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MSE 510 , Autumn 2003
Bonding, symmetry, crystallography and properties of materials
Schedule
Introduction
Note: This will only serve as a guideline. The actual course will vary depending on the background of the students enrolled in the course.
1. Introduction and Overview (GRch1; DPch1;SAETch1)
Definitions - cohesive energy, metallicity, electronegativity etc. Periodic trends in materials structure & properties. Katelaar's triangle. Pettifor's AB map, Symmetry/structure of crystals. radius ratios. Descriptors
Lectures 2/3
2/3. Introduction and Review of Quantum Mechanics (DPch2, BDch6, WAHch1)
Wave-particle duality, phoelectric effect, electron diffraction
Heissenberg's uncertainty principle
The wave function in QM and its properties
The Schrodinger equation : time-dependent and time-independent
Examples: Particle in a box; Free electron gas
The Fermi energy
Free Atoms – the hydrogen atom
Electronic structure of the elements
Alloys
Lecture 4
4. Secondary Bonding (GRch6, CKch3)
van der Waals , hydrogen and dipolar bonds
van der Waals-London interaction
Leonard-Jones potentials
Cohesive energy in secondary bonding
Example: Self- assembly in nanoscale structures
Lecture 5
5. The ionic bond (GRch7, CKch3, WAHch13)
Introduction
Born-haber cycle
Electronegativity and the ionic bond
Ionization energy
Electrostatic interactions
Madelung energy and methods for calculating the same
Critical radius ratios
Practical Application: The Bulk modulus
Lecture 6
6. The Covalent Bond (GRch9, DPch3, BDch6,WAHch1)
Key features of the covalent bond
8-N rule of elementary chemistry
Hybridization
The hydrogen molecular ion
Creation of MO from AO
The Be2 molecule
Quantum mechanics revisited
Linear Combination of Atomic Orbitals
The homopolar diatomic molecule
The polar covalent bond
Lecture 7
7. Covalent crystals (GRch9, DPch3, BDch7,WAHch3)
Generalized LCAO formulation
Bloch’s theorem and concept of Brillouin zones
Dispersion of electrons in a square lattice
Parameterization coefficients for elements in selected structures
Heteropolar crystal with covalent bonding
Comparison of LCAO with experiments
Band gaps and opto-electronic applications
Evolution of semiconducting crystal structures
Lecture 8
8. The metallic bond and the structure of alloys (GRch8, BDch8, CKch6/9)
Free electron theory of metals
The density of states
Bloch’s theorem (revisited) and Bloch states
Nearly-free electron approximation
Reduced and periodic zone schemes
Diffraction: Bragg’s laws and Laue equations
The band gap
Metals and insulators
The Fermi surface
Cohesion of metals in the free-electron model
Introduction to Pseudopotentials
Metallic solid solutions, Hume-Rothary rules; electron compounds
Metal-Insulator transitions
Lecture 9
9. The crystalline state: introduction to symmetry (GRch2/3, SAETch3, MJB, DS, HH)
Symmetry is everywhere
Handedness, chirality, enantiomorphs
Geometric approach to symmetry and crystallography
Introductory definitions: translation symmetry, lattices, primitive/multiple cells, reflection, glide, the plane point group
Intersecting mirrors
Example: Chirality and microstructure – magnetic domains
Lecture 10
10. 2D Crystallography: point and space groups (GRch2/3, SAETch3, MJB, DS, HH)
Intersecting mirrors – examples,
The 10 plane point groups
Combination of rotation and translation; specialized plane nets
The 5 Plane lattices
Reflection and translations
The 17 crystallographic Plane groups
The international table of crystallography (ITC)
Examples
Mid Term
Lecture 12
12. 2D Crystallography continued (GRch2/3, SAETch3, MJB, DS, HH)
Review of the 17 plane groups
Asymmetric unit
Description of 2D structures
Using the ITC
The inverse problem: determining the plane group of 2D patterns
Example: New materials with tailored bandgaps
Lecture 13
13. Introduction to 3D crystallography (GRch2/3, SAETch3, MJB, DS, HH)
Review- designation of points, lines and planes in a space lattice, Miller indices
The zonal equation, Miller-Bravais notation
Comparison of 2D and 3D crystallography
Additional geometric operations: inversion, rotoinversion, rotoreflection, screw axis
Helical symmetry
Spherical geometry and trigonometry
The stereographic projection: basics and applications
Lecture 14
14. The 3D Point groups (GRch2/3, SAETch3, MJB, DS, HH)
Spherical geometry revisited
Euler’s construction and rules applied to crystallography
Allowed axial combinations:
The Schoenflies notation
Examples of point groups
Decomposition of rotoinversion and rotoreflection
Introduction to the 32 crystallographic point groups
Lecture 15
15. Point groups and Bravais lattices (GRch2/3, SAETch3, MJB, DS, HH)
The dihedral and isometric groups
Derivation of the 32 point groups: Axial combinations + extenders
External symmetry of crystals
The 14 Bravais lattices: primitive and centered
Friedel’s laws and diffraction
Laue groups
Lecture 16
16. Space groups and Crystal structures (GRch2/3, SAETch3, MJB, DS, HH)
Crystal systems, point groups and symmetry-dependent properties
Glide planes in 3D
Space groups and the ITC - The effect of symmetry operations, general/special positions
Trial structures
A detailed example of using the ITC: P4/m3(bar)2/m
Perovskites
Other examples of space groups: bcc, fcc, NaCl, diamond, zincblende, hexagonal
Another example: High Tc superconductors
The inverse problem: determining the simplest ITC description of a 3D pattern
Lecture 17
17. Crystal Chemistry: Structures , Diffraction and Defects (GR ch2/5, SAET ch3, WBO ch11-13)
Hard Sphere Packing
Close-packing in 2D and 3D, FCC & HCP
Interstitial sites
Ionic crystals & coordination polyhedra
Linking of polyhedra
Polymorphism & crystal transformations – Dilation, displacive and reconstructive
Order-disorder transformations and superlattices
Polytypes and polytypoids (example – aluminum oxynitride)
Diffraction and the reciprocal lattice
Ewald sphere construction
Overview of defects: point, line and planar
Holiday
Lecture 19
19. Introduction to Crystal Physics (JFNch1/2, SAET ch3)
Symmetry constraints on materials properties
Basics of scalars, vectors, tensors and matrices
Examples of second-rank tensor properties
Transformations of second-rank tensors
The representation quadric
Principal Axis
The Mohr circle construction
Introduction to Neumann’s principle
Matrices for various symmetry elements
Lecture 20
20. Symmetry and materials properties (JFNch1/2, SAET ch3)
Detailed discussion of Neumann’s principle
Effect of symmetry on materials properties: Cubic, Uniaxial, Biaxial cases
Example: electrical conductivity
Symmetry constraints on second-rank tensor properties
Example: Monoclinic crystals
Curies principle
Example: electric polarization of ADP crystals
The stress tensor
Piezoelectric, pyroelectric and ferroelectric materials
Lecture 21
21. Course Summary and Discussions of Term Projects etc.
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