Lecture 4: Diffraction from real 3D crystals.
4.1 The Geometry of the X-ray diffraction experiment
4.2 Scattering by an atom
4.3 Scattering by a collection of atoms (e.g. a molecule, or the unit cell of a crystal)
4.4 Scattering by a crystal.
4.4.1 The Reciprocal Lattice in three dimensions
4.4.2 The Laue conditions and Miller indices
4.4.3 The Structure Factors for the Crystal. Fourier Analysis and Fourier Synthesis.
4.4.2 Symmetry and its effect on the Structure factor.
4.5 Bragg’s treatment of diffraction
4.5.1 Why diffraction peaks are called “reflections”
4.6 A Small Refinement : Temperature Factors / Atomic Displacement Parameters
4.7 Anomalous Dispersion (Resonant Scattering) Effects (Not taught in 2015)
Monday, 16 March 15
1
X-ray diffraction from 3D crystals
•The principles of diffraction illustrated in the last lecture, using 1D and 2D examples,
remain true when we consider 3D crystals
• One important thing to comprehend is that the complete diffraction pattern does not
immediately appear on a 2D detector when you illuminate a crystal with
monochromatic X-rays. The crystal must be moved into different orientations to allow
the complete diffraction pattern to be measured. We’ll discuss the practicalities of doing
this in the next lecture.
•Now we have to develop the theory of scattering a little more thoroughly.
The
objective is to develop a physical model of scattering by a crystal. We’ll
begin by considering the scattering from an isolated atom, then from a group of atoms
(like a molecule, or the contents of the unit cell) and finally from a 3D crystal. This is
necessary if we want to ultimately work backwards and determine the arrangement of
atoms (“the structure”) that gives rise to a measurable diffraction pattern.
•One fundamental approximation of this treatment is that the scattering of an isolated
atom is the same as the scattering from a bonded atom (i.e. that the covalent linkage of
two atoms makes no difference to their ability to scatter X-rays). This is a pretty good
approximation. Only in a few cases has high enough resolution data been collected from
biological specimens to show distortions in the electron clouds due to chemical
bonding.
Monday, 16 March 15
2
The Geometry of the X-ray diffraction experiment
The scattering vector s ... of fundamental importance in X-ray diffraction
Define two vectors So and S (length 1/λ) which lie along the directions of the incident and scattered
beams, respectively. s = S - So
So
N.B. if you want to prove to
yourself that s = 2sinθ/λ , you can
do this using the law of cosines
(the cosine rule) as well as the
trigonometric identity cos2θ = 1 2sin2θ
2θ
s
S
The vector s is highly significant in describing the scattering process. The amplitude of s, s = 2sinθ/λ, is
generally reported in Å-1 in biological crystallography.
The amplitude of s can vary:
from zero
to 2/λ
Monday, 16 March 15
So
S
So
S
[2θ =0° ∴ sin(θ) = sin(0) = 0]
[2θ =180° ∴ sin(θ) = sin(90°) = 1]
3
The amplitude of s and “the resolution”
So
2θ
s
S
The term resolution, in protein crystallography, refers specifically to the minimum
wavelength of the terms used in the Fourier synthesis to compute the electron
density.
This is simply the inverse of the length of the scattering vector.
Hence the quantity 1/|s| = λ/2sinθ is termed the resolution
We’ll return to this concept later, and discuss an operational definition of
resolution - i.e. what does a resolution of 2 Å allow us to “see”
Monday, 16 March 15
4
Scattering from an atom
As we learned in Lecture 1, X-rays are scattered by the electrons which surround the nucleus. These
are distributed over a finite volume, so atoms don’t behave as simple “point scatterers”. We can get an
idea of what’s going to happen, by analogy with the scattering of visible light by large particles.
