Mike Glazer MathCryst School Bogota 2018 Applications of Group Theory & Phase transitions Mike Glazer Universities of Oxford and Warwick
Mike Glazer MathCryst School Bogota 2018
Applications of Group Theory & Phase transitions
Mike GlazerUniversities of Oxford and Warwick
Mike Glazer MathCryst School Bogota 2018
PHASE
TRANSITIONS
DISRUPTIVE
NON-
DISRUPTIVE
FERROIC
NONFERROIC
NONDISRUPTIVE
FERROELASTIC
NONFERROELASTIC-
FERROIC
TRANSLATIONAL
NONDISRUPTIVE
ISOSTRUCTURAL
3BaTiO 3 4Pm m P mm
3Cu Au 3 3Fm m Pm m
3 3Ce 6 / 6 /P mmc P mmc
1 4QUARTZ 3 21 6 22P P
3Ti, Zr, Hf P6 / 3mmc Im m
A crystallographic classification of phase transitions
Mike Glazer MathCryst School Bogota 2018
Zn
Cu 0.5(Cu+Zn)
0.5(Cu+Zn)
Displacive
Order-disorder
Mike Glazer MathCryst School Bogota 2018
EhrenfestHysteresis
L=00 0
p
GL T T S
T
= − =
1st order
2nd order, continuous
T0
T0
T0
T0
T0
T0
Paul Ehrenfest1880-1933
Mike Glazer MathCryst School Bogota 2018
Landau theory of phase transitions
Lev Landau 1908-1968
2 4 6
0
1 1 1( ) ....
2 4 6= + + + G G a T b c
0 order parameter = 0 for T>T =
2 4
0
1 1For 0 assume ( )
2 4b G G a T b = + +
Find stable solution for 30 ( )
dGa T b
d= = +
0Assume ( ) '( )a T a T T −
( )
1 112 22
0
( ) '0 and
a T aT T
b b
= = − = −
Critical exponent b
Assume group-subgroup relationship,define quantity called the order parameter
Mike Glazer MathCryst School Bogota 2018
0G G−
2 0T T=
0T T
0T T
0T T
0( ) '( )a T a T T −
Mike Glazer MathCryst School Bogota 2018
0G G−
2
0T T=
0T T0T T
0T T
i.e. with b > 0 we get a 2nd order transition
Mike Glazer MathCryst School Bogota 2018
b < 02 4 6
0
1 1 1( )
2 4 6G G a T b c= + + +
Find stable solution for
3 50 ( )dG
a T b cd
= = + +
0Assume ( ) '( )a T a T T −
Mike Glazer MathCryst School Bogota 2018
0G G−
2 0T T=
0T T
0T T
0T T
2 4 6
0
1 1 1( )
2 4 6G G a T b c= + + +
Mike Glazer MathCryst School Bogota 2018
0G G−
2
No minimum0T T=
0T T
0T T
0T T
2 4 6
0
1 1 1( )
2 4 6G G a T b c= + + +
Mike Glazer MathCryst School Bogota 2018
0G G−
2
0T T=
cT T=
cT T
0cT T T
i.e. with b < 0 we get a 1st order transition
2 4 6
0
1 1 1( )
2 4 6G G a T b c= + + +
Mike Glazer MathCryst School Bogota 2018
T
P
( )1
40Prove that if 0 then b T T= −
Critical Point
C
gas
liquid
solid
ice water steam− −
Mike Glazer MathCryst School Bogota 2018
cT
( )1
40Prove that if 0 then b T T= −
Critical Point
Mike Glazer MathCryst School Bogota 2018
3( )dG
E a T P bPdP
= = +
21( ) 2
dEa T bP
dP= = +
0 0
1 0 ( ) '( )T T P a T a T T = = = −
( )
1122
0 0 0
' 1 2 '( )
aT T P T T a T T
b
= − = −
Dielectric susceptibility
Li2Ge7O15:0.7%Bi
Curie-Weiss Law (Curie Law in magnetism)
A.K. Bain, Prem Chand & K. VeerabhadraRao, Ferroelectrics- Physical Effects(2011)
Mike Glazer MathCryst School Bogota 2018
2 4 6 2 2
0
1 1 1 1 1( )
2 4 6 2 2G G a T b c C= + + + + +
Strain coupling
Exercise
Mike Glazer MathCryst School Bogota 2018
2 4 6 2 2
0
1 1 1 1 1( )
2 4 6 2 2G G a T b c C= + + + + +
210
2
dGC
d= = +
2
2C
= −
2 4 6 4 4
0
1 1 1 1 1( )
2 4 6 4 8G G a T b c
C C
= + + + − +
22 4 6
0
1 1 1( )
2 4 2 6
= + + − +
G G a T b c
C
Strain coupling
Can be positive or negative
Mike Glazer MathCryst School Bogota 2018
Effect of strain on domain growth and phase transition order
How many domains? How many ferroelastic and ferroelectric domains?
