Lecture 3: Understanding Diffraction
3.1 Fourier Synthesis, Fourier Analysis and the Fourier Transform
3.2 Diffraction from 1D arrays
3.3 Some important general points regarding diffraction
3.3.1 The Phase problem
3.3.2 Friedel’s law
3.3.3 Centrosymmetry and Phase angles
3.3.3 Fundamental resolution limits.
3.4 Diffraction from 2D arrays
3.5 The Convolution Theorem and Diffraction from crystals
3.6 The Effects of Limited Resolution and Missing Data
3.7 Crystals aren’t Perfect: The Effects of Disorder.
Wednesday, 11 March 15
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Lecture 3: Understanding Diffraction
In this lecture we’ll try to illustrate the basic principles
involved in diffraction from crystals using objects periodic in
1 and 2 dimensions. This is a bit easier to to visualize than
scattering from real 3D crystals, which we’ll consider in the
next lecture.
But before getting to the diagrams we’ll start with a little
math.
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Wednesday, 11 March 15
3
Fourier Synthesis, Fourier Analysis and the
Fourier Transform
Let’s begin with some mathematical results which are absolutely
central to all theoretical descriptions of diffraction from crystals.
Josef Fourier showed
that any periodic
function can be
mathematically
represented by a
sum of a series of
sinusoidal waves.
The figure shows how
this works for a simple
1D function.
From Eisenberg & Crothers (1979)
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Fourier Synthesis, Fourier Analysis and the
Fourier Transform
Here’s a hint as to why we are
interested in Fourier series
expansion. Notice how it can
be used to represent the
density in this one-dimensional
“crystal”.
Note also, how the waves of
small number define the
general location of the atoms in
the cell, while the waves of high
number define the molecular
details (i.e. adding higher terms
increases the resolution of the
structure).
From Eisenberg & Crothers (1979)
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Fourier Synthesis, Fourier Analysis and the
Fourier Transform
Fourier decomposing
Here’s a final example
which shows
the approximation
functions
plays
a big role
of a square wave (green) by a Fourier series expansion
in optics.
with one, two and three terms.
Mathematical combination
of waves
to wave
produce
Here, we write
a square
as aa sum
sine waves
of different
frequency:
periodic function isofcalled
Fourier
synthesis,
while
the opposite process, the decomposition of the periodic
function into its component waves is called Fourier
analysis.
In fact:
ANY ARBITRARY WAVE FORM
can be written as a sum of sines
and cosines, if you choose the
amplitudes of each term in the
sum correctly.
Credit: Daniel Mittleman, Rice University
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Fourier Synthesis, Fourier
Analysis and the Fourier
Transform
A square wave
approximated by a
Fourier series with
even more terms
From Lattman and Loll (2008)
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Fourier Synthesis, Fourier Analysis and the
Fourier Transform
Fourier Analysis involves decomposition of a periodic function into waves with
progressively shorter wavelengths, each with its own amplitude (An) and phase (ϕn). Fourier
synthesis reverses this process Using complex numbers to represent the waves we can
compactly represent Fourier synthesis.
A0 exp[i(0⋅2πx + ϕ0)]
+ A1 exp[i(1⋅2πx + ϕ1)]
+ A2 exp[i(2⋅2πx + ϕ2)]
+ A3 exp[i(3⋅2πx + ϕ3)]
Text
From Eisenberg &
Crothers (1979)
+ A4 exp[i(4⋅2πx + ϕ4)]
+ A5 exp[i(5⋅2πx + ϕ5)]
= f(x)
f (x) = / A n exp "i Q2rnx + z nV%
3
n=0
3
f (x) = / iθA n exp 7i ^2rnx + zn hA
NB: sincen =e0 = cosθ + isinθ we could eliminate the complex exponential term and write the synthesis in
terms of sine and cosine functions. But while more familiar, that’s actually more cumbersome in many ways
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Representation of waves by complex
numbers, redux.
real
1⋅2π
imaginary
ϕ1
A1
n=1
real
A1 exp[i(1⋅2πx + ϕ1)]
imaginary
2⋅2π
A2 exp[i(2⋅2πx + ϕ2)]
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ϕ2
n=2
A2
0
1
x
2
9
A minor rearrangement of the Fourier
series expansion
"
%
A
i
2
r
nx
+
z
Q
n exp
n
V
f (x)f=(x)/= /
A
exp
i
2rnx
+
z
7
^
n
n hA
n=0
3
3
n=0
Using elementary complex arithmetic (eiθ1 x eiθ2 = ei(θ1+θ2)) we can rewrite this
!
