CHE20031 Structural Inorganic Chemistry Xray Diffraction Crystallography lecture

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

Resources • Textbook: ‘Shriver & Atkins: Inorganic Chemistry 5 th edition’ (Atkins, Overton, Rourke, Weller & Armstrong) OUP, ISBN 978 -0 -19 -923617 -6 • 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 • 11 th February, 17: 00 -18: 00 lecture 1 • 13 th February, 15: 00 -17: 00 workshop 1 (problem class) • 18 th February, 17: 00 -18: 00 lecture 2 • 20 th February, 15: 00 -17: 00 workshop 2 (computational workshop) • 25 th 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 lectures will explain how 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. che-20031: XRD & Crystallography lecture 1 6

Unit cells and crystal class • Unit cells are the ‘building blocks’ 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 x α β a b γ 3 sides of length a, b, and c Convention: a on x-axis, b on y-axis, c on z-axis 3 angles α, β and γ between faces (α opposite a, β opposite b, γ opposite c) che-20031: XRD & Crystallography lecture 1 8

Crystal class summarised Length a=b=c a = b ≠c a≠b≠c Angle α = β = γ = 90° Crystal Class Cubic Tetragonal Orthorhombic a≠b≠c a=b=c a≠b≠c α = β = 90° γ = 90° α = β = 90° γ = 120° α = β = γ ≠ 90° α ≠ β ≠ γ ≠ 90° Monoclinic Hexagonal Trigonal/Rhombohedral Triclinic 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 (5 th 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 (½, ½, ½) • 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 z c y a b x • 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 che-20031: XRD & Crystallography lecture 1 15

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 a x c b b y y a 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 ∞ z – Miller index is 1/∞ = 0 y • Parallel to a: • Parallel to b: • Parallel to c: 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. ‘one bar’ 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 z -x y • Consider the two 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. che-20031: XRD & Crystallography lecture 1 22

d-spacing z d-spacing is the perpendicular distance from the origin to the nearest plane. -x dhkl y It is necessary to be able to calculate the d-spacing for use in the Bragg equation. 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: 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 Å • 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): • Tetragonal (a = b ≠ c): • Orthorhombic (a ≠ b ≠ c): che-20031: XRD & Crystallography lecture 1 26

The Bragg equation and X-ray diffraction • The Bragg equation relates the wavelength of the X-rays, λ to the d-spacing dhkl and the angle of incidence on a plane (hkl), θ : nλ = 2 dhkl sin θ • 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