1 colored symmetry applications in: crystal surfaces & crystal growth strained materials...
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Colored symmetry
Applications in:crystal surfaces & crystal growthstrained materialstwinningmagnetic structuresferroelectricsother physical propstextures
(see Colored Symmetry by Shubnikov & Belov
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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
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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
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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
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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
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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'
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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
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Colored symmetry
Polar & neutral figures –
polar white outside, black inside
black outside, white inside
neutral black inside & outside orwhite inside & outside
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Colored symmetry
Operations –
mirror in xy plane - anti-symmetric2'
anti-reflection
1 0 0 00 1 0 00 0 -1 00 0 0 -1
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Colored symmetry
Operations –
anti-rotation of 180° along x-axis2'
1 0 0 00 -1 0 00 0 -1 00 0 0 -1
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Colored symmetry
Operations –
anti-inversion1'
-1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1
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Colored symmetry
Operations –
4'
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Colored symmetry
Operations –
4'
6'
blackunderneath
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Colored symmetry
Operations –
2/m'
blackunderneath
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Colored symmetry
Operations –
2/m'
2'/m whiteunderneath
blackunderneath
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Colored symmetry
Operations –
2'/m'
whiteunderneath
blackunderneath
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Colored symmetry
Operations –
2'/m'
2'm'm
whiteunderneath
blackunderneath
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Colored symmetry
Operations –
2m'm'
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Colored symmetry
Operations –
2m'm'
2'2'2
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Colored symmetry
Operations –
2/mm'm'
whiteunderneath
blackunderneath
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Colored symmetry
Glides –
g
g'
m=a'
a=m'
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The 36 lattice translations for colored groups
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The 36 lattice translations for colored groups
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The 46 dichromatic plane groups
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The 46 dichromatic plane groups
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The 46 dichromatic plane groups
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The 46 dichromatic plane groups
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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
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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
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Comments
4. Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis
2 t||' ––> 2 & 21'
21 t||' ––> 21 & 2'
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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'
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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