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