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

3 Magnetic Space Groups (1651) OG Superfamily of G BNS Superfamily of G 1. Space group G (230) II. Grey group G+G (230) III. Black-and-white groups H+(G-H) (674) H: a t-subgroup of G of index 2 IV. Magnetic groups with black-and-white lattices (517) 1 1 H+(G-H) 1 H: a k-subgroup of index 2 1 G+Gt t TG, t 2 TG

4 Space groups G Coset decomposition G:TG (I,0) (W2,w2)... (Wm,wm)... (Wi,wi) (I,t1) (W2,w2+t1)... (Wm,wm+t1)... (Wi,wi+t1) (I,t2) (W2,w2+t2)... (Wm,wm+t2)... (Wi,wi+t2) (I,tj) (W2,w2+tj)... (Wm,wm+tj)... (Wi,wi+tj) Factor group G/TG isomorphic to the point group PG of G Point group PG = {I, W1, W2,...,Wi}

5 Grey Magnetic Space groups: G 1 = G+G1 1 Coset decomposition G :TG (I,0) (W2,w2)... (Wm,wm)... (I,0) (W2,w2),..., (I,t1) (W2,w2+t1)... (Wm,wm+t1)... (I,t1) (W2,w2+t1),..., (I,t2) (W2,w2+t2)... (Wm,wm+t2)... (I,t2) (W2,w2+t2),..., (I,tj) (W2,w2+tj)... (Wm,wm+tj)... (I,tj) (W2,w2+tj),..., Factor group G /TG isomorphic to the point group PG of G Point group PG = {I, W2,...,Wi,..., I, W2,...,Wi,... }

6 Example: P222 1 P2221 =P P222 Coset decomposition G :TG TG TG2z... 1 TG1 1 TG2z (I,0) (2z,0)... (I,0) (2z,0) Factor group G /TG PG (I,t1) (2z,t1)... (I, t1) (2z,t1) PG = {1, 2z, 2y, 2x,, 2z, 2y, 2x } (I,t2) (2z,t2)... (I, t2) (2z,t2) (I,tj) (2z,tj)... (I, tj) (2z,tj) International Tables Vol. A D.Litvin, Magnetic Group Tables, 2011

7 Type III Black-and-white Magnetic Space Groups 674 Magnetic Space-group types of Type III G(H) = H+(G-H) 1 H: t-subgroup of G of index 2 PG>PH, TG=TH

8 Example: P12/m1 Coset decomposition G:TG Factor group G/TG PG PG = {1, 2, 1, m} TG TG 2 1 TG TG m 1 (I,0) (2,0) (,0) (m,0) 1 (I,t1) (2,t1) (, t1) (m, t1) 1 (I,t2) (2,t2) (, t2) (m,t2) (I,tj) (2,tj) (, tj) (m, tj) inversion centres (1,t): n n n 1 at n n n

9 Subgroups of space groups Translationengleche subgroups H<G: Example: P2/m { TH= TG PH<PG TG TG 2 1 TG TG m Coset decomposition 1 (I,0) (2,0) (,0) (m,0) 1 (I,t1) (2,t1) (, t1) (m, t1) 1 (I,t2) (2,t2) (, t2) (m,t2) (I,tj) (2,tj) (, tj) (m, tj) t-subgroups: H1=TG TG2 P2 H2=TG TG 1 H3=TG TG m Pm

10 Black-and-white Magnetic Space Groups G(H) = H+(G-H) 1 Type III (674 types): H is a t-subgroup, TG=TH Coset decomposition G(H) :TG (I,0) (W2,w2)... (Wm,wm)... (I,0) (W2,w2),..., (I,t1) (W2,w2+t1)... (Wm,wm+t1)... (I,t1) (W2,w2+t1),..., (I,t2) (W2,w2+t2)... (Wm,wm+t2)... (I,t2) (W2,w2+t2),..., (I,tj) (W2,w2+tj)... (Wm,wm+tj)... (I,tj) (W2,w2+tj),..., Factor group G(H)/TG isomorphic to the point group PG(PH) Point group PG(PH) = {I,...,Wm,..., W2,...,Wk,... }

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12 Derivation of Type III Magnetic space groups by irreducible representations at k=0 Indenbom (1959), Bertaut (1968)

