Role of Magnetic Symmetry in the Description and Determination of Magnetic Structures

IUCR Congress Satellite Workshop 14-16 August Hamilton, Canada

MAGNETIC POINT GROUPS

Bilbao Crystallographic Server

http://www.cryst.ehu.es

Cesar Capillas, UPV/EHU 1

Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

Historical Briefs

Heesch: 4-dim groups in 3-dim space: 122 anti-symmetry point groups

1945

Shubnikov: describes and illustrates all two-color point groups

1953 Zamorzaev: derives the magnetic space groups

1955 Belov, Neronova and Smirnova: complete listing of the magnetic space groups; BNS notation

1965 Opechowski and Guccione: complete listing of the magnetic space groups; OG notation

2001 Litvin: corrected OG notation2009-11 Litvin: tables of magnetic subperiodic and space groups

1951

Shubnikov: re-introduces the concept of ‘anti-symmetry’

1963-4 Birss: tensor properties of crystals with magnetic group symmetry

1966 Koptsik: diagrams of magnetic space groups

1929-30

1st type(32): M=G, 1’∉ M (classical crystallographic groups)

Heesch-Shubnikov groups

2nd type(32): M=G + 1’G, 1’∈ M (grey groups)

3rd type(58): M=H + 1’(G-H) |G|/|H|=2

(black-and-white groups)

Example

general position symmetry elements

Stereographic Projections of 4mm

Black-and-white groups

The group of the square 4mm (C4v)

Symmetry operations of 4mm:

{e 4z 4z 2z mx my m+ m-}

G=4mm: {e 4z 4z 2z mx my m+ m-}

Example Black-and-white groups

H1=4: {e 4z 4z 2z }

M=G(H)=H+1’(G-H)

M=4mm(4) = 4mm : {e 4z 4z 2z mx my m+ m-}

Black-and-white group

G = 4mm: {e 4z 4z 2z mx my m+ m-}

Example Black-and-white groupsM=G(H)=H+1’(G-H)

M=4mm(2mm) = 4mm : {e 4z 4z 2z mx my m+ m-}

Black-and-white groupH = 2m+m-: {e 2z m+ m-}

G = 4mm: {e 4z 4z 2z mx my m+ m-}

Example Black-and-white groupsM=G(H)=H+1’(G-H)

M=4mm(2mm) = 4mm : {e 4z 4z 2z mx my m+ m-}

Black-and-white group

H = 2mxmy: {e 2z mx my}

Black-and- white point

groups

Bradley and

Cracknell

The mathematical theory of

symmetry in solids

Magnetic point groups (types I and III)

International Tables for Crystallography (2006). Vol. D.!Borovic-Romanov, Grimmer. Chapter 1.5 Magnetic properties

magnetic groups

4mm

4m’m’

4’mm’

4’m’m

——

Magnetic point groups derived from the representations of 4mm(C4v)

Indenbom (1959), Bertaut (1968)

http://www.cryst.ehu.esBilbao Crystallographic Server

D.B. Litvin Magnetic Space Groups v. V3.02 http://www.bk.psu.edu/faculty/litvin/Download.html

H. Stokes, B.J. Campbell Magnetic Space-group Data http://stokes.byu.edu/magneticspacegroups.html

Magnetic Point Groups

Bilbao Crystallographic Server

(under development)

Bilbao !Crystallographic !

Server

Geometric interpretation

coordinate triplets

axial-vector

coefficientsmatrix-column presentation

Curie’s principle

characteristic symmetry of a phenomena

the maximum symmetry compatible with the phenomena(the invariance group of a phenomena)

A phenomenon can exist in a system which possesses either the characteristic symmetry of the phenomenon Pphen or the symmetry of one of the subgroups of Pphen

ferromagnetism (spontaneous magnetization Ms - axial vector): ∞/m2’/m’

ferroelectricity (spontaneous polarization Ps - polar vector): ∞ m1’

PG Pphen≤PG Pphen1 ⋂ Pphen2 ⋂ …≤

Pphen=

Pphen=

Polar and axial vectors

polar axial

∞m ∞ /mMarc De Graef

©2009 IUCr"

polar vector

polar vector axial vector

Transformation of polar and axial vectors under space and time inversion

polar∞m1’

axial∞/m2’/m’

