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.)
> -----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 <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
>
