X-ray Diffraction & Crystallography Lecture Overview

CHE-20031 (Structural Inorganic Chemistry) X-ray Diffraction & Crystallography lecture 1 Dr Rob Jackson LJ1.16, 01782 733042 r.a.jackson@keele.ac.uk www.facebook.com/robjteaching Twitter: #che20031

Resources Textbook: Shriver & Atkins: Inorganic Chemistry 5th edition Rourke, Weller & Armstrong) OUP, ISBN 978-0-19-923617-6 (Atkins, Overton, Web resources: Many! Links on FB teaching page or in lecture notes. A particularly good site is: http://www.doitpoms.ac.uk/ che-20031: XRD & Crystallography lecture 1 2

Lectures and Workshops 11th February, 17:00-18:00 lecture 1 13th February, 15:00-17:00 workshop 1 (problem class) 18th February, 17:00-18:00 lecture 2 20th February, 15:00-17:00 workshop 2 (computational workshop) 25th February, 17:00-1800 lecture 3 che-20031: XRD & Crystallography lecture 1 3

X-ray Diffraction & Crystallography: lecture 1 plan Introduction to Crystallography Unit cells: crystal class and lattice type, Bravais lattices Crystal planes and Miller indices d-spacing Introducing the Bragg equation che-20031: XRD & Crystallography lecture 1 4

Introduction to Crystallography - 1 2014: The International Year of Crystallography Many sites on the web are devoted to this! e.g. see http://learn.crystallography.org.uk Crystallography is a powerful technique that can be used to look inside materials and generate a three-dimensional picture of the arrangement of atoms and molecules inside a crystal . (taken from the above site) che-20031: XRD & Crystallography lecture 1 5

Introduction to Crystallography - 2 The crystallography experiments work and what information they produce. First we have to look at the background theory so we can understand what is going on in these experiments. lectures will explain how che-20031: XRD & Crystallography lecture 1 6

Unit cells and crystal class Unit cells are the buildingblocks of crystals. They are defined by two properties, the crystal class and the lattice type. Crystal class defines the relative lengths of the sides of the cell and the angles between them (a, b, c, , , ). che-20031: XRD & Crystallography lecture 1 7

Crystal class explained c z y b a x 3 sides of length a, b, and c Convention: a on x-axis, b on y-axis, c on z-axis 3 angles , and between faces ( opposite a, opposite b, opposite c) che-20031: XRD & Crystallography lecture 1 8

Crystal class summarised Length a = b = c a = b c a b c a b c a = b c a = b = c a b c Angle Crystal Class Cubic Tetragonal Orthorhombic Monoclinic Hexagonal Trigonal/Rhombohedral Triclinic = = = 90 = = = 90 = = = 90 = = 90 = 90 = = 90 = 120 = = 90 90 So there are 7 crystal classes with different relationships between cell parameters. che-20031: XRD & Crystallography lecture 1 9

Crystal class: diagrams The 7 crystal classes from slide 9. See also Shriver & Atkins (5th edn.) p 67 10 che-20031: XRD & Crystallography lecture 1

Lattice types (4) Primitive, P 1 atom/cell Atoms located at the corners of the parallelepiped 8 corners, 8 atoms, 1/8 contribution to cell Body Centred, I 2 atoms/cell Atom at corners, 1 atom in centre of unit cell 8 corners: (8 x 1/8) = 1 atom, 1 atom in centre at ( , , ) Face Centred, F 4 atoms/cell, Atoms at each corner, Atoms on each face 8 corners: (8 x 1/8) = 1 atom, 6 faces: (6 x 1/2) = 3 atoms Face (Side) Centred, C 2 atoms/cell Atoms at each corner, Atoms on one set of parallel faces 8 corners: (8 x 1/8) = 1 atom, 2 faces: (2 x 1/2) = 1 atom che-20031: XRD & Crystallography lecture 1 11

Bravais lattices By combining the 7 crystal classes and the 4 lattice types, 14 Bravais lattices are obtained. They are also shown on the next slide: che-20031: XRD & Crystallography lecture 1 12

