Representational analysis of magnetic structures Branton J. Campbell Harold T. Stokes Department of Physics & Astronomy Brigham Young University New Trends in Magnetic Structure Determination Institute Laue Langevin Grenoble France, 12-16 December 2016
Applications of Representation Analysis Molecular vibrations Hybridized and molecular orbitals Crystal-field splitting Electronic-transition selection rules Crystal band structure Landau theory of phase transitions Parameterize crystal distortions magnetic displacive occupational lattice strain
Group representations Discovered by Ferdinand Frobenius (Germany, 1897). Point group: 222 I 2 x 2 y 2 z I 1 0 2 0 1 I I 2 x 2 y 2 z x 1 0 0 1 2 2 x 2 x I 2 z 2 y y 1 0 1 0 2 y 2 y 2 z I 2 x 2 1 0 z 2 z 2 z 2 y 2 x I 0 1 Representations map group elements onto matrices that obey the same multiplication table as the group. 1 0 1 0 1 0 2 x 2 y = 2 z 0 1 0 1 0 1
Irreducible representations (irreps) � 2 � → 1 2 � → 1 � �1: 1 → 1 2 � → 1 � 2 � → 1 � �2: 1 → 1 2 � → 1 2 � → 1 � � �3: 1 → 1 0 1 2 � → 1 0 1 2 � → 1 0 � 2 � → 1 0 � 0 0 1 0 0 1 �3 � �1 ⊕ �2 � �1 0 Reducible representation: 0 �2 Irreducible representations can’t be separated into smaller pieces! Irreps are recipes for symmetry breaking ! michaeldepippo.com
Wonderful orthogonality theorem (WOT) � � � � � � � � � 1 1 1 1 � � 1 �1 �1 1 � � 1 1 �1 �1 Beautiful Computable � � 1 �1 1 �1 Issai Schur Orthogonality and completeness relations (1904-07) Irreps provide a symmetry-based coordinate system (parameter set) for describing deviations from symmetry.
A rose by any other name … Irrep basis function Irrep mode Order parameter component Symmetry-adapted basis function Symmetry mode Distortion modes (too vague) Normal modes (superposition of modes of same irrep)
Familiar symmetry modes Action of point group = 222 on � � orbital 1 2 � 2 � � z z z z � + y y y y x x x x � � � � � � � Under the group operations, 1 1 1 1 � � a � � orbital transforms like � � 1 �1 �1 1 � � which irrep? � � 1 1 �1 �1 � � � � � � 1 �1 1 �1
Familiar symmetry modes Action of point group = 222 on � �� orbital 1 2 � 2 � � z z z z y y y y x x x x � � � � � � � Under the group operations, 1 1 1 1 � � � � � �� � , � � � a � �� orbital transforms like � � 1 �1 �1 1 � � , � �� which irrep? � � 1 1 �1 �1 � � , � �� � � 1 �1 1 �1 � � , � ��
Familiar symmetry modes Spherical � � 0 harmonics! Irreps of the symmetry group � � 1 of a sphere: O(3) � � 2 � � 3 Irreps of the translational group of a periodic signal Fourier harmonics!
Irrep � of 2D magnetic irrep � �̅� � � �� 1 � � � � � 2 � 4 � 4 � � � � � 1 1 1 0 0 � 0 1 0 0 1 � 1 0 1 1 0 0 0 1 0 0 1 � � 0 0 1 1 1 0 0 1 1 1 0 � � � � 1 1 0 � 1 0 0 0 1 0 0 1 1 0 � 1 0 1 0 1 0 0 1 � � 0 1 0 1 1 0 0 1 0 1 1 � � �� 1 � � �̅� � ′ � �� ′ � � ′ 2 � � � ′ 4 � 4 � Multiply the matrix of an unprimed operator by 1 to obtain the matrix of the corresponding primed operator.
Irrep � of 2D magnetic irrep � is a 90° CW rotation 4 � 0 1 0 � � � � 0 1 � � � �� � 4 � 4 � 1 0 0 1 0 0 0 1 In 3D real space: � � 0 1 0 � � �̅ � 1 0 0 � � 0 0 1 � In 2D carrier space: � � � 0 1 � � � 1 0 �
Irrep � of 2D magnetic irrep � � 1 � 1 � � ′ � �� � � � 4 � � �� � � ′ 4 � � 2 � 2 � �� � 4 � � � ′ � �̅� � � 4 � � �̅� ′ � � , � � � � � � � � 1 0 1 1 0 � 0 1 0 0 1 � 1 1 1 0 0 0 0 1 0 0 1 � � 0 0 1 1 1 0 0 1 1 1 0 � � � � � � �̅ �̅ � � � � � � � �̅ � � � �̅
Distortion space For a given subgroup symmetry and cell size, the collection of all variable structural parameters spans a vector space that contains all possible distortions. �4��1’ general � �� � �� � �� � �� � �� � �� � �� � �� 2 � 2 supercell parent 4 unique atoms in supercell → 8 structural parameters in the xy plane
Symmetry modes: a new parameter set � � �0,0,0� � � �½, ½, 0� �Γ � ��, �� �� � ��, �� � � 2 �2′ � � ½, 0,0 � � ½, 0,0 � � �0, ½, 0� � � �0, ½, 0� �� � ��, �� �� � ��, �� � � ��2 � � ��2 Symmetry modes yield an orthogonal basis for distortion space.