BaTiO3 : P4mm3Pm m
Mike Glazer MathCryst School Bogota 2018
Effect of strain on domain growth and phase transition order
BaTiO3 : P4mm3Pm m
486
8i = =
Number of ferroelectric states in each ferroelastic state = 2
Number of ferroelastic states = 3
Mike Glazer MathCryst School Bogota 2018
What’s it all about?
Eigenvalues
Eigenvectors
Mike Glazer MathCryst School Bogota 2018
S3
65 = 6-1
SCHOENFLIES VS INTERNATIONAL
Mike Glazer MathCryst School Bogota 2018
Symmetry OperationsSchoenflies International
Rotation -
inversion
S65
S64 = C3
2
S63 = i
S62 = C3
S6
S35
S34 = C3
S33 = s
S32 = C3
2
S3
3
32 = 32
33 = 1
34 = 3
35
6
62 = 3
63 = m
64 = 32
65
Mike Glazer MathCryst School Bogota 2018
Class
bxxa 1−=
Each element is said to be in a class if there is a similarity transform
a is said to be conjugate to b
a is always conjugate to itself
e.g. 2/m (C2h)
2112__
11 === −− mmmmSame
class
Mike Glazer MathCryst School Bogota 2018
110 ( )dm s
_
110
( )dm s
100 ( )vm s
010 ( )vm s
)(2)(4 2
22
4CC =
)(1 E1 1 3
110 001 110 110 010 0014 4m m m m− −= =
But in 4mm (C4v) the elements are
)(4 4C
)(4 33
4C
5 classes
1
4 43
2m100 m010
m110 m110
_
Mike Glazer MathCryst School Bogota 2018
Representations of a group
•A set of matrices G, each corresponding to an operation in the group
e.g. consider 2/m (C2h)
_
001 2 0011( ) 2 ( ) 1( ) ( )hE C i m s
+
+
+
100
010
001
−
−
100
010
001
−
−
−
100
010
001
−
+
+
100
010
001
Mike Glazer MathCryst School Bogota 2018
Let Y = Basis Function
Suppose
11 =R 22 =R
Consider
2121 = RRR
But R1 operates on each element of Y individually
2 is a matrix that permutes elements of Y
1221 RRR = 1221 =RR
Better to use transposed matrices =R
Mike Glazer MathCryst School Bogota 2018
Note that in group theory it is usual to specify operations according to
Cartesian axes rather than crystal axes as used so far
−
100
0cossin
0sincos
These representations are 3-dimensional representations and can in
general be decomposed into so-called irreducible representations
Mike Glazer MathCryst School Bogota 2018
Irreducible representations
AXXA 1' −=
Consider matrices A, B, C and D forming a representation of a group
Suppose we perform a similarity transformation:
BXXB 1' −=
CXXC 1' −=
DXXD 1' −=
DAB = if Now
BXAXXXBA 11'' −−= ABXX 1−= '1 DDXX == −
So the set of matrices A’, B’, C’ and D’ is also a representation of the group
Mike Glazer MathCryst School Bogota 2018
Suppose A can be written in matrix form e.g. 6 X 6 matrix
This is a 6-dimensional reducible representation
2X2
1X1
3X3
6 X 6
AXXA 1' −=
Mike Glazer MathCryst School Bogota 2018
2X2
1X1
3X3
Suppose A’ can be written in matrix form as a BLOCKED matrix e.g.
