I think we are just discussing different ways of saying the same thing now.
But that can be interesting, too. If not, read no farther.
herman.schreu...@sanofi.com wrote:
Dear Edward,
Now I am getting a little confused: If you look at a "higher order" 2n reflection, you will also
get diffraction from the intermediate "1n" layers, so the structure factor you are looking at is in
fact the "1n" structure factor. I think your original post was correct.
Yes- I think the original poster's question about diffraction from
the 2n planes, and whether that contributes to diffraction in the
1n reflection, has been answered- physically they are the same thing.
My question now is whether it is useful to consider Braggs-law "n" to
have values other than one, and whether it is useful to tie Braggs law
to the unit cell, or better to derive it for a set of equally spaced
planes (as I think it originally was derived) and later put conditions
on when those planes will diffract.
In addition to Bragg's law one also talks about the "Bragg condition",
as somewhat related to the "diffraction condition" although maybe that
is closer to "Laue condition".
But anyway, the motivation for presenting Braggs law is to decide where
(as a function of lambda and theta) diffraction will be observed.
And in a continuos crystal (admittedly not what Braggs law was derived
for, but what the students are interested in) you don't get diffraction
without periodicity, and the spacing of the planes has to be related to
the unit cell for braggs law to help (as you say, periodicity of the planes
must match periodicity of the crystal).
When Bragg's condition is met, points separated by d scatter in phase.
Diffraction occurs when d matches the periodicity of the material, so that
crystallographically-equivalent-by-translation points scatter in phase,
and the resultants from each unit layer (1-D unit cell) scatter in phase.
If we are just considering equal planes separated by d with nothing between,
then the periodicity is just d, and bragg condition gives diffraction
condition.
If we are considering a crystal with continuous density, if d is equal to
a unit cell dimension and the planes are perpendicular to that axis, then
then the periodicity is d and brags law gives the (1-dimensional) diffraction
condition.
If d is some arbitrary spacing not related to periodicity of the matter,
brag condition still tells you that points separated by d along S scatter
in phase but if d has no relation to the periodicity, diffraction conditions
are not met and the different slabs thickness d will not scatter in phase.
If d is an integral submultiple of the periodicity, we get diffraction.
What is the best way to explain this?
1. if points separated by d scatter in phase (actually out of phase by one
wavelength),
then spots separated by an integral multiple n of d will scatter in phase
(out of phase by n wavelengths). Now if n*d is the unit cell spacing, spots
separated by nd will be crystallographically equivalent, and scatter in
phase (actually out of phase by n wavelengths).
But this is more elegantly expressed by using braggs law with d' = the unit
cell spacing, nd, and n'= n. The right hand side of braggs law is calculating
the phase difference, and the left hand is saying this must be = n lambda.
That's what n is there for!
2. the periodicity of the set of planes must match the periodicity of the
crystal-
if d is a submultiple of the unit cell spacing, points separated by d will
scatter in
phase, but there is no relation between what exists at those points, so they
will
not interfere constructively. each slab of thickness d will have resultant phase
(and amplitude) different from the slab above or below it.
But if d is an integral submultiple of unit cell spacing, there will be
periodicity to these slabs- the sixth slab will have the same content as
the first (or the fifth will be the same as the zero'th may be more comfortable)
so each stack of five slabs will interfere constructively with the 5 slabs above
it and so on throughout the crystal.
And as in the answer to original poster's question, it is the same diffraction
whether you consider it to be the first order diffraction of planes with d=c/5
or the fifth-order diffraction from the unit cell spacing.
I think these are equivalent in terms of the underlying physics
so this is semantics, or choosing the most intuitive explanation.
I will consider introducing Bragg's law for arbitrary planes in space
and introducing diffraction condition later with Laue condition.
And of course I should look again at some of the excellent textbooks
that are available in coming up with a plan.
But when a colleague studying 2D crystals in cryo-EM gloats:
"I got diffraction out to the fifth order", we don't want to pour
cold water by saying, "sorry, those are first order diffraction from
from planes at 1/5 spacing!" even though it means the same thing in
terms of resolution. (OK, order of diffraction doesn't have to be
equated to "n" in braggs law, but it is convenient to do so)
To summarize how I see it:
1) Braggs law has nothing to do with crystals or unit cells, it only describes
diffraction from sets of planes.
2) However, to get constructive interference from all unit cells in the
crystal, the periodicity of the set of planes must match the periodicity of the
crystal, which means that only sets of planes with integer miller indices are
allowed.
So the unit cell dictates which sets of planes are able to constructively
diffract. However, there might not be anything physically present in the
crystal with that periodicity. In this case the corresponding reflection will
be weak or absent. This is the kind of information we use to calculate our
wonderful electron density maps.
Best,
Herman
-----Ursprüngliche Nachricht-----
Von: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] Im Auftrag von Edward
A. Berry
Gesendet: Freitag, 23. August 2013 01:01
An: CCP4BB@JISCMAIL.AC.UK
Betreff: Re: [ccp4bb] AW: [ccp4bb] Dependency of theta on n/d in Bragg's law
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
>
