Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law

[Edward A. Berry]

OK, I see my mistake. n has nothing to do with higher-order
reflections or planes at closer spacing than unit cell dimensions.
n >1 implies larger d, like the double layer mentioned by the original
poster, and those turn out to give the same structure factor as the
n=1 reflection so we only consider n=1 (for monochromatic).
The higher order reflection from closer spaced miller planes
of course do not satisfy bragg lawat the same lambda and theta.
So I hope people will disregard my confused post (but I think the
one before was somewhat in the right direction)
The higher order diffractions come from finding planes through
the latticethat intersect a large number of points? no- planes
corresponding to 0,0,5 in an orthorhombic crystal do not  all
intersect lattice points, and anyway protein crystals aren't
made of lattice points, they havecontinuous density.

Applying Braggs law to these closer-spaced miller planes  
will tell you that points in one plane will diffract in phase.
But since the protein in the five layers between the planes  
will be different, in fact the layers will not diffract in
phase  and diffraction condition will not be met.

You could say OK, each of the 5 layesr will diffract
with different amplitude and out of phase, but their
vector-sum resultant will be the same as that of
every other five layers, so diffraction from points
through the whole crystal  will interfere constructively.

Or you could say that this theta and lambda satisfy the
bragg equation with d= c axis and n=5, so that points
separated by cell dimensions, which are equal due to
the periodicity of the crystal, will diffract in phase.
That would be a use for n>1 with monochromatic light.
The points separated by the small d-spacing scatter in
phase, but that is irrelevant since they are not
crystallographically equivalent. But they also scatter in phase
(actually out of phase by 5 wavelengths) with points separated
by one unit cell, because they satisfy braggs law with
d=c and n=5 (for 0,0,5 reflection still).
So then the higher-order reflections do involve n,
but it is the small d-spacing that corresponds to n=1
and the unit cell spacing which corresponds to the higher n.
The latter results in the diffraction condition being met.
(or am I still confused?)
(and I hope I've got my line-wrapping under control now so this won't be so 
hard to read)






















Ethan Merritt wrote:
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

  >