The 14 Bravais lattices che-20031: XRD & Crystallography lecture 1 13

Fractional Coordinates (i) Describe the position of atoms within a unit cell (x,y,z) Each atom is displaced by x a, parallel to a y b, parallel to b z c, parallel to c x z c y b a All with respect to the origin of the unit cell che-20031: XRD & Crystallography lecture 1 14

Fractional Coordinates - (ii) For example, what are the fractional coordinates of these atoms? z y x 15 che-20031: XRD & Crystallography lecture 1

Crystal planes and Miller indices An excellent demo is available at: http://www.doitpoms.ac.uk/tlplib/miller_indices/index.php Work through this again later to help you understand this section! A crystal plane is a plane of atoms within a crystal. A typical plane is shown on the next slide: che-20031: XRD & Crystallography lecture 1 16

An example of a crystal plane Draw the plane cuts the x axis at a/2, the y-axis at b and the z-axis at c/2 (other examples will be given) z z c c b b y y a a x x che-20031: XRD & Crystallography lecture 1 17

Miller indices Miller Indices are used to label lattice planes. If the intercepts are: a/h on x-axis b/k on y-axis c/l on z-axis Then the Miller Indices are (hkl) What are the Miller indices for the plane on slide 17? che-20031: XRD & Crystallography lecture 1 18

Planes parallel to axes If a plane doesn t intersect a unit cell axis, it is said to intercept that axis at infinity. If a plane intercepts unit cell axis at Miller index is 1/ = 0 Parallel to a: Parallel to b: Parallel to c: z y x che-20031: XRD & Crystallography lecture 1 19

Negative intercepts For planes which cut the axes at a negative value CONVENTION: negative signs put above numbers as a bar (e.g. onebar or bar one ) che-20031: XRD & Crystallography lecture 1 20

Why is all this important? X-rays interact with planes of atoms Their wavelength is comparable with the distances between parallel planes of atoms. So we need to consider parallel lattice planes and the distance between them. che-20031: XRD & Crystallography lecture 1 21

Parallel lattice planes Consider parallel lattice planes shown: Calculate the Miller indices for each plane and comment on the result. A set of Miller indices defines a family of planes. the two -x z y 22 che-20031: XRD & Crystallography lecture 1

d-spacing d-spacing perpendicular from the origin to the nearest plane. is the distance z -x It is necessary to be able to calculate the d-spacing for use in the Bragg equation. dhkl dhkl y che-20031: XRD & Crystallography lecture 1 23

Calculation of d-spacing The d spacing depends on the Miller indices, so is sometime labelled dhkl: Use Miller Indices (hkl) and lattice parameters (a, b, c) to calculate the separation of the planes For an orthogonal system (all angles 90 ), this expression is used: 1 d 2 2 2 h a k b l c = + + 2 hkl che-20031: XRD & Crystallography lecture 1 24

d-spacing: worked example Calculate the d-spacing for the ( 122) family of planes in a cubic unit cell of length 5 2 2 2 1 h a k b l c = + + 2 hkl d To be done in the lecture. Other examples will be given for you to try yourselves. che-20031: XRD & Crystallography lecture 1 25

d-spacing: expression for different crystal classes Relationship between d-spacing and lattice parameters can be determined by geometry and depends on crystal system General case: Cubic (a = b = c): 2 2 2 1 h a k b l c = + + 2 hkl 2 2 2 d + + 2 2 2 1 ( ) h k a l = 2 hkl 1 d 2 d Tetragonal (a = b c): + 2 2 2 ( ) h k l c = + 2 hkl 2 2 a Orthorhombic (a b c): 2 2 2 1 h a k b l c = + + 2 hkl 2 2 2 d che-20031: XRD & Crystallography lecture 1 26

The Bragg equation and X-ray diffraction The Bragg equation relates the wavelength of the X-rays, to the d-spacing dhkl and the angle of incidence on a plane (hkl), : n = 2 dhkl sin It forms the basis for X-ray diffraction, and will be explained in next week s lecture. che-20031: XRD & Crystallography lecture 1 27

Summary and learning objectives Having attended this lecture, and read and understood the notes, you should be able to: Draw unit cells, and define crystal class and lattice type Understand how Bravais lattices are obtained Draw crystal planes and calculate Miller indices Calculate d-spacing for different crystal systems che-20031: XRD & Crystallography lecture 1 28