Role of Magnetic Symmetry in the Description
and Determination of Magnetic Structures
IUCR Congress Satellite Workshop
14-16 August Hamilton, Canada
MAGNETIC POINT
GROUPS
Mois I. Aroyo
Universidad del Pais Vasco, Bilbao, Spain
Historical Briefs
1929-30 Heesch: 4-dim groups in 3-dim space: 122 anti-symmetry point groups
Shubnikov: re-introduces the concept of ‘anti-symmetry’
1945
1951
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
Birss: tensor properties of crystals with magnetic group symmetry
Opechowski and Guccione: complete listing of the magnetic space groups; OG notation
Koptsik: diagrams of magnetic space groups
1963-4
1965
1966
2001
2009-11
Litvin: corrected OG notation
Litvin: tables of magnetic subperiodic and space groups
Heesch-Shubnikov groups
1st type(32): M=G, 1’∉ M
(classical crystallographic 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)
Black-and-white groups
Example
The group of the square 4mm (C4v)
Symmetry operations
of 4mm:
{e 4z 4z 2z mx my m+ m-}
Stereographic Projections of 4mm
general position
symmetry elements
Black-and-white groups
Example
M=G(H)=H+1’(G-H)
G=4mm:
H1=4:
{e 4z 4z 2z mx my m+ m-}
{e 4z 4z 2z }
Black-and-white group
M=4mm(4) = 4mm : {e 4z 4z 2z mx my m+ m-}
Black-and-white groups
Example
M=G(H)=H+1’(G-H)
G = 4mm:
{e 4z 4z 2z mx my m+ m-}
H = 2m+m-: {e
2z
m+ m-}
Black-and-white group
M=4mm(2mm) = 4mm : {e 4z 4z 2z mx my m+ m-}
Black-and-white groups
Example
M=G(H)=H+1’(G-H)
G = 4mm:
{e 4z 4z 2z mx my m+ m-}
H = 2mxmy: {e
2z mx my}
Black-and-white group
M=4mm(2mm) = 4mm : {e 4z 4z 2z mx my m+ m-}
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 point groups derived from the representations of 4mm(C4v)
magnetic
groups
4mm
4m’m’
4’mm’
4’m’m
——
Indenbom (1959), Bertaut (1968)
Bilbao Crystallographic Server
http://www.cryst.ehu.es
H. Stokes, B.J. Campbell Magnetic Space-group Data
http://stokes.byu.edu/magneticspacegroups.html
D.B. Litvin Magnetic Space Groups v. V3.02 http://www.bk.psu.edu/faculty/litvin/Download.html
Bilbao Crystallographic Server
Magnetic Point Groups
(under development)
coordinate
triplets
axialvector
coefficients
Geometric
interpretation
matrixcolumn
presentation
Bilbao !
Crystallographic !
Server
Curie’s principle
characteristic symmetry of a phenomena
(the invariance group of a phenomena)
the maximum symmetry compatible with the 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
PG ≤ Pphen
PG ≤ Pphen1 ⋂ Pphen2 ⋂ …
ferromagnetism (spontaneous magnetization Ms - axial vector):
Pphen= ∞/m2’/m’
ferroelectricity (spontaneous polarization Ps - polar vector):
Pphen= ∞ m1’
Polar and axial vectors
polar vector
polar
axial
∞m
∞ /m
Marc De Graef
©2009 IUCr"
Transformation of polar and axial vectors under space and time inversion
polar vector
polar
∞ m1’
axial vector
axial
∞/m2’/m’
The groups in red are compatible with both phenomena
Grimmer, Leuven 2006
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’
Marc De Graef
©2009 IUCr"