These form 3 irreducible representations 1d, 2d and 3d
Mike Glazer MathCryst School Bogota 2018
2X2
1X1
3X3
6 X 6
2X2
1X1
3X3
AXXA 1' −=
A A’
Reduction of Reducible
Representations
Mike Glazer MathCryst School Bogota 2018
Characters
In general, it is difficult to do this process to find the correct matrix X, unless the
representations are 1-dimensional. Fortunately, the traces of the matrices,
called CHARACTERS , can be used, since they behave in the same way and
are by definition 1-dimensional.
_
001 2 0011( ) 2 ( ) 1( ) ( )hE C i m s
+
+
+
100
010
001
−
−
100
010
001
−
−
−
100
010
001
−
+
+
100
010
001
1 2 1 m
3 1− 3− 1
Mike Glazer MathCryst School Bogota 2018
Character Tables
group oforder ......2
3
2
2
2
1
2 hlllli =+++=
rep eirreduciblth i' ofdimension Let (1) =il
ji
jihRR ij
R
ji
=
===
for 0
for )()(ity Orthogonal (2)
identical are class
same in the matrices all of characters therep,any In (3)
classes ofnumber reps eirreducibl ofnumber (4) =
reps d-1four are there
repsfour bemust there(4) From
reps d-1four or rep d-2 oneeither (1) From
classes 4 are thereand 4h )(C 2/min e.g. 2h
=
Mike Glazer MathCryst School Bogota 2018
)(1 E )(2 2C )(1 i )( hm s
gB 1 1− 1 1−
uA 1 1 1− 1−
uB 1 1− 1− 1
g = gerade (even) u = ungerade (odd)
gA 1 1 1 1
)(/2 2hCm
Totally symmetric
representationhttp://www3.uji.es/~planelle/APUNTS/TGS/taules_TG_oxford.pdf
Mike Glazer MathCryst School Bogota 2018
)(3 3vCm
)(1 E
)(3
)(3
2
3
2
3
C
C 110
120
210
( )
( )
( )
v
v
v
m
m
m
s
s
s
Order h = 6
Number of classes
(conjugate elements) = 3
6211 (1) From 222 =++
reps 3 classes 3 (4) From =
Mike Glazer MathCryst School Bogota 2018
0))(1(3))(1(2)2)(1(1 =++= zyEA
0))(1(3))(1(2)2)(1(2 =−++= zyEA
232 −=+ zy 232 −=− zy0 1 =−= zy
Check group oforder 6)0(3)1(22 222 =+−+
ji
jihRR ij
R
ji
=
===
for 0
for )()(ity Orthogonal (2)
)(1 E )2(32 3C )3(3 vm s)(3 3vCm
1A 1 1 1
2A 1 1 1−
E 2 y z
Totally symmetric rep
Mike Glazer MathCryst School Bogota 2018
−= 2 1Group { , , ,............... }n
n n n nC E C C C
There are n irreducible reps each of which is 1-dimensional
EC n
n =
1)( =n
nC
nm
eCmC nmi
nn
,......3,2,1
)( /2
=
=
IR 'th the in of Character
Characters in Cyclic Groups
Mike Glazer MathCryst School Bogota 2018
For group 3 (C3) we can write the effect of rotation thus
= 3/2
3)(3 ieC
*)(3 3/23/42
3
2 == − ii eeC
In this group there h = 3 and so there are three 1-d irreducible reps
)(1 E )(3 3C )(3 2
3
2 C)(3 3C
A 1 1 1
1 1 *
2 1 *
E =
Mike Glazer MathCryst School Bogota 2018
)(1 E )(3 3C )(3 2
3
2 C)(3 3C
A 1 1 1
E 23
2cos2
3
2cos2
Or for analysis of physical problems we can use
http://www3.uji.es/~planelle/APUNTS/TGS/taules_TG_oxford.pdf
https://tinyurl.com/y83xyynn
Mike Glazer MathCryst School Bogota 2018
Reduction of reducible reps: Magic formula
typeof reps eirreducibl ofnumber =n
igi classin elements ofnumber =
=i
i RRgh
n )()(1