$
= /AAn exp
exp "7
iziz% exp
2
r
inx
5
?
f (x)f (x)
=/
exp
2rinx
A
n
3
3
n
n
n=0
n=0
The complex numbers Anexp(iϕn) are the Fourier coefficients. An
is the amplitude of the Fourier coefficient and ϕn is its phase. Writing the
complex Fourier coefficients as Fn we get.
?
f (x) = /f (x)Fn=exp
!
exp
2
r
inx$
/ F52rinx
n
3
n=0
3
n=0
Fourier analysis and Fourier synthesis are readily generalized to 2 and 3
dimensions.
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Decomposition of a 2D density function into
component waves
Here’s the idea in 2 dimensions ...
From Jefferey (1972)
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The Fourier Transform
Fourier also went on to generalize his results to non-periodic functions and introduced what is now
known as the Fourier integral transform, or more simply the Fourier transform. For a 1D function f(x)
we can write the Fourier transform like this:
3
FT (p) =
#
-3
f (x) exp "- 2ripx% dx
And the inverse Fourier transform like this:
3
f (x) =
#
-3
FT (p) exp "2ripx% dp
These may look a little fearsome, but they are just the analogs of Fourier analysis and Fourier synthesis,
respectively, for periodic functions. Note the close similarity between the inverse Fourier transform, and
the expression for the Fourier synthesis presented 2 slides back:
3
f (x) = / Fn exp !2rinx$
n=0
The summation (over n) used for the periodic function has been replaced by an integration (over xi)
for an arbitrary function. Again ... the Fourier transform can be readily generalized to 2D and 3D.
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Some 1D functions and their Fourier transforms
Note the reciprocity in
dimensions of the functions
and their Fourier transforms
From Blow (2002)
Note ... the Fourier transform of a real function is a complex function ... we are displaying
only the amplitude of the transform.
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The Fourier transform of a 2D function
Actually of a circle .... nice and simple. The relationship to the Fourier transform of the 1D Top Hat
function should be apparent.
Thank you CatSynth. http://www.ptank.com/blog/category/mathematics/
Note ... the Fourier transform of a real function is a complex function ... we are displaying
only the amplitude of the transform
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Why is the Fourier transform so central to X-ray
diffraction?
This is because the X-ray scattering from any object (the Fraunhofer diffraction pattern of
the object) is described by the Fourier transform of its electron density function.
(For now we will simply accept this as a fact. In the next lecture, concerned with diffraction from 3D
crystals, we will demonstrate why this is so).
TeLet’s think about what this means for a periodic crystal ...
We’ve learned that
Fourier Analysis involves decomposition of a periodic function into waves with progressively shorter
wavelengths, each with its own amplitude (An) and phase (ϕn).
Consequently ...
The X-ray diffraction experiment amounts to decomposing the electron density
function of the crystal into waves with progressively shorter wavelengths, each
with its own amplitude and phase.
xt
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Diffraction from 1D arrays
This is getting a little theoretical so let’s consider the simplest possible “crystal”.
That’s a line of regularly spaced scattering points. Let’s get a feeling for what’s
going to happen from an optical diffraction experiment, which is analogous to Xray diffraction.
Direct beam position
From Taylor and Lipson (1964)
A row of regularly-spaced small apertures
Optical diffraction pattern
Only at specific angles is the scattering significant.
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Path Difference 0λ
Path Difference 1λ
Diffraction from 1D arrays
Let’s assume we have a line of regularly spaced
scattering points, and that these points elastically
scatter the incoming radiation (which is parallel
and monochromatic) in all directions.
Path Difference 2λ
Path Difference 3λ
The key result. The resulting wave
interference pattern has negligible
values everywhere, except in the
special directions for which the path
difference is nλ
Complete Destructive
interference
Path Difference 1/2λ
From Lattman and Loll (2008)
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Diffraction from 1D arrays
Since it’s important, here’s a second diagram illustrating the same point.
