c Symmetry b  a   a b The unit cell in three dimensions. The unit cell is defined by three edge vectors a, b, and c, with , , , corresponding to the angles between b-c, a–c, and a-b, respectively. Crystals are regular periodic arrays, i.e. they have long range translational symmetry. Crystals are often considered to have essentially infinite dimensions. b a One Unit Cell b  a   c Unit cell = The smallest volume from which the entire crystal can be constructed by translation only. All crystals have translational symmetry, with the translational vectors equal to edges of the unit cell. Unit cells are defined in terms of the lengths of the three vectors and the three angles. For example, a=94.2Å, b=72.6Å, c=30.1Å, =90°, =102.1°, =90°.

Symmetry Operators and Elements Symmetry Operator = an operation that leaves the structure unchanged. Apart from the identity and translational symmetry, protein crystals can only contain the following symmetry elements: Proper rotation: Rotate by 360°/n. Screw rotation: Rotate by 360°/n & translate by d(m/n); d= unit cell edge. Screw Rotations Symbol (nm) 21 31, 32 41, 42, 43 61, 62, 63, 64, 65 Proper Rotations Two-fold Three-fold Four-fold Six-fold Symbol (n) 2 3 4 6

21 Example of a 2-fold screw axis Forbidden Symmetry Crystals can not contain 5-fold rotations, or any rotation axis other than those listed on the previous slide, because they are incompatible with translational symmetry. Note that it is impossible to tile a floor with pentagons without leaving gaps. Crystals of proteins, or other chiral molecules, can not contain mirror planes or centers of inversion, since these require that the mirror image molecule also packs into the crystal.

Crystallographic Symmetry = Symmetry operators, such as rotation axes, that apply over the entire crystal. Non-crystallographic Symmetry (NCS) – also called local symmetry = Symmetry operators that apply to a local region of the crystal, but do not apply over the entire crystal. For example, two molecules in an asymmetric unit may be related to each other by an NCS 2-fold, but the same operation will not superimpose more distant parts of the structure onto equivalent sites. NCS elements can include rotation axis that are not compatible with translational symmetry, such as five-fold axes. NCS applies only here

Asymmetric Unit = The smallest volume from which the unit cell can be constructed by application of the crystallographic symmetry.

Space Groups Because crystallographic symmetry must be compatible with translational symmetry (i.e. a crystal), symmetry elements can only occur in certain combinations. Combinations of symmetry elements that are compatible with translational symmetry in three dimensions are called space groups. The figure illustrates plane group P2. Assuming that the third unit cell axis was normal to the page, this would be a projection of Space Group P2.

Projection of Space Group P4

Point Group = Collection (group) of symmetry operators that all pass through the same point. The group must be closed, have an identity element, and every element must have an inverse. Plane Group = Group of symmetry operators that are compatible with two-dimensional symmetry in a plane. Space Groups = Collections of symmetry operators that are compatible with three-dimensional crystallographic (i.e. translational) symmetry. There are 230 space groups. Because protein and nucleic acid molecules are chiral, there are only 65 “biological” space groups.

a b Fractional Coordinates. Positions in the unit cell are often given in fractional coordinates, i.e. the full length along the a edge corresponds to x = 1.0. The fractional distances along b and c = y and z. Because of lattice (translational) symmetry, the coordinates x = 0.5, x = 1.5, x = -0.5, are identical. at x=0.5, y=0.5 Final coordinate files, such as from the PDB, are given in orthogonal Å. (Which have a defined relationship to the unit cell) at x=0.25, y=0.5 Origins. The origin of the coordinate system (drawn as the top left-hand corner in standard figures) is defined with respect to symmetry axes. Space group P212121. Origin at mid point of mutually perpendicular 2-fold screw axis 1/4 Space group P2. Origin on 2-fold axis Relevant information on space group symmetry is given in the International Tables

Equivalent Positions -x, y +1/2, -z -x, y, -z P2 P21 x, y, z x, y, z Every space group has a characteristic set of equivalent positions. These always include x, y, and z. Because of crystallographic symmetry, for an atom at position x, y, z, there will generally be other equivalent atoms that have identical environments. For example, in space group P2 the equivalent positions are x, y, z and -x, y, -z. +1/2 +1/2 -x, y +1/2, -z -x, y, -z P2 P21 x, y, z x, y, z

Primitive Cubic Body Centered Cubic Face Centered Cubic The Seven Crystal Systems The 230 space groups can be grouped into seven crystal systems and 14 Bravais lattices Name Possible Bravais Lattices Lattice constraints Triclinic P a = b = c  =  =  Monoclinic P, C a = b = c  =  =90°=  Orthorhombic P, C, I, F a = b = c  =  = =90° Tetragonal P, I a = b = c  =  = =90° Trigonal P, R a = b = c  =  = =90° Hexagonal P a = b = c  =  = 90°; =120° Cubic P, I, F a = b = c  =  = =90° Lattice points are a set of points that define the symmetry of the lattice and all have identical environments. All crystal lattices have lattice points at the corners of the unit cells. Lattices that only have lattice points in the corners of the unit cell are call Primitive (P) lattices. Some lattices additionally have lattice points in the center of the unit cell (Body or I Centered), or in the middle of all faces (Face or F centered), or in the middle of two opposing faces (C Centered). Primitive Cubic Body Centered Cubic Face Centered Cubic

