1.
Application of Fourier
transform in X-ray
crystallography

2.
The Process: An Overview
why?? for
structure
function
relationship

3.
A basic understanding of diffraction physics
is required if crystal structure solution and
refinement is to be understood.

4.
Diffraction
• Diffraction is a phenomenon by which wave fronts of propagating
waves bend in the neighborhood of obstacles.
•The amount of bending depends on the relative size of
the wavelength of light to the size of the opening.
•If the opening is much larger than the light's
wavelength, the bending will be almost unnoticeable

5.
Diffraction at a single slit

6.
Destructive interference:
Constructive interference

7.
When X-ray photons collide with matter, the oscillating electric
field of the radiation causes the charged particles of the
object to oscillate with the same frequency as the incident
radiation. Each oscillating dipole returns to a less energetic
state by emitting an electromagnetic photon that can, in
general, travel in any outward direction. Thus, the emitted
photons have the same energy as the incident photons. This
process is known as coherent scattering.

8.
When do waves scatter in phase?
• the two waves will be in phase if the pathlengths differ
by any multiple of the wavelength

9.
• Constructive interference (diffraction) will only occur if CB + BD = 2CB = n
d sin = CB CB = n/2
Bragg’s Law 2dsin = n sin  1/d (reciprocal space)
sin = (n/2)/d

10.
• sin(θ)/l = 1/(2 λ)
• λ = l / (2 sin(θ))
• the bigger the angle of diffraction, the smaller the spacing to which
the diffraction pattern is sensitive

11.
(1 1 0) planes
(1 -1 0) planes
(2 1 0) planes

12.
15

13.
• For every family of plane (h k l), a vector can be drawn from a
common origin having the direction of the plane normal and a length
1/d (d is inter planner distance).This new coordinate space is reciprocal
space.Any reciprocal lattice vector is defined by h k l.
• Reciprocal lattice is Fourier transform of the real
lattice

14.
Lattice plane

15.
Electron source
Electron
detector
Ewald
sphere
Reciprocal
lattice
3D diffraction data
collection (high resolution)

16.
• The mathematical description of the
diffracted X-ray can be described by the
structure factor equation
• FT as lens

17.
Fourier transform Mathematical lens
Fourier transform describes
precisely the mathematical
relationship between an object and
its diffraction pattern
X-ray
diffraction
Diffraction
pattern
Crystal
Fourier transform
(Lens)
Protein structure
Fourier transform is the lens-simulating
operation that a computer performs to
produce an image of molecule in a crystal

18.
24

19.
Two Electron system
• Phase difference = 2π r.S
• Wave can be regarded as being reflected against a plane
with as the reflection angle.
• S is perpendicular to the imaginary “reflecting plane”
• Resultant vector (T sum of 1+2)
= 1+1 exp[2π i r.S]
• Phase diff = path diff / x 2 p
• T = 1 +2 = 1+1 exp[2π i r.S]
s0 – incident wave vector {|s0| = 1/ λ }
s – diffracted wave vector { |s| = 1/ λ }
Path difference = p + q
p = λ.r. s0
q = -λ.r. s
PD = λ.r (s0 - s) or λ.r.S
S = s-s0
S=2sinθ/λ

20.
Shift of Origin by -R
• Shift of origin causes an increase of all phase angle by 2π R.S.
• Amplitude and intensity of resultant vector (T) do not change.
2π (r+R).S
2πR.S

21.
Consequences of sin  1/d (Reciprocal Space)
http://www.uni-wuerzburg.de/mineralogie/crystal/teaching/
Oblique Lattice
Crystal
Translation

22.
Scattering by an atom
• Electron cloud scatters X-ray.
• Number and position of electrons affect the scattering.
• Assume origin at nucleus.
atomic scattering factor (f)
Electron cloud is assumed spherically symmetric.
So the atomic scattering factor is independent of the
direction of S, but does depend on the length.

23.
Scattering by a unit cell
• n atoms at position rj.
• Diffraction origin as unit cell origin.
• So
• Total scattering from unit cell
• F(S) is called structure factor as it depends on the
arrangement (structure) of the atoms in unit cell.

24.
Scattering by a crystal
• Transition vectors a b c
n1  a, n2  b, n3  c
• Add scattering by all unit cells with respect to single origin.
• If unit cell has origin at (t.a+u.b+v.c), then its scattering is
• Total scattering by crystal i.e. K(S)

25.
Electron density from structure
factor
• Electron density of a crystal is a complicated
periodic function and can be described as Fourier
series.
• Do structure factor equation have any connection
with the Fourier series?
Our goal is to find electron density
m
x
y
z
Is there any way to solve these
equations for the function ρ(x,y,z) ?