Tensor properties of non-magnetic crystals
(brief summary)
Tensor representation of physical properties
︷n
Tijk…l (3n components)
physical property
crystallographic
symmetry
intrinsic symmetry
pyroelectric coefficients
pyroelectricity:
electric dipole moment change
electric conductivity:
current
density
∆Pi = pi ∆T
i=1,2,3
electrical conductivity
ji =
X
ij Ej
temperature
change
applied
electric field
j
i,j=1,2,3
piezoelectric modula
piezoelectric effect:
polarization
pi =
X
jk
dijk Sjk
stress tensor
Tensor properties of non-magnetic crystals
Crystallographic symmetry
Transformation properties under W ∈ PG
X
polar tensor: d0ijk...n =
Wip Wjq ...Wnu dpqr...u
p,q,r,...,u
0
d
axial tensor: ijk...n = |W |
Neumann´s principle
X
Wip Wjq ...Wnu dpqr...u
p,q,r,...,u
The symmetry operations of any physical property of a crystal must
include the symmetry operations of the point group of the crystal
X
Wip Wjq ...Wnu dpqr...u
polar tensor: dijk...n =
p,q,r,...,u
axial tensor: dijk...n = |W |
X
p,q,r,...,u
Wip Wjq ...Wnu dpqr...u
Tensor properties of non-magnetic crystals
polar tensor:
dijk...n =
X
Wip Wjq ...Wnu dpqr...u
p,q,r,...,u
axial tensor:
dijk...n = |W |
Simple examples: W=
-1 0 0
0 -1 0
0 0 -1
polar tensors: if n=2k+1
axial tensors: if n=2k
Tabulations:
X
Wip Wjq ...Wnu dpqr...u
p,q,r,...,u
dij...m ⌘ 0
dij...m ⌘ 0
Nye (1957): Physical Properties of Crystals
Birss (1966): Symmetry and Magnetism
Sirotin, Shaskolskaya (1979): Fundamentals of Crystal Physics
Intrinsic symmetry
Tensor isomers
Tensor properties of non-magnetic crystals
0
Ti1 i2 ...ip k1 k2 ...kp
= Tk1 k2 ...kp i1 i2 ...ip
symmetrization: arithmetic average of all isomers of A
1
A[ik] = (Aik + Aki )
2
1
A[ijk] = (Aijk + Akij + Ajki + Ajik + Akji + Aikj )
6
antisymmetrization: arithmetic average of all isomers of A
(+) cyclic permutations
(-) non-cyclic permutations
1
A{ik} = (Aik Aki )
2
1
A{ijk} = (Aijk + Akij + Ajki Ajik Akji Aikj )
6
partial symmetrization/antisymmetrization:
Bijk = Ai[jk] : Bijk = Bikj
Bijkl = A[ij][kl] : Bijkl = Bjikl = Bijlk = Bjilk
Symmetric polar tensor of rank two
generators
-1 0 0
0 -1 0
0 0 -1
¯1
-1 0 0
0 1 0
0 0 -1
¯1, 2y
-1 0 0
0 -1 0
0 0 1
0 -1 0
1 0 0
0 0 1
0 0 1
1 0 0
0 1 0
Nye notation
¯1, 2y , 2z
¯1, 2y , 2z , 4z
¯1, 2y , 2z , 4z , 3+
xxx
Piezoelectric effect
electric polarization p produced by mechanical stress S
polar vector
pi =
X
polar symmetric tensor of second rank
dijk Sjk
jk
polar tensor of third
rank symmetric in the
last two indices
matrix presentation:
i k
11
22
33
23
13
12
↵"
1
2
3
4
5
6
pi =
i = 1, 2, 3
pi
=
"
(3x1)
X
di↵ S↵
↵
↵ = 1, ..., 6
"
di↵
S"↵
"
(3x6)
"
(6x1)
Symmetry restrictions on form of piezoelectric tensor
Grimmer, Leuven 2006
Tensor properties of magnetic crystals
M=H+S’H |M|/|H| = 2
magnetic point group
subgroup of non-primed
H = {Wi}
symmetry operations
additional generator
S’=S1’ 1’
(+) i tensor
(-) c tensor
non-primed symmetry operations X
polar tensor:
dijk...n =