Mike Glazer MathCryst School Bogota 2018
=i
i RRgh
n )()(1
_
001 2 0011( ) 2 ( ) 1( ) ( )hE C i m s
0)]1)(1(1)1)(3(1)1)(1(1)1)(3(1[4
1=+−+−+=gA
n
0)]1)(1(1)1)(3(1)1)(1(1)1)(3(1[4
1=−+−+−−+=gB
n
1)]1)(1(1)1)(3(1)1)(1(1)1)(3(1[4
1=−+−−+−+=uA
n
2)]1)(1(1)1)(3(1)1)(1(1)1)(3(1[4
1=+−−+−−+=uB
n
uured BA 2+=
)(1 E )(2 2C )(1 i )( hm s
gB 1 1− 1 1−
uA 1 1 1− 1−
uB 1 1− 1− 1
gA 1 1 1 1
)(/2 2hCm
+
+
+
100
010
001
−
−
100
010
001
−
−
−
100
010
001
−
+
+
100
010
001
)2/m(Cin tion representa reducible a had Earlier we 2h
Mike Glazer MathCryst School Bogota 2018
• Find group symmetry for the crystal/molecule being studied
• Mathematical functions to describe property in question
•Apply each symmetry operator in group to each function to obtain a
Reducible Representation
•Reduce Representation to a sum of Irreducible Representations
•Understand the meaning of each Irreducible Representation, e.g. using
Projection Operator Techniques
•Do experiment to find eigenvalues
•Interpret using eigenvectors
Procedure
Mike Glazer MathCryst School Bogota 2018
Vibrations of H2O
)(2 2C
x
y
z
Mike Glazer MathCryst School Bogota 2018
Vibrations of H2O
)(2 2C
x
y
z
( )y
y vm s
Mike Glazer MathCryst School Bogota 2018
)(2 2C
x
y
z
( )y
y vm s
( )x
x vm s
)(2 2vCmm
VIBRATIONS OF H2O
Mike Glazer MathCryst School Bogota 2018
)(2 2C ( )v
y
ym s ( )x
x vm s
2A 1 1 1− 1−
1B 1 1− 1 1−
2B 1 1− 1− 1
1A 1 1 1 1
)(2 2vCmm )(1 E
Mike Glazer MathCryst School Bogota 2018
x3
y3
z3
x2
y2
z2
x1
y1
z1
Mike Glazer MathCryst School Bogota 2018
1
1
1
1
1
1
1
1
1
)(1
3
3
3
2
2
2
1
1
1
333222111
z
y
x
z
y
x
z
y
x
zyxzyxzyxEx3
y3
z3
x2
y2
z2
x1
y1
z1
9=
Mike Glazer MathCryst School Bogota 2018
1
1
1
1
1
1
1
1
1
)(2
3
3
3
2
2
2
1
1
1
3332221112
z
y
x
z
y
x
z
y
x
zyxzyxzyxC
−
−
−
−
−
−
x3y3
z3
x2
y2
z2
x1
y1
z1
1−=
Mike Glazer MathCryst School Bogota 2018
1 1 1 2 2 2 3 3 3
1
1
1
2
2
2
3
3
3
( )
1
1
1
1
1
1
1
1
1
y
y vm x y z x y z x y z
x
y
z
x
y
z
x
y
z
s
−
−
−
x3y3
z3
x2
y2
z2
x1
y1
z1
1=
Mike Glazer MathCryst School Bogota 2018
1 1 1 2 2 2 3 3 3
1
1
1
2
2
2
3
3
3
( )
1
1
1
1
1
1
1
1
1
x
x vm x y z x y z x y z
x
y
z
x
y
z
x
y
z
s
−
−
−
x3y3
z3
x2
y2
z2
x1
y1
z1
3=
Mike Glazer MathCryst School Bogota 2018
=i
i RRgh
n )()(1
3 1 1- 9
)( )( C E vv2
red
yzxz
ss
331194
1n 1A
=++−=
131194
1n 2A
=−−−=
231194
1n 1B
=−++=
2Bn ????=
)(2 2C ( )v
y
ym s ( )x
x vm s
2A 1 1 1− 1−
1B 1 1− 1 1−
2B 1 1− 1− 1
1A 1 1 1 1
)(2 2vCmm )(1 E
Mike Glazer MathCryst School Bogota 2018
=i
i RRgh
n )()(1
3 1 1- 9
)( )( C E vv2
red
yzxz
ss
331194
1n 1A
=++−=
131194
1n 2A
=−−−=
231194
1n 1B
=−++=
331194
1n 2B
=+−+=
2121 323 BBAA +++=red
)(2 2C ( )v
y
ym s ( )x
x vm s
2A 1 1 1− 1−
1B 1 1− 1 1−
2B 1 1− 1− 1
1A 1 1 1 1
)(2 2vCmm )(1 E
Mike Glazer MathCryst School Bogota 2018
2121 323 BBAA +++=red
)(2 2C ( )v
y
ym s ( )x
x vm s
2A 1 1 1− 1−
1B 1 1− 1 1−
2B 1 1− 1− 1
1A 1 1 1 1
)(2 2vCmm )(1 E
Tz
RzTx Ry
Ty Rx
212 BAvib +=
