1/12/2015phy 752 spring 2015 -- lecture 11 phy 752 electrodynamics 11-11:50 am mwf olin 107 plan for...
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PHY 752 Spring 2015 -- Lecture 1 11/12/2015
PHY 752 Electrodynamics11-11:50 AM MWF Olin 107
Plan for Lecture 1:
Reading: Chapters 1-2 in Marder’s text1. Course structure and expectations2. Crystal structures
a. Real and reciprocal space lattice vectors
b. Some examplesc. Point symmetry propertiesd. Introduction to group theory
PHY 752 Spring 2015 -- Lecture 1 21/12/2015
http://users.wfu.edu/natalie/s15phy752/
PHY 752 Spring 2015 -- Lecture 1 31/12/2015
PHY 752 Spring 2015 -- Lecture 1 41/12/2015
PHY 752 Spring 2015 -- Lecture 1 51/12/2015
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
PHY 752 Spring 2015 -- Lecture 1 61/12/2015
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.
PHY 752 Spring 2015 -- Lecture 1 71/12/2015
Introduction to crystalline solids
• An ideal crystal fills all spaceo 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
PHY 752 Spring 2015 -- Lecture 1 81/12/2015
Two-dimensional crystals
PHY 752 Spring 2015 -- Lecture 1 91/12/2015
Some details – square lattice example
1 1 2 2i i in nr a a
21 ˆ ˆ a a x ya a
PHY 752 Spring 2015 -- Lecture 1 101/12/2015
Energy associated with crystal configuration Simple pair potential models
r
12 6
/6
Lennard-Jones model: ( )
Buckingham model: ( ) r
A Br
r rB
r Aer
PHY 752 Spring 2015 -- Lecture 1 111/12/2015
Energy of square lattice
( )
For square lattice with lattice constant :
4
( )
( ) 4 ( 2 ) 4 (2 ) 8 ( 5 ) ...
j i jijr
a aE a a
E
a
( )r
E(a)
Note: This is the energy per atom assuming all atoms are identical.
PHY 752 Spring 2015 -- Lecture 1 121/12/2015
Two-dimensional crystals -- continued
1 1 2 2i i in nr a a
21
1 3ˆ ˆ ˆ
2 2a a
a ax x y
For hexagonal lattice with lattice c
( ) 6 ( 3 ) 6 (2 ) ...
onstant :
6 a a a
a
E
PHY 752 Spring 2015 -- Lecture 1 131/12/2015
Two-dimensional crystals -- continued
Honeycomb lattice (graphene sheet)
Example of lattice with a “basis”
PHY 752 Spring 2015 -- Lecture 1 141/12/2015
Honeycomb lattice (graphene sheet)
red 1 1 2
1
2
bl 21 2ue
red atoms:
blue atoms:i i i
i i i
n n
n n
τ a a
τ ar a
ra
1
red blue
2
1 3ˆ ˆ ˆ3 3
2 2
ˆ0
a a
a
a x y
τ τ y
ax
PHY 752 Spring 2015 -- Lecture 1 151/12/2015
1
2 ˆ
0
ˆ
ˆ
2 2ˆ
A
B
c
a c
a
a
τ
x
y
τ
x
a
y
1
2 ˆ2
ˆ
ˆ2
a c
a
a x
a x y
“Conventional” versus “Primitive” unit cell
PHY 752 Spring 2015 -- Lecture 1 161/12/2015
Wigner-Seitz cell
Note that primitive cell and Wigner-Seitz cell have the same volume.
PHY 752 Spring 2015 -- Lecture 1 171/12/2015
Generalization to 3-dimensions
1 1type 2 32 3i i i in n n ar τ a a
basis vector
Bravais lattice
PHY 752 Spring 2015 -- Lecture 1 181/12/2015
Fromhttp://web.mit.edu/6.730
PHY 752 Spring 2015 -- Lecture 1 191/12/2015
Simple cubic lattice:
PHY 752 Spring 2015 -- Lecture 1 201/12/2015
Face-centered cubic lattice
PHY 752 Spring 2015 -- Lecture 1 211/12/2015
Body centered cubic lattice
21 31 1 1 1 1 1 1 1 1 2 2 2
a a aa a a ������������������������������������������
Wigner-Seitz cell