When light is scattered by particles that are small
relative to the wavelength of light, the scattered
radiation has approximately the same intensity in
all directions. When it is scattered by larger
particles, the radiation scattered from different
regions of the particle will still be in phase in the
forward direction, but at higher scattering angles
there is interference from radiations scattered
from various parts of the particle. The intensity of
radiation scattered at higher angles is thus less
than for that scattered in the forward direction.
The effect is greater the larger the particle relative
to the wavelength.
From Glusker and Trueblood (1985)
As it is for light so it is for X-rays. As depicted in the bottom part of the figure, what matters is the
angle of scattering, or equivalently, the amplitude of the scattering vector s = 2sinθ/λ.
Monday, 16 March 15
5
Scattering from an atom
Obviously some calculus is needed ... we will skip to the result.
Monday, 16 March 15
6
Atomic scattering factors
( |s| )
From Lattman and Loll (2008)
•The atomic scattering factor represents the X-ray scattering from a single stationary atom with its
nucleus located at the origin.
•Values for the atomic scattering factors, as a function of s = 2sinθ/λ, can be looked up in tables, and
are encoded in all modern computer programs used for analyzing X-ray diffraction data.
•Note that the atomic scattering factors drop fairly sharply with increasing scattering angle - that’s
unfortunate. As we’ve learned it’s the scattering at high angles which results in “high resolution”
images. We can see already its going to be weaker, and hence harder to measure.
Monday, 16 March 15
7
Scattering from a collection of atoms
Now we need to figure out how to calculate the X-ray scattering from a group of atoms. This group
might be a molecule, or it might be whatever is in the unit cell of a crystal. Each of the atoms will
scatter X-rays, as specified by its atomic scattering factor. How do we calculate the resultant
scattering? Obviously the waves scattered by the individual atoms are going to interfere.
Let’s begin simply with just two atoms (1 and 2). We’ll put one at the origin, and the
second at a position r. Now our scattering diagram looks like this ...
S
p
2
r
So
q
1
The X-rays that are scattered by atom 2 have a different path to the detector than the X-rays
scattered by atom 1, causing a relative phase difference in the scattered radiation. The path difference
clearly depends on the relative position of the two atoms, and the direction of scattering.
Monday, 16 March 15
8
Mathematical aside : The Vector dot
product
The dot product of two vectors a and b is simple to calculate
if a = [xa ya za] and b = [xb yb zb] then a.b = xaxb + yayb +zazb
The scalar projection of a onto b, is the length of the orthogonal
projection of a onto b
a
b
The scalar projection of a onto b = a.b / |b|
Monday, 16 March 15
9
Calculating the path difference and the phase difference
We can easily calculate the path difference (q-p) by employing the vector dot product
First Recall that So and S have length 1/λ.
Hence λSo and λS have length 1.
p = r.λSo (= the projection of r on to So , the incident beam direction)
q = r.λS (= the projection of r on to S, the scattered beam direction)
Hence the path difference = q - p = r.λS - r.λSo = λr.(S - So) = λr.s
(where s is the scattering vector)
S
p
2
r
So
q
1
Monday, 16 March 15
10
Calculating the path difference and the phase difference
So the path difference for the X-rays scattered by the two atoms is λr.s
To convert this into a phase difference, remember that the phase of a wave cycles through 2π in a
distance of 1 wavelength
2π radians / 360°
Hence the corresponding phase difference is (2π/λ) x λr.s = 2πr.s
Monday, 16 March 15
11
Summary:
Scattering from
two atoms: one at
the origin
2
r
So
The phase of the wave scattered by atom 1 is 0
S
The phase of the wave scattered by atom 2 is 2πr.s
The phase difference depends on the relative position
of the two scattering atoms (defined by r) and the
direction of scattering (defined by the scattering
vector s = S - So )
1
To get the X-ray scattering for this system we need to sum the scattered waves. Once again
we employ complex numbers to represent the waves.
F(s) = f1 exp(i x 0) + f2 exp(i x 2πr.s)
where f1 and f2 are the atomic scattering factors of the atoms.