x3y3
z3
x2
y2
z2
x1
y1
z1
3N-6 vibrations
To find out the types of vibration we use projection operators
Mike Glazer MathCryst School Bogota 2018
Projection Operators
• If a number of normal modes transform as a particular irreducible representation projection operator techniques only give a linear combination of the eigenvectors involved. These are called symmetry-adapted vectors.
R
RRV )(
• Define the character projection operator:
Mike Glazer MathCryst School Bogota 2018
R
RRV )(
1 2A
z y xV E m m= + + +2 2
A
z y xV E m m= + − −1 2
B
z y xV E m m= − + −2 2
B
z y xV E m m= − − +
A1 modes
0)or or ( 3211 = xxxV
A
0)( 122111 =−+−= xxxxxV
A
x3y3
z3
x2
y2
z2
x1
y1
z1
)(2)( 21122111 yyyyyyyV
A−=+−−=
0)( 31 =yV
A
)(2)( 2111 zzzV
A+=
33 4)(1 zzVA
=For A1 modes
oxygen does not
move in x and y
Mike Glazer MathCryst School Bogota 2018
Mike Glazer MathCryst School Bogota 2018
A1 modes
Mike Glazer MathCryst School Bogota 2018
R
RRV )(
B2 modes
0)(2 =xVB
x3y3
z3
x2
y2
z2
x1
y1
z1
)(2)( 21122112 yyyyyyyV
B+=+++=
33 4)(2 yyVB
=
)(2)( 2112 zzzV
B−= 0)( 3
2 =zVB
1 2A
z y xV E m m= + + +2 2
A
z y xV E m m= + − −1 2
B
z y xV E m m= − + −2 2
B
z y xV E m m= − − +
Mike Glazer MathCryst School Bogota 2018
B2 mode
Mike Glazer MathCryst School Bogota 2018
Triangular molecule
E 32C23C hs
32S vs3'
1A'
2A
'E"
1A"
2A"E
hD3
1
1
2
1
1
2
1 1 1 1 1
1 1− 1 1 1−1− 0 2 1− 0
1 1 1− 1− 1−
1 1− 1− 1− 11− 0 2− 1 0
zRxyT
zT
yxRR
z
x
y
62m
Mike Glazer MathCryst School Bogota 2018
C1: 1
C2: 2010, 2110, 2100
C3: m210, m1-10, m120
C4: -6+001, -6-
001
C5: 3-001, 3+
001
C6: m001
62m
Mike Glazer MathCryst School Bogota 2018
1 0 3 1- 0 9
3 2S 3C 2C E v3h23
red
ss
1 2
3
130330912
1n 1A'
=+++−+=
130330912
1n 2A'
=−++++=
261812
1n
E'=+=
030330912
1n 1A"
=−+−−+=
130330912
1n 2A"
=++−++=
161812
1n
E"=−=
""'2'' 221irr EAEAA ++++=
Mike Glazer MathCryst School Bogota 2018
0 1- 2 0 1- 2
1- 2
1- 1 1-
2
1- 1
1 2
1- 1 1
2
1- 1
3 2S 3C 2C E v3h23
xy
y
x
T
T
T
ss
−−
−→
−
2
1
2
32
3
2
1
cossin
sincos
For example, 3-fold rotation C3 on Tx and Ty
=E’
''1 EAvib +=
Mike Glazer MathCryst School Bogota 2018
A’1 mode
Mike Glazer MathCryst School Bogota 2018
These are all the same mode as molecular orientation does not matter
Bending mode
Asymmetric modeThese two can be taken to be E’ as
the two partners are orthogonal
motion
Mike Glazer MathCryst School Bogota 2018
Kronecker Products
Let G be a group with subgroups H and K such that
khhkK then k H,h (1) =
hkg as expressed becan G g all (2) =
{E} K H (3) =
G is the outer direct product of H and K
KHHKG ==
H and K are invariant subgroups of G
Number of classes in G = number of classes in H x number of
classes in K
If H and K are not commutative – semi-direct product of H and K
Mike Glazer MathCryst School Bogota 2018
KHG =
G oftion representa andK and H of tionsrepresenta are and If ijji =
jiij = Kronecker product
bacge = ..