The groups in red are compatible with both phenomena

Grimmer, Leuven 2006

Marc De Graef©2009 IUCr"

Transformation of an axial vector parallel to the 2-fold axis

point group 2/m grey point group 2/m1’

Marc De Graef©2009 IUCr"

Transformation of an axial vector parallel to the 2-fold axis

point group 2’/m point group 2/m’ point group 2’/m’

Marc De Graef©2009 IUCr"

Transformation of an axial-vector parallel to the mirror plane under "the operations of the point group 2/m and 2’/m’

Tensor properties of non-magnetic crystals(brief summary)

Tensor representation of physical properties

pyroelectricity: ∆Pi = pi ∆Tpyroelectric coefficients

temperature change

electric dipole moment change

electric conductivity:

electrical conductivity

applied electric field

current density

ji =X

j

�ijEj

piezoelectric effect:

piezoelectric modula

stress tensorpolarization pi =X

jk

dijkSjk

i=1,2,3

i,j=1,2,3

physical property ⇒ Tijk…l (3n components)︷n

crystallographic symmetry

intrinsic symmetry

Tensor properties of non-magnetic crystals

Neumann´s principle

The symmetry operations of any physical property of a crystal must include the symmetry operations of the point group of the crystal

dijk...n =X

p,q,r,...,u

WipWjq...Wnudpqr...u

d0ijk...n =X

p,q,r,...,u

WipWjq...Wnudpqr...u

d0ijk...n = |W |X

p,q,r,...,u

WipWjq...Wnudpqr...u

polar tensor:

axial tensor:

Transformation properties under W ∈ PG

dijk...n = |W |X

p,q,r,...,u

WipWjq...Wnudpqr...u

polar tensor:

axial tensor:

Crystallographic symmetry

Tensor properties of non-magnetic crystals

Simple examples: W=

dijk...n =X

p,q,r,...,u

WipWjq...Wnudpqr...u

dijk...n = |W |X

p,q,r,...,u

WipWjq...Wnudpqr...u

polar tensor:

axial tensor:

-1 0 00 -1 00 0 -1

polar tensors: if n=2k+1⇒ dij...m ⌘ 0

axial tensors: if n=2k ⇒ dij...m ⌘ 0

Tabulations: Nye (1957): Physical Properties of CrystalsBirss (1966): Symmetry and MagnetismSirotin, Shaskolskaya (1979): Fundamentals of Crystal Physics

Intrinsic symmetry

Tensor isomers T 0i1i2...ipk1k2...kp

= Tk1k2...kpi1i2...ip

symmetrization: arithmetic average of all isomers of A

A[ik] =1

2(Aik +Aki)

A{ik} =1

2(Aik �Aki)

A{ijk} =1

6(Aijk +Akij +Ajki �Ajik �Akji �Aikj)

A[ijk] =1

6(Aijk +Akij +Ajki +Ajik +Akji +Aikj)

antisymmetrization: arithmetic average of all isomers of A(+) cyclic permutations (-) non-cyclic permutations

partial symmetrization/antisymmetrization:

Tensor properties of non-magnetic crystals

Bijk = Ai[jk] : Bijk = Bikj

Bijkl = A[ij][kl] : Bijkl = Bjikl = Bijlk = Bjilk

Symmetric polar tensor of rank two

-1 0 00 -1 00 0 -1

-1 0 00 1 00 0 -1

-1 0 00 -1 00 0 1

0 -1 01 0 00 0 1

0 0 11 0 00 1 0

generators

1

1, 2y

1, 2y, 2z

1, 2y, 2z, 4z

1, 2y

, 2z

, 4z

, 3+xxx

Nye notation

pi =X

jk

dijkSjk

Piezoelectric effect

polar tensor of third rank symmetric in the last two indices

polar symmetric tensor of second rankpolar vector

matrix presentation:

electric polarization p produced by mechanical stress S

= "

"(3x6)

pi "

(3x1)

"

