise by referring to reflection indices as Miller indices is well illustrated if you try to explain Bragg's equation to a novice, because the d in the equation (i.e. n lambda = 2d sin[theta]) is the interplanar separation for planes as calculated from their Miller indices, whereas the theta is of course the theta angle as calculated from the corresponding reflection indices. If you say that Miller reflection indices are the same thing you have a hard time explaining the equation! One obvious way out of the dilemma is to drop the n term (so now lambda = 2d sin[theta]) and then redefine d as d/n so the new d is calculated from the same reflection indices as theta, and the Miller indices don't enter into it. But then you have to explain to your novice why you know better than a Nobel prizewinner! As you say Bragg no doubt had a good reason to include the n (i.e. to make the connection between the macroscopic properties of a crystal and its diffraction pattern). Sorry for coming into this discussion somewhat late! Cheers -- Ian
Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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 n1 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
[ccp4bb] AW: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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. 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 n1 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
Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
Hi, Despite the not so large size of the pdf (256 kbs), the file does not want to get through. Since a reasonable amount of people seem to have liked a copy for their students, following some smart suggestions I have put the booklet on Dropbox. Here's the link: https://www.dropbox.com/s/gljckhw7ui6df6c/Booklet.pdf?m Best, D -Original Message- From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of Dom Bellini Sent: 22 August 2013 23:38 To: ccp4bb Subject: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law Dear Community, I have attached a short booklet written some 6 years ago during rainy evenings to teach principle of crystal diffraction with biologist students in mind, never used it as I don't have students, but I now believe its mission was this thread ;-) It uses lots of real space diffraction examples, easily to picture them in the head, so to stick with the students for good. It was written with the philadelphia philosophy of explain it to me as if I was a 5 yo. I hope it can save some students the time of going to look for many different sources as it is probably a nice summary of the diffraction process. A short answer, same as many of the other answers but in different words, to the original post would be: each hkl family of planes generates one and only one structure factor or diffraction spot without any contributions from other families (talking of monochromatic experiments). Perhaps doubts may arise due to the fact that, e.g., every other 002 plane superpose/aligns with one 001 plane, but since their spacing d is half of the other and lambda is fixed, from 2d sin(theta)=lambda it will result that 002, despite perfectly superposing with (same inclination of) 001, will reflect in a different direction with sin(theta) twice as that for 001. Despite 001 and 002 superpose/align with one another the diffraction angle changes because it is not a real reflection phenomenon (as if they were mirrors), keeping in mind that the planes are only imaginary and a way to help us to visualize the process. Probably many people from the bb that could have given a better explanation than mine might have been put off by the length of the email that might have been required. Since I had already written it and ready to go I decided to attach it with the best of intentions. Hopefully it may turn up to be useful for some one. D From: CCP4 bulletin board [CCP4BB@JISCMAIL.AC.UK] on behalf of Ethan Merritt [merr...@u.washington.edu] Sent: 22 August 2013 22:57 To: ccp4bb Subject: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law 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
Re: [ccp4bb] AW: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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
Re: [ccp4bb] AW: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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
[ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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 -Ursprüngliche 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 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 not a harmonic 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
Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
Adding to what James wrote: I see this as follows: Bragg's law is only a necessary but not a sufficient criterion for occurrence of a diffraction peak if viewed as a reflection. The problem imho comes from not considering the structure factor as the actual quantifier of (single photon) diffraction. Take for example the reflection condition rules for I centered cells: Sum of the three reflection indices all even. 111 333 missing but 222 is there... The problem does not arise if you treat the diffraction pattern as a probability distribution where the intensities are proportional to the (Square of) the structure factors which includes the hkls and thus automatically and necessarily the Bragg ns. Taking the simple 2-wave Bragg reflection picture verbatim does not do justice to the diffraction process (e.g. the non-coherence between incoming photons and necessity for a single photon process have been discussed here before). James has already suggested a few historic reasons. Best, BR -Original Message- From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of Dom Bellini Sent: Dienstag, 20. August 2013 16:49 To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] Dependency of theta on n/d in Bragg's law Dear Pietro, Ladd Palmer book does explain it, just first example that springs to mind. HTH D -Original Message- From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of Pietro Roversi Sent: 20 August 2013 15:37 To: ccp4bb Subject: [ccp4bb] Dependency of theta on n/d in Bragg's law 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 -- This e-mail and any attachments may contain confidential, copyright and or privileged material, and are for the use of the intended addressee only. If you are not the intended addressee or an authorised recipient of the addressee please notify us of receipt by returning the e-mail and do not use, copy, retain, distribute or disclose the information in or attached to the e-mail. Any opinions expressed within this e-mail are those of the individual and not necessarily of Diamond Light Source Ltd. Diamond Light Source Ltd. cannot guarantee that this e-mail or any attachments are free from viruses and we cannot accept liability for any damage which you may sustain as a result of software viruses which may be transmitted in or with the message. Diamond Light Source Limited (company no. 4375679). Registered in England and Wales with its registered office at Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire, OX11 0DE, United Kingdom
Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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.) -Ursprüngliche 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 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
Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
