Electrodynamics Lecture 1 Overview: Crystal Structures and Group Theory Fundamentals
Explore the foundations of crystal structures and group theory in Electrodynamics Lecture 1, covering topics such as real and reciprocal lattice vectors, point symmetry properties, and the role of group theory in understanding crystal structures. Dive into fundamental concepts essential for a deeper understanding of the course material.
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PHY 752 Electrodynamics 11-11:50 AM MWF Olin 107 Plan for Lecture 1: Reading: Chapters 1-2 in Marder s text 1. Course structure and expectations 2. Crystal structures a. Real and reciprocal space lattice vectors b. Some examples c. Point symmetry properties d. Introduction to group theory 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 1
http://users.wfu.edu/natalie/s15phy752/ 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 2
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Course syllabus: In the formal portion of the course, the following topics are usually covered: Crystalline structures Translation vectors and reciprocal lattice vectors Symmetry properties of crystals; brief introduction to group theory analysis Brief survey of common structures and their properties X-ray and neutron scattering analysis of crystals Electronic structure analysis Simple model examples Linear combinations of atomic orbital analyses Hartree-Fock approximations Density functional formalism Numerical approximations and methods Surfaces and interfaces Relationship of interface and bulk properties Low energy electron diffraction analysis Scanning probe analysis of surfaces Defects and disorder effects 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 5
Course syllabus continued: Depending on time available the following additional formal topics are often covered: Mechanical properties of solids Cohesive properties Deformation properties Vibrational modes Dislocations and cracks Electronic transport in solids Semi-classical approximations and comparison with experiment Optical properties in solids Magnetic properties of solids Feedback on topics choice appreciated. 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 6
Introduction to crystalline solids An ideal crystal fills all space o Limited possibilities for crystalline forms Only 14 Bravais lattices Only 32 crystallographic point groups Only 230 distinct crystallographic structures Quantitative descriptions of crystals in terms of lattice translations vectors and basis vectors Use of group theory in the study of crystal structures 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 7
Two-dimensional crystals 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 8
Some details square lattice example = + r a a n n 1 1 2 2 i i i = = a x a y a a 1 2 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 9
Energy associated with crystal configuration Simple pair potential models r A r B r = Lennard-Jones model: ( ) r 12 6 B r = / r Buckingham model: ( ) r Ae 6 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 10
Energy of square lattice ( ) r E(a) = ( ) E ijr Note: This is the energy per atom assuming all atoms are identical. ( ) j i j For square lattice with lattice constant : + a + 4 (2 ) + 8 ( 5 ) ... + 4 ( ) a 4 ( 2 ) E a a a 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 11
Two-dimensional crystals -- continued = + r a a n n 1 1 2 2 i i i a 1 2 3 = = + a x a x y a 1 2 2 onstant : For hexagonal lattice with lattice c + a + 6 (2 ) ... a + 6 ( ) a 6 ( 3 ) E a 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 12
Two-dimensional crystals -- continued Honeycomb lattice (graphene sheet) Example of lattice with a basis 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 13
Honeycomb lattice (graphene sheet) red atoms: = r + + n n a a a red 1 1 2 2 i i i blue atoms: = r + + a n n a bl ue 1 1 2 2 i i i 1 2 3 = = + a x a x y 3 3 a a 1 2 2 = = y 0 a red blue 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 14
Conventional versus Primitive unit cell = = = a a x a = a x a 1 1 y c 2 a c = + a x y 0 a 2 A 2 2 c = + x y B 2 2 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 15
Wigner-Seitz cell Note that primitive cell and Wigner-Seitz cell have the same volume. 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 16
Generalization to 3-dimensions = + + + r a n n n a a type 1 1 2 2 3 3 i i i i basis vector Bravais lattice 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 17
From http://web.mit.edu/6.730 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 18
Simple cubic lattice: 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 19
Face-centered cubic lattice 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 20
Body centered cubic lattice Wigner-Seitz cell a a a ( ) ( ) ( ) = 1 1 1 = 1 1 1 = 1 1 1 a a a 1 2 3 2 2 2 1/12/2015 PHY 752 Spring 2015 -- Lecture 1 21
