Role of Magnetic Symmetry in the Description and Determination of Magnetic Structures. IUCR Congress Satellite Workshop August Hamilton, Canada

Transcription

1 Role of Magnetic Symmetry in the Description and Determination of Magnetic Structures IUCR Congress Satellite Workshop August Hamilton, Canada

2 MAGNETIC POINT GROUPS Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain

3 Historical Briefs Heesch: 4-dim groups in 3-dim space: 122 anti-symmetry point groups Shubnikov: re-introduces the concept of anti-symmetry 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 1965 Opechowski and Guccione: complete listing of the magnetic space groups; OG notation 1966 Koptsik: diagrams of magnetic space groups 2001 Litvin: corrected OG notation Litvin: tables of magnetic subperiodic and space groups

4

5 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)

6 Example Black-and-white groups 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

7 Example Black-and-white groups M=G(H)=H1 (G-H) G=4mm: {e 4z 4z 2z mx my m m-} H1=4: {e 4z 4z 2z } Black-and-white group M=4mm(4) = 4mm : {e 4z 4z 2z mx my m m-}

8 Example Black-and-white groups M=G(H)=H1 (G-H) G = 4mm: {e 4z 4z 2z mx my m m-} H = 2mm-: {e 2z m m-} Black-and-white group M=4mm(2mm) = 4mm : {e 4z 4z 2z mx my m m-}

9 Example Black-and-white groups M=G(H)=H1 (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-}

10 Black-andwhite point groups Bradley and Cracknell The mathematical theory of symmetry in solids

11 Magnetic point groups (types I and III) International Tables for Crystallography (2006). Vol. D.! Borovic-Romanov, Grimmer. Chapter 1.5 Magnetic properties

12 Magnetic point groups derived from the representations of 4mm(C4v) magnetic groups 4mm 4m m 4 mm 4 m m Indenbom (1959), Bertaut (1968)

13 Bilbao Crystallographic Server H. Stokes, B.J. Campbell Magnetic Space-group Data D.B. Litvin Magnetic Space Groups v. V3.02

14 Bilbao Crystallographic Server Magnetic Point Groups (under development)

15 coordinate triplets Geometric interpretation axialvector coefficients matrixcolumn presentation Bilbao! Crystallographic! Server

16 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 PG Pphen Pphen1 Pphen2 ferromagnetism (spontaneous magnetization M s - axial vector): Pphen= /m2 /m ferroelectricity (spontaneous polarization P s - polar vector): Pphen= m1

17 Polar and axial vectors polar vector polar axial m /m Marc De Graef 2009 IUCr"

18 Transformation of polar and axial vectors under space and time inversion polar vector axial vector polar m1 axial /m2 /m

19 The groups in red are compatible with both phenomena Grimmer, Leuven 2006

20

21 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"

22 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

23 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"

24 Tensor properties of non-magnetic crystals (brief summary) Tensor representation of physical properties crystallographic symmetry { n intrinsic symmetry physical property Tijk l (3 n components) pyroelectricity: electric conductivity: electric dipole moment change current density pyroelectric coefficients Pi = pi T i=1,2,3 electrical conductivity j i = X j ije j i,j=1,2,3 piezoelectric modula temperature change applied electric field piezoelectric effect: polarization p i = X jk d ijk S jk stress tensor

25 Crystallographic symmetry Neumann s principle Tensor properties of non-magnetic crystals Transformation properties d 0 X under W PG polar tensor: ijk...n = W ip W jq...w nu d pqr...u p,q,r,...,u d 0 X axial tensor: ijk...n = W W ip W jq...w nu d pqr...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 polar tensor: d ijk...n = W ip W jq...w nu d pqr...u axial tensor: d ijk...n = W p,q,r,...,u X p,q,r,...,u W ip W jq...w nu d pqr...u

26 polar tensor: axial tensor: d ijk...n = Tensor properties of non-magnetic crystals X p,q,r,...,u d ijk...n = W W ip W jq...w nu d pqr...u X p,q,r,...,u W ip W jq...w nu d pqr...u Simple examples: W= polar tensors: if n=2k1 d ij...m 0 axial tensors: if n=2k d ij...m 0 Tabulations: Nye (1957): Physical Properties of Crystals Birss (1966): Symmetry and Magnetism Sirotin, Shaskolskaya (1979): Fundamentals of Crystal Physics