Dear Pietro, The n in Bragg's Law is indeed most interesting for teachers and a most delicate matter for those enquiring about it. The diffraction grating equation, from which W L Bragg got the idea, a 'cheap accolade' he said to have it named after him in his Scientific American article, has each order at its own specific theta. n=1 at one angle, n=2 the next order of diffraction at higher angle, n=3 the next order at higher angle still and so on. This is how physicists usually first meet the effect and use monochromatic laser light and a periodically ruled, line, grating to see the laser diffraction pattern in the modern physics teaching labs. In crystal structure analysis the ruled line of the above is now the unit cell of the crystal and the contents are of chemical and biological interest, unlike the inside of a ruled line! Thus the switch to using lamba = 2d sin theta form of the equation and the n subsumed into the interplanar spacing. d=1 is the unit cell edge, d/2 half the unit cell and so on. Each has its own reflection intensity. The highest resolution molecular detail we get of the insides of the unit cell arising from the highest n diffraction reflection with a 'measurable' intensity. The use of polychromatic light, or white X-rays, we need not consider just now. Suffice to say at this point that, eg historically, the W H Bragg X-ray spectrometer provided monochromatic X-rays to illuminate a single crystal and so, as his son W L Bragg put it, immediately enabled a clear and more powerful analysis of crystal structure and thereby allowed the first detailed atomic X-ray crystal structure, sodium chloride, to be resolved. Several other Xray crystal structures immediately followed from the Braggs, using the Xray spectrometer, before 1914 ie when the Great War pretty much put all basic research and development 'on hold'. When one does come to the question of 'Laue diffraction' the so called multiplicity distribution of Bragg reflections in Laue pattern spots has been treated in detail by Cruickshank et al 1987 Acta Cryst A, as pointed out by Tim. Prime numbers are pivotal to the analysis, as James pointed out. Best wishes, 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 20 Aug 2013, at 15:36, Pietro Roversi pietro.rove...@bioch.ox.ac.uk 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
Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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 ns cancel out. (Of course, Im 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 mailto:herman.schreu...@sanofi.com 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.) -Ursprüngliche Nachricht- Von: CCP4 bulletin board [ mailto:CCP4BB@JISCMAIL.AC.UK mailto:CCP4BB@JISCMAIL.AC.UK] Im Auftrag von James Holton Gesendet: Donnerstag, 22. August 2013 08:55 An: mailto:CCP4BB@JISCMAIL.AC.UK 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
Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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. 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? 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. 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.) -Ursprüngliche 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:
Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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
Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
Dear Community, I have attached a short booklet written some 6 years ago during rainy evenings to teach principle of crystal diffraction with biologist students in mind, never used it as I don't have students, but I now believe its mission was this thread ;-) It uses lots of real space diffraction examples, easily to picture them in the head, so to stick with the students for good. It was written with the philadelphia philosophy of explain it to me as if I was a 5 yo. I hope it can save some students the time of going to look for many different sources as it is probably a nice summary of the diffraction process. A short answer, same as many of the other answers but in different words, to the original post would be: each hkl family of planes generates one and only one structure factor or diffraction spot without any contributions from other families (talking of monochromatic experiments). Perhaps doubts may arise due to the fact that, e.g., every other 002 plane superpose/aligns with one 001 plane, but since their spacing d is half of the other and lambda is fixed, from 2d sin(theta)=lambda it will result that 002, despite perfectly superposing with (same inclination of) 001, will reflect in a different direction with sin(theta) twice as that for 001. Despite 001 and 002 superpose/align with one another the diffraction angle changes because it is not a real reflection phenomenon (as if they were mirrors), keeping in mind that the planes are only imaginary and a way to help us to visualize the process. Probably many people from the bb that could have given a better explanation than mine might have been put off by the length of the email that might have been required. Since I had already written it and ready to go I decided to attach it with the best of intentions. Hopefully it may turn up to be useful for some one. D From: CCP4 bulletin board [CCP4BB@JISCMAIL.AC.UK] on behalf of Ethan Merritt [merr...@u.washington.edu] Sent: 22 August 2013 22:57 To: ccp4bb Subject: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law 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
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 n1 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
Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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 n1 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
[ccp4bb] Dependency of theta on n/d in Bragg's law
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
Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
-BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Dear Pietro, You may take a textbook into account which deals with Laue diffraction. If you search for the keyword Laue and the author Helliwell at the IUCR journals, you will get a large number of hits, indicating that this is by no means a 'trivial' issue (e.g. http://dx.doi.org/10.1107/S0909049599006366 for an overview or http://dx.doi.org/10.1107/S0108767387098763 for the treatment of harmonics). As far as I understand, certain scaling programs take the lambda/2 contribution of monochromators into account. Regards, Tim Gruene On 08/20/2013 04:36 PM, 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 - -- - -- Dr Tim Gruene Institut fuer anorganische Chemie Tammannstr. 4 D-37077 Goettingen GPG Key ID = A46BEE1A -BEGIN PGP SIGNATURE- Version: GnuPG v1.4.14 (GNU/Linux) Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org/ iD8DBQFSE4EgUxlJ7aRr7hoRAomhAJ94WrXRCTx8gevMAzrhenVri2EkhwCghyC1 UuckhqtUEG0uB9hheG1uxz0= =+cAo -END PGP SIGNATURE-
Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
Dear Pietro, Ladd Palmer book does explain it, just first example that springs to mind. HTH D -Original Message- From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of Pietro Roversi Sent: 20 August 2013 15:37 To: ccp4bb Subject: [ccp4bb] Dependency of theta on n/d in Bragg's law 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 -- This e-mail and any attachments may contain confidential, copyright and or privileged material, and are for the use of the intended addressee only. If you are not the intended addressee or an authorised recipient of the addressee please notify us of receipt by returning the e-mail and do not use, copy, retain, distribute or disclose the information in or attached to the e-mail. Any opinions expressed within this e-mail are those of the individual and not necessarily of Diamond Light Source Ltd. Diamond Light Source Ltd. cannot guarantee that this e-mail or any attachments are free from viruses and we cannot accept liability for any damage which you may sustain as a result of software viruses which may be transmitted in or with the message. Diamond Light Source Limited (company no. 4375679). Registered in England and Wales with its registered office at Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire, OX11 0DE, United Kingdom
