Lecture 3:Diffraction and Symmetry

Diffraction A characteristic of wave phenomena, where whenever a wavefront encounters an obstruction that alters the amplitude or phase of a part of the wavefront, diffraction will occur. The components of the wavefront, both the unaffected and the altered, will interfere with one another, causing an observable energy-density distribution referred to as the diffraction pattern.

Electrons are scatterers. The removal of energy from an incident wave and the subsequent re-emission of some portion of that energy is known as scattering. Hecht, Eugene; OPTICS, 2nd ED.; Addison-Wesley; Reading, Massachusetts.

When an incident x-ray beam hits a scatterer, scattered x-rays are emitted in all directions. Most of the scattering wavefronts are out of phase interfere destructively. Some sets of wavefronts are in phase and interfere constructively. A crystal is composed of many repeating unit cells in 3-dimensions, and therefore, acts like a 3- dimensional diffraction grating. The constructive interference from a diffracting crystal is observed as a pattern of points on the detector. The relative positions of these points are related mathematically to the crystal’s unit cell dimensions. Destructive Interference Constructive Interference

Diffraction gratings Diffraction patterns Notice - when the diffraction grating gets smaller, the pattern spacing gets larger (inverse relationship)

Diffraction pattern from a protein crystal.

Bragg’s Law 2d sin  = n where = wavelength of incident x-rays  = angle of incidence d = lattice spacing n = integer Diffractions spots are observed when the following conditions are met: 1.The angle of incidence = angle of scattering. 2.The spacing between lattice planes is equal to an integer number of wavelengths.

Reciprocal Lattice For a crystal lattice with unit cell dimensions a, b, and c, the reciprocal lattice with axes a*, b*, and c*, are related in such a way that: a*  b and c b*  a and c c*  a and b and the repeat distance between points in a particular row of the reciprocal lattice is inversely proportional to the crystal lattice spacing (d). The incoming ray (1) and the diffracted ray (2) are both at an angle  from a set of Bragg planes in the crystal. The difference vector (shown in red) between the direct beam passing undeflected through the crystal (3) and the diffracted ray is perpendicular to the Bragg planes. The sides corresponding to the two halves of the red vector each have a length of sin  /, which (by Bragg's law) is equal to 1/2d. The red vector is the reciprocal space vector with a length of 1/d. (That is why we chose to give the X-ray vectors a length of 1/ in this illustration.) Diagram from Randy Read’s section of The University of Cambridge on-line Protein Crystallography Course

The Ewald Sphere A tool to visualize the conditions under which Bragg’s law is satisfied and therefore a reflection (diffraction spot) will be observable. This occurs when the surface of a sphere centered about the crystal with radius = 1/ intersects with a point on the reciprocal lattice. Movie downloaded from An Interactive Course on Symmetry and Analysis of Crystal Structure by Diffraction By: Gervais Chapuis and Wes Hardaker

Unit Cell A crystal’s unit cell dimensions are defined by six numbers, the lengths of the 3 axes, a, b, and c, and the three interaxial angles, ,  and . The convention for designating the reciprocal lattice defines its axes as a*, b*, and c*, and its interaxial angles as  *,  * and  *.

Miller Indices Method of identifying the lattice planes of a crystal. The plane with Miller indices h, k, and l, makes intercepts a/h, b/k, and c/l with the unit cell axes a, b, and c. Crystallographic planes parallel to one of the 3 axes have indices (0kl), (h0l), or (hk0). Planes parallel to one of the 3 faces of the unit cell have indices (h00), (0k0), or (00l). Rules for determining Miller Indices: 1. Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions. 2. Take the reciprocals 3. Clear fractions 4. Reduce to lowest terms An example of the (111) plane (h=1, k=1, l=1) is shown on the right. You can find this diagram and others like it at this URL:

Symmetry a state in which parts on opposite sides of a plane, line, or point display arrangements that are related to one another via a symmetry operation such as translation, rotation, reflection or inversion.

Asymmetric unit Recall that the unit cell of a crystal is the smallest 3-D geometric figure that can be stacked without rotation to form the lattice. The asymmetric unit is the smallest part of a crystal structure from which the complete structure can be built using space group symmetry. The asymmetric unit may consist of only a part of a molecule, or it can contain more than one molecule, if the molecules not related by symmetry. Protein Crystal Contacts by Eric Martz, April

Symmetry Elements Rotation Good introductory symmetry websites turns all the points in the asymmetric unit around one point, the center of rotation. A rotation does not change the handedness of figures in the plane. The center of rotation is the only invariant point (point that maps onto itself).

Symmetry Elements Reflection flips all points in the asymmetric unit over a line, which is called the mirror, and thereby changes the handedness of any figures in the asymmetric unit. The points along the mirror line are all invariant points (points that map onto themselves) under a reflection.

Symmetry Elements Translation moves all the points in the asymmetric unit the same distance in the same direction. This has no effect on the handedness of figures in the plane. There are no invariant points (points that map onto themselves) under a translation.

Symmetry Elements Inversion every point on one side of a center of symmetry has a similar point at an equal distance on the opposite side of the center of symmetry.

Symmetry Elements Glide reflections reflects the asymmetric unit across a mirror line and then translates parallel to the mirror. A glide reflection changes the handedness of figures in the asymmetric unit. There are no invariant points (points that map onto themselves) under a glide reflection.

Symmetry Elements Screw axes rotation about the axis of symmetry by 360  /n, followed by a translation parallel to the axis by r/n of the unit cell length in that direction. Diagram from:

A lesson in symmetry from M. C. Escher

Crystal SystemExternal Minimum SymmetryUnit Cell Properties TriclinicNonea, b, c, al, be, ga, MonoclinicOne 2-fold axis, || to b (b unique) a, b, c, 90, be, 90 OrthorhombicThree perpendicular 2-foldsa, b, c, 90, 90, 90 TetragonalOne 4-fold axis, parallel ca, a, c, 90, 90, 90 TrigonalOne 3-fold axisa, a, c, 90, 90, 120 HexagonalOne 6-fold axisa, a, c, 90, 90, 120 CubicFour 3-folds along space diagonala, a,,a, 90, 90, 90 triclinic trigonal hexagonal cubic tetragonal monoclinic orthorhombic 7 Crystal Systems

The combination of all available symmetry operations (point groups plus glides and screws) within the seven crystal systems equals 230 combinations, called the 230 Space Groups. The International Tables list those by symbol and number, together with symmetry operators, origins, reflection conditions, and space group projection diagrams. Inversion symmetry elements are not allowed when dealing with protein crystals (all amino acids present in proteins have the L stereochemical configuration; the inverse, the D configuration, can’t be found in proteins.) Therefore, the number of space groups is reduced from 230 for small molecules to 65 for proteins. Page 151, Vol. I. from International Tables for X-Ray Crystallography, 1965 edition

Identification of the Space Group is called indexing the crystal. The International Tables for X-ray Crystallography tell us a huge amount of information about any given space group. For instance, If we look up space group P2, we find it has a 2-fold rotation axis and the following symmetry equivalent positions: X, Y, Z -X, Y, -Z and an asymmetric unit defined by: 0 ≤ x ≤ 1 0 ≤ y ≤ 1 0 ≤ z ≤ 1/2 An interactive tutorial on Space Groups can be found on-line in Bernhard Rupp’s Crystallography 101 Course: