3/1/2017
PHY 745 Group Theory
11-11:50 AM MWF Olin 102
Plan for Lecture 21:
Symmetry of lattice vibrations
Chapter 11 in DDJ
1. Review of vibrations in a one-dimensional
lattice
2. Vibrations in a three-dimensional lattice
3. Lattice modes and “molecular” modes
Some materials taken from DDJ and also Solid State Physics
text by Grosso and Parravicini (2014)
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Thanks to the Born-Oppenheimer approximation, the
nuclei of a material in equilibrium move in the potential
field provided by the ground electronic state of the system
The ground electronic state depends on the nuclear positions

E0 R a 

Suppose R a  R a 0 +ua
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For one-dimensional case:
Relationships:
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Classical equations of motion:
Solution
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Analytic result for monoatomic chain with only nearest
neighbor interactions
Dnn   0 if | n  n | 1
D (q)   Dnn 'e
iqa ( n  n ')
 C 2  e
iqa
e
 iqa
  4C sin  qa / 2
2
n'

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velocity of sound in the
material
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One-dimensional diatomic lattice
Equations of motion for this case:
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Solution:
Necessary condition for non-trivial solution:
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convenient
constant
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One-dimensional diatomic lattice -- continued
Normal modes
M* 
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M 1M 2
M1  M 2
11
Lattice modes of general three-dimensional crystals
Relationships:
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Lattice modes of general three-dimensional crystals -continued
Equations of motion
Solution
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Lattice modes of general three-dimensional crystals -continued
Some special values
Orthogonality of normal modes
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Example for fcc Al lattice
3 branches
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Example for Si and Ge lattices
6 branches
6 branches
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Example for LiF
6 branches
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6 branches
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Notation for Oh symmetry
BSW
Molecular
G1
A1g
G2
A2g
G12
Eg
G15’
T1g
G25’
T2g
G1’
A1u
G2’
A2u
G12’
Eu
G15
T1u
G25
T2u
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How many phonon branches?
# atoms in unit cell x dimension of lattice
Example -- NaCl
2 atoms per primitive
cell
 6 phonon branches
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Symmetry associated with k=0 phonons
Oh symmetry
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