Very Important. The quantity F(s) is known as the Structure Factor. It
represents the amplitude and phase of the scattered radiation from our
two-atom system.
Monday, 16 March 15
12
Vector diagram for calculating the structure factor
Here’s the equation for the structure factor for our two atom system, with
with one atom at the origin
F(s) = f1 exp(i x 0) + f2 exp(i x 2πr.s)
where f1 and f2 are the atomic scattering factors of the atoms.
As we learned in Lecture 1, we can also represent the summation of the
two waves in the complex plane (on an Argand diagram)
F(s)
f1
f2
2πr.s
real
F(s) is the resultant
scattered wave for our two
atom system
imaginary
Monday, 16 March 15
13
Scattering from
two atoms in
arbitrary positions
S
2
We cheated a little bit in placing one of the atoms at
the origin. In general, the atoms will be in arbitrary
positions. This is the situation depicted on the left.
With respect to our new origin, atom 1 is at the
position specified by r1 and atom 2 is at the position
specified by r2.
The result we need, with we state without proof, is
that the scattering of an atom, at an arbitrary position
r is given by f exp(i x 2πr.s)
So
r2
1
r1
Arbitrary origin
Consequently, the X-ray scattering for this system
F(s) = f1 exp(i x 2πr1.s) + f2 exp(i x 2πr2.s)
where f1 and f2 are the atomic scattering factors of the atoms.
Monday, 16 March 15
14
Vector diagram for calculating the structure factor
Here’s the equation for the structure factor for our two atom system, with
both atoms in arbitrary positions
F(s) = f1exp(i x 2πr1.s) + f2exp(i x 2πr2.s)
Here’s the representation in the complex plane
F(s) is the resultant
scattered wave for our two
atom system
F(s) f2
f1
imaginary
Monday, 16 March 15
2πr2.s
2πr1.s
real
Note ... comparing with our
previous result, shifting the
origin has not changed the
amplitude of the resultant
Structure Factor, just it’s phase
15
Scattering from a collection of atoms
Now we can extend this treatment to as many atoms as we want. If we have a
collection of n atoms, with positions r1, r2, ... ,rn. then the resultant structure
factor is
F(s) = f1 exp(i x 2πr1.s) + f2 exp(i x 2πr2.s) + ... + fn exp(i x 2πrn.s)
Monday, 16 March 15
16
Scattering from a collection of atoms
2πr3.s
n=3 atoms.
f3
r3
r1
F(s)
f2
f1
r2
2πr2.s
2πr1.s
real
imaginary
Structure
Structure Factor
Specifies Atomic
Positions
Specifies the Amplitude and Phase of the scattered Xrays
Monday, 16 March 15
17
Scattering from a collection of atoms
So now we know how to calculate the X-ray scattering for a collection of atoms.
That’s most of the theory we need. We can apply it to calculate the X-ray
scattering from a molecule (the molecular transform), or the contents
of the unit cell in a crystal
But before moving on to finish the discussion of diffraction from 3D crystals,
notice something about the form of the expression for the structure factor.
F (s) = / f j exp !2rir j .s$
j=n
j=1
It is of course ... a Fourier transform.
Monday, 16 March 15
18
Scattering by a crystal
We have already noted the essential point in the last lecture. A crystal can be considered to be the
convolution of a motif (the unit cell contents) with a lattice. By employing the convolution theorem
we can see why the diffraction from a crystal is discrete. Just to remind you here’s the basic idea,
slightly schematized
Duck
Structure Factor
Fourier
transformation
Duck
(we just learned how to
calculate this, provided
our duck is made of
atoms)
F(s)
a
⊗
x
x
s
1/a
Reciprocal
Lattice
Lattice
x
=
=
s
Crystal
Structure Factor
F(s)
Crystal
x
(we know what this
looks like ... at least in
1D and 2D)
s
The only problem ... the 3D reciprocal lattice is a little mind bending.
