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PHY 752 Fall 2015 -- Lecture 8 19/11/2015
PHY 752 Solid State Physics11-11:50 AM MWF Olin 103
Plan for Lecture 8:
Reading: Chap. 2 in GGGPP;
Conclude brief introduction to group theory
1. Compatibility relations for representations
2. Crystal field splitting
3. International Tables for Crystallography
PHY 752 Fall 2015 -- Lecture 8 29/11/2015
PHY 752 Fall 2015 -- Lecture 8 39/11/2015
The great orthogonality theorem
order of the group
element of the group
( ) th representation of
Notation:
denote matrix i ndices
dim
i
i
R
R i
h
l
R
ension of the representation
*( ) ( )j
R
iij
i
Rl
Rh
Results from last time --
PHY 752 Fall 2015 -- Lecture 8 49/11/2015
2
iil h
Results from last time -- continued
*( ) ( )ji
RijhR R
Character orthogonality theorem: 1
( ) ( )jl
j jR R
*
In terms of cla
( ) (
sse :
)
s
i jijN h C
C
C C
Number of elements in class C
PHY 752 Fall 2015 -- Lecture 8 59/11/2015
Summary of relationships between the characters and classes of a group which follow from the great orthogonality theorem
*
*
( ) ( )
( ) ( )a
i jij
i ia b ab
h
N
h
N
CC
i C
C C
C C
These results also imply that the number of classes is the same as the number of characters in a group.
PHY 752 Fall 2015 -- Lecture 8 69/11/2015
Example character table for cubic group corresponding to point symmetry of Brillouin zone at k=0
*( ) ( )i j
ijhN CC
C C
PHY 752 Fall 2015 -- Lecture 8 79/11/2015
Symmetry of Bloch wave function: in ne u k rk kr r
purely periodic
Example at :
Examp
point
, ,
lle along :
ine
, ,x
n n
ik xn n
x y z
x y z
u
e u
k k
k k
r
r
PHY 752 Fall 2015 -- Lecture 8 89/11/2015
Example at :
48
E
point
, ,
linxample along :
e
, 8,x
n n
ik xn n
u h
e
x y z
y zu x h
k k
k k
r
r
How do states at k=0 “connect” with states with ˆ ?xkk x
PHY 752 Fall 2015 -- Lecture 8 99/11/2015
Band structure diagram for fcc Cu (Burdick, PR 129 138-159 (1963))
1 1
25 ' 5
2 ' +
12 1
2 +
PHY 752 Fall 2015 -- Lecture 8 109/11/2015
PHY 752 Fall 2015 -- Lecture 8 119/11/2015
Example
D E C42 2C4 2JC4
2 2JC2
G25’ 3 -1 -1 -1 1
D2’ 1 1 -1 -1 1
D5 2 -2 0 0 0
PHY 752 Fall 2015 -- Lecture 8 129/11/2015
Use of character table analysis in crystal field splitting Question: What happens to a spherical atom when placed in a crystal?
In a spherical environment, an atomic wave function
ha
ˆ( ) ( ) ( )
with , 1,...0,1,..., 1, 2 1 values
s the form:
nlm nl lmR r Y
m l l l l l
r r
The group which describes the general rotations in 3-dimensions has an infinite number of members, but an important representation of this group is the matrix which rotates to coordinate system about the origin R, transforming ˆ ˆ( ) ( ').lm lmY Yr r
'ˆ ˆIt can be shown that: (( ()) )l
lm mm lmY Yr rR R
PHY 752 Fall 2015 -- Lecture 8 139/11/2015
Analysis of the 3-dimensional rotation group -- continued
'
12
2
ˆ ˆ( ) ( )
( )
Note that for corresponding to a
rotation of about the axis,
sin( )
sin
Note that the character for invers
(
ion i
)
( )
)
s
(
llm mm lm
ll l
mmm l
l immm mm
ll im
m l
l
Y Y
e
e
r r
z
R R
R R
R
R
R
12
2
an
( ) (2 1)( 1)
sin( )d ( 1)
si
n
l l
l l
l
l
J
J R
PHY 752 Fall 2015 -- Lecture 8 149/11/2015
Compatibility of continuous rotation group with the cubic group:
l=0 1 1 1 1 1 1 1 1 1 1 l=1 3 -1 1 -1 0 -3 1 -1 1 0l=2 5 1 -1 1 -1 5 1 -1 1 -1
PHY 752 Fall 2015 -- Lecture 8 159/11/2015
l=0 1 1 1 1 1 1 1 1 1 1 l=1 3 -1 1 -1 0 -3 1 -1 1 0l=2 5 1 -1 1 -1 5 1 -1 1 -1
G1
G15
G12+G25’
PHY 752 Fall 2015 -- Lecture 8 169/11/2015
Visualization of l=2 orbitals from http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Crystal_Field_Theory/Crystal_Field_Theory
G12 G25’
Octahedral symmetry
PHY 752 Fall 2015 -- Lecture 8 179/11/2015
Octahedral symmetry effects on d-orbitals
Energy diagram
G12 =eg G25’=t2g
PHY 752 Fall 2015 -- Lecture 8 189/11/2015
Comment on representation notations for cubic group
PHY 752 Fall 2015 -- Lecture 8 199/11/2015
Tetrahedral symmetry effects on d-orbitals
G12 =eg G25’=t2g
PHY 752 Fall 2015 -- Lecture 8 209/11/2015
Symmetry information available in the International Tables for Crystallographyhttp://it.iucr.org/
PHY 752 Fall 2015 -- Lecture 8 219/11/2015