Path Difference 2λ
Path Difference 1/2λ
From Blow (2002)
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Diffraction from 1D arrays
Just one more ...
From Glusker and Trueblood (1985)
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Diffraction from 1D arrays
“Bragg’s law” for the 1D crystal
From Lattman and Loll (2008)
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Diffraction from 1D arrays
The structure can be more complicated than a simple line of regularly spaced scatterers. A 1D
crystal with varying scattering density within the unit cell still scatters radiation only in directions
identified by the orders of diffraction (n=0,1,2,3 ...).
Path Difference 2λ
However, because of interference between the
waves scattered from different positions in the
unit cell, the diffraction pattern becomes more
complicated. Let’s see what happens using
optical diffraction.
Path Difference 1/2λ
From Blow (2002)
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Diffraction from 1D arrays
The effect of having a density distribution within the unit cell studied using optical diffraction
Mask
Optical diffraction pattern
Path Difference 2λ
Path Difference 1/2λ
From Taylor and Lipson (1964)
Scattering remains restricted to certain special directions, but the intensities of each order of
diffraction are dictated by the density distribution in the 1D unit cell.
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Diffraction from 1D arrays
So what are the Amplitude
and Phases of the scattered
waves from a 1D crystal?
These are exactly the
Amplitudes and Phases of the
Fourier series expansion of
the scattering density.
Returning to the first diagram
we showed in this lecture. The
density function is at the
bottom. The amplitudes and
phases of the various orders
of diffraction (n=0,1,2,3,4,5 ...)
which would be obtained
from this 1D cr ystal, are
shown above.
From Eisenberg & Crothers (1979)
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n=0
A note on the “Zeroeth”
order of diffraction (n=0)
This corresponds to scattering
in the direction of the incident
beam.
Experimentally this is very
hard to measure, because it’s
tangled up with the
unscattered radiation.
The effect of the
corresponding term in the
Fourier series expansion is to
set the mean value of the
function. If it is omitted (as it
usually is) the Four ier
synthesis will look just the
same, but will be on an
arbitrary scale.
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From Eisenberg & Crothers (1979)
24
Diffraction from 1D arrays
Ke e p c l e a r o n t h i s
point.
The scattered X-rays arriving
at the detector(order s
n=1,2,3,4 ...) have the same
wavelength as the incident
r a d i a t i o n . T h e y h ave a n
Amplitude and Phase dictated
by the density distribution in
the unit cell.
The terms in the Fourier
synthesis (n=1,2,3,4 ...) have
the Amplitude and Phase of
the scattered X-rays of the
same order, but are of steadily
decreasing wavelength. These
“density waves” should not be
confused with the actual
scattered waves.
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From Eisenberg & Crothers (1979)
25
Diffraction from 1D arrays
Let’s review the things we’ve just learned ...
•Periodicity in a structure leads to “discreteness” in the diffraction pattern, and the emergence of orders
of diffraction. This is the big difference between diffraction from periodic and non-periodic objects. The
scattering from crystals is restricted to particular directions, whereas the
scattering from non-repetitive objects is a continuous function.
•In generating a discrete diffraction pattern, all that matters is the periodicity of the scattering object. Any
repetitive structure will have its diffraction restricted to particular directions, specified by the wavelength
and the repeat distance. However the density distribution within the unit cell dictates
the amplitude and phase of the the scattered radiation.
•Applied to a crystal, the diffraction process amounts to Fourier analysis of the scattering density. Each of
the orders of diffraction corresponds to a term in the Fourier series. For each order of
diffraction, the amplitude and phase of the scattered wave are the amplitude and
phase of the corresponding term in the Fourier series.
•Although
illustrated with a 1D example, these statements remain true for crystals
periodic in 2 or 3 dimensions
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Some additional general points regarding diffraction
We’ll stick with 1D crystals for a while to illustrate some additional general points regarding
diffraction.
The Phase problem
•We’ve shown how the X-ray diffraction experiment amounts to performing Fourier Analysis
on a crystal structure - decomposing the scattering density into a series of simple sinusoidal waves.
•However
the crystallographer wants to reverse the process ... starting with the experimental
observations, perform a Fourier synthesis to reveal the (unknown) density distribution within
the crystal.
