The Fascination of Crystals and Symmetry

Unit 2.9

by Frank Hoffmann & Michael Sartor

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

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

Rhombohedron

cube

Assembling Rhombohedra

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

Transforming the Rhombohedral Lattice

3-fold axis of rotation

c‘

0

⅔

⅓

1

projectionalong c‘

a’

b’

Transforming the Rhombohedral Lattice

3-fold axis of rotation

c‘

0

⅔

⅓

1

projectionalong c‘

a’

b’

new centered cell

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

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

Trigonal – Rhombohedral – Hexagonal-R

6-fold axis of rotation?

Trigonal – Rhombohedral – Hexagonal-R

Trigonal – Rhombohedral – Hexagonal-R

Trigonal – Rhombohedral – Hexagonal-R

Trigonal – Rhombohedral – Hexagonal-R

Trigonal – Rhombohedral – Hexagonal-R

Trigonal – Rhombohedral – Hexagonal-R

!

!

Trigonal – Rhombohedral – Hexagonal-R

the trigonal crystal system is characterized by its 3-fold rotational symmetry

3-fold axis of rotation!

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