Translation
Lecture 4 of the “Crystallography” course
1. Translation
It represents a symmetry operation in which an object is moved along the directions of
translation:
A) x - points toward the observer, the translation length is denoted by a
B) y - parallel to the observer, the translation length is denoted by b
C) z - always vertical, the translation length is denoted by c
The inclination of the translation directions relative to each other depends on the angles
between them:
A) 𝞪 - located between the translations along y and z, opposite to translation a (x)
B) 𝛃 - located between the translation along x and z, opposite to translation b (y)
C) 𝝲 - located between the translations along x and y, opposite to translation c (z)
We take a piece of the crystal structure (a motif) and reduce it to a point (a node). By
translating the node in one direction, we form a row of nodes, by translating the node in
two directions, we form a crystallographic net, and by translating it in three directions,
we form a crystal lattice.
When translating in two directions, five types of crystallographic nets are formed:
A) Those where a=b (translation lengths are equal)
– square net, with an angle between a,b = 90
– triangular net, with an angle between a,b = 60
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, – hexagonal net, with an angle between a,b = 120
B) those where a≠b (translation lengths are different)
– rectangular net, with an angle between a,b = 90
– non-rectangular net (oblique) with an arbitrary angle between a,b ≠ 90
The smallest volume in the lattice enclosed by the unit translations a, b and c is called
Bravais cell
Only the following axes of symmetry appear in crystals: monogyre, digyre, trigyre,
tetragyre, hexagyre.
The crystal lattice (translation along x, y and z) is a way of representing the periodic
repetition in space of different material particles and the gaps between them.
Structure is a periodic arrangement of real material particles.
2. Deriving primitive Bravais cells
A) for translations a≠b≠c (those with different lengths) three possible primitive Bravais
cells are formed:
– 𝜶 = 𝛃 = 𝜸 = 90
– 𝜶 = 𝜸 = 90, 𝛃 ≠ 90
– 𝜶 ≠ 𝛃≠ 𝜸 ≠ 90
Cells can also be derived by translating crystallographic nets along z.
When a rectangular crystallographic net, where a ≠ b and the angle between them is 90,
is translated perpendicularly along z, an Orthorhombic Bravais cell is formed, in which:
– a ≠ b ≠ c, 𝜶 = 𝛃 = 𝜸 = 90
– the symmetry elements are: a center of symmetry, three digyres, and three mirror
planes, written as 2/m 2/m 2/m (where the twos are digyres and the m’s are planes)
When the same rectangular net is translated along the z-axis at an arbitrary angle, a
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Lecture 4 of the “Crystallography” course
1. Translation
It represents a symmetry operation in which an object is moved along the directions of
translation:
A) x - points toward the observer, the translation length is denoted by a
B) y - parallel to the observer, the translation length is denoted by b
C) z - always vertical, the translation length is denoted by c
The inclination of the translation directions relative to each other depends on the angles
between them:
A) 𝞪 - located between the translations along y and z, opposite to translation a (x)
B) 𝛃 - located between the translation along x and z, opposite to translation b (y)
C) 𝝲 - located between the translations along x and y, opposite to translation c (z)
We take a piece of the crystal structure (a motif) and reduce it to a point (a node). By
translating the node in one direction, we form a row of nodes, by translating the node in
two directions, we form a crystallographic net, and by translating it in three directions,
we form a crystal lattice.
When translating in two directions, five types of crystallographic nets are formed:
A) Those where a=b (translation lengths are equal)
– square net, with an angle between a,b = 90
– triangular net, with an angle between a,b = 60
1
, – hexagonal net, with an angle between a,b = 120
B) those where a≠b (translation lengths are different)
– rectangular net, with an angle between a,b = 90
– non-rectangular net (oblique) with an arbitrary angle between a,b ≠ 90
The smallest volume in the lattice enclosed by the unit translations a, b and c is called
Bravais cell
Only the following axes of symmetry appear in crystals: monogyre, digyre, trigyre,
tetragyre, hexagyre.
The crystal lattice (translation along x, y and z) is a way of representing the periodic
repetition in space of different material particles and the gaps between them.
Structure is a periodic arrangement of real material particles.
2. Deriving primitive Bravais cells
A) for translations a≠b≠c (those with different lengths) three possible primitive Bravais
cells are formed:
– 𝜶 = 𝛃 = 𝜸 = 90
– 𝜶 = 𝜸 = 90, 𝛃 ≠ 90
– 𝜶 ≠ 𝛃≠ 𝜸 ≠ 90
Cells can also be derived by translating crystallographic nets along z.
When a rectangular crystallographic net, where a ≠ b and the angle between them is 90,
is translated perpendicularly along z, an Orthorhombic Bravais cell is formed, in which:
– a ≠ b ≠ c, 𝜶 = 𝛃 = 𝜸 = 90
– the symmetry elements are: a center of symmetry, three digyres, and three mirror
planes, written as 2/m 2/m 2/m (where the twos are digyres and the m’s are planes)
When the same rectangular net is translated along the z-axis at an arbitrary angle, a
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