13 Derivation of Type III Magnetic space groups by irreducible representations at k=0 Example: Pnma (62) OG! No Indenbom (1959), Bertaut (1968)

14 Type IV Black-and-white Magnetic Space Groups 517 Magnetic Space-group types of Type IV G(H) = H+(G-H) 1 H: k-subgroup of G of index 2 PG=PH, TG>TH

15 Klassengleiche subgroups H<G: Example: P1 { Subgroups of space groups TH< TG PH=PG t=ua+vb+wc Coset decomposition Te={t(u=2n,v,w)} Te (I,0) (I,t1) Te ta (I,ta) (I,t1+ta) ta(a,0,0) (I,t2) (I,t2+ta) (I,tj) (I,tj+ta) isomorphic k-subgroups: P1(2a,b,c) H=Te

16 Subgroups of space groups Klassengleiche subgroups H<G: non-isomorphic { TH< TG PH=PG Example: C2 Coset decomposition ti=integer tc=1/2,1/2,0 Ti Titc Ti 2 Titc 2 (I,0) (I,tc) (2, 0) (2,tc) (I,t1) (I,t1+tc) (2, t1) (2, t1+tc) (I,t2) (I,t2+tc) (2, t2) (2, t2+tc) (I,tj) (I,tj+tc) (2, tj) (2, tj+tc) k-subgroups: H1=Ti Ti2 H2=Ti Titc 2 P2 P21

17 Black-and-white Magnetic Space Groups G(H) = H+(G-H) 1 Type IV (517 types): H is a k-subgroup, PG=PH Coset decomposition G(H) :TH tc=(0,0,1) 2tc=(0,0,2) (1,2tc) (I,0) (I,t1) (I,t2) (I,tj) (1,tc)1 (I,tc) (I,t1+tc) (I,t2+tc) (I,tj+tc) (W2,2tc) (W2,tc) (W2,0) (W2,tc) Factor group G(H)/TH isomorphic to the point group PG1 1 Point group PG = {I, W2,...,Wi,..., I, W2,...,Wi,... }

18 The black-and-white Bravais lattice 3dim: 36 Bravais lattices

19 WYCKOFF POSITIONS OF MAGNETIC SPACE GROUPS

20 Wyckoff positions of Magnetic groups The distribution of points in direct space into orbits with respect to a Shubnikov magnetic group follows directly from the distribution of points of its isomorphic space group The Wyckoff positions of Shubnikov magnetic groups are derived from the Wyckoff positions of their isomorphic space groups The space group isomorphic to a Shubnikov magnetic group is obtained by changing the primed elements into non-primed

21 Magnetic space group P2s-1 OG description Isomorphic space group ITA description of P-1

22 Magnetic space group P2s-1 OG ITA Note: unit cell (a,b,2c): t(001), t(002) WP multiplicities doubled

23 Wyckoff positions site-symmetry groups, multiplicities, unit cell representatives Group P2s-1 General position: 4i 1 (x,y,z mx,my,mz) (-x,-y,-z mx,my,mz) ( ) (x,y,z+1 -mx,-my,-mz) m x m y m z = m -x m -y m -z (-x,-y,-z+1 -mx,-my,-mz) ( ) m m m x y z = -m -m -m -x -y -z+1 Site symmetry: S={(W,w): (W,w)(Xo)= (Xo)} Si={(1,0)} Axial-vector field: Mi={mx,my,mz} S {1}

24 Wyckoff positions site-symmetry groups, multiplicities, unit cell representatives 1 Special position: 2f Site symmetry: Group P2s-1 Sf={(W,w): (W,w)(Xf)= (Xf)} (1/2,0,1/2 0,0,0) (-1/2,0,-1/2 0,0,0) ( ) Sf={(1,0), /2 (-1,101)Xf = Xf} 1/2 = Sf {1, -1 } /2 1/2 isomorphic Axial-vector field: Mf: (W,w)f(Mf Xf)= (Mf Xf), (W,w)f Sf} ( ) 1 m 1/2 0 m 0 1 m 1/2 = -m -m -m 1/2 0 1/2 m m m 1/2 0 1/2 = /2 0 1/2

25 pages of crystallographic data

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27 Survey of Magnetic Group Types Extensive tabulations of magnetic point groups magnetic subperiodic and space groups D. Litvin (2011). Magnetic space groups

