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 [[log in to unmask]] on behalf of Edward A. Berry [[log in to unmask]]
> Sent: 23 August 2013 00:01
> To: ccp4bb
> Subject: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
>
> OK, I see my mistake. n has nothing to do with higher-order
> reflections or planes at closer spacing than unit cell dimensions.
> n >1 implies larger d, like the double layer mentioned by the original
> poster, and those turn out to give the same structure factor as the
> n=1 reflection so we only consider n=1 (for monochromatic).
> The higher order reflection from closer spaced miller planes
> of course do not satisfy bragg lawat the same lambda and theta.
> So I hope people will disregard my confused post (but I think the
> one before was somewhat in the right direction)
>
> The higher order diffractions come from finding planes through
> the latticethat intersect a large number of points? no- planes
> corresponding to 0,0,5 in an orthorhombic crystal do not all
> intersect lattice points, and anyway protein crystals aren't
> made of lattice points, they havecontinuous density.
>
> Applying Braggs law to these closer-spaced miller planes
> will tell you that points in one plane will diffract in phase.
> But since the protein in the five layers between the planes
> will be different, in fact the layers will not diffract in
> phase and diffraction condition will not be met.
>
> You could say OK, each of the 5 layesr will diffract
> with different amplitude and out of phase, but their
> vector-sum resultant will be the same as that of
> every other five layers, so diffraction from points
> through the whole crystal will interfere constructively.
>
> Or you could say that this theta and lambda satisfy the
> bragg equation with d= c axis and n=5, so that points
> separated by cell dimensions, which are equal due to
> the periodicity of the crystal, will diffract in phase.
> That would be a use for n>1 with monochromatic light.
> The points separated by the small d-spacing scatter in
> phase, but that is irrelevant since they are not
> crystallographically equivalent. But they also scatter in phase
> (actually out of phase by 5 wavelengths) with points separated
> by one unit cell, because they satisfy braggs law with
> d=c and n=5 (for 0,0,5 reflection still).
> So then the higher-order reflections do involve n,
> but it is the small d-spacing that corresponds to n=1
> and the unit cell spacing which corresponds to the higher n.
> The latter results in the diffraction condition being met.
> (or am I still confused?)
> (and I hope I've got my line-wrapping under control now so this won't be so hard to read)
>
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> 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:[log in to unmask]] On Behalf Of Edward A. Berry
>>>> Sent: Thursday, August 22, 2013 2:16 PM
>>>> To: [log in to unmask]
>>>> Subject: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
>>>>
>>>> [log in to unmask] <mailto:[log in to unmask]> 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:[log in to unmask]] Im Auftrag von
>>>>
>>>> > James Holton
>>>>
>>>> > Gesendet: Donnerstag, 22. August 2013 08:55
>>>>
>>>> > An: [log in to unmask] <mailto:[log in to unmask]>
>>>>
>>>> > Betreff: Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
>>>>
>>>> >
>>>>
>>>> > Well, yes, but that's something of an anachronism. Technically, a
>>>>
>>>> > "Miller index" of h,k,l can only be a triplet of prime numbers (Miller, W. (1839). A treatise on crystallography.
>>>> For J. & JJ Deighton.). This is because Miller was trying to explain crystal facets, and facets don't have
>>>> "harmonics". This might be why Bragg decided to put an "n" in there. But it seems that fairly rapidly after people
>>>> starting diffracting x-rays off of crystals, the "Miller Index" became generalized to h,k,l as integers, and we never
>>>> looked back.
>>>>
>>>> >
>>>>
>>>> > It is a mistake, however, to think that there are contributions from different structure factors in a given spot.
>>>> That does not happen. The "harmonics" you are thinking of are actually part of the Fourier transform. Once you do the
>>>> FFT, each h,k,l has a unique "F" and the intensity of a spot is proportional to just one F.
>>>>
>>>> >
>>>>
>>>> > The only way you CAN get multiple Fs in the same spot is in Laue diffraction. Note that the "n" is next to lambda,
>>>> not "d". And yes, in Laue you do get single spots with multiple hkl indices (and therefore multiple structure factors)
>>>> coming off the crystal in exactly the same direction. Despite being at different wavelengths they land in exactly the
>>>> same place on the detector. This is one of the more annoying things you have to deal with in Laue.
