the fascination of crystals and symmetry · 2016. 6. 13. · the fascination of crystals and...
TRANSCRIPT
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The Fascination of Crystals and Symmetry
Unit 2.9
by Frank Hoffmann & Michael Sartor
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Trigonal – Rhombohedral – Hexagonal-R
The trigonal crystal system is characterized by its 3-fold rotational symmetry
There are two different kinds of crystals belonging to this trigonal crystal system
1. those with a primitive rhombohedral unit cell/lattice
2. those with a primitive hexagonal unit cell/lattice
primitive hexagonal unit cellprimitive rhombohedral unit cell
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Trigonal – Rhombohedral – Hexagonal-R
rhombohedral metric
a = b = c
a = b = g ≠ 90 °
rhombohedral cells
3-fold axis of rotation
≠ 90 °
a b c
primitive rhombohedral unit cell
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Rhombohedron
cube
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Assembling Rhombohedra
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Rhombohedron
3-fold axis of rotation
a b c
Rule: The axis of rotation with the highest order should run parallel to the c-axis
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Transforming the Rhombohedral Lattice
3-fold axis of rotation
c‘
0
⅔
⅓
1
projectionalong c‘
a’
b’
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Transforming the Rhombohedral Lattice
3-fold axis of rotation
c‘
0
⅔
⅓
1
projectionalong c‘
a’
b’
new centered cell
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Trigonal – Rhombohedral – Hexagonal-R
a
b
120°
⅔
⅓
hexagonal metric (a = b, g = 120°)rhombohedral centered (hR)
three times the volume of the primitive cell
new centered cell
The trigonal crystal system is characterized by its 3-fold rotational symmetry
There are two different kinds of crystals belonging to this trigonal crystal system
1. those with a primitive rhombohedral unit cell/lattice
2. those with a primitive hexagonal unit cell/lattice
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Trigonal – Rhombohedral – Hexagonal-R
hexagonal metric (a = b, g = 120°)rhombohedral centered (hR)
ab
c
a
b
120°
⅔
⅓
(⅓, ⅔, ⅔)
(⅔, ⅓, ⅓)
The trigonal crystal system is characterized by its 3-fold rotational symmetry
There are two different kinds of crystals belonging to this trigonal crystal system
1. those with a primitive rhombohedral unit cell/lattice
2. those with a primitive hexagonal unit cell/lattice
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Trigonal – Rhombohedral – Hexagonal-R
6-fold axis of rotation?
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Trigonal – Rhombohedral – Hexagonal-R
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Trigonal – Rhombohedral – Hexagonal-R
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Trigonal – Rhombohedral – Hexagonal-R
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Trigonal – Rhombohedral – Hexagonal-R
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Trigonal – Rhombohedral – Hexagonal-R
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Trigonal – Rhombohedral – Hexagonal-R
!
!
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Trigonal – Rhombohedral – Hexagonal-R
the trigonal crystal system is characterized by its 3-fold rotational symmetry
3-fold axis of rotation!
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Trigonal – Rhombohedral – Hexagonal-R
ab
c A rhombohedral lattice exists
But there is no rhombohedral
crystal system
The advantage of transforming
the primitive rhombohedral cell
into its R-centered counterpart is
that the 3-fold axis of rotation is
running along the c-direction