applications of group theory & phase transitions mike glazer universities of...

Mike Glazer MathCryst School Bogota 2018

Applications of Group Theory & Phase transitions

Mike GlazerUniversities of Oxford and Warwick


Seitz operator

{ | }R R= +v r r v

The set of Seitz operators defines a group

001 001{2 | (0,0,1/ 2} 2 (0,0,1/ 2)= +r r

001 001{ | (1/ 2,0,0} (1/ 2,0,0)m m= +r r

21 along [001]

a glide perp. to [001]


{ | }{ | } { | }( )R S R S= +v u r v r u

RS R= + +r u v

{ | }RS R= +u v r

uR is a translation

Product of two Seitz operators is a third

(1)


(2) There is an identity element

}|{}|1{ 00 E or

(3) The inverse is1 1 1{ | } { | }R R R− − −= −v t

1 1 1 1{ | }{ | } { | }( )R R R R R R− − − −− = −v v r v r v

1 1= − +r v v

r0}|1{=

Because

(4) From (1) we can prove associativity


Translation Group

Consider 1-d lattice. Symmetry operations given

by }|1{ nt

Each operation is in a class by itself

}|1{}|1}{|1{}|1}{|1{}|1{ 1

nmnmmnm tttttt == +−

−

So there are as many irreducible representations as

elements in the groupUse periodic boundary conditions with N=number of

lattice points

}|1{}|1{ nNn tt =+


We can now express the entire group in terms of

powers of generating elements }|1{ a

Nnn

n

+== }|1{}|1{}|1{ aat

}|1{}|1{ 0a =N But

So a representation of the group consists of the N

roots of unityNimNimN ee /2/12/1 ][]1[}|1{ === a

The values of m are restricted since there are only as

many irreducible representations as classes = N


10 − Nm Let

11

12

11

11110

}|1{}|1{}|1{}|1{}|1{

2221

2)1(242

12

12

=−

==

==

=

=

−−−

−

−

−

NNNN

NN

NN

NN

εεεN

εεεεm

εεεεm

m

....................................

............. 0aaaa

m is a label for the irreducible representation


For a 3-d lattice the translations in the 3 directions commute and therefore the

irreducible rep is the direct product of 1-d reps.

Nimnn

n e /2}|1{}|1{ == at

321 TTTT =

333222111

321

/2/2/2}|1{

NnimNnimNnim

nmmm eee

= t

)(

21

cba

cbg

•

=

a)(cb

acg

•

=

22

b)(ac

bag

•

=

23

Conjugate to the real lattice we can define a reciprocal lattice

332211 gggK nnnn ++=

Note that Kn is a vector perpendicular to plane with Miller indices (n1 n2 n3)

0 =•=• bgag 11 2


Periodic Boundary Conditions

Circumference

= L

= .....2 3 4

i

L L LL

= =

2 2 4 6 8 ....... i

i

kL L L L

0 2

a

−

2

a


b

2|| 2 =g

a

2|| 1 =g

332211 gggk ppp ++=


b

2|| 2 =g

a

2|| 1 =g

A B

CD

A

C

B

Wigner-Seitz Construction


Contains allowed values of k that label the irreducible representations of the

translation group

2

1

2

1−− ip

Wigner-Seitz construction always gives a primitive unit cell.


Brillouin zone is described by a unit cell in reciprocal space


M

R

X

0 0 0

R ½ ½ ½

M ½ ½ 0

X 0 ½ 0


Wigner-Seitz construction always gives a primitive unit cell which at the same time

displays the full symmetry of the lattice.


Wigner-Seitz construction always gives a primitive unit cell which at the same time

displays the full symmetry of the lattice.

But note that for low-symmetry crystals the Wigner-Seitz construction may not be

the most appropriate one to use.


Unit cell in k-space contains all allowed wave states. This cell is known as

the Brillouin Zone


CHAINPLOT www.amg122.com/programs


Diamond

Graphite


Bloch Theorem

Consider operator T operating on a functionrk•ie

rkrk •• → iTi eTe

Consider a scalar product. This gives a scalar and therefore is not affected by

operator.

qpqp TT •=•

)()( 11 rkrk −− •=• TTTT

rk •=T

)()( 1rr

−= TfTf general In


Now consider operator T is translation nt

)( nii eetrkrk −•• →

rktk ••−= ii

ee n

rk•= iTe

Thus basis functions for translation can be taken to berk•i

e

)()( rurrk

k

i

k e •= Generally

)()( nkk truru += whereThe wave function for a particle moving in a periodic potential V(r) with

periodicity T is given by

)()( rurrk

k

i

k e •=


Readers’ Digest Version of Space Group Multiplier Representations

(ray representations, Lyubarsky, Kovalev)

G = space group

= representation of subgroup T of all translations of G

can be decomposed into 1-dimensional representations of T characterised by k.

The set of k characterising the representations of T in is called the star of the

representation – this is invariant with respect to the elements of G.


Gk = subgroup of G = group of k or little group

Gives small representations })|{;( τtk + nRKovalev’s Tables

P6/mmm

Gk = Pm


=kG Point group of k or little co-group (in our example

point group is m)

Consists of all elements R

With each element we can associate an operator T(k;R)

)(})|{;();(

τtkτtkk

+•−+= ni

n eRRT

This definition is unique despite the fact that to every element R, there corresponds

an infinite number of different operations in Gk.


