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TRANSCRIPT
Lecture 4
Bragg’s LawHandling Diffraction Data
Dr. Susan YatesTuesday, 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 dsinθafter reflection
• For the upper and lower beams to be in phase at the detector 2dsinθ = nλ (Bragg’s Law)
• BC = CD = dsinθ• If BC +CD = λ then wave 1 and wave 2 would be in
phase and result in constructive diffraction
B
C
D
dsinθdsinθ
Bragg’s Law• If the pathlength difference is nλ then constructive
diffraction will occur and…
2dsinθ = nλor 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 andDiffraction Experiment
• Given three variables • Distance (A)• Diffraction angle
(2θ)• Detector radius (r)
tan2θ = r/A
Diffraction Geometry• Combine Bragg’s law and
diffraction equation2d 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?• 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 1-n)• 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)Electron Density ρ(x,y,z)
Reciprocal space (hkl)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
MoleculeElectron Density
“real space”
Fourier Transform
“reciprocal space”
Crystals Amplify Diffraction Signal
FourierTransform
Crystal
• 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
Fourieranalysis
Fouriersynthesis
Diffraction Data to Electron Density
Contributions of One Reflection
Contributions of a Second Reflection
The Combined Contributions ofTwo 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