Colored symmetry

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 Mn2 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'
ma'
am'
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' gt 2 21'
• 21 t' gt 21 2'

31
Comments

• 4. Axis translation () colored translation
  makes axis colored screw or uncolored, depending
  on the axis
• 2 t' gt 2 21'
• 21 t' gt 21 2'
• 4 t' gt 4 42'
• 41 t' gt 41 43'

32
Comments

• n odd axes cannot be black-white
• Two intersecting 2-fold axes
• 30 gt new 6 45 gt new 4 60 gt new 3 etc.
• new axis uncolored if 2-folds are same
  color.otherwise, new axis is colored

uncolored 4
colored 4