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

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http://users.wfu.edu/natalie/s15phy752/

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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

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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.

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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

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Two-dimensional crystals

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Some details – square lattice example

1 1 2 2i i in nr a a

21 ˆ ˆ a a x ya a

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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

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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.

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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

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Two-dimensional crystals -- continued

Honeycomb lattice (graphene sheet)

Example of lattice with a “basis”

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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

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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

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Wigner-Seitz cell

Note that primitive cell and Wigner-Seitz cell have the same volume.

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Generalization to 3-dimensions

1 1type 2 32 3i i i in n n ar τ a a

basis vector

Bravais lattice

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Fromhttp://web.mit.edu/6.730

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Simple cubic lattice:

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Face-centered cubic lattice

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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