=
baba
baba
bb
bb
aa
aa
2221
1211
2221
1211
2221
1211
=
2222212222212121
1222112212211121
2212211222112111
1212111212111111
babababa
babababa
babababa
babababa
)()()( bac =
Mike Glazer MathCryst School Bogota 2018
Kronecker Square
ji
i
oftion representa therequire wei.e.
tionrepresenta particular a of square i.e.
VV tohappens what know wish toWe
V. space ain function a have weSuppose
We can produce two functions
symmetric 2
1 +
ij
ijji
ricantisymmet 2
1 −
ij
ijji
Mike Glazer MathCryst School Bogota 2018
The result for characters is
)()(2
1)( 22
2 RRR =
For a 1-dimensional representation
)()( 22 RR
=
2v 2 2
E 2 m m
. . in C (1 1) (1 1) (-1 -1) (-1 -1)
z y x
e g A A =
1 1 1 1 =
1A=
Mike Glazer MathCryst School Bogota 2018
2-dimensional example --- E” representation
3h 3 2 h 3 v
2
D E 2C 3C 2S 3
2 -1 0 -2 1 0
4 1 0 4 1 0
s s
2 (R ) 2 -1 2 2 -1 2
Symmetric square
Antiymmetric square
3 0 1 3 0 1
1 1 -1 1 1 -1
Symmetrised square = A’1+ E’
Antiymmetrised square = A’2
62m
Mike Glazer MathCryst School Bogota 2018
Selection Rules
0|| ifonly possible Transition
iHf
state ground for symmetric totally be i| of tionrepresenta Let i
elementmatrix zero-nonFor
0 ity orthogonal otherwise |
i| →=
fH
tionrepresenta symmetric totally the i||f =
H
= H
|f if possibleonly is This
General Principle:
The Kronecker product of the representations of the three terms
must contain the totally symmetric representation
Mike Glazer MathCryst School Bogota 2018
Infra-red Absorption
onperturbatidependent -time moment dipole
with couples field py thespectrosco absorptionIn
→p
Ε
p.E=
'H
0|'| if possible Transition
iHf
acquiredenergy additional '=
H
==
ifeifiHf |||||'| r.Ep.E
}||||||{ ++= izfEiyfEixfEe zyx
Mike Glazer MathCryst School Bogota 2018
Vibrational spectraSolution of harmonic oscillator problem in quantum mechanics
hnEn )2
1( +=
2/2
)()( −= eHA nn
constant gnormalisin=nA
polynomial Hermite)( =nH
xm
2/1
=
24)( 2)( 1)( 2
210 −=== HHH
nodes no symmetric totally is )(0 H
symmetric totally is state ground
Mike Glazer MathCryst School Bogota 2018
Dipole transition has symmetry properties of translation
Therefore we must consider functions such as
etc. || iTf x
dxxf )(function Consider
= 0)( dxxf
+
-
ricantisymmet odd,
)(xff(x) R −⎯→⎯
0)( dxxf
++
symmetricy totall
)(xff(x) R⎯→⎯
Mike Glazer MathCryst School Bogota 2018
symmetric totally is 0 dT ixf
species symmetric totally contains
then assymmetry same has If ixfxfi TT
An infra-red transition is allowed if the product of the
states before and after transition transforms with
respect to the relevant symmetry operations like a
coordinate axis.
