Lecture 4
Bragg’s Law
Handling Diffraction Data
Dr. Susan Yates
Tuesday, Feb. 8, 2011
Steps in Solving an X-ray Structure
Diffraction Principles:
Scattering of Radiation
• X-rays
• High energy electromagnetic radiation that interacts
with charged particles as a function of charge/mass
ratio of the particle
• Electrons make the dominant contribution to X-ray
scattering, which occurs when the radiation passes
through a non-homogeneous distribution of electrons
• The atoms in matter provide localized concentrations
of electron density, causing some of the X-radiation
to be deflected at different angles
• The scattering power of a particular atom is a simple
function of the number of electrons
Scattering from a
Regular Array of Points
• When radiation is scattered from a regular, equally
spaced array of identical points the resulting
diffraction pattern has the same spacing as for two
points but with sharper more intense peaks
• Peaks at larger deflection angles easier to detect
Scattering from a Complex Molecule
• Individual peaks have contributions from all the
scattering points
• Each scatterer in the array contributes to all the peaks
• Individual peaks do not correlate with individual
scatterers in the array of points
Interference
• When two waves are summed, they cancel to zero
when exactly ½λ out of phase
• With multiple scatterers in an array, the sum of many
waves cancels to zero for any fractional phase
difference
• Reinforcement only occurs when the waves are
exactly in phase
• The diffraction pattern is sharpened as the number of
points in the array increases
Scattering from a
Regular Array of Molecules
• Combines the diffraction pattern of the molecules
with the diffraction pattern of the array
• Interatomic distances within the molecule are smaller
than intermolecular distances in the array
• Molecular pattern spread over wider spacings than
array pattern
• Reciprocal relationship of object distance and pattern
spacing
• Large number of molecules in array gives a strong
signal to noise ratio
• Diffraction from a single molecule is good for
understanding but impractical experimentally
Scattering from a
Regular Array of Molecules
Reciprocal Relationship
• Peaks of intensity appearing in the diffraction pattern
correlate with whole number values of n
• nλ correlates with waves in phase
• Spacings in the interference pattern are reciprocally
related to distance between the scattering points (d)
d
d
Summary
• A molecule is a collection of scattering points
variously arranged about the center of mass
• The overall diffraction pattern is the sum of the
individual contributions, with many superimposed
sets of spacings corresponding to the different
interatomic distances in the molecule
• The spacings in the pattern are reciprocally related to
distances in the molecule so the diffraction pattern
has no outwardly obvious visual relationship to the
molecule
• Nevertheless, the diffraction pattern contains the
information needed to reproduce the structure of the
object
For a Reflection to Occur…
• The angle of incidence equals the angle of reflection
(mirror law)
• The pathlength difference between the reflection from
the adjacent plane must be a whole number of
wavelengths (nλ) so that reflection from all
equivalent planes are in phase
Lattice and Planes
• Assembly of a set of planes
Bragg’s Law
• Planes in a crystal are separated by distance (d)
• Incident beam meets the plane at angle θ
• Diffraction spots are called reflections, because
crystal is composed of lots of “mirrors” that reflect
the X-rays
• When light (in our case X-rays) is reflected from a
mirror, the angle of incidence is equal to the angle of
reflection
Bragg’s Law
• Lower beam travels an additional distance dsinθ
before meeting the lower plane and another dsinθ
after reflection
• For the upper and lower beams to be in phase at the
detector 2dsinθ = nλ (Bragg’s Law)
B
dsinθ
D
C
dsinθ
• BC = CD = dsinθ
• If BC +CD = λ then wave 1 and wave 2 would be in
phase and result in constructive diffraction
Bragg’s Law
• If the pathlength difference is nλ then constructive
diffraction will occur and…
2dsinθ = nλ
or simply
2dsinθ = λ
• The goal of diffraction experiments is to enable
constructive diffraction
Bragg’s Law
• Reflections only occur at specific values of θ (whole
number values of n)
• For any angle of incidence of the beam, only a subset
of planes meet the Bragg law conditions
• Crystal must be rotated to vary the angle of incidence
of the beam
• As each system of planes reaches its Bragg angle, a
reflection is recorded at a corresponding point in a
detection plane
Diffraction Resolution
• d is resolution
• How fine and how much detail we can see in the
determined structure?