28 D. Litvin (2011). Magnetic space groups

29 GENERAL LAYOUT: LEFT-HAND PAGE Lattice diagram Headline block Symmetry-element! diagram General-position! diagram Origin Asymmetric unit Symmetry operations! block D. Litvin (2011). Magnetic space groups

30 Lattice diagram D. Litvin (2011). Magnetic space groups

31 Headline block Headline of the magnetic space group P4/m mm Short Hermann- Mauguin symbol Crystal class (point group) Number of magnetic space group Full Hermann- Mauguin symbol Crystal system D. Litvin (2011). Magnetic space groups

32 Diagrams General-position diagram and symmetry-elements diagram of the! magnetic space group P2bm ma D. Litvin (2011). Magnetic space groups

33 Symmetry operations Symmetry-operations block of! magnetic space group P2bmma D. Litvin (2011). Magnetic space groups

34 Symmetry operations General-position block of! magnetic space group P2bmma D. Litvin (2011). Magnetic space groups

35 General Layout: Right-hand page Abbreviated headline Generators selected General and special! Wyckoff positions! with data on: Multiplicity Wyckoff letter Site symmetry Coordinates Magnetic moment! coefficients Special projections D. Litvin (2011). Magnetic space groups

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

37 Transformation of the basis Example MGENPOS

38 Example MGENPOS: Magnetic Superfamilies of P1 BNS Superfamily of G OG Superfamily of G

39 Example MGENPOS: PS-1 (BNS) or P2s-1(OG) OG Symmetry operations BNS Matrix-column presentation Seitz notation Geometric interpretation

40 Example MWYCKPOS: P2s-1(OG)

41 BNS AND OG DESCRIPTIONS OF TYPE IV MAGNETIC GROUPS

42 Type IV Magnetic Space Groups (OG) G(H) = H+(G-H) 1 H is a k-subgroup, PG=PH TG>TH (index 2) Coset decomposition G(H) :TH tc=(0,0,1) 2tc=(0,0,2) (1,2tc) (I,0) (I,t1) (I,t2) (I,tj) (1,tc)1 (I,tc) (I,t1+tc) (I,t2+tc) (I,tj+tc) (W2,2tc) (W2,tc) (W2,0) (W2,tc) Factor group G(H)/TH isomorphic to the point group PG1 1 Point group PG = {I, W2,...,Wi,..., I, W2,...,Wi,... }

43 Type IV Magnetic Space Groups (OG) G(H) = H+(G-H) 1 H is a k-subgroup, PG=PH TG>TH (index 2) OG symbols of Type IV Magnetic Space groups HM symbol of the magnetic group Gs indicates the sublattice of unprimed translations Example: P2s1 G=P1(a,b,c) H=P1(a,b,2c)

44 1 TG TG(W2,w2) TGt (I,0) (I,t1) (I,t2) Type IV Magnetic Space Groups (BNS) S(G)=G+Gt1 S(G) is a k-supergroup of G, PS=PG TS>TG(index 2) Coset decomposition S(G) :TG (W2,w2) (W2,w2+t1) isomorphic to the point group PG1 Point group 1 PG = {I, W2,...,Wi,..., I, W2,...,Wi,... } Factor group S(G)/TG......(I,t) (I,t1+t) (I,t2+t) 1 1 TG t1 (W2,w2) t1 (W2,w2) t TG t 2 TG t1 (W2,w2+t1) (I,tj) (W2,w2+tj) (I,tj+t) 1 t1 (W2,w2+tj)

45 Type IV Magnetic Space Groups (BNS) S(G)=G+Gt1 BNS symbols of Type IV Magnetic Space groups HM symbol of the monochrome subgroup Gt additional primed translations symbol a, b or c t a/2, b/2 or c/2 A, B or C 1/2(b+c), 1/2(a+c) or 1/2(a+b) I 1/2(a+b+c)

46 OG and BNS symbols for the black-and-white lattices VH=2 n VG Grimmer, Acta Cryst. 66, 284 (2010)

47 Example 1: Superfamily of P-1 OG BNS P-1 P-11 P-1 Minimal k-supergroups of P-1

48 data BCS: Ps-1 (P2s-1) Example 1: PS-1 (BNS) or P2s-1(OG) OG (description with respect to the group G cell) P=(a,b,2c) BNS (description with respect to the subgroup H cell)