26.
From Structure Factor to Electron
density
I  Fhkl
2
The contribution of scattering from a very short length of scattering
matter dx , so short that ρ(x) can be considered constant within it ,
will be proportional to ρ(x)dx.
Each structure factor equation can be written as a sum in which each
term describes the diffraction by the electron in one volume element
Fhkl = f( ρ1) + f( ρ2) + … + f( ρm) + … + f( ρn) + f( ρ1)

27.
Total scattering from unit cell
This summation is over j atoms in unit cell, instead of summing over
all separate atoms we can integrate over all electrons in unit cell
F(S)=

28.
Diffraction data to structure
•A simple wave like that of a visible light or x-ray
can be described as a periodic function
•F(x)= F cos 2π ( hx+α)
•F(x)= F sin2π ( hx+α)

29.
Fourier synthesis
Figure 1: Fourier series approximation to sq(t) . The number of
terms in the Fourier sum is indicated in each plot, and the square
wave is shown as a dashed line over two periods.
Jean Baptiste Joseph Fourier (1768 – 1830
Most intricate periodic function can be
described as the sum of simple sine and
cosine function
Fourier synthesis is used to comput the
sine and cosine terms that describe a
complex wave

30.
Structure factor- wave description of x-ray reflection
• Each diffracted ray arrives at the film to
produce a recorded reflection can be
described as sum of contribution of all
scatterer.
• Sum that describe a diffracted ray is
called a structure factor
• Fhkl = fA + fB +…….. fA’ + fB’ + fF’
• Every atom in the unit cell contributes to
every reflection in the diffraction pattern

31.
Electron-density maps
• Diffraction reveals the distribution of the
electron density of the molecule
• Electron density reflects the molecule shape
• Electron density of proteins in crystal can be
described mathematicaly by a periodic
function
• ρ(x,y,z)
• Graph of the function is an image of the
electron cloud that sourrounds the molecule in
unit cell
•
The goal of crystallography is to
obtain the mathematical function
whose graph is the desired electron
density map!

32.
Fourier series
Simple wave form: f (x) = F0 cos 2π ( h +α 0)
f (x) = F0 cos 2π ( 0x +α 0) + F1 cos 2π ( 1x+α 1) + F2 cos 2π ( 2x+α 2 )……..+
Fn cos 2π ( nx+α n)
Or equivalently
f (x) = F0 cos 2π ( 0 x+α 0)
Waveform of cosine and sine are combined to make a complex number
cos 2π (hx) + i sin 2π ( hx)
f (x)= Fh [cos 2π (hx) + i sin 2π ( hx)]

33.
Three dimensional waves
Each term is a simple wave with its own amplitude Fh , its own frequency h, and
its own phase α
Complex number in square brackets can be express exponentially,
Cos θ + i sin θ =
so the fourier series becomes
f (x)= Fh
A three dimensional wave has three frequencies one along
each of the x, y, and z-axes. So, a general Fourier series for the
wave f(x,y,z) is written as
f (x,y,z)= Fhkl
Fourier series representation of three dimensional wave f(x,y,z)
Each term is a simple wave with its own amplitude Fh , its own frequency h, and
its own phase α
Complex number in square brackets can be express exponentially,
Cos θ + i sin θ =
so the fourier series becomes
f (x)= Fh
A three dimensional wave has three frequencies one along
each of the x, y, and z-axes. So, a general Fourier series for the
wave f(x,y,z) is written as
f (x,y,z)= Fhkl

34.
The fourier transform
Fourier demonstrated that for any function f(x,y,z) there exist the function F(h,k,l)
such that
F(h,k,l)= f(x,y,z ) dx dy dz.
F(h,k,l) is called Fourier transform of f(x,y,z) and in turn, f(x,y,z) is Fourier
transform of F(h,k,l) as follows
f(x,y,z)= F(h,k,l) dh dk dl.
Information about real space f(x,y,z) can be obtain
from information about reciprocal space, F(h,k,l).