Mike Glazer MathCryst School Bogota 2018
e.g. H2O molecule
)(2 2C ( )v
y
ym s ( )x
x vm s
2A 1 1 1− 1−
1B 1 1− 1 1−
2B 1 1− 1− 1
1A 1 1 1 1
)(2 2vCmm )(1 E
z
Rz
x Ry
y Rx
Modes A1 B1 B2 are infra-red active
Thus all three modes seen in IR spectrum
Mike Glazer MathCryst School Bogota 2018
E 32C23C hs
32S vs3'
1A'
2A
'E"
1A"
2A"E
hD3
1
1
2
1
1
2
1 1 1 1 1
1 1− 1 1 1−1− 0 2 1− 0
1 1 1− 1− 1−
1 1− 1− 1− 11− 0 2− 1 0
zRxyT
zT
yxRR
'
molecule Triangular
1 EAvib += active red-infra is 'only E
Mike Glazer MathCryst School Bogota 2018
Raman Scattering
2|k| 0 =
In a crystal the electric field of a monochromatic
wave propagating with wave-vector
].[0 0tieEΕ
−=
rk0
This induces a dipole moment by exciting radiation
rlity tensopolarisabi == Ep
Mike Glazer MathCryst School Bogota 2018
].[0 0tieEΕ
−=
rk0
E
Mike Glazer MathCryst School Bogota 2018
+
-
rlity tensopolarisabi == Ep
The polarisability is modulated by vibrations
Let Qj = normal coordinate of the jth mode
E
Mike Glazer MathCryst School Bogota 2018
....2
1'
'0'
2
0
0 +
+
+= jj
j j j jj
j
j
QQQQ
][ ti
jjjeAQ
−=
.rk j
...'])()[(0
0
][0
00000 +
+=
−−
ti
j
j
j
ti jeEAQ
eEp
.rkk.rk j
j 0frequency oflight scattered
jkkk s += 0
js += 0
js += 0
onannihilatiantiStokes
jkkk s −= 0
js −= 0
creationStokes
js −= 0
Mike Glazer MathCryst School Bogota 2018
o o+1o−1
jkkk s += 0
js += 0
js += 0
onannihilatiantiStokes
jkkk s −= 0
js −= 0
creationStokes
js −= 0
Mike Glazer MathCryst School Bogota 2018
0|| if b
Consider symmetry of polarisability tensor
zzzyzx
yzyyyx
xzxyxx
field electricincident of
direction is when direction in inducedon polarisati yxxy =
yxyx EP = yxxy TT like is
Mike Glazer MathCryst School Bogota 2018
e.g. H2O
1 1 1 1
2 -1 -1 1
1 -1 -1
-1 1 -1
x y
z
xz
xy
T T
m
m
=
=
=
=
2 like s transform Axy
212 BAvib +=
All modes are Raman active
Note: to observe B2 mode light must be polarised parallel to
y direction and will be radiated polarised parallel to z. This is
particularly important for crystals.
Mike Glazer MathCryst School Bogota 2018
Triangular molecule
'1 EAvib +=
E’ is infra-red active; A1 and E’ are Raman active
A mode is Raman active if the product of the states before
and after transition transforms with respect to the relevant
symmetry operations like the product of a pair of
coordinate axes.