49 Magnetic space group P2s-1 D. Litvin (2011). Magnetic space groups

50 Magnetic space group Ps-1 c a Koptsik: Shubnikovskie grupi, MGU, Moskva,1966

51 Example 2: Type IV magnetic space groups of Cmme(67) OG BNS

52 Type IV magnetic space groups of Cmme(67) OG

53 Type IV magnetic space groups of Cmme(67) Minimal supergroups of Cmme (67) BNS

54 Superfamily 67 Criticism to the OG symbolism Symbols of Type IV groups different centering types no primes the same centering types primes distinguish different groups

55 Criticism to the OG symbolism Cmma (67) International Tables, vol. A (2002) CPmma ( ) D. Litvin (2011). Magnetic space groups

56 ! EXAMPLES OF BNS AND OG DESCRIPTIONS OF TYPE IV MAGNETIC GROUPS

57 Type IV Magnetic Groups Example 1: PP4 (OG) or PC4(BNS) OG BNS Superfamily of P4 magnetic space groups (OG) Litvin, Magnetic groups, 2011

58 Magnetic space group PP4 Litvin, Magnetic groups, 2011

59 OG General Positions PP4 BCS, Magnetic groups

60 Superfamily of magnetic groups of P4 Litvin, Magnetic groups, 2011

61 Maximal subgroups of P4 ITA1, Subgroups of space groups

62 Maximal subgroups of P4 Family of magnetic groups of P4

63 Magnetic space group PP4 translation subgroup of P4 T(a,b,c) T(2a,2b,c) t(110) T(2a,2b,c) t(100) T(2a,2b,c) t(010) T(2a,2b,c) C(2a,2b,c) k-subgroup t(100)c(2a,2b,c) magnetic group PC4=C(2a,2b)4 t(100)c(2a,2b)4

64 Magnetic space group PP4 Why PP4 and not PC4? 1/2a+1/2b, -1/2a+1/2b,c C4 P4 b a a b

65 Magnetic space group PP4 a 2a Proposal: Complete description: unit cell (2a, 2b, c) b (0,0,0)+ (1,1,0)+ (1,0,0)+ (0,1,0)+ 2b 16 d 1

66 BCS Magnetic groups

67 Magnetic space group PP4 Special Wyckoff positions Litvin, Magnetic groups, 2011 ITA data P4 Proposal: Change of multiplicities 8 c 2.. 0,1/2,z [u,v,0] 1/2,0,z [u,-v,0] 4 b 4.. 1/2,1/2,z [0,0,0] 4 a 4.. 0,0,z [0,0,w]

68 BNS General Positions PC4 BCS, Magnetic groups

69 BNS and OG descriptions BNS OG a 2a a 2a -a+b b a+b b 2b 2b PC4 P(a+b,-a+b,c) P=(a+b,-a+b,c) PC4(PP4) C(2a,2b,c)

70 BNS and OG descriptions T(2a,2b)4 t(110)t(2a,2b)4 t(100)t(2a,2b)4 t(010)t(2a,2b)4 OG: PP4=C(2a,2b)4 t(100)c(2a,2b)4 a=a+b, b=-a+b,c BNS: PC4=T(a,b )4 t(½½0)t(a,b )4

71 BNS OG P=(a-b,a+b,c)

72 BNS: Wyckoff Positions PC4 BCS, Magnetic groups ITA data P4

73 Wyckoff positions site-symmetry groups, multiplicities, unit cell representatives Special position: 4c 2.. Site symmetry: Group PC4 (3/4,1/4,0 m x,my,0) (1/4,3/4,0 -mx,-my,0) Sf={(W,w): (W,w)(Xf)= (Xf)} (1/4,1/4,0 my,-mx,0) (3/4,3/4,0 -my,mx,0) ( ) /2 1/2 0 3/4 1/4 0 = 3/4 1/4 0 Sf={(1,0), (2, 3/2 1/2 0)Xf = Xf} Sf {1, 2 } isomorphic Axial-vector field: Mf: (W,w)f(Mf Xf)= (Mf Xf), (W,w)f Sf} ( ) 3/2 m 3/4 1/2 m 1/4 0 m 0 = m m -m 3/4 1/4 0 m m m 3/4 1/4 0 = m m 0 3/4 1/4 0