Introduction to magnetic symmetry III. Magnetic space groups vs. Iirreps J. Manuel Perez-Mato Facultad de Ciencia y Tecnología Universidad del País Vasco, UPV-EHU BILBAO, SPAIN
Distortions transforming according to irreducible representations of the symmetry group of the undistorted structure Simplest example: some operations keep it invariant some change its sign Distortion in the structure Distortion after application of g i g i d 1 -d 1 d 1 g i -d 1 Q = Q 1 d 1 Q ‘= Q’ 1 d 1 Irreducible representation T(g i ) = -1 of G (irrep) T(g i ) Q = Q ’ (matrices)
Distortions transforming according to irreducible representations of the symmetry group of the undistorted structure example: some operations transform the distortion into another independent one, or in a linear combination with it Distortion in the structure Distortion after application of g i g i d 1 d 2 d 1 g i d 2 Q = Q 1 d 1 +Q 2 d 2 Q ‘= Q’ 1 d 1 +Q’ 2 d 2 Irreducible representation 0 1 T(g i ) = of G (irrep) T(g i ) Q = Q ’ 1 0 (matrices)
Phase Transition / Symmetry break / Order Parameter High symmetry group G o 1’ = {g i } Key concept of a symmetry break: order parameter Distortion in the structure Distortion after application of g i g i Q Q’ Q = Q 1 d 1 + … +Q n d n Q ‘= Q’ 1 d 1 + … +Q’ n d n T(g) Q = Q ’ Irreducible representation T(g) : one nxn matrix for each operation g of G of G (irrep) (matrices) distortions: Vectors in a multidimensional space
Phase Transition / Symmetry break / Order Parameter group-subgroup relation: High symmetry group G = {g} G F F: isotropy subgroup High symmetry Low symmetry T(g) Q = Q For special Irreducible directions of representation g belongs to F Q . F of higher of G (irrep) symmetry: (matrices) epikernels T(g) Q = Q ’ ≠ Q For general Q 2 direction of Q , the lowest g does not belong to F: Q ’ equivalent F: kernel but distinguishable state (domain) amplitude Q 1 Order parameter Q = (Q 1 ,Q 2 ) = ρ (a 1 ,a 2 ) Key concept of a symmetry break a 1 2 +a 2 2 =1
epikernels of the irrep, depending on isotropy subgroups: the direction (a,a, … ) ,(a,0, … ), Invariance equation : etc … a ( R, θ | t ) is a T[( R, θ | t )] = b b conserved .. .. by the magnetic kernel of the irrep: .. arrangement .. nxn matrix of irrep operations represented by the unit matrix. MSG kept by any direction (a,b, … )
Single irrep assignment vs. magnetic space groups (MSG) in commensurate structures. Cases 1) 1-dim. irrep: irrep and MSG assignment are equivalent for defining the constraints on the atomic magn. moments
Description in terms of irreps (Irrep = irreducible representation) Pn’ma’ === one irrep k=0 1’ -1 Pnma -1 Pn’m’a 2 z ’ 2 y ’ 2 x ’ -1’ m z ’ m y ’ m x ’ -1 1 -1 1 -1 1 -1 1 Pn’ma’ -1 Pnm’a’ -1 Pn’m’a’ -1 Pnma’ -1 Pnm’a -1 Pn’ma
Example: parent space group Pnma (Pnma1’) k=0 8 possible irreps, all 1-dim obtained with k-SUBGROUPSMAG One to one correspondence between each irrep and one MSG
Irrep decomposition of the magnetic degrees of freedom obtained with k-SUBGROUPSMAG
subgroups allowing non-zero magnetic moment at site 4b are coloured Only non-zero moments for the MSGs associated with the irreps present in the magnetic representation mGM4+ mGM3+ mGM2+ mGM1+
Space Group: Pn'ma' irrep basis spin modes La equivalent to Wyckoff Mn position constraints F y mode along y G z mode along z A x mode along x weak ferromagnet
Single irrep assignment vs. magnetic space groups (MSG) in commensurate structures. Cases 1) 1-dim. irrep: irrep and MSG assignment are equivalent for spin relations. It includes the case of 1k-structures witth k ≠ 0 and –k equivalent to k, and the small irrep active being 1-dim
1k magn. structure with -k equiv. to k and small irrep 1-dim: MSG and irrep assignment equivalent for spin constraints Paramagnetic symmetry: P6 3 mc1’ ErAuGe k =(1/2,0,0) (point M in the BZ) Magnetic phase symmetry: P C na21 (#33.154) Magndata 1.33 irreps mM i irrep star: 3 k dim. extended small irrep: 1 mM1 mM3 mM4 mM2 dim. full irrep: 3 One to one correspondence MSG : irrep However it is convenient to know that the magnetic point group is mm21 ’ … . a nd the effective space group for atomic positions in case of magnetostructural non-negligible effects is: Cmc2 1 … .