Note: a totally symmetric mode is always Raman active
Mike Glazer MathCryst School Bogota 2018
)(2 2C ( )v
y
ym s ( )x
x vm s
2A 1 1 1− 1−
1B 1 1− 1 1−
2B 1 1− 1− 1
1A 1 1 1 1
)(2 2vCmm )(1 E
z
Rz
x Ry
y Rx
E 32C23C hs
32S vs3'
1A'
2A
'E"
1A"
2A"E
hD3
1
1
2
1
1
2
1 1 1 1 1
1 1− 1 1 1−1− 0 2 1− 0
1 1 1− 1− 1−
1 1− 1− 1− 11− 0 2− 1 0
zRxyT
zT
yxRR
Subgroup to Group
A1+B2=E’
Mike Glazer MathCryst School Bogota 2018
Infra-Red Raman
Equilateral E’ A1+E’
Isosceles 2A1+B2 2A1+B2
D3h
C2v
A1+B2=E’
Correlation Table
Mike Glazer MathCryst School Bogota 2018
Lattice Vibrations (Sorry Massimo!)( )=
ls
l
ssuMT 2
2
1 crystal ofenergy Kinetic
componentsCartesian cell, in the atoms cells, sl
Potential energy V is expanded in a power series of displacements
( ) ( ) ( ) ( ) ( ) ...2
1 '
'
''
'
'0 +++= l
s
l
s
slls
ll
ss
l
s
ls
l
sVV b
b
b
uuu
.....210 +++= VVVV
( )( )
ionconfigurat mequilibriu 0
0
=
=
l
ss
l
u
V
( )( ) ( )
0
'
'
2'
'
=
l
s
l
s
ll
ssuu
V
b
b
If higher order terms neglected, harmonic approximation
b
b
along displaced is
)''( atom when the)( atomon direction in the force theis ofcomponent slls
Mike Glazer MathCryst School Bogota 2018
irrelevant is and termstatic is 0V
( ) m.equilibriufor zero is and
),( atom on the acting force theof negative theis because 01 slV l
s=
( ) ( ) ( ) ( ) ++=
b
bb ls ls sl
l
s
l
s
ll
ss
l
ss uuuMVH''
'
'
'
'
2
02
1
2
1
is crystalfor n Hamiltonia Total
( )( )
( ) ( )−=
−=
''
'
'
is )( atomfor motion ofequation The
sl
l
s
ll
ssl
s
l
ss uu
VuM
ls
This gives an infinite number of coupled differential equations to solve. This can be simplified by making use of lattice periodicity i.e. the motions in different unit cells must be identical in amplitude and direction but not necessarily in phase.
Mike Glazer MathCryst School Bogota 2018
( ) cellunit th ' ofector position v is )( )(.2/1 lleeUM liti
ss
l
s rurk−−=
( ) )(.2/12 liti
ss
l
s eeUMu rk −−−=
( ) )'(.2/1
'
''
'
'
)(.2/12 liti
ss
sl
ll
ss
liti
ss eeUMeeUM rkrk −−−− −=−
='
'
2 ),'(s
ss UssDUb
bb k
( ) )]'()'(.['
'
2/1
')(),'( lli
l
ll
ssss eMMss rrkkD
−=
matrix Dynamical ),'( =kD ss
So problem is reduced to solution of 3n equations, since as s runs over 3n values. D is a 3n X 3n matrix.
Mike Glazer MathCryst School Bogota 2018
0),'( '
2 =− ssssD bb k
Non-trivial solution given by
Normal coordinates
3n values of 2 at given k k2
Problem can be made even simpler. T is diagonal, whereas V contains cross-products. So replace
s
lu by new parameters
These correspond to independent oscillators
Normal modes
Mike Glazer MathCryst School Bogota 2018
So, why is this too difficult in practice?
Consider just 2 atoms and the possible force constants.
Mike Glazer MathCryst School Bogota 2018
So, why is this too difficult in practice?
Consider just 2 atoms and the possible force constants.
Internuclear force constant
Mike Glazer MathCryst School Bogota 2018
So, why is this too difficult in practice?
Consider just 2 atoms and the possible force constants.
Electron-electron force constant e.g. van der Waals
Mike Glazer MathCryst School Bogota 2018
So, why is this too difficult in practice?
Consider just 2 atoms and the possible force constants.
Nuclear-electron force constant
Mike Glazer MathCryst School Bogota 2018
So, why is this too difficult in practice?
Consider just 2 atoms and the possible force constants.
Nuclear-electron force constant
Mike Glazer MathCryst School Bogota 2018
So, why is this too difficult in practice?
Consider just 2 atoms and the possible force constants.
Polarizability
Mike Glazer MathCryst School Bogota 2018
So, why is this too difficult in practice?
Consider just 2 atoms and the possible force constants.
Polarizability