On Thursday, August 22, 2013 02:19:11 pm Edward A. Berry wrote:
> One thing I find confusing is the different ways in which d is used.
> In deriving Braggs law, d is often presented as a unit cell dimension,
> and "n" accounts for the higher order miller planes within the cell.
It's already been pointed out above, and you sort of paraphrase it later,
but let me give my spin on a non-confusing order of presentation.
I think it is best to tightly associate n and lambda in your mind
(and in the mind of a student). If you solve the Bragg's law equation for
the wavelength, you don't get a unique answer because you are actually
solving for n*lambda rather than lambda.
There is no ambiguity about the d-spacing, only about the wavelength
that d and theta jointly select for.
That's why, as James Holton mentioned, when dealing with a white radiation
source you need to do something to get rid of the harmonics of the wavelength
you are interested in.
> But then when you ask a student to use Braggs law to calculate the resolution
> of a spot at 150 mm from the beam center at given camera length and
> wavelength,
> without mentioning any unit cell, they ask, "do you mean the first order
> reflection?"
I would answer that with "Assume a true monochromatic beam, so n is necessarily
equal to 1".
> Yes, it would be the first order reflection from planes whose spacing is the
> answer i am looking for, but going back to Braggs law derived with the unit
> cell
> it would be a high order reflection for any reasonable sized protein crystal.
For what it's worth, when I present Bragg's law I do it in three stages.
1) Explain the periodicity of the lattice (use a 2D lattice for clarity).
2) Show that a pair of indices hk defines some set of planes (lines)
through the lattice.
3) Take some arbitrary set of planes and use it to draw the Bragg construction.
This way the Bragg diagram refers to a particular set of planes,
d refers to the resolution of that set of planes, and n=1 for a
monochromatic X-ray source. The unit cell comes back into it only if you
try to interpret the Bragg indices belonging to that set of planes.
Ethan
> Maybe the mistake is in bringing the unit cell into the derivation in the
> first place, just define it in terms of
> planes. But it is the periodicity of the crystal that results in the
> diffraction condition, so we need the unit cell
> there. The protein is not periodic at the higher d-spacing we are talking
> about now (one of its fourier components is,
> and that is what this reflection is probing.)
> eab
>
> Gregg Crichlow wrote:
> > I thank everybody for the interesting thread. (I'm sort of a nerd; I find
> > this interesting.) I generally would always
> > ignore that �n� in Bragg's Law when performing calculations on data, but
> > its presence was always looming in the back of
> > my head. But now that the issue arises, I find it interesting to return to
> > the derivation of Bragg's Law that mimics
> > reflection geometry from parallel planes. Please let me know whether this
> > analysis is correct.
> >
> > To obtain constructive 'interference', the extra distance travelled by the
> > photon from one plane relative to the other
> > must be a multiple of the wavelength.
> >
> > ________\_/_________
> >
> > ________\|/_________
> >
> > The vertical line is the spacing "d" between planes, and theta is the angle
> > of incidence of the photons to the planes
> > (slanted lines for incident and diffracted photon - hard to draw in an
> > email window). The extra distance travelled by
> > the photon is 2*d*sin(theta), so this must be some multiple of the
> > wavelength: 2dsin(theta)=n*lambda.
> >
> > But from this derivation, �d� just represents the distance between /any/
> > two parallel planes that meet this Bragg
> > condition � not only consecutive planes in a set of Miller planes. However,
> > when we mention d-spacing with regards to a
> > data set, we usually are referring to the spacing between /consecutive/
> > planes. [The (200) spot represents d=a/2
> > although there are also planes that are spaced by a, 3a/2, 2a, etc]. So the
> > minimum d-spacing for any spot would be the
> > n=1 case. The n=2,3,4 etc, correspond to planes farther apart, also
> > represented by d in the Bragg eq (based on this
> > derivation) but really are 2d, 3d, 4d etc, by the way we define �d�. So we
> > are really dealing with
> > 2*n*d*sin(theta)=n*lambda, and so the �n�s� cancel out. (Of course, I�m
> > dealing with the monochromatic case.)
> >
> > I never really saw it this way until I was forced to think about it
> > by this new thread � does this makes sense?
> >
> > Gregg
> >
> > -----Original Message-----
> > From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of
> > Edward A. Berry
> > Sent: Thursday, August 22, 2013 2:16 PM
> > To: CCP4BB@JISCMAIL.AC.UK
> > Subject: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
> >
> > herman.schreu...@sanofi.com <mailto:herman.schreu...@sanofi.com> wrote:
> >
> > > Dear James,
> >
> > > thank you very much for this answer. I had also been wondering about it.
> > To clearify it for myself, and maybe for a
> > few other bulletin board readers, I reworked the Bragg formula to:
> >
> > >
> >
> > > sin(theta) = n*Lamda / 2*d
> >
> > >
> >
> > > which means that if we take n=2, for the same sin(theta) d becomes twice
> > as big as well, which means that we describe
> > interference with a wave from a second layer of the same stack of planes,
> > which means that we are still looking at the
> > same structure factor.
> >
> > >
> >
> > > Best,
> >
> > > Herman
> >
> > >
> >
> > >
> >
> > This is how I see it as well- if you do a Bragg-law construct with two
> > periods of d and consider the second order
> > diffraction from the double layer, and compare it to the single-layer case
> > you will see it is the same wave traveling
> > the same path with the same phase at each point. When you integrate rho(r)
> > dot S dr, the complex exponential will have
> > a factor of 2 because it is second order, so the spatial frequency is the
> > same. (I haven't actually shown this, being a
> > math-challenged biologist, but put it on my list of things to do).