Consider two operations

2)2(1)1(

)2(2)1(1 }|{}|{

τttτtt

tt

+=+= nn

RR

and where

}|{}|}{|{ )2(1)1(21)2(2)1(1 tttt RRRRR +=

)2(1

1)1(})|}{|{;();( )2(2)1(121

tktkttkk

•−•− −

=iRi

eRRRRT

)(})|{;();(

τtkτtkk

+•−+= ni

n eRRT

)(

)2(2)1(121)2(1)1(})|}{|{;();(

ttkttkk

RieRRRRT

+•−=

)()()( 2121 RRRR =

)2(1

1 )(

2121 );();();(tkk

kk•−− −

=Ri

eRkTRTRRT

Using

1

1

−R either leaves k unchanged or transforms it into an equivalent vector by

adding a vector K of the reciprocal lattice.


Kkk +=−1

1R

2);();();( 2121

•=

Kkk

ieRRTRkTRT

Using the fact that integer=• 2ntK

2);( 21

•=

Kk

ieRR multiplier

The matrices T(k,R) form a unitary multiplier representation


From

)(})|{;();(

τtkτtkk

+•−+= ni

n eRRT

to each irreducible representation of Gk there corresponds an irreducible

representation of iG ˆ →kwith matrices in ith irreducible representation)(ˆ ri

group a in As rqp RRR =

);()(ˆ)(ˆ)(ˆ qp

iii RRqpr k=

to form character table

Note that if k lies within the Brillouin zone the vector K=0 and so = 1

Also if space group is symmorphic = 1

The effect is to introduce a degeneracy on some Brillouin zone boundary of a

nonsymmorphic crystal


Use Representations SG in Bilbao server to get irreps at Z point for P2/m and P21/c


P2/m Ag Au Bg Bu


P21/c


( )( ;{ | }) ni

nR e − • + +

k tk t


Perovskite – a flexible friend

ABX3


Coordination polyhedra


Two origin choices in Pm3m

A 1b ½, ½, ½

B 1a 0, 0, 0

X 3d ½, 0, 0

A 1a 0, 0, 0

B 1b ½, ½, ½

X 3c 0, ½, ½


Importance of pseudosymmetry

Octahedral

tilts

CaTiO3 Naray-Szabo

1943

Cation

displacements

BaTiO3 Megaw 1945

Distortions KCuF3 Okazaki &

Suemune 1961


(Mg,Fe)SiO3 (also known

as bridgmanite) perovskite may

form up to 93% of the lower

mantle, and the magnesium iron

form is considered to be the most

abundant mineral in the Earth,

making up 38% of its volume.

Percy Bridgman awarded Nobel

Prize in Physics 1946


Barium Titanate

O Ti

Amm2 P4mm

R3m

Pm3m-

red=4T1u+T2u


Vibrate!

https://www.amg122.com/programs/vibrate.html


AMPLIMODES - Bilbao Crystallographic Server

J. Appl. Cryst. (2009) 42, 820-833


4

−

Amm2

5

−

O Ba Ti


2 ( )cT T −

Cochran-Anderson Soft Mode

Note that condensation at the zone boundary e.g. at R, M, X

results in a reciprocal lattice point, thus doubling the unit cell

repeat in that wave-vector direction. In diffraction terms this

means introduction of superstructure peaks.


No octahedral tilts

Cubic a0a0a0

a

bc


Octahedral tilt about one axis

Tetragonal a0a0c+

Doubled

unit cell

2a x 2a x

c

a

bc


Tetragonal a0a0c-

Octahedral tilt about one axis

Doubled

unit cell

2a x 2a x

2c

a

bc


Cubic a+a+a+

Equal octahedral tilts about three axes

a

b

c


Equal octahedral tilts about three axes

Rhombohedral a-a-a-

a

b

c


TWO TILTSTHREE TILTS NO TILTS

ONE TILT


SrTiO3 Phase transition at 105K

a0a0a0a0a0c-


Irreducible Representations & Brillouin Zone

1 1 1R

2 2 2

1 1M 0

2 2

000

1X 00

2


SrTiO3 Phase transition at 105K


SrTiO3


4R+


Irreducible Representations & Brillouin Zone

Using Bilbao Server or ISODISTORT each

Irrep can be further labelled to represent

thermal or static displacements of atomic

arrays

But be aware that for irreps away from

Brillouin zone centre, the label actually

depends on the choice of origin used to

describe the crystal structure!

In this example R4+ describes this

arrangement of tilted octahedra with the origin

chosen on the atom at the centre of an

octahedron. If the origin is chosen to lie on

the atom between the octahedra the label

changes to R5-

Sometimes in publications the origin is not specified or the label

does not match the choice of origin!

5R−


BUT WARNING –

Atención!


WARNING – Atención!


i ijk jkP d=

ij ijk kd E =converse

direct

Discovered in 1880 by French physicists Jacques and Pierre Curie

Conversion between electrical and mechanical energy

Pierre

Curie 1859-

1906

Jacques

Curie 1856-

1941

Gabriel

Lippmann 1845-

1911

The Piezoelectric Effect


ASDIC SONAR


Quartz crystals - left and right

http://www.quartzpage.de/intro.html


1

2

cf

l =

1 1 1 1

2 2

df dc dl d

f dT c dT l dT dT

= − −

frequency

l = thickness

c = shear stiffness coefficient

=f

Why do we use quartz in our watches?

MgO Tc11 = -2.3 Tc44 = -1.0

SrTiO3 Tc11 = -2.6 Tc44 = -1.1

Quartz Tc11 = -0.5 Tc33 = -2.1

Tc44 = -1.6 Tc66 = +1.6

Temperature coefficients of elastic stiffness in units of 10-4 K-1

AT cut q = 35o 25’


P3221 P6222846K


Glazer, A.M. (2018) Confusion over the description of the quartz structure yet again. J. Appl. Cryst.51,

915-918

applications of group theory & phase transitions mike glazer universities of...

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