•Unfortunately
all X-ray detectors are phase-insensitive. They record only the
intensity of the scattered radiation, which is proportional to the square of its amplitude.
So we only can only directly measure half the quantities required for the Fourier synthesis. This
difficulty in determining the phase of the scattered waves is termed the Phase Problem. We’ll
discuss ways of overcoming it in Lecture 6.
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Illustration of the phase problem
(A) Shows a 1D density function.
(B) shows the Fourier Analysis - the decomposition of
the function into its component waves. This function is such
that only 4 component waves are needed to represent it
exactly
A
Distance
Each of the component waves has Amplitude and Phase.
n=1, Amplitude 20, Phase 10°
n=2, Amplitude 40, Phase 40°
n=3, Amplitude 30, Phase 0°
n=4, Amplitude 10, Phase -20°
B
If we reverse the process and perform a Fourier
synthesis with the correct amplitudes, but phases that
aren’t quite correct (all set to zero), the function shown in
(C) results. It’s not too bad, but contains significant
distortions
Distance
Note that the phase errors in this case are quite small
n=1, phase error 10°
n=2, phase error 40°
n=3, phase error 0°
n=4, phase error 20°
Conclusion: To perform the Fourier synthesis
usefully we need accurate phase information.
C
Distance
Adapted from Nölting (2005)
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Friedel’s law
Direct beam position
From Taylor and Lipson (1964)
h -6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
Mask
Optical diffraction pattern
An integer, h, is used to identify each of the different scattered waves forming the diffraction pattern,
counting from 0 at the undiffracted beam position. We use negative indices to identify the diffraction
maxima on the other side of the direct beam. In normal diffraction, the intensities of the diffraction of
orders -h and h are identical (i.e the scattered waves have the same amplitude). However the waves
actually have a different phase. We don’t “see” this in the illustration above because the detector is
responding only to the amplitude of the scattered waves.
Although we have used optical diffraction to illustrate this idea, it’s true for X-ray diffraction as well, and
it holds in 2 and 3 dimensions. This is Friedel’s law, and we’ll see later that it is connected with some
elementary properties of the Fourier transform.
For X-ray diffraction in certain circumstances Friedel’s law breaks down. This
phenomenon, which arises from Anomalous (or Resonant) scattering forms the basis for
a powerful method for phase determination, which we’ll discuss in Lecture 6.
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Centrosymmetric and non-centrosymmetric structures
Consider these two structures, and their Fourier series expansion. The structure of the left is centrosymmetric (it can be inverted through the origin and it looks the same). The structure on the right is not.
(NB Mathematicians would call the function on the left an “even” function)
From Blow (2002)
Tex (Note crystallographers commonly use the index h, rather than n, to refer to the order of diffraction, and the corresponding terms in the
Fourier series expansion).
The component waves of the structure on the left have phase either 0 or 180°. The component waves of
the structure on the right have arbitrary phases (0-360°)
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Centro-symmetric and non-centrosymmetric structures
The Fourier series expansion of a
centrosymmetric function
contains only waves, which are
themselves symmetric about the
origin.
There are only two choices for
the phase of this kind of wave.This
greatly simplifies the phase
problem.
Unfor tunately, as we learned
earlier, naturally produced proteins
exist as a single enantiomer, and
inversion symmetry doesn’t exist
in protein crystals. So we can’t
generally take advantage of this.
Wednesday, 11 March 15
From Blow (2002)
31
Fundamental Resolution Limits
The wavelength and the repeat
distance in the crystal set a limit on the
number of orders of diffraction that
can be observed.
The path difference d can never
exceed the repeat distance a.
d < a ∴ n λ < a ∴ n < a/λ
Thus if the repeat distance is 50 Å and
the wavelength is 1.54 Å, only n= 32
orders of diffraction could be
observed.
The longer the wavelength or the
smaller the repeat distance, the fewer
orders of diffraction that can be
observed.
From Lattman and Loll (2008)
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Practically - biological crystals are not
likely to be perfect enough to allow
data collection to this limit regardless.
