Colored symmetry see Colored Symmetry by Shubnikov Belov

Colored symmetry (see Colored Symmetry by Shubnikov & Belov Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props textures 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 Spin moment structure in Mn. O determined by neutron diffrn only Mn+2 ions shown 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 Will consider here only two colors - black & white 3

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 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 if right- & left-handed, can get: mirror equality - nothing congruent anti-equality - turn one inside-out 5

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' 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' c 11 c 21 c 31 0 c 12 c 22 c 32 0 c 13 0 c 23 0 c 33 0 0 ± 1 7

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

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

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

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

Colored symmetry Operations – 4' 12

Colored symmetry Operations – 4' 6' black underneath 13

Colored symmetry Operations – 2/m' black underneath 14

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

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

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

Colored symmetry Operations – 2 m'm' 18

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

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

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

The 36 lattice translations for colored groups 22

The 36 lattice translations for colored groups 23

The 46 dichromatic plane groups 24

The 46 dichromatic plane groups 25

The 46 dichromatic plane groups 26

The 46 dichromatic plane groups 27

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 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 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 29

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

Comments 5. n = odd axes cannot be black-white 6. Two intersecting 2 -fold axes: 7. 30° ––> new 6 45° ––> new 4 60° ––> new 3 etc. 8. new axis uncolored if 2 -folds are same color……. otherwise, new axis is colored uncolored 4 32