Theory and Practice of X-ray Crystal Structure Determination Basic Crystallography Part 1 Theory and Practice of X-ray Crystal Structure Determination Charles Campana, Ph.D. Senior Applications Scientist Bruker AXS

Course Overview Introduction: Crystals and Crystallography Basic Crystallography – Part 1 Introduction: Crystals and Crystallography Crystal Lattices and Unit Cells Generation and Properties of X-rays Bragg's Law and Reciprocal Space X-ray Diffraction Patterns from Crystals Basic Crystallography – Part 2 Review of Part 1 Selection and Mounting of Samples Unit Cell Determination Intensity Data Collection Data Reduction Structure Solution and Refinement Analysis and Interpretation of Results This is an outline of the topics which will be covered in the two sections of this course. Our emphasis will be to develop a conceptual understanding of the concepts without using a detailed mathematical approach. The most important data collection techniques and methods of data reduction, structure solution and refinement will be discussed from a modern, practical point of view.

Introduction to Crystallography We will begin with a discussion of crystals and crystal lattices.

What are Crystals? What are crystals? We all know what crystals are. They are all around us in nature – such familiar things as diamonds, rubies, snowflakes, salt, sugar, etc., but also in less obvious places such as kidney stones and the residue in the bottom of a wine barrel.

Examples of Crystals Here are a few more pictures of some crystals found in nature.

Examples of Protein Crystals Some materials studied using crystallography do not occur naturally as crystals, for example , proteins. Typically, such molecules are placed in solution and allowed to crystallize over days, weeks, or months through vapor diffusion or other methods Here are a few examples of protein crystals still in the mother liquor. These crystals are so small that they must be viewed through a high-powered microscope

Kirsten Böttcher and Thomas Pape Growing Crystals This is a movie, prepared at the U. of Göttingen, showing the actual growth of a crystal of lysozyme in solution. The movie was made by taking exposures over a period of several days. Kirsten Böttcher and Thomas Pape

Crystal Systems and Crystal Lattices Let’s begin by talking about the properties of crystals

What are Crystals? A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly, repeating pattern extending in all three spatial dimensions. How do we define a crystal? A crystal or crystalline solid is a solid material, whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions.

Foundations of Crystallography Crystallography is the study of crystals. Scientists who specialize in the study of crystals are called crystallographers. Early studies of crystals were carried out by mineralogists who studied the symmetries and shapes (morphology) of naturally-occurring mineral specimens. This led to the correct idea that crystals are regular three-dimensional arrays (Bravais lattices) of atoms and molecules; a single unit cell is repeated indefinitely along three principal directions that are not necessarily perpendicular. Crystals have long been admired for their regularity and symmetry, but they were not investigated scientifically until the 17th century. Crystal symmetry was first studied by mineralogists who showed that the angles between the faces are the same in every exemplar of a particular type of crystal. They also discovered that every face of a crystal can be described by simple stacking patterns of blocks of the same shape and size. William H. Miller was able to give each face a unique label of three small integers, the Miller indices which are still used today for identifying crystal faces. This led to the correct idea that crystals are regular three-dimensional arrays (Bravais lattices) of atoms and molecules; a single unit cell is repeated indefinitely along three principal directions that are not necessarily perpendicular. In the 19th century, a complete catalog of the possible symmetries of a crystal was worked out.

The Unit Cell Concept Ralph Krätzner One of the fundamental concepts of crystallography is that crystals are made up of millions of idential motifs or unit cells. This is a two-dimensional unit cell. This is similar to the motif in a wallpaper pattern. Note that the environment at each corner of the cell is the same. If we ignore the color differences, there is also 4-fold symmetry at each corner and also in the center and at the mid-point of each edge. As we will see symmetry is also important in the study of crystals. Ralph Krätzner

Unit Cell Description in terms of Lattice Parameters a ,b, and c define the edge lengths and are referred to as the crystallographic axes. a, b, and g give the angles between these axes. Lattice parameters  dimensions of the unit cell. a b c The fundamental repeating unit in a crystal is the crystallographic unit cell, which has the properties of size, shape and dimensions. a, b, and c define the edge lengths, and alpha, beta, and gamma define the angles between the crystallographic axes. These are called lattice parameters. The unit cell is given by its lattice parameters, the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi  , yi  , zi) measured from a lattice point. To a first approximation, each crystalline material has its own unique set of unit cell parameters. Unit cell parameters for known structures are stored in databases and may be used to match with unknown samples.

Choice of the Unit Cell The unit cell is the smallest unit that can generate the entire structure by translation operations alone. Within the cell there can be several symmetry-related ‘asymmetric units’ with identical contents, but—in general—in different orientations. The fractional coordinates of the atoms in the asymmetric unit may be used to generate all of the other atoms through crystallographic symmetry operations. The cell is always chosen so that the symmetry elements are positioned in accord with Volume A of the International Tables for Crystallography.

Choice of the Unit Cell A A B B C C D When the symmetry is low, there can be a wide choice of possible unit-cells. In certain cases it is better to choose a non-primitive, centered cell in order to show the full symmetry of the structure more clearly. When there is no symmetry, the position and shape but not volume of the cell may be chosen freely. Here a primitive cell with angles close to 90º (C or D) is preferable. Here the conventional C-centered cell C has 90º angles, but one of the primitive cells (B) has two equal sides. No symmetry - many possible unit cells. A primitive cell with angles close to 90º (C or D) is preferable. The conventional C-centered cell (C) has 90º angles, but one of the primitive cells (B) has two equal sides.

7 Crystal Systems - Metric Constraints Triclinic - none Monoclinic -  =  = 90,   90 Orthorhombic -  =  =  = 90 Tetragonal -  =  =  = 90, a = b Cubic -  =  =  = 90, a = b = c Trigonal -  =  = 90,  = 120, a = b (hexagonal setting) or  =  =  , a = b = c (rhombohedral setting) Hexagonal -  =  = 90,  = 120, a = b The crystal lattices are classified into 7 crystal systems. The names of the crystal systems are part of the ‘jargon’ of crystallography that all chemists should know. I should point out that these metric constraints on unit cell dimensions are necessary, but not sufficient conditions for the crystal systems. In addition to the metric conditions, there are also symmetry requirements that must be met.

Bravais Lattices Within each crystal system, different types of centering produce a total of 14 different lattices. P – Simple I – Body-centered F – Face-centered B – Base-centered (A, B, or C-centered) All crystalline materials can have their crystal structure described by one of these Bravais lattices. Within each crystal system different types of centering (consistent with symmetry) exist which add up to a total of 14

Bravais Lattices Within each crystal system, a total of 14 different types of centering (consistent with symmetry) exist. Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, 3rd Ed., Addison-Wesley

Bravais Lattices Within each crystal system, a total of 14 different types of centering (consistent with symmetry) exist. Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, 3rd Ed., Addison-Wesley

Bravais Lattices The distribution of the 14 Bravais lattice types into 7 lattice systems is shown here.

Bravais Lattices The distribution of the 14 Bravais lattice types into 7 lattice systems is shown here.

Crystal Families, Crystal Systems, and Lattice Systems In three dimensions there are seven crystal families: triclinic, monoclinic, orthorhombic, tetragonal, trigonal (rhombohedral), hexagonal, and cubic. When all possible combinations of point group symmetry and translational symmetry are combined we have a total of 230 unique crystallographic space groups. In order to solve a crystal structure on must first determine the correct space group for the specimen.