32
On to periodicity in 2 dimensions
From Harburn, Taylor and Wellberry (1975)
Mask
Optical diffraction pattern
The diffraction pattern of a lattice is a lattice
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Diffraction
pattern
Grating
TheThe
reciprocal
lattice
reciprocal
lattice
The diffraction pattern of a lattice is
another lattice ... termed the
reciprocal lattice.
a
Let’s consider the idealized 2dimensional case
K/a
The real lattice is defined by generating
vectors a and b. The angle between
these vectors is γ
b
K/b
a* is orthogonal (at 90°) to b
b* is orthogonal (at 90°) to a
a
b
Ka*
a
Kb*
γ
γ*
b
After Glusker and Trueblood (1985)
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The reciprocal lattice is defined by
generating vectors a* and b*, where
γ* , the angle between a* and b* is
180-γ
(the scale factor K indicated on the
figure depends on the wavelength of
the incident radiation and the
experimental geometry)
34
Real
Space
Reciprocal
Space
a*
γ*
a
b
γ
b* 1/b
1/a
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The reciprocal
lattice
Now let’s focus on the unit
cell alone.
If the unit cell of the real
lattice has dimensions a and
b, with an angle of γ between
them,
then
The unit cell of the of
reciprocal lattice has
dimensions a* = 1/a.sinγ and
b* = 1/b.sinγ with an angle of
γ* = 180-γ between them.
35
More complicated 2D crystals
More optical diffraction results using masks.
Just as for the one-dimensional case, we see that the
periodicity of the structure restricts diffraction to certain
angles. The contents of the unit cell are responsible for
the variation in the diffracted intensity (and the phase,
although we don’t detect it)
For the 2D “crystal”, we can still index the orders of
diffraction with integers, starting from the direct beam
position, but now we need two integers, h and k, to
describe the maxima, which are arranged on the
reciprocal lattice.
Freidel’s law still seems to be working. Diffraction of
orders h and k, has the same intensity as Diffraction with
orders -h and -k.
From Taylor and Lipson (1964)
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Fourier a nalysis a nd
synthesis in 2-dimensions
Just as for the 1D case, each of the
order s of diffraction (h,k)
corresponds to a term in the
Fourier series expansion of the
scattering density ... providing the
amplitude and phase of the wave.
The diagram on the right shows
how the density is built up from
the individual terms in the Fourier
series expansion, which specify
density waves of steadily decreasing
wavelength.
A simple
structure
(bottom) and its
Fourier
transform (top)
Adpated from Rhodes (2006)
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Understanding diffraction from crystals:
The convolution theorum
Recall from Lecture 2, that we can think of a crystal as
being a convolution of a lattice with a motif.
From Holmes and Blow (1965)
The convolution of two functions f1 and f2 is denoted
f1*f2.
So (the crystal) = (the lattice)*(the motif)
From Taylor and Lipson (1964)
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Understanding diffraction from crystals:
The convolution theorem
The convolution theorem states that the
Fourier transform of the convolution of
two functions is the product of their
Fourier transforms
FT(f1*f2) = FT(f1) x FT(f2)
So FT [crystal]
= FT [(the lattice)*(the motif)]
= FT [the lattice] x FT [the motif]
The diffraction patterns shown in (c) through (f) are the
transform of a single motif (shown in a), multiplied
(sampled) by the transform of four different lattices.
The continuous transform of the motif is sampled
differently in each case. Only in case (e) is the underlying
transform of the motif readily discernible. You might
want to consider why this is so ...
From Taylor and Lipson (1964)
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The effects of limited resolution
The penultimate thing we consider in
today’s lecture is the effect of limited
resolution on Fourier transformation.
Firstly for a non-periodic specimen with
a continuous transform ...
From McPherson (2003)
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The effects of limited resolution
... and secondly for a periodic specimen
with a discrete transform.
The implications for structure
determination using diffraction methods
should be readily apparent.
From McPherson (2003)
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The effects of crystal disorder
Finally, examine what happens if the crystal is imperfect
and exhibits various types of disorder. This breakdown
in periodicity is reflected in the diffraction patterns,
which start to exhibit lower resolution Bragg (periodic)
diffraction - and become semi-continuous.
There is a branch of X-ray theory which attempts to
interpret such “diffuse scattering” patterns ... we
will not consider them further. However the presence
of such effects is diagnostic of crystal disorder.
From Taylor and Lipson (1964)
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