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Hand calculations of magnetic modes

Hand calculations of magnetic modes. A. Daoud-Aladine (ISIS). Physical problem. Ideal cubic Perovskite. Site A. LaMnO 3 T N =150K k=(0 0 0). La/Tb. TbMnO 3 T N =41K k=(0.28 0 0). O. Mn. P nma. m. [010] O [001] C. [101] O [010] C. n. m. n. a. a. Physical problem.

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Hand calculations of magnetic modes

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  1. Hand calculations of magnetic modes A. Daoud-Aladine (ISIS)

  2. Physical problem Ideal cubic Perovskite Site A LaMnO3 TN=150K k=(0 0 0) La/Tb TbMnO3 TN=41K k=(0.28 0 0) O Mn Pnma m [010]O [001]C [101]O [010]C n m n a a

  3. Physical problem LaMnO3 TN=150K k=(0 0 0) (a) TbMnO3 TN=41K k=(0.28 0 0) (b) Pnma m [101]O [010]C m

  4. Linear algebra problem (a) (b) e1z Sk1 e2y Sk2 Sk3 a b e3x Pnma c Sk4

  5. Linear algebra problem e1z Sk1 Hermitian vector space (complex) of dimension 12 e2y Sk2 Sk3 a b e3x Pnma c Sk4

  6. The group G=Pnma Zero blocks elements

  7. The group of k: Gk (b) Gk= P21ma (a) Gk = Pnma 2x k G = Pnma a (x) b (y) c (z) mxy mxz

  8. Irreps of Gk (a) (b)

  9. Representation of gGk Sk1 Sk2 Sk3 Sk4 c b a (a and b) Trivial Ex1: Identity operation 1

  10. Representation of gGk (a and b) Trivial Ex1: Identity operation 1

  11. Representation of gGk Sk1 Sk2 Sk3 Sk4 c b a (a only) Trivial Ex2: Inversion -1

  12. Representation of gGk (a only) Trivial Ex2: Inversion -1

  13. Representation of gGk Sk1 Sk2 Sk3 Sk4 (a and b) Ex1: 21 axis along x Point Mn1: 1/2, 1/2, 0 = Mn3 0,0,1/2 c b Sk Spin . Sk a

  14. Representation of gGk e1z Sk1 e1y e1x 1 1’ Sk1’ (a and b) Ex1: 21 axis along x e1z e2y ap(21,3) c b e3x e3x a

  15. Representation of gGk (a and b) Ex1: 21 axis along x

  16. Representation of gGk (a only) Ex2: Glide plane n axis  x e1z e2y e3x

  17. Representation of gGk (a only) Ex2: Glide plane n axis  x e1z e2y e3x

  18. Representation of gGk (a only) Ex2: Glide plane n axis  x

  19. Decomposition

  20. Decomposition = 12

  21. Decomposition = 0 etc…

  22. Decomposition

  23. Projection

  24. Projection

  25. Projection

  26. Projection

  27. Magnetic structure (b) TbMnO3 TN=41K k=(0.28 0 0)

  28. Magnetic structure

  29. Magnetic structure (b)

  30. Magnetic structure Example (a)

  31. Magnetic structure

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