Single irrep assignment vs. magnetic space groups (MSG) in commensurate structures. Cases 1) 1-dim. irrep: irrep and MSG assignment are equivalent for spin relations . 2) N dim. irrep, N>1: several MSG (epikernels or isotropy subgroups of the irrep) are possible for the same irrep. The MSG depends on the way the spin basis functions are combined. The assignment of a MSG restricts the magnetic configuration beyond the restrictions coming from the irrep.
Single irrep assignment vs. magnetic space groups (MSG) in commensurate structures. Cases 1) 1-dim. irrep: irrep and MSG assignment are equivalent for spin relations. 2) N dim. irrep, N>1: several MSG (epikernels or isotropy subgroups of the irrep) are possible for the same irrep. The assignment of a MSG restricts the magnetic configuration beyond the restrictions coming from the irrep. case 2.1: The MSG is a k-maximal subgroup: it only allows a spin ordering according to a single irrep (further restricted to fulfill the MSG constraints). No other irrep arrangements are compatible with the MSG.
Parent space group: P4 2 /mnm (N. 136) Propagation vector: k = (0,0,0) Magnetic site: Cr 4e (0,0,z)
K-SUBGROUPSMAG: maximal subgroups
coloured groups obtained allow non-zero with MAXMAGN magnetic or moment at (at k-SUBGROUPSMAG least some) & MAGMODELIZE atoms from the parent site 4e (0,0,z)
two 2-dim irreps obtained with k-SUBGROUPSMAG four 1-dim irreps
k-maximal subgroups = irrep epikernels four 1-dim irreps one irrep <> one MSG two 2-dim irreps irrep epikernels: four MSGs
Programs that determine the epikernels and kernel of any irrep, and produce magnetic structural models complying with them. Program for mode analysis: http://stokes.byu.edu/iso/isotropy.php Stokes & Campbell, Provo Both programs also support incommensurate cases, deriving epikernels and kernel of the irreps in the form of MSSGs, and corresponding magnetic models Program for structure refinement: http://jana.fzu.cz/ V. Petricek, Prague
Programs that determine the epikernels and kernel of any irrep, and produce magnetic structural models complying with them. filter by irreps
Output of option “Get irreps” of k-subgroupsmag for epikernel Cmm’m’: special fixed direction, 1 d. freedom per each irrep only 1 magn. irrep mGM5+ in the irrep decomposition
Epikernels of the two possible 2-dim irreps obtained with k-SUBGROUPSMAG Irrep mGM5+ : Irrep mGM5-: 2 d.f. 2 d.f. 2 d.f. 2 d.f. 2 basis f. 2 basis f. 2 basis f. 2 basis f. 4 degrees of freed. 4 degrees of freed. 4 basis functions 4 basis functions Invariance equation : possible MSGs ( R, θ | t ) is a a depending on the T[( R, θ | t )] = b b conserved direction of the by the magnetic order parameter (a,b) arrangement 2x2 matrix of irrep
epikernels of the irrep, depending on isotropy subgroups: the direction (a,a) ,(a,0), etc … Invariance equation : ( R, θ | t ) is a a T[( R, θ | t )] = b b conserved by the magnetic kernel of the irrep: arrangement 2x2 matrix of irrep operations represented by the unit matrix. MSG kept by any direction (a,b)
Cr1_2 Cr1_1 Cr1_1 (mx,-mx,0) Cr1_2 (mx’,-mx’,0) Cr1_2 Magnetic site splits into two independent sites Cr1_1 Two spin parameters to be fit
Cr1_1 (mx,my,0) Only ONE independent magnetic site. But two independent spin components. Spin canting is symmetry allowed Two spin parameters to be fit
Single irrep assignment vs. magnetic space groups (MSG) in commensurate structures. Cases 1) 1-dim. irrep: irrep and MSG assignment are equivalent for spin relations. 2) N dim. irrep, N>1: several MSG (epikernels or isotropy subgroups) are possible for the same irrep. The assignment of a MSG restricts the magnetic configuration beyond the restrictions coming from the irrep. case 2.1: The MSG is a k-maximal subgroup: it only allows a spin ordering according to a single irrep (further restricted to fulfill the MSG constraints). No other irrep arrangements are compatible with the MSG . case 2.2: The MSG is NOT a k-maximal subgroup: it allows the presence of other irreps (secondary). Other irreps are compatible with the MSG. (for simple propagation vectors (2k=reciprocal lattice) not frequent).
Parent space group: Fm-3m NiO k = (1/2,1/2,1/2) – point L in the BZ MSG: C c 2/c M Ni =m(1,1,-2)
k =(1/2,1/2,1/2) k =(1/2,1/2,1/2) magnetic site 4a
k = (1/2,1/2,1/2) – point L in the BZ magnetic site 4a mL2+ M rep (site 4a )= mL2 + + mL3 + 1-dim 2-dim mL3+ (a,0) mL3+ (a,a) this MSG is an irrep epikernel but it is not k-maximal mL3+ (a,b) mL3+ mL2+ (a,a) (a,0) (a,b)
Irreps and order parameter directions compatible with the subgroup C c 2/c: secondary irrep (1 d. of freedom) primary irrep (1 d. of freedom) The MSG allows two deg. of freedom for the spin arrangement, but only one corresponds to the active primary irrep
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