Monday, 16 March 15
19
The reciprocal lattice in 3 dimensions
Just as was the case in 1 and 2 dimensions, the Fourier transform of a 3D lattice is another lattice the reciprocal lattice.
We invoke a unit cell when we describe the original (direct) lattice
From Drenth (2002)
If a, b and c are the vectors defining the unit cell then the reciprocal unit cell is defined by three
vectors a*, b* and c*.
a* is perpendicular to b and c
b* is perpendicular to a and c
c* is perpendicular to a and b
This can be a little hard to visualize but we can try ...
Monday, 16 March 15
20
The real and reciprocal unit cell
From Cantor and Schimmel (1980)
Monday, 16 March 15
21
The real and reciprocal unit cell
From McPherson (2009)
Monday, 16 March 15
22
Relationships among direct and reciprocal lattice
parameters
From Giacovazzo (2002)
These expressions look rather fearsome, but simplify considerably for some classes of unit cell ...
e.g. orthorhombic cells, where α=β=γ=90°
a* = 1/a, b*=1/b, and c*=1/c : α*=β*=γ*=90°
Monday, 16 March 15
23
A brief aside - an undistorted view of the reciprocal
lattice
It’s not easy to illustrate the actual
diffraction pattern from a 3D
crystal because it is a three
dimensional pattern. However,
there is a data collection
technique (the precession
method) which allows the
collection of undistorted slices
through the 3D diffraction
pattern. A precession image
shows directly one plane of the
3D reciprocal lattice . Here’s an
example
From Cantor and Schimmel (1980)
Monday, 16 March 15
24
Miller Indices
The concept of the reciprocal lattice allows us to understand how the
X-ray scattering due to the unit cell contents is going to get sampled
But what’s the connection with the scattering vector s we employed
when calculating the Structure Factor of the unit cell contents ?
So
2θ
s
S
We aren’t going to prove it, but for a 3D crystals with unit cell vectors a, b, and c, diffraction can only
occur if
s.a = h
s.b = k
s.c = l
Where h,k,l are integers (positive or negative). These are known as the Laue conditions. The
points at which the Laue conditions are satisfied are the points of the reciprocal
lattice. Hence s = ha* + kb* + lc*
The integers h,k,l are known as Miller indices. They serve to index the diffraction pattern just as we
could index the diffraction pattern from a 1D crystal (with a single index h) or from a 2D crystal (with
two indices, h & k).
Monday, 16 March 15
25
Miller Indices and resolution
So
2θ
s
s = ha* + kb* + lc*
S
(only at the points of the reciprocal lattice will scattering by a crystal be significant).
Recalling that 1/|s| is the resolution, its useful to be able to calculate this quantity, given the Miller indices
(h,k,l) of a reciprocal lattice point. Here’s how we do that for the various crystal systems
|s|2
Adapted from Giacovazzo (2002)
Monday, 16 March 15
26
Structure Factor from a crystal
Now at last we are ready to calculate the X-ray scattering from a 3D
crystal since ...
(1) We know how to calculate the scattering from the
contents of the unit cell
(II) We know how the scattering will be sampled due to
the translational periodicity of the crystal.
We just have to do a little algebra to put the two together.
Monday, 16 March 15
27
Structure Factor from a crystal
Let’s consider our expression for the scattering for the scattering of the n atoms the unit cell
FF
(s)(s=) /
f
exp
2rir
.s
6
@ir j .s$
j
j
=
f j exp !2r
j= n
j= 1
j=n
/
j=1
We just need to work at
the term rj.s to make the
expression more practical
It’s convenient to express the position of the atoms rj in terms of the unit
cell vectors a, b and c.
rj = xj a + yj b + zj c
Now rj.s = (xj a + yj b + zj c).s = xj a.s + yj b.s + zj c.s
Finally if we incorporate the Laue conditions (the Structure Factor is significant only
when a.s = h, b.s = k and c.s = l ), we get ...
rj.s = hxj + kyj + lzj
Monday, 16 March 15
28
Structure Factor from a crystal, continued
Which leads to this ...