axial tensor:
dijk...n
Wip Wjq ...Wnu dpqr...u
p,q,r,...,u
X
= |W |
Wip Wjq ...Wnu dpqr...u
p,q,r,...,u
additional primed generator S’
polar c tensor:
dijk...n = ( 1)
X
Sip Sjq ...Snu dpqr,...,u
p,q,r,...,u
axial c tensor:
dijk...n = ( 1) |S|
X
p,q,r,...,u
Sip Sjq ...Snu dpqr,...,u
Tensor properties of magnetic crystals
Example: magnetic group 4’22’=222+4’z 222
polar tensor of rank 2: ij
non-primed subgroup 222:
ij
=
X
Wip Wjq
pq
pq
additional primed generator: 4z=
i tensor
ij
=
X
Sip Sjq
pq
pq
0
-1
0
1
0
0
0
0
1
11
0
0
0
22
0
0
0
33
c tensor
ij
= ( 1)
X
Sip Sjq
pq
11
0
0
0
11
0
0
0
33
11
0
11
0
0
0
0
0
33
pq
Tensor properties of magnetic crystals
n even
i tensor c tensor
n odd
i tensor c tensor
polar
axial
polar
axial
polar
axial
polar
¯1
+
—
+
—
—
+
—
+
0
1
+
+
—
—
+
—
—
10 ¯1
+
—
—
+
—
+
+
+
—
magnetization Ms - axial c tensor of rank 1
polarization Ps - polar i tensor of rank 1
axial
Tensor properties of magnetic crystals
Symmetry-adapted forms of the spontaneous magnetization M
axial c tensor of rank 1
non-primed operations
Mi = |W |
primed operations
X
Wip Mp
p
Mi = ( 1)|S|
X
Sip Mp
p
ferromagnetic (pyromagnetic) effect in 31 magnetic point groups
Tensor properties of magnetic crystals
Grey groups: M=G+1’G
n even
i tensor
polar
¯1
+
0
1 +
10 ¯1 +
n odd
c tensor
i tensor
c tensor
axial
polar
axial
polar
axial
polar
axial
—
+
—
—
—
+
+
—
—
+
—
—
—
—
+
—
+
+
+
+
—
c tensors: must be null in any grey group
i tensors: the form of any tensor in M is identical to that of G
0¯
grey groups with 1 1
polar i tensors of rank 2k+1: null
axial i tensors of rank 2k: null
Tensor properties of magnetic crystals
Black-white groups: M=H+W’H
n even
i tensor
polar
¯1
+
0
1 +
10 ¯1 +
n odd
c tensor
i tensor
c tensor
axial
polar
axial
polar
axial
polar
axial
—
+
—
—
—
+
+
—
—
+
—
—
—
—
+
—
+
+
+
+
—
Birss, 1966: i and c tensors of ranks up to four
i tensors: the form of any tensor in M is identical to that of
G=H+1’W’H
c tensors: more complicated relation to classical groups
axial c tensors of even rank and polar c tensors of odd rank are null for 21 M ∋ ¯
1
polar c tensors of even rank and axial c tensors of odd rank are null for 21 M ∋ 10 ¯
1
Magnetoelectric effect
indiced magnetization X
applied
electric field
linear effect
Mi =
Qij Ei
(electrically induced)
j
polar i tensor
axial c vector
magnetoelectric tensor Q:
non-primed Qij
symmetry operations
Curie, 1894
Astrov, 1960
axial c tensor of rank 2
X
= |W |
Wip Wjq Qpq
pq X
primed Q = ( 1)|S|
symmetry operations ij
Sip Sjq Qpq
pq
the effect can occur in 58 type I and III
no effect in type II (grey) groups
Indenbom, 1960
Birss, 1966
X
T
‘inverse’ magnetoelectric
Pi =
(Q )ij Hj
effect (magnetically induced)
Xj
X
higher-order Mi ⇠
Qij Hj +
Rikl Hk Hl + . . .
magnetoelectric effects
j
kl
Bilbao Crystallographic Server
Symmetry-adapted form of the magnetoelectric tensor for
all magnetic point groups
Grimmer, Leuven 2006
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