>>>>
>>>> >
>>>>
>>>> > A common example of this is the "harmonic contamination" problem in
>>>>
>>>> > beamline x-ray beams. Most beamlines use the h,k,l = 1,1,1 reflection
>>>>
>>>> > from a large single crystal of silicon as a diffraction grating to
>>>>
>>>> > select the wavelength for the experiment. This crystal is exposed to
>>>>
>>>> > "white" beam, so in every monochromator you are actually doing a Laue
>>>>
>>>> > diffraction experiment on a "small molecule" crystal. One good reason
>>>>
>>>> > for using Si(111) is because Si(222) is a systematic absence, so you
>>>>
>>>> > don't have to worry about the lambda/2 x-rays going down the pipe at
>>>>
>>>> > the same angle as the "lambda" you selected. However, Si(333) is not
>>>>
>>>> > absent, and unfortunately also corresponds to the 3rd peak in the
>>>>
>>>> > emission spectrum of an undulator set to have the fundamental coincide
>>>>
>>>> > with the Si(111)-reflected wavelength. This is probably why the
>>>>
>>>> > "third harmonic" is often the term used to describe the reflection
>>>>
>>>> > from Si(333), even for beamlines that don't have an undulator. But,
>>>>
>>>> > technically, Si(333) is n
>>>>
>>>> ot a "har
>>>>
>>>> monic" of Si(111). They are different reciprocal lattice points and each has its own structure factor. It is only the
>>>> undulator that has "harmonics".
>>>>
>>>> >
>>>>
>>>> > However, after the monochromator you generally don't worry too much about the n=2 situation for:
>>>>
>>>> > n*lambda = 2*d*sin(theta)
>>>>
>>>> > because there just aren't any photons at that wavelength. Hope that makes sense.
>>>>
>>>> >
>>>>
>>>> > -James Holton
>>>>
>>>> > MAD Scientist
>>>>
>>>> >
>>>>
>>>> >
>>>>
>>>> > On 8/20/2013 7:36 AM, Pietro Roversi wrote:
>>>>
>>>> >> Dear all,
>>>>
>>>> >>
>>>>
>>>> >> I am shocked by my own ignorance, and you feel free to do the same,
>>>>
>>>> >> but do you agree with me that according to Bragg's Law a diffraction
>>>>
>>>> >> maximum at an angle theta has contributions to its intensity from
>>>>
>>>> >> planes at a spacing d for order 1, planes of spacing 2*d for order
>>>>
>>>> >> n=2, etc. etc.?
>>>>
>>>> >>
>>>>
>>>> >> In other words as the diffraction angle is a function of n/d:
>>>>
>>>> >>
>>>>
>>>> >> theta=arcsin(lambda/2 * n/d)
>>>>
>>>> >>
>>>>
>>>> >> several indices are associated with diffraction at the same angle?
>>>>
>>>> >>
>>>>
>>>> >> (I guess one could also prove the same result by a number of Ewald
>>>>
>>>> >> constructions using Ewald spheres of radius (1/n*lambda with n=1,2,3
>>>>
>>>> >> ...)
>>>>
>>>> >>
>>>>
>>>> >> All textbooks I know on the argument neglect to mention this and in
>>>>
>>>> >> fact only n=1 is ever considered.
>>>>
>>>> >>
>>>>
>>>> >> Does anybody know a book where this trivial issue is discussed?
>>>>
>>>> >>
>>>>
>>>> >> Thanks!
>>>>
>>>> >>
>>>>
>>>> >> Ciao
>>>>
>>>> >>
>>>>
>>>> >> Pietro
>>>>
>>>> >>
>>>>
>>>> >>
>>>>
>>>> >>
>>>>
>>>> >> Sent from my Desktop
>>>>
>>>> >>
>>>>
>>>> >> Dr. Pietro Roversi
>>>>
>>>> >> Oxford University Biochemistry Department - Glycobiology Division
>>>>
>>>> >> South Parks Road Oxford OX1 3QU England - UK Tel. 0044 1865 275339
>>>>
>>>> >
>>>>
> >
--
Petr Leiman
EPFL
BSP 415
CH-1015 Lausanne
Switzerand
Office: +41 21 69 30 441
Mobile: +41 79 538 7647
Fax: +41 21 69 30 422
|