Atomic scattering factor
Atom location
F (h, k, l) = / f j exp !2ri (hx j + ky j + lz j)$
j=n
j=1
Resultant
Wave
Sum over all atoms in the unit cell
This is one of the key results in X-ray crystallography. It is called the structure
factor equation. It represents the X-ray scattering from the unit cell, sampled at
the reciprocal lattice points h, k, and l. The structure factor is a wave with
amplitude |F(h,k,l)| and phase αh,k,l
The structure factor equation provides a direct way to calculate the the Xray diffraction from a crystal, provided the contents of the unit
cell are known.
Monday, 16 March 15
29
Reversing the Process ... Calculating electron density
from the Structure Factors of a crystal
That is interesting and important. But we also need to go the other way. We observe the diffraction
pattern of a crystal and want to calculate the electron density distribution which gave rise to it.
We know from our consideration of 1D and 2D crystals how this is going to work. The diffraction
experiment amounts to Fourier analysis of the electron density in the crystal. The
amplitude and phase of each of the resulting structure factors F(h,k,l) is the amplitude and phase of a term
in the Fourier series expansion of the density. To recover the electron density we need to
perform the reverse operation ... Fourier synthesis.
Fourier analysis
(X-ray diffraction)
F(h,k,l)
Fourier synthesis
(Computing)
Of course, as we learned in the last lecture, we can only measure the amplitude of the
Structure Factor. We have to get the phases by some less direct means, as we’ll
discuss in lecture 6.
Monday, 16 March 15
30
Reversing the Process ... Calculating electron density
from the Structure Factors of the crystal
Here’s the expression for performing the Fourier synthesis and calculating the electron
density from a 3D crystal. This is a second key result in X-ray crystallography.
1
t Q x, y, zV = V
3
3
3
/ / /
h =- 3 k =- 3 l =- 3
F (h, k, l) exp !ia Qh, k, lV$ exp !- 2ri (hx + ky + lz)$
V is the volume of the unit cell
h,k, l are the Miller indices, which index the diffraction data. Each term in the Fourier summation
corresponds to a sinusoidal density wave passing through the crystal. h,k,l specify its direction and
wavelength
|F(h,k,l)| is the amplitude of the Structure Factor, which is obtained directly from the
diffraction experiment.
α(h,k,l) is the phase of the Structure Factor, which we have to estimate by some means (Lecture 6).
ρ(x,y,z) is what we want to recover - the electron density function - an image which will show where the
atoms are located.
You should compare this with the expressions for the 1D Fourier synthesis given earlier and note the
fundamental similarity.
Monday, 16 March 15
31
Friedel’s law and an alternative way of
writing the equation for Fourier synthesis
The Fourier synthesis involves summation of complex quantities - the
3
3
3 Factors. ext
crystallographic
1 Structure
t ^x, y, z h= V / / / F (h, k, l) exp 6ia ^h, k, lh@exp 6- 2ri (hx + ky + lz) @
h = - 33k = - 3 l = - 3
3
3
3
3
3
1
t ^x, y, z h= V / 1/ / F (h, k, l) exp 6ia ^h, k, lh@exp 6- 2ri (hx + ky + lz) @
t Q x, y, zVh ==- 3 k = - 3 l = - 3
F (h, k, l) exp !ia Qh, k, lV$ exp !- 2ri (hx + ky + lz)$
V
/ / /
h =- 3 k =- 3 l =- 3
T However the electron density is a real function. How does this work ??
