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 (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
>