35.
Structure factor as Fourier series
Fhkl = f( ρ1) + f( ρ2) + … + f( ρm) + … + f( ρn) + f( ρ1)
Or equivalently,
F(hkl) = ρ(x,y,z) dV
Thus, Fhkl is the transform of ρ(x,y,z) on the set of real lattice plane (hkl)
Applying reverse fourier transform to the above function
ρ(x,y,z)= 1/V F(h,k,l)
V is the volume of unit cell
This equation tell us how to obtain the
ρ(x,y,z)

36.
• The mathematical process of connecting the diffraction of x-rays with the crystal structure is based in
Fourier analysis.
• crystal lattice function, which
• is composed of one point ( Dirac delta function) at a fixed location in each repeating periodic unitof
the crystal structure. Its Fourier transform is called the reciprocal lattice function, in which
• each point (again a Dirac delta function) represents the wave-vector of a wave of electron density in
• the crystal, of a given wavelength and orientation. The second element is the Fourier transform of
• the contents (electron density) of one unit cell of the crystal structure, called the basis function
• of the structure. This fourier transform is called the structure factor function. As you will
• learn, because we can describe a periodic crystal structure as the convolution of the basis function
• with the crystal lattice function, the Fourier transform of the crystal structure is the product of
• the structure factor function times the reciprocal lattice function. That is, the Fourier transform
• of the basis function is “sampled” at each reciprocal lattice point.

37.
• validity of the proposed structure must be
tested by comparison of the calculated
values of the amplitudes of the structure
factor Fc with the observed amplitudes |F0|.
This is done by calculating a reliability
index or R factor defined by
•

38.
ATOMIC SCATTERING FACTOR
ATOMIC SCATTERING FACTOR f is - ratio of the
amplitude of the wave scattered by the atom to that of an
electron under identical conditions.

I 
Ione4
2p 2
m2
c4
(1 cos2
)
Thomson’s theory of X-ray scattering
Where
I = scattered intensity
Io = incident intensity
n = number of electrons/unit volume
m =mass
c = speed of light

39.
Another version of atomic scattering factor
What happens near the absorption edge -you get
resonance scattering - a phase shift of 90o
What happens due to thermal motion

40.
STRUCTURE FACTOR - scattering from planes

41.
FOURIER TRANSFORM
A complex wave
Simple wave with
Frequence of 2
Simple wave with
Frequence of 3
Simple wave with
Frequency of 5
ADD them all up

42.
ELECTRON DENSITY - the result of a
crystallographic experiment.

43.
Let us recall what we actually measure. The
intensity I of a reflection is proportional to the
square of the complex structure factor F: I ~ F2 , and
F2 = |F|2
NOW CAN WE DETERMINE THE STRUCTURE?
THE PHASE PROBLEM

44.
Structure Factors to Electron Density
• The structure factor F(h,k,l) reflects distribution of electrons
in crystal; positions of electrons described by x,y,z
• If amplitude and phase of structure factors are known, can
determine distribution of electrons in crystal,r (x,y,z)
where V is volume of unit cell

45.
Reflecting (or Scattering) Planes in Protein Crystals
• Lower resolution - whole molecules; units of secondary
structure
• Higher resolution - residues; atoms

46.
Solving the Phase Problem
• Molecular replacement
Existing molecular model for initial guess on electronic distribution in crystal
• Isomorphous replacement (Perutz)
multiple heavy-atom derivatives
must not perturb protein conformation
• Anomalous dispersion (MAD phasing)
also relies on heavy atoms (typically Se in Se-Met)
only one derivative needed
relies on absorption of x-rays by heavy atoms
leads to changes in hkl intensities vs. x-ray wavelength
Intensity variations related to positions of heavy atoms in crystal
need variable wavelength x-rays (synchrotron sources)

47.
Convolution theory
f(X)
X X
g(X)
c(X) = f(X) * g(X)
X
c(X)

48.
Electron Density Calculation
• I(hkl) is proportional to square of F(hkl)
• The summation is over all atoms in unit cell. Instead of
summing over all separate atoms we can integrate over all
electrons in unit cell….
and
r.S = (a.x + b.y + c.z). S = a.S.x + b.S.y + c.S.z
= hx + ky + lz
• So F(S) can be written as F(hkl)….

49.
• F(hkl) is the Fourier transform of ρ(xyz)…. So
• Since
|F(hkl)| can be derived from I(hkl) but can not be derived
straight forward.
The Phase Problem

50.
Why Phase Problem
• Very high frequency of X-rays.
– Changes have to monitored at the time scale
10-18 sec.
• Wave length of X-rays.
– Measurement have to be at 10-11 m scale.
• Actual nature of scattering.
– The incident X-ray is incoherent so there is
no well defined phase relation between
incident and reflected wave.
F(hkl) is a complex number and can be represented as:

51.
Technique for solving the phase problem
1. The Isomorphous Replacement Method.
• Requires the attachment of heavy atoms to the protein molecules
in the crystal.
2. The Multiple Wavelength Anomalous Diffraction
Method.
• Depends on strong anomalously scattering atoms present in
protein.
• Anomalous scattering occurs if electron can not be regarded as
free electron.
3. The Molecular Replacement Method.
• Based on the known homologous structure.
4. Direct Method.
• Method of future 