27 Intrinsic symmetry Tensor properties of non-magnetic crystals Tensor isomers T 0 i 1 i 2...i p k 1 k 2...k p = T k1 k 2...k p i 1 i 2...i p symmetrization: arithmetic average of all isomers of A A [ik] = 1 2 (A ik A ki ) A [ijk] = 1 6 (A ijk A kij A jki A jik A kji A ikj ) antisymmetrization: arithmetic average of all isomers of A () cyclic permutations A {ik} = 1 (-) non-cyclic permutations 2 (A ik A ki ) A {ijk} = 1 6 (A ijk A kij A jki A jik A kji A ikj ) partial symmetrization/antisymmetrization: B ijk = A i[jk] : B ijk = B ikj B ijkl = A [ij][kl] : B ijkl = B jikl = B ijlk = B jilk

28 Symmetric polar tensor of rank two generators , 2 y , 2 y, 2 z , 2 y, 2 z, 4 z , 2 y, 2 z, 4 z, 3 xxx Nye notation

29 Piezoelectric effect electric polarization p produced by mechanical stress S polar vector p i = X jk d ijk S jk polar symmetric tensor of second rank polar tensor of third rank symmetric in the last two indices matrix presentation: p i = X d i S =1,...,6 i k " i =1, 2, pi " (3x1) d i = " " (3x6) S" " (6x1)

30 Symmetry restrictions on form of piezoelectric tensor Grimmer, Leuven 2006

31 Tensor properties of magnetic crystals magnetic point group subgroup of non-primed symmetry operations additional generator non-primed symmetry operations polar tensor: axial tensor: d ijk...n = M=HS H X p,q,r,...,u d ijk...n = W additional primed generator S polar c tensor: axial c tensor: d ijk...n =( 1) d ijk...n =( H = {Wi} S =S1 1 W ip W jq...w nu d pqr...u X p,q,r,...,u 1) S X p,q,r,...,u X p,q,r,...,u M / H = 2 () i tensor (-) c tensor W ip W jq...w nu d pqr...u S ip S jq...s nu d pqr,...,u S ip S jq...s nu d pqr,...,u

32 Tensor properties of magnetic crystals Example: magnetic group 4 22 =2224 z 222 polar tensor of rank 2: ij non-primed subgroup 222: ij = X pq W ip W jq pq additional primed generator: 4z= i tensor c tensor X ij = S ip S X jq pq ij =( 1) S ip S jq pq pq pq

33 Tensor properties of magnetic crystals n even n odd i tensor c tensor i tensor c tensor polar axial polar axial polar axial polar axial magnetization M s - axial c tensor of rank 1 polarization P s - polar i tensor of rank 1

34 Tensor properties of magnetic crystals Symmetry-adapted forms of the spontaneous magnetization M axial c tensor of rank 1 non-primed operations M i = W X p W ip M p primed operations M i =( 1) S X p S ip M p ferromagnetic (pyromagnetic) effect in 31 magnetic point groups

35 Grey groups: M=G1 G Tensor properties of magnetic crystals i tensor n even c tensor i tensor n odd c tensor polar axial polar axial polar axial polar axial c tensors: i tensors: must be null in any grey group the form of any tensor in M is identical to that of G grey groups with polar i tensors of rank 2k1: null axial i tensors of rank 2k: null

36 Black-white groups: M=HW H Tensor properties of magnetic crystals i tensor n even c tensor i tensor n odd c tensor polar axial polar axial polar axial polar axial Birss, 1966: i and c tensors of ranks up to four i tensors: c tensors: the form of any tensor in M is identical to that of G=H1 W H 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 1 0 1

37 Magnetoelectric effect indiced magnetization linear effect (electrically induced) axial c vector M i = X j Q ij E i applied electric field polar i tensor Curie, 1894 Astrov, 1960 magnetoelectric tensor Q: axial c tensor of rank 2 non-primed symmetry operations primed symmetry operations X Q ij = W W ip W jq Q pq pq X Q ij =( 1) S S ip S jq Q pq pq the effect can occur in 58 type I and III Indenbom, 1960 no effect in type II (grey) groups Birss, 1966 inverse magnetoelectric effect (magnetically induced) higher-order magnetoelectric effects P i = X j M i X j (Q T ) ij H j Q ij H j X kl R ikl H k H l...

38 Bilbao Crystallographic Server Symmetry-adapted form of the magnetoelectric tensor for all magnetic point groups

39 Grimmer, Leuven 2006