• The smallest spacing that will be resolved
• Measured in Å
• Although d is a variable in Bragg’s equation, in reality
it is dictated by the crystal
Distance between
Detector and Crystal
• Distance between crystal and detector can be readily
changed by moving detector
• Capturing diffractions
• Distance between crystal and detector
• Size of the detector
Bragg’s Law and
Diffraction Experiment
• Given three variables
• Distance (A)
• Diffraction angle
(2θ)
• Detector radius (r)
tan2θ = r/A
Diffraction Geometry
• Combine Bragg’s law and
diffraction equation
2d sinθ = λ
and tan2θ = r/A
• Solve any variable if
sufficient parameters are
known
• e.g. If d (resolution) and
r (radius) are known, you
can move detector to
capture all reflections in
the most appropriate way
(though d is dictated by
crystal, for calculation
purposes d can be
considered a variable)
Detector Distance
At Home Experiment
• Materials
• CD/DVD
• Laser pointer
• Method
• Shoot the laser pointer at the
CD/DVD and look for the
reflections on the wall
• Make some observations!
What Dictates Resolution Limits?
•
•
•
•
•
•
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Crystal size
Molecular weight
Solvent content
Packing
Protein flexibility
Strength of the X-ray beam
Detector sensitivity
• Question
• Why would a plate-shaped crystal generate good
diffraction in some rotation angles but not in others?
Miller Index (hkl)
• Each set of planes in a crystal is identified by its
Miller Index
• Each reflection assigned a unique 3D address based
on its underlying diffraction geometry
• Three integers, h, k, l, define reflection
• Requires knowledge of the cell parameters and crystal
orientation
Reflections
Indexing, Merging and Scaling
• The number of photons that hit the detector for each
reflection is used to calculate the intensity for that
reflection
• In principle, the intensities, along with the cell
dimensions encodes all the information needed to
solve the protein structure
Diffraction versus Microscopy
Fourier Transform
• Reciprocal relationship of real distance in the
molecule and position in the diffraction pattern
• Reflections and their associated Fhkl’s correlate with the
reciprocal of distance (periodicities)
• Fourier demonstrated that any repetitive property can
be represented as the sum of a series of periodic (i.e.
sine and cosine or exponential) functions whose
wavelengths are integral fractions of the overall
repeat
Fourier Transform
• Distribution of electrons in the molecule is the overall
repetitive property
• It can be shown that the Fhkl’s are members of the
Fourier series representing that distribution
• The diffraction pattern is a Fourier transform of the
three-dimensional crystal with the unit cell making up
the fundamental repeat unit
Fourier Transform
• Summation for each atom j in the molecule (atom 1n)
• hkl = index from diffraction pattern
• f(j) = atomic scattering factor for atom j
• xyz = coordinates of an atom
Going Backwards with Fourier
• We have the diffraction pattern and want to calculate
the structure
• To reconstruct the crystal structure, just do a second
Fourier transform (synthesis)
• Diffraction: real space vs. reciprocal space
Real space (x,y,z)
Reciprocal space (hkl)
Electron Density ρ(x,y,z) Diffracted Waves Fhkl, αhkl
Fourier Backwards
• Determine electron density (ρ) at each coordinate
(x,y,z) in space within the unit cell by summing the
contribution from every Fhkl available
• The triple summation means sum for every hkl value
available in the data set (well over 1010 calculations in
total)
• Result of this calculation is a contour map of electron
density in the unit cell
Diffraction as Fourier Transform
• Diffracted waves are Fourier transforms of electron
density
• A backward transform (synthesis) will bring us back to
electron density
• Another words… once we know the amplitudes and
phase of diffracted waves we can calculate the electron
density!
Couple of Complications
• We don’t know a priori what atoms are present so we
can’t do an atom by atom calculation
• We scan blindly across the unit cell in each of the three
axes stepping in small increments of x, y and z
• Before electron density can be calculated we need
phase information (next lecture)
Fourier Tour in Two Dimensions
Light/dark:
Intensities
Colours:
Phase
Molecule
Electron Density
Fourier Transform
“real space”
“reciprocal space”
Crystals Amplify Diffraction Signal
Crystal
Fourier
Transform
• The signal from a single molecule too weak to detect
• The signals from molecules in a crystal add up
because the molecules are in identical orientation
• Diffraction results in a pattern with discrete spots and
empty areas
The Fourier Transform is Reversible
Fourier
analysis
Fourier
synthesis
Diffraction Data to Electron Density
Contributions of One Reflection
Contributions of a Second Reflection
The Combined Contributions of
Two Reflections
Contribution of 5 Reflections
Increasing the Number of Reflections
Implies Increasing Resolution
Fourier Transform for Calculating
Electron Density
Next Time…
• Let’s Solve the Phase Problem