> >
> > So we could calculate the structure factor as either first order
> > diffraction from the conventional d or second order
> > diffraction from spacing of 2d and get the same result. by convention we
> > use first order diffraction only.
> >
> > (same would hold for 3'd order diffraction from 3 layers etc.)
> >
> > > -----Urspr�ngliche Nachricht-----
> >
> > > Von: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] Im Auftrag von
> >
> > > James Holton
> >
> > > Gesendet: Donnerstag, 22. August 2013 08:55
> >
> > > An: CCP4BB@JISCMAIL.AC.UK <mailto:CCP4BB@JISCMAIL.AC.UK>
> >
> > > Betreff: Re: [ccp4bb] Dependency of theta on n/d in Bragg's law
> >
> > >
> >
> > > Well, yes, but that's something of an anachronism. Technically, a
> >
> > > "Miller index" of h,k,l can only be a triplet of prime numbers (Miller,
> > W. (1839). A treatise on crystallography.
> > For J. & JJ Deighton.). This is because Miller was trying to explain
> > crystal facets, and facets don't have
> > "harmonics". This might be why Bragg decided to put an "n" in there. But
> > it seems that fairly rapidly after people
> > starting diffracting x-rays off of crystals, the "Miller Index" became
> > generalized to h,k,l as integers, and we never
> > looked back.
> >
> > >
> >
> > > It is a mistake, however, to think that there are contributions from
> > different structure factors in a given spot.
> > That does not happen. The "harmonics" you are thinking of are actually
> > part of the Fourier transform. Once you do the
> > FFT, each h,k,l has a unique "F" and the intensity of a spot is
> > proportional to just one F.
> >
> > >
> >
> > > The only way you CAN get multiple Fs in the same spot is in Laue
> > diffraction. Note that the "n" is next to lambda,
> > not "d". And yes, in Laue you do get single spots with multiple hkl
> > indices (and therefore multiple structure factors)
> > coming off the crystal in exactly the same direction. Despite being at
> > different wavelengths they land in exactly the
> > same place on the detector. This is one of the more annoying things you
> > have to deal with in Laue.
> >
> > >
> >
> > > A common example of this is the "harmonic contamination" problem in
> >
> > > beamline x-ray beams. Most beamlines use the h,k,l = 1,1,1 reflection
> >
> > > from a large single crystal of silicon as a diffraction grating to
> >
> > > select the wavelength for the experiment. This crystal is exposed to
> >
> > > "white" beam, so in every monochromator you are actually doing a Laue
> >
> > > diffraction experiment on a "small molecule" crystal. One good reason
> >
> > > for using Si(111) is because Si(222) is a systematic absence, so you
> >
> > > don't have to worry about the lambda/2 x-rays going down the pipe at
> >
> > > the same angle as the "lambda" you selected. However, Si(333) is not
> >
> > > absent, and unfortunately also corresponds to the 3rd peak in the
> >
> > > emission spectrum of an undulator set to have the fundamental coincide
> >
> > > with the Si(111)-reflected wavelength. This is probably why the
> >
> > > "third harmonic" is often the term used to describe the reflection
> >
> > > from Si(333), even for beamlines that don't have an undulator. But,
> >
> > > technically, Si(333) is n
> >
> > ot a "har
> >
> > monic" of Si(111). They are different reciprocal lattice points and each
> > has its own structure factor. It is only the
> > undulator that has "harmonics".
> >
> > >
> >
> > > However, after the monochromator you generally don't worry too much
> > about the n=2 situation for:
> >
> > > n*lambda = 2*d*sin(theta)
> >
> > > because there just aren't any photons at that wavelength. Hope that
> > makes sense.
> >
> > >
> >
> > > -James Holton
> >
> > > MAD Scientist
> >
> > >
> >
> > >
> >
> > > On 8/20/2013 7:36 AM, Pietro Roversi wrote:
> >
> > >> Dear all,
> >
> > >>
> >
> > >> I am shocked by my own ignorance, and you feel free to do the same,
> >
> > >> but do you agree with me that according to Bragg's Law a diffraction
> >
> > >> maximum at an angle theta has contributions to its intensity from
> >
> > >> planes at a spacing d for order 1, planes of spacing 2*d for order
> >
> > >> n=2, etc. etc.?
> >
> > >>
> >
> > >> In other words as the diffraction angle is a function of n/d:
> >
> > >>
> >
> > >> theta=arcsin(lambda/2 * n/d)
> >
> > >>
> >
> > >> several indices are associated with diffraction at the same angle?
> >
> > >>
> >
> > >> (I guess one could also prove the same result by a number of Ewald
> >
> > >> constructions using Ewald spheres of radius (1/n*lambda with n=1,2,3
> >
> > >> ...)
> >
> > >>
> >
> > >> All textbooks I know on the argument neglect to mention this and in
> >
> > >> fact only n=1 is ever considered.
> >
> > >>
> >
> > >> Does anybody know a book where this trivial issue is discussed?
> >
> > >>
> >
> > >> Thanks!
> >
> > >>
> >
> > >> Ciao
> >
> > >>
> >
> > >> Pietro
> >
> > >>
> >
> > >>
> >
> > >>
> >
> > >> Sent from my Desktop
> >
> > >>
> >
> > >> Dr. Pietro Roversi
> >
> > >> Oxford University Biochemistry Department - Glycobiology Division
> >
> > >> South Parks Road Oxford OX1 3QU England - UK Tel. 0044 1865 275339
> >
> > >
> >
>
--
Ethan A Merritt
Biomolecular Structure Center, K-428 Health Sciences Bldg
University of Washington, Seattle 98195-7742
