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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