Crystal Geometry

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Classification of lattice The Seven Crystal
System And The Fourteen Bravais Lattices
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7 crystal Systems
System Unit Cell Shape 1. Cubic abc,
???90?
3
7 crystal Systems
System Unit Cell Shape 2. Tetragonal ab?c,
???90?
4
7 crystal Systems
System Unit Cell Shape 3 Orthorhombic a?b?c,
???90?
5
7 crystal Systems
System Unit Cell Shape 4. Hexagonal ab?c,
?? 90?, ?120? 5. Rhombohedral abc,
????90? 6. Monoclinic a?b?c, ??90??? 7.
Triclinic a?b?c, ?????
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14 Bravais lattices divided into seven crystal
systems

• Crystal system Bravais lattices
• Cubic P I F

Simple cubicPrimitive cubicCubic P
Body-centred cubicCubic I
Face-centred cubicCubic F
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14 Bravais lattices divided into seven crystal
systems

• Crystal system Bravais lattices
• Cubic P I F
• Tetragonal P I
• Orthorhombic P I F C
• Hexagonal P
• Trigonal P
• Monoclinic P C
• Triclinic P

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Orthorhombic CEnd-centred orthorhombicBase-centr
ed orthorhombic
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14 Bravais lattices divided into seven crystal
systems

• Crystal system Bravais lattices
• Cubic P I F
• Tetragonal P I
• Orthorhombic P I F C
• Hexagonal P
• Trigonal P
• Monoclinic P C
• Triclinic P

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End-centred cubic not in the Bravais list ?
End-centred cubic Simple Tetragonal
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14 Bravais lattices divided into seven crystal
systems

• Crystal system Bravais lattices
• Cubic P I F C
• Tetragonal P I
• Orthorhombic P I F C
• Hexagonal P
• Trigonal P
• Monoclinic P C
• Triclinic P

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Face-centred cubic in the Bravais list ?
Problem 3.1
Cubic F Tetragonal I
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14 Bravais lattices divided into seven crystal
systems

• Crystal system Bravais lattices
• Cubic P I F C
• Tetragonal P I
• Orthorhombic P I F C
• Hexagonal P
• Trigonal P
• Monoclinic P C
• Triclinic P

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What is the basis for classification of lattices
into 7 crystal systems and 14 Bravais lattices?
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Lattices are classified on the basis of their
symmetry
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What is symmetry?
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Symmetry
If an object is brought into self-coincidence
after some operation it said to possess symmetry
with respect to that operation.
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Rotational symmetry
A rectangle comes into self-coincidence by 180
degrees rotation
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Rotation Axis
If an object come into self-coincidence through
smallest non-zero rotation angle of ? then it is
said to have an n-fold rotation axis where
?180?
2-fold rotation axis
n2
?90?
n4
4-fold rotation axis
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Reflection (or mirror symmetry)
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Translational symmetry
Lattices also have translational symmetry
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Symmetry of lattices
Lattices have
Translational symmetry
Rotational symmetry
Reflection symmetry
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Symmetry classification of lattices

• Based on rotational and reflection symmetry
  alone ? 7 types of lattices
• ? 7 crystal systems

Based on complete symmetry, i.e., rotational,
reflection and translational symmetry ? 14
types of lattices ? 14 Bravais lattices
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7 crystal Systems

• System Required symmetry
• Cubic Three 4-fold axis
• Tetragonal one 4-fold axis
• Orthorhombic three 2-fold axis
• Hexagonal one 6-fold axis
• Rhombohedral one 3-fold axis
• Monoclinic one 2-fold axis
• Triclinic none

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Tetragonal symmetry
Cubic symmetry
Cubic C Tetragonal P
Cubic F ? Tetragonal I
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The three Bravais lattices in the cubic crystal
system have the same rotational symmetry but
different translational symmetry.
Simple cubicPrimitive cubicCubic P
Body-centred cubicCubic I
Face-centred cubicCubic F
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Symmetry classification of lattices

• Based on rotational and reflection symmetry
  alone ? 7 types of lattices
• ? 7 crystal systems

Based on complete symmetry, i.e., rotational,
reflection and translational symmetry ? 14
types of lattices ? 14 Bravais lattices
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Notation P Primitive (lattice points only at
the corners of the unit cell) I Body-centred
(lattice points at the corners one lattice
point at the centre of the unit cell) F
Face-centred (lattice points at the corners
lattice points at centres of all faces of the
unit cell) C End-centred or base-centred
(lattice points at the corners two lattice
points at the centres of a pair of opposite
faces)