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Introduction to Crystallography
Dr. Mohsin Ali Raza
Lattice
The periodic nature of the crystal structure is represented with a lattice.
A lattice is a regular array of geometrical points each of which has
same environment.
In space lattice every point has the same surrounding.
Lattice can be generated with translation operator.
Unit cell in 3 dimensions
Unit cell
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Crystal
Arrangement of atoms in a pattern that repeats
periodically in three dimensions
A crystal has two characteristics:
• Translational symmetry
• Symmetry
• An ideal crystal is a periodic array of structural units, ions, atoms or
molecules
• A crystal can be constructed by infinite repetition of identical structural
units in space
• A structure can be described in terms of lattice, with group of atoms
attached to each lattice. The group of atom is basis.
Dr. Mohsin Ali Raza
1D, 2D lattices
A lattice vector is a vector joining any two lattice points
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1D, 2D lattices
A lattice vector is a vector joining any two lattice points
t = U𝐚 + V𝐛 + W𝐜
In short hand, lattice vectors are written in the form
t = UVW
If the lattice vectors are negative then
ഥV
t= U ഥW
ഥ
Dr. Mohsin Ali Raza
1D, 2D lattices
A lattice vector is a vector joining any two lattice points
Dr. Mohsin Ali Raza
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1D, 2D lattices
A lattice vector is a vector joining any two lattice points
Dr. Mohsin Ali Raza
3D lattice/ Space lattice
a, b and c are
fundamental
vectors,
Crystal axes or
basis vectors
• Using translational vector, every point of space lattice can be generated.
• Two types of translation vectors.
• Primitive translation vector having integral coefficient of basis vectors
• Non-primitive translation vector having fractional
Dr. Mohsin Ali Raza coefficient of basis vectors.
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Unit Cell
A unit cell of lattice is a volume which can describe lattice using only
translations. In 3-D, this volume is parallelepiped. A volume (unit cell) can
be defined by 6 numbers. Repeating unit cell will generate lattice.
Primitive unit cell
Non-primitive unit cell
Lattice vectors, a,b,c (length of three sides)
Angle between vectors α,β,γ
Dr. Mohsin Ali Raza
Basis
Basis is set of atoms associated with each lattice point
Is one atom is associated with one lattice point???
Basis can be complex.
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2D lattices and Unit Cell
Five different types of unit cells: Four of them are primitive and one is centred
rectangular
Primitive unit cell has effective point only 1 (each point is shared with others).
Points are only at corners.
Non-primitive unit cell has more than one point.
Dr. Mohsin Ali Raza
2D lattices and Unit Cell
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Identifying Lattice
In 2-D lattice there are six choices for unit cell
The best choice is one in which
• Edges of unit cell coincide with the symmetry of lattice
• The edges of unit cell should be related with the symmetry of the
lattice
• The smallest possible cell that contains all elements
Dr. Mohsin Ali Raza
Identifying Lattice
• Choose an arbitrary reference point and see its environment.
• Mark all those points which have same environment.
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Crystallography
2-D Lattice Types
• Square
• Rectangular
• Centered Rectangular
• Oblique
• Hexagonal
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2-D Lattice Types
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2-D Lattice Types
Rules:
(i) If a=b, and γ= 90°, then Square
(ii) If a is not equal to b and γ= 90°, then Rectangle
(iii) If a=b, and γ= 120°, then hexagonal
(iv) If a is not equal to b and γ is not equal to 90°, then oblique
(v) If a is not equal to b and γ= 90°, then Rectangle centered, if
there comes a point at the center of a rectangle. It can be taken
as oblique, because then it would have lower symmetry.
(vi) Choose smallest possible unit cell
Dr. Mohsin Ali Raza
Crystallography
2-D Crystal Lattices
➢ oblique
➢ each point
➢ A pair of atoms
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Crystallography
➢ A square lattice
➢ two atoms per lattice point
Dr. Mohsin Ali Raza
Crystallography
➢ A rectangular lattice
➢ three atoms per lattice point
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Crystallography
Square The square lattice
Rectangle
The rectangle lattice
Dr. Mohsin Ali Raza
Crystallography
Centered rectangle
Centered rectangle lattice
The oblique
The oblique lattice
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Crystallography
The hexagonal
The hexagonal lattice
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