Part 4. Diffraction as Reflection Miller Planes and Miller Indices (hkl) The idea that crystals can be described in terms of planes follows from the observation of facets in the external morphology of crystals. Miller defined an indexing system for planes that is related to the geometry of the unit cell. It is important to understand the system of Miller indices because diffraction appears to behave like reflection from Miller planes. Note that the Miller planes are not real things, they are constructs that are drawn or imagined to exist with respect to the crystal lattice. Start at a point on one plane and move along the a direction to the next plane. h = 1/fraction of a to next plane. Start at the same point and move along the b direction to the same plane as before. k = 1/fraction of b to next plane. Same for the third direction. l = 1/fraction of c to next plane. a b h = 1/(1/2) = 2 k = 1/(1/2) = 2 Convention,  > 90. Sorry

Examples of Miller Planes b h=1, k=1 h=1, k=0 h=1, k=2 h=4, k=1 h=2, k=-1 Note that the 2, 2 and -2, -2 planes are identical

The diffraction pattern also forms a lattice The diffraction pattern forms a lattice that is related to the crystal lattice. The lattice of diffracted x-rays is very obvious in a precession photograph (a camera geometry that used to be popular). Precession photograph. a* h Indexing is the processes of assigning hkl indices to the reflections. In a precession photograph this is done by counting out from the direct beam position. b* k This reflection has indices h=10 , k=7, l=0. Its intensity is I(10,7,0) = |F(10,7,0)|2

The diffraction pattern also forms a lattice Most contemporary x-ray data collection used the rotation geometry, in which the crystal makes a simple rotation of a degree or so while the image is being collected. The geometry of the diffraction pattern is less obvious than for a precession photograph, although data collection is more efficient. Oscillation (rotation) photograph. X-rays Crystal rotates during exposure

Crystal lattice & Diffraction lattice Real lattice & Reciprocal lattice Real space & Reciprocal space The lattice of diffracted x-rays has an inverse or reciprocal relationship to the crystal lattice. For this reason the lattice of the diffraction pattern is called the reciprocal lattice, while the crystal is said to form the real or direct lattice. Real lattice Reciprocal lattice a* b* a b It is important to remember that the crystal lattice, the reciprocal lattice, and Miller planes are not actual physical objects – assuming you could see objects that small, you would not see an actual dot at the corner of a unit cell, etc. Nevertheless, these concepts are extremely helpful for crystal structure determination.

Symmetry of the Diffraction Pattern The diffraction pattern has almost the same symmetry as the crystal. One important difference is that the diffraction pattern also contains a center of inversion (Friedel symmetry). The combination of rotational symmetry and a center of inversion can give rise to mirror plane symmetry in the diffraction pattern – which of course is not possible in the crystal. 2-fold Friedel symmetry. This precession photograph is a slice through the center of the reciprocal lattice 2-fold

The first x-ray diffraction pattern (1912) von Laue, was the first person to perform a diffraction experiment with a crystal. He hypothesized that if atoms really existed, and if x-rays really were waves, then the wavelength of x-rays would approximate the distance between atoms in a crystal and diffraction would be observed. His brilliant insight thus proved both atomicity and the wave character of x-rays. This remarkable achievement was recognized by the Nobel prize. Laue did not actually do the experiment himself. Rather, he persuaded a couple of graduate students to do the experiment for him. Laue then set an example that has inspired PIs ever since – he was given all the credit! The first x-ray diffraction picture, which was taken from a crystal of copper sulfate by von Laue’s students, and dubbed the “beerstein” pattern. W. L. (Lawrence) Bragg realized that von Laue’s diffraction pattern could be modeled as reflection from Miller planes -- i.e. the angle of incidence wrt Miller planes = the angle of reflection. This is the reason why diffracted x-rays are generally called reflections. Bragg published this work in 1913. His insight further substantiated the wave character of x-rays (thereby contradicting his father’s theory that x-rays are particles). This work also allowed the Bragg’s (father and son) to determine the first atomic resolution structures. The Bragg’s were jointly awarded the Nobel prize in 1915 – at which time W.L. Bragg was 25 years old and fighting in the trenches of world war I.

Bragg’s Law  = 2.d.sin  d d.sin Diffraction of a given x-ray beam can occur when the path difference for reflection from successive planes is equal to a whole number of wavelengths  d Notice the reciprocal relationship between interplanar spacing (d) and Bragg angle (). d.sin Reflection from the hkl Miller planes gives rise to the hkl diffracted x-ray beam (hkl reflection).