This comes about because of Friedel’s law. For each Friedel pair included in
the summation, the imaginary terms cancel out. With a few trigonometric
manipulations, the Fourier synthesis can be written in the following alternate
form - which is explicitly real:
2
t ^ x, y, z h =
V
3
+3
+3
// /
h = 0 k =- 3 l =- 3
ext
F (h, k, l) cos 62r ^hx + ky + lz) - a (h, k, l)h@
See the Appendix of Rupp (2010) for a fuller explanation
ext
Monday, 16 March 15
32
Back to protein crystallography in overview
Now we have the tools to do protein crystallography
Given a diffraction pattern and some phases, we can calculate an electron density
map using Fourier synthesis.
With luck we’ll be able to interpret this map in terms of an atomic model.
From the atomic model, we can calculate the diffraction pattern, using the
structure factor equation (“theory”). We will need to adjust the model to
obtain the best agreement with the experimental diffraction pattern - this is the
process of model refinement.
Monday, 16 March 15
33
Density waves in 2D and 3D
The terms in the Fourier series expansion are easy to
visualize in 1-dimension; a bit tricker to see in 2dimensions; and quite hard to visualize in 3dimensions.
Top
2D Fourier series expansion
Schematic representation of a density wave for the
h=3, k=2 Fourier term
Bottom
3D Fourier series expansion
Schematic representation of a density wave for the
h=3, k=1, l=2 Fourier term
As depicted here, the phase of each of these waves
is zero and the amplitude is 1. The peaks of the
wave are light, and the troughs are dark
From Woolfson (1978)
Monday, 16 March 15
34
Symmetry and the Structure Factor Amplitudes
•As noted in Lecture 2, the symmetry of the diffraction pattern is the
rotational symmetry of the Space group, plus inversion, and is
termed the Laue symmetry.
•The Inversion symmetry is a consequence of Friedel’s law which holds for 3D
crystals under normal conditions.
Concisely stated Friedel’s law says that |F(h,k,l)| = |F(-h,-k,-l)|).
•This has an important practical consequences. Just as there is an asymmetric unit
in the crystal, there is an asymmetric unit in the diffraction pattern.
•We don’t necessarily need to measure every diffraction peak h,k,l because some
will be related by symmetry. We will return to this in the next lecture.
Monday, 16 March 15
35
Bragg’s law ... an alternate way of looking at diffraction
from crystals
The individual diffraction maxima (spots) in the diffraction
pattern - each having an index h,k,l - are commonly called
“reflections” by crystallographers. You might ask, why is this ??
This has its origin in an alternate way of looking at diffraction
from 3D crystals, developed by William Lawrence Bragg and his
father, shortly after von Laue’s discovery of X-ray diffraction by
crystals
William Lawrence Bragg
Monday, 16 March 15
36
Bragg’s law ... an alternate way of looking at diffraction
from crystals
In the Bragg model the crystal contains
families of equally-spaced parallel
planes of atoms, running in different
directions through the crystal. These
planes always intersect at least one
lattice point.
The families are named by three
indices h,k and l (sound familiar ??)
b
c
a
The indices specify how many times
the planes intersect the a, b and and c
axes of the crystal within a unit cell.
From Cantor and Schimmel (1980)
E a s i e s t t o s e e fi r s t i n t w o
dimensions ...
Monday, 16 March 15
37
Bragg Planes in 3D.
... before proceeding to 3 dimensions
b
a
c
From Lattman and Loll (2008)
Monday, 16 March 15
38
Bragg’s law ... an alternate way of looking at diffraction
from crystals
Bragg showed that diffraction from crystals could be treated *as if* it resulted
from the reflection of X-rays from these planes. Perhaps surprisingly, this works.
“Reflection” of X-rays from any set of planes h,k,l results in the diffraction peak with index h,k,l.
We will not pursue this treatment
any further.
