The Muppet’s Guide to:

1. The Muppet’s Guide to: The Structure and Dynamics of Solids Thomas Hase Room MAS4.02 E-mail: T.P.A.Hase@warwick.ac.uk

2. My Research X-ray (and neutron) scattering from thin films and multilayers….

4. Objectives for part 1 • Understand how thermodynamics and bonding are responsible for crystal structure and material properties • Have knowledge of simple crystal structures and how they relate to bonding requirements. Be able to reproduce the bcc, fcc, diamond and perovskite structures and be able to explain the concepts of the lattice and the basis. • Understand the structural origin of the ferroelectric effect. • Be able to explain the difference between a first and second order phase transition and justify why they occur from a thermodynamic point of view. • Be able to read and use a unary and binary phase diagram and calculate phase compositions and relative amounts. Understand the diffusive mechanisms in determining structure and predict the evolution of structure using a eutectic phase diagram. • Have an understanding of different crystal growth methods and the types of materials they produce. Be able to describe simple defect structures and how these relate to material properties. • Understand how x-rays, neutrons and electrons interact with matter. Appreciate the differences between probes and be able to compare x-ray and neutron scattering . • Be able to apply basic scattering concepts to the study of single crystal and powder materials. Be able to predict the likely scattering from simple structures and describe the changes it scattering as a material undergoes a phase transition. • Understand how scattering is related to reciprocal space. Know how to determine material properties such as lattice parameters, strain and particle size from scattering data.

5. Recommended Books – Part 1 Materials Science and Engineering - An Introduction, 7th EditionWilliam D. Callister, Jr., ISBN: 978-0-471-73696-7, Wiley, Hardback 2007£37.95 / €58.50 (Wiley) The Physics and Chemistry of Solids Stephen Elliott ISBN: 978-0-471-98195-4, Wiley, Paperback June 1998 £47.50 / €71.30 (Wiley) Extended notes which cover all the course content and more are available on-line and a printed copy will be handed out soon

6. Feedback • Please give honest and critical feedback during the course: • During the lectures • After the lectures

7. Exam Structure For this course: • 1 compulsory question: • covering all 3 parts (“bite size”) • 2 from 3 questions • one each from the three sections (detailed)

8. The Muppet’s Guide to: The Structure and Dynamics of Solids 1. Structure from a Thermodynamic Viewpoint

9. Objectives • Understand how thermodynamics and bonding are responsible for crystal structure and material properties • Have knowledge of simple crystal structures and how they relate to bonding requirements. Be able to reproduce the bcc, fcc, diamond and perovskite structures and be able to explain the concepts of the lattice and the basis. • Understand the structural origin of the ferroelectric effect. • Be able to explain the difference between a first and second order phase transition and justify why they occur from a thermodynamic point of view. • Be able to read and use a unary and binary phase diagram and calculate phase compositions and relative amounts. Understand the diffusive mechanisms in determining structure and predict the evolution of structure using a eutectic phase diagram. • Have an understanding of different crystal growth methods and the types of materials they produce. Be able to describe simple defect structures and how these relate to material properties. • Understand how x-rays, neutrons and electrons interact with matter. Appreciate the differences between probes and be able to compare x-ray and neutron scattering . • Be able to apply basic scattering concepts to the study of single crystal and powder materials. Be able to predict the likely scattering from simple structures and describe the changes it scattering as a material undergoes a phase transition. • Understand how scattering is related to reciprocal space. Know how to determine material properties such as lattice parameters, strain and particle size from scattering data.

10. Bonding in Solids Arrangement of atoms in a solid (on a local atomic level) is driven by energy considerations Require a thermodynamical treatment of atomic positions and bonding….

11. Entropy: A measure of disorder or randomness in a system: Enthalpy: The total energy of a thermodynamic system. It is the sum of the internal energy of a body and the energy associated with displacing it from its environment: Thermodynamics U is the Internal energy of crystal

12. The Gibbs Free Energy • Is a measure of the energy that depends on both enthalpy and entropy: Also known as the free enthalpy

13. Re-expression of 1st Law • Recall 1st law of thermodynamics: • Which using the definition of entropy, DS=DQ/T, becomes • Consider small change in Gibbs free energy (G= H-TS)

14. Enthalpy, H • Largest contribution comes from the internal energy U: • Potential due to bonding • Lattice vibrations (phonons and magnons) • Entropy, S • Disorder in atomic positions / moments • Thermal vibrations about average position Initially we will assume that the lattice vibrations do not contribute to H

15. Gibbs free energy “Every system seeks to achieve a minimum of free energy.” Stable at x=0 Unstable at x=0

16. Stable Phases Solid Liquid DISORDERED ORDERED At low temperatures the Gibbs free energy is lowered by minimising the enthalpy (H=U-PV) and this is associated with an ordered ground state. As the temperature rises it becomes more important to maximise the entropy (S).

17. Low Temp, TS < H Minimise enthalpy High Temp, TS > H Maximise entropy Solid Phase Minimum G when H is at optimum value U stabilised by bonding - ORDERED Liquid Phase: Disorder becomes more important Bonding requirements loosened Gas Phase: No-longer any real bonding requirements dominated by S and high disorder - DISORDERED

18. SOLIDS - Internal Energy • Assume that the contribution to the internal energy of the lattice arising from vibrations is low (meV) Energy, U proportional to position of atoms in unit cell Position determined by bonding potential, fij which only depends on separation, rij. Expect fij to depend on the type of bonding and to be different for ionic, covalent, etc.

19. U(P,V) - Ice Structures L. Pauling considered the packing of water molecules in ice in 1923. Developed the ‘Ice rules’ S. Klotz et al.Europhys. Lett., 72 (4), p. 576 (2005)

20. ICE Ice VII Cubic

21. Bonding For stable bonds balance attractive and repulsive forces: Bond energy: Stable bonds have a minimum in the total energy Figure adapted from Callister, Materials science and engineering, 7th Ed.

22. Bonding or EA is bonding dependent Figure adapted from Callister, Materials science and engineering, 7th Ed.

23. van der Waals Temporary time varying electric dipole created in neutral materials through zero point motion of the electrons. e=Potential energy at equilibrium separation s=distance between the two atoms at zero energy Lennard-Jones Figure adapted from Callister, Materials science and engineering, 7th Ed.

24. van der Waals Materials Inert gases and Molecular crystals He, Ar, Ne, Xe, N2, O2, H2, Cl2, Graphitic Carbon Only nearest neighbours attract Fast fall-off Low melting temperatures Weak bond No restriction on bond angles Want to maximise the number of nearest neighbours Maximise Packing Density: Simple Cubic Structures

25. Ionic Bonds Small lattice parameters

26. Ionic Materials Range of materials such as NaCl, CsCl, MgO etc. Moderate fall-off First and second nearest neighbours attract Strong bond High melting temperatures Non-directional – bond strength same in all directions brittle and hard materials All positive ions surrounded by negative ions and want to maximise the number of nearest neighbours Simple Cubic Structures