1

Colored symmetry

Applications in:crystal surfaces & crystal growthstrained materialstwinningmagnetic structuresferroelectricsother physical propstextures

(see Colored Symmetry by Shubnikov & Belov

2

Colored symmetry

Applications in:crystal surfaces & crystal growthstrained materialstwinningmagnetic structuresferroelectricsother physical propstextures

Spin moment structure in MnOdetermined by neutron diffrn -only Mn+2 ions shown

(see Colored Symmetry by Shubnikov & Belov

3

Colored symmetry

Applications in:crystal surfaces & crystal growthstrained materialstwinningmagnetic structuresferroelectricsother physical propstextures

Will consider here only two colors - black & white

(see Colored Symmetry by Shubnikov & Belov

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 nothingmirror 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 nothingmirror anti-equality - turn one inside-

out

if right- & left-handed, can get:mirror equality - nothingcongruent 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 orwhite inside & outside

9

Colored symmetry

Operations –

mirror in xy plane - anti-symmetric2'

anti-reflection

1 0 0 00 1 0 00 0 -1 00 0 0 -1

10

Colored symmetry

Operations –

anti-rotation of 180° along x-axis2'

1 0 0 00 -1 0 00 0 -1 00 0 0 -1

11

Colored symmetry

Operations –

anti-inversion1'

-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'

blackunderneath

14

Colored symmetry

Operations –

2/m'

blackunderneath

15

Colored symmetry

Operations –

2/m'

2'/m whiteunderneath

blackunderneath

16

Colored symmetry

Operations –

2'/m'

whiteunderneath

blackunderneath

17

Colored symmetry

Operations –

2'/m'

2'm'm

whiteunderneath

blackunderneath

18

Colored symmetry

Operations –

2m'm'

19

Colored symmetry

Operations –

2m'm'

2'2'2

20

Colored symmetry

Operations –

2/mm'm'

whiteunderneath

blackunderneath

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

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

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

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

5. n = odd axes cannot be black-white

6. Two intersecting 2-fold axes:

30° ––> new 6 45° ––> new 4 60° ––> new 3etc.

new axis uncolored if 2-folds are same color…….otherwise, new axis is colored

uncolored 4 colored 4