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