" (6x1)

i k 11 22 33 23 13 12

"1 2 3 4 5 6

di↵ S↵

pi =X

↵

di↵S↵

↵ = 1, ..., 6↵ i = 1, 2, 3

Symmetry restrictions on form of piezoelectric tensor

Grimmer, Leuven 2006

Tensor properties of magnetic crystals

M=H+S’Hsubgroup of non-primed

symmetry operationsadditional generator

non-primed symmetry operationsdijk...n =

X

p,q,r,...,u

WipWjq...Wnudpqr...u

dijk...n = |W |X

p,q,r,...,u

WipWjq...Wnudpqr...u

polar tensor:

axial tensor:

additional primed generator S’polar c tensor:

axial c tensor:

magnetic point group

H = {Wi}

|M|/|H| = 2

S’=S1’ 1’(+) i tensor

(-) c tensor

dijk...n = (�1)X

p,q,r,...,u

SipSjq...Snudpqr,...,u

dijk...n = (�1) |S|X

p,q,r,...,u

SipSjq...Snudpqr,...,u

Example: magnetic group 4’22’=222+4’z 222polar tensor of rank 2:

non-primed subgroup 222:

�ij

�11

�22

�33

0 0

0 0

0 0

additional primed generator:

i tensor c tensor

�11

�11

�33

0 0

0 0

0 0

�11

��11

�33

0 0

0 0

0 0

0 -1 0

1 0 0

0 0 1

4z=

Tensor properties of magnetic crystals

�ij =X

pq

WipWjq�pq

�ij =X

pq

SipSjq�pq �ij = (�1)X

pq

SipSjq�pq

1

10

101

n even n odd

i tensor i tensorc tensor c tensor

polar axial polar polar polaraxial axial axial

+

+

+

+

+

+

+

+

+

+ +

+

—

—

—

—

—

—

—

—

—

— —

—

Tensor properties of magnetic crystals

magnetization Ms - axial c tensor of rank 1polarization Ps - polar i tensor of rank 1

Symmetry-adapted forms of

the spontaneous magnetization M

Tensor properties of magnetic crystals

axial c tensor of rank 1

Mi = (�1)|S|X

p

SipMp

Mi = |W |X

p

WipMp

non-primed operations

primed operations

ferromagnetic (pyromagnetic) effect in 31 magnetic point groups

Grey groups: M=G+1’G

c tensors:the form of any tensor in M is identical to that of G

must be null in any grey group

i tensors:

polar i tensors of rank 2k+1: nullaxial i tensors of rank 2k: null

1

10

101

n even n odd

i tensor i tensorc tensor c tensor

polar axial polar polar polaraxial axial axial

+++

++

++

+++ +

+

—

—

—

—

—

—

—

—

—

— —

—

Tensor properties of magnetic crystals

grey groups with 101

Black-white groups:

c tensors:

the form of any tensor in M is identical to that of G=H+1’W’H

i tensors:

axial c tensors of even rank and polar c tensors of odd rank are null for 21 M ∋

1

10

101

n even n odd

i tensor i tensorc tensor c tensor

polar axial polar polar polaraxial axial axial

+++

++

++

+++ +

+

—

—

—

—

—

—

—

—

—

— —

—

Tensor properties of magnetic crystalsM=H+W’H

more complicated relation to classical groups

1polar c tensors of even rank and axial c tensors of odd rank are null for 21 M ∋ 101

Birss, 1966: i and c tensors of ranks up to four

Magnetoelectric effect

Mi =X

j

QijEi

indiced magnetization

applied electric field

Curie, 1894Astrov, 1960

axial c vector polar i tensor

magnetoelectric tensor Q: axial c tensor of rank 2

non-primed symmetry operations Qij = |W |

X

pq

WipWjqQpq

Qij = (�1)|S|X

pq

SipSjqQpqprimed

symmetry operations

no effect in type II (grey) groupsthe effect can occur in 58 type I and III Indenbom, 1960

Birss, 1966

‘inverse’ magnetoelectric effect (magnetically induced)

higher-order magnetoelectric effects

Pi =X

j

(QT )ijHj

Mi ⇠X

j

QijHj +X

kl

RiklHkHl + . . .

linear effect (electrically induced)

Symmetry-adapted form of the magnetoelectric tensor for all magnetic point groups

Bilbao Crystallographic Server

Grimmer, Leuven 2006