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Exploring Colored Symmetry Operations in Materials Science

Understand the concept of Colored Symmetry, including its applications in crystal surfaces, growth, and physical properties. Discover new operations and figures through antireflection, inversion, and more.

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Exploring Colored Symmetry Operations in Materials Science

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  1. Colored symmetry (see Colored Symmetry by Shubnikov & Belov Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props textures

  2. Colored symmetry (see Colored Symmetry by Shubnikov & Belov Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props textures Spin moment structure in MnO determined by neutron diffrn - only Mn+2 ions shown

  3. Colored symmetry (see Colored Symmetry by Shubnikov & Belov Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props textures Will consider here only two colors - black & white

  4. Colored symmetry Anti-equality – Pair of gloves which are, say, black on outside & white on inside if both left-handed, can get: congruent equality - do nothing mirror anti-equality - turn one inside-out

  5. Colored symmetry Anti-equality – Pair of gloves which are, say, black on outside & white on inside if both left-handed, can get: congruent equality - do nothing mirror anti-equality - turn one inside-out if right- & left-handed, can get: mirror equality - nothing congruent anti-equality - turn one inside-out

  6. Colored symmetry New operations – anti-rotations (rotation + "sign change") 1' 2' 3' 4' 6' mirror anti-rotations (mirror rotation + "sign change") 1' 2' 3' 4' 6'

  7. Colored symmetry New operations – anti-rotations (rotation + "sign change") 1' 2' 3' 4' 6' mirror anti-rotations (mirror rotation + "sign change") 1' 2' 3' 4' 6' c11 c12 c13 0 c21 c22 c23 0 c31 c32 c33 0 0 0 0 ±1

  8. Colored symmetry Polar & neutral figures – polar white outside, black inside black outside, white inside neutral black inside & outside or white inside & outside

  9. Colored symmetry Operations – mirror in xy plane - anti-symmetric 2' anti-reflection 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1

  10. Colored symmetry Operations – anti-rotation of 180° along x-axis 2' 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1

  11. Colored symmetry Operations – anti-inversion 1' -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1

  12. Colored symmetry Operations – 4'

  13. Colored symmetry Operations – 4' 6' black underneath

  14. Colored symmetry Operations – 2/m' black underneath

  15. Colored symmetry Operations – 2/m' 2'/m white underneath black underneath

  16. Colored symmetry Operations – 2'/m' black underneath white underneath

  17. Colored symmetry Operations – 2'/m' 2'm'm black underneath white underneath

  18. Colored symmetry Operations – 2m'm'

  19. Colored symmetry Operations – 2m'm' 2'2'2

  20. Colored symmetry Operations – 2/mm'm' black underneath white underneath

  21. Colored symmetry Glides – g g' m=a' a=m'

  22. The 36 lattice translations for colored groups

  23. The 36 lattice translations for colored groups

  24. The 46 dichromatic plane groups

  25. The 46 dichromatic plane groups

  26. The 46 dichromatic plane groups

  27. The 46 dichromatic plane groups

  28. Comments Mirror translation ( ) – gives another mirror of same character if uncolored. If colored, new mirror is colored Mirror translation (||) – colored translation gives colored glide

  29. Comments Mirror translation ( ) – gives another mirror of same character if uncolored. If colored, new mirror is colored Mirror translation (||) – colored translation gives colored glide Axis translation ( ) – new axis has same charater if uncolored. If colored, new axis is colored or uncolored, depending on the axis. Character is unchanged by colored translation Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis

  30. Comments 4. Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis 2 t||' ––> 2 & 21' 21 t||' ––> 21 & 2'

  31. Comments 4. Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis 2 t||' ––> 2 & 21' 21 t||' ––> 21 & 2' 4 t||' ––> 4 & 42' 41 t||' ––> 41 & 43'

  32. Comments n = odd axes cannot be black-white Two intersecting 2-fold axes: 30° ––> new 6 45° ––> new 4 60° ––> new 3 etc. new axis uncolored if 2-folds are same color…….otherwise, new axis is colored uncolored 4 colored 4

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