Bragg’s method provides a simple
way to figure out *where*
diffraction will be observed from a
3D crystal i.e. for understanding
the geometry of diffraction
However, it provides little insight
into the process of crystallographic
image formation, and it isn’t
connected with the physical basis
of scattering (X-rays are not
being reflected)
From Serdyuk, Zaccai and Zaccai (2007)
Monday, 16 March 15
39
Formal equivalence of Bragg and Laue
treatments
Bragg
Consider “Reflection” from the 1 0 0 plane of a simple
orthorhombic crystal
real space
“Reflected”
Incident
θ
θ
interplanar
spacing d = a in
this instance
Constructive interference if λ= 2dsinθ = 2asinθ
Monday, 16 March 15
40
Formal equivalence of Bragg and Laue
treatments
Laue
Laue condition for the 1 0 0 observation
s=ha* +kb* +lc* = 1a* + 0b* +0c* = a*
Now |s| = | a*| = 1/a (since the crystal is orthorhombic)
Also |s| = 2sinθ/λ, from the definition of s
Combining these two statements we get λ = 2asinθ ...
Bragg’s law.
Monday, 16 March 15
41
Temperature factors
There’s a small refinement to our model of X-ray diffraction that you need to
be aware of.
•The
atomic scattering factors which describe X-ray scattering by atoms,
apply to stationary atoms.
•In
reality, the atoms in a crystal are not stationary, but vibrate about some
mean position. This temperature-dependent vibration diminishes the X-ray
scattering by an atom.
Monday, 16 March 15
42
Temperature factors
The simplest assumption that can be made is that the motion of the atom is
isotropic (the same in all directions).
To account for this, the atomic scattering factors are multiplied by a Gaussian
function:
exp(-B/4 x s2)
B is the Temperature factor and has units Å2.
s is the magnitude of the scattering vector, and has units Å-1
B is related to the mean-square displacement of the atom.
The larger B is, the greater the attenuation of the atomic
scattering factor.
Monday, 16 March 15
43
The effect of the temperature factor on atomic
scattering
s=
From Drenth (2002)
These corrective factors, are variously known as B factors, B values,
Temperature Factors, or Atomic Displacement Parameters. The
effect of the Temperature factor in real space, is to smear the electron density of
the associated atom (that’s the convolution theorem at work, once again).
Monday, 16 March 15
44
The Temperature Factor
Temperature factors are included in
our models of protein structure, and
are estimated during the process of
model refinement (Lecture 7)
Typically surface loops and the ends
of the polypeptide chain are more
mobile than interior regions, and
hence have higher temperature
factors.
You can see this in the figure on the
left - a schematic diagram of a
protein structure, colored by it’s
Temperature Factors.
Image from the Polyview 3D website. http://polyview.cchmc.org/
White is low, Red is High.
Monday, 16 March 15
45
Summary
The X-ray diffraction experiment amounts to Fourier analysis of the density in a crystal
X-ray diffraction
Each spot in the diffraction pattern corresponds to a term in the Fourier series expansion of the
electron density - or in crystallographic parlance - a Structure Factor
Three indices h,k and l specify the order of the Structure Factor, which has amplitude |F(h,k,l)| and
phase α(h,k,l)
h k l
|F(h,k,l)| α(h,k,l)
Adapted from Rupp (2010)
•If |F(h,k,l)| and α(h,k,l) are both known, we can visualize the electron density using Fourier synthesis.
•Unfortunately X-ray detectors are phase-insensitive and we can only directly measure |F(h,k,l)|.
•Determining the unknown α(h,k,l) constitutes the phase problem.
Monday, 16 March 15
46
Summary
What we do in crystallography is simply an extension of Fourier analysis and synthesis of a 1D
function - which is easy to visualize:
From Eisenberg & Crothers (1979)
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47
Summary
From Rupp (2010)
Furthermore ... if we can interpret the electron density map in terms
of an atomic model, we can calculate the Structure Factors (i.e. the
diffraction pattern) using the Structure Factor equation
Then we can make adjustments the atomic model to get best
agreement with the experimental observations.
Monday, 16 March 15
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