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 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