Symmetry modes: a new parameter set � � �0,0,0� � � �½, ½, 0� �Γ � ��, �� �� � ��, �� � � 2 �2′ � � ½, 0,0 � � ½, 0,0 � � �0, ½, 0� � � �0, ½, 0� �� � ��, �� �� � ��, �� � � ��2 � � ��2 Symmetry modes yield an orthogonal basis for distortion space.
Displacive/magnetic/occupancy/strain La 2 CoRuO 6 Displacive � 1/2, 1/2, 0 M � � 1/2, 0, 0 X � � 1/4, 1/4, 1/4 R � Site order (Co/Ru) � 1/2, 1/2, 1/2 � � Magnetic (Co) 1/4, 1/4, 1/4 �Λ � J.W. Bos and J.P. Attfield, J. Mater. Chem. 15, 715–720 (2005).
Non-magnetic case: WO 3 Describe the structures of 1 � 1 � 1 Pm 3 m each of the phases with a Symmetry common parameter set. relationships Demonstrates generality. � ��, 0,0� � � 2 � 2 � 1 Phase P 4/ nmm transitions 1173 K � �, 0,0 � � Irreps/OPDs 2 � 2 � 2 � ��, 0,0� P 4/ ncc � � � �, �, 0 � � � �, 0,0 � ��, 0,0� � � � � 993 K � �0,0,0,0, �, � Pbcn P 2 1 / c � ��, �, �� � � �� � � 2 � 2 � 2 2 � 2 � 2 � �, 0,0 623 K � � � �, �, 0 � � � ��, �, 0� � � � �, �, � 2 � 2 � 2 2 � 2 � 2 � � P 2 1 / n Pc �� � �0,0, �, �, �, � � � � ��, �, �� à � � �, �, 0 � � 290 K � ��, �, �� � � � �0,0, �, �, �, �� � � 2 � 2 � 2 P 1
Traditional vs symmetry-mode parameters One mode can affect many symmetry-distinct atoms. One atom can be affected by many modes. Symmetry-modes span the same configurational space as traditional coordinates if all relevant k -points, irreps, and OPD components are considered simultaneously. Number of free variables is conserved! The relationship between traditional and symmetry-mode coordinate systems is linear! Related by a square numerical invertible matrix derived from group representation theory. Symmetry modes often provide the more natural/efficient basis. Even complicated magnetic structures are often described by the modes of a single irrep!
A recipe for symmetry breaking Example: �� � irrep of �4��1 � 1 � 1 � � �� � ′ � � � 4 � � �� � � ′ 4 � � 2 � 2 � �� � � � ′ � �̅� 4 � � � 4 � � �̅� ′ � � , � � � � � 1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � � � � � � 1 0 1 1 0 � 0 1 0 0 1 � 1 0 1 1 0 0 0 1 0 0 1 � � 0 0 1 1 1 0 0 1 1 1 0 Find the group elements whose matrices leave some vector invariant. � � � � ⇒ �� � , � � � ⇒ �� � � 0 ⇒ �� � , � � � � � � ��2 � � 2 � � ��2 The vector used is called the order parameter direction or OPD. The resulting symmetry is called an isotropy subgroup of the parent.
A recipe for symmetry breaking Example: �� � irrep of �4��1 � 1 � 1 � � �� � � ′ � �� � � ′ � � 4 � 4 � � 2 � �� � � � � � ′ 2 � � �̅� 4 � 4 � � �̅� ′ � � , � � , � � � 1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � � � � � � 1 0 1 1 0 � 0 1 0 0 1 � 1 0 1 1 0 0 0 0 0 1 1 � � 0 0 1 1 1 0 0 1 1 1 0 Find the group elements whose matrices leave some vector invariant. � � � � ⇒ �� � , � � � ⇒ �� � � 0 ⇒ �� � , � � � � �� � �2′ ��′�2′ �2′ The vector used is called the order parameter direction or OPD. The resulting symmetry is called an isotropy subgroup of the parent.
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