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1 Colored symmetry Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props.

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Presentation on theme: "1 Colored symmetry Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props."— Presentation transcript:

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

2 2 Colored symmetry 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 (see Colored Symmetry by Shubnikov & Belov

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

4 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 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 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 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' c 11 c 12 c 13 0 c 21 c 22 c 23 0 c 31 c 32 c ±1

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

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

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

11 11 Colored symmetry Operations – anti-inversion 1'

12 12 Colored symmetry Operations – 4'

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

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

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

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

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

18 18 Colored symmetry Operations – 2m'm'

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

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

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

22 22 The 36 lattice translations for colored groups

23 23 The 36 lattice translations for colored groups

24 24 The 46 dichromatic plane groups

25 25 The 46 dichromatic plane groups

26 26 The 46 dichromatic plane groups

27 27 The 46 dichromatic plane groups

28 28 Comments 1. Mirror translation ( ) – gives another mirror of same character if uncolored. If colored, new mirror is colored 2. Mirror translation (||) – colored translation gives colored glide

29 29 Comments 1. Mirror translation ( ) – gives another mirror of same character if uncolored. If colored, new mirror is colored 2. Mirror translation (||) – colored translation gives colored glide 3. 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 4. Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis

30 30 Comments 4.Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis 2 t || ' ––> 2 & 2 1 ' 2 1 t || ' ––> 2 1 & 2'

31 31 Comments 4.Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis 2 t || ' ––> 2 & 2 1 ' 2 1 t || ' ––> 2 1 & 2' 4 t || ' ––> 4 & 4 2 ' 4 1 t || ' ––> 4 1 & 4 3 '

32 32 Comments 5. n = odd axes cannot be black-white 6. Two intersecting 2-fold axes: 30° ––> new 645° ––> new 460° ––> new 3etc. new axis uncolored if 2-folds are same color…….otherwise, new axis is colored uncolored 4 colored 4


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