Space lattices

1. Crystallography

2. •In 1669 Nicolaus Steno found angles
between adjacent prism faces of quartz
crystal (interfacial angle), to be 120°.
•In 1780 Carangeot invented the
goniometer, a protactor like device used to
measure interfacial angles on crystals.
•Law of “constancy of interfacial angels”:
angles between equivalent faces of
crystals of the same mineral are always the
same.The law acknowledges that the size
and shape of the crystal may vary.

3. •In 1784 Rene’Hauy hypothesized the
existence of basic building blocks of
crystals called integral molecules and
argued that large crystals formed when
many integral molecules bonded together.

4. Old view
•Crystals are made of small
building blocks
•The blocks stack together in a
regular way, creating the whole
crystal.
•Each block contains a small
number of atoms
•All building blocks have the
same atomic composition
•The building block has shape
and symmetry of the entire
crystal.
We now accept that:
•Crystals have basic building
blocks called unit cells
•The unit cells are arranged in
a pattern described by points
in a lattice.
•The relative proportions of
elements in a unit cell are
given by the chemical formula
of a mineral.
•Crystals belong to one of the
seven crystal systems. Unit
cells of distinct shape and
symmetry characterize each
crystal system.
•Total crystal symmetry
depends on Unit cell
symmetry and lattice
symmetry.

5. Crystal Geometry
1.Crystals
2.Lattice
3.Lattice points, lattice
translations
4.Cell--Primitive & non primitive
5.Lattice parameters
6.Crystal=lattice+motif

6. Matter
Solid Liquid Gas
Crystalline Amorphous

7. Crystal?

8. A 3D translationally
periodic
arrangement of
atoms in space is
called a crystal.

9. A two-dimensional periodic pattern
by a Dutch artist M.C. Escher

10. Lattice?

11. A 3D translationally
periodic
arrangement of
points in space is
called a lattice.

12. A 3D
translationally
periodic
arrangement
of atoms
Crystal
A 3D
translationally
periodic
arrangement of
points
Lattice

13. What is the relation between
the two?
Crystal = Lattice + Motif
Motif or basis: an atom or a
group of atoms associated
with each lattice point

14. Crystal=lattice+basis
Lattice: the underlying periodicity of
the crystal,
Basis: atom or group of atoms
associated with each lattice points
Lattice: how to repeat
Motif: what to repeat

15. +
Love PatternLove Lattice+ Heart=

16. Space Lattice
A discrete array of points in 3-d space
such that every point has identical
surroundings

17. Lattice
Finite or infinite?

18. Primitive
cell
Primitive
cell
Nonprimitive
cell

19. Cells
 A cell is a finite representation of the
infinite lattice
 A cell is a parallelogram (2D) or a
parallelopiped (3D) with lattice points at
their corners.
 If the lattice points are only at the
corners, the cell is primitive.
 If there are lattice points in the cell other
than the corners, the cell is nonprimitive.

20. Lattice Parameters
Lengths of the three sides of
the parallelopiped :
a, b and c.
The three angles between the
sides: a, b, g

21. Convention
a parallel to x-axis
b parallel to y-axis
c parallel to z-axis
a Angle between y and z
b Angle between z and x
g Angle between x and y

22. The six lattice parameters a, b, c, a, b, g
The cell of the lattice
lattice
crystal
+ Motif

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24. In order to define
translations in 3-d
space, we need 3
non-coplanar
vectors
Conventionally, the
fundamental
translation vector is
taken from one
lattice point to the
next in the chosen
direction

25. With the help of
these three
vectors, it is
possible to
construct a
parallelopiped
called a CELL

26. The smallest cell
with lattice points
at its eight
corners has
effectively only
one lattice point in
the volume of the
cell.
Such a cell is called
PRIMITIVE CELL

27. Bravais Space Lattices
Conventionally, the finite representation
of space lattices is done using unit
cells which show maximum possible
symmetries with the smallest size.
Symmetries:1.Translation
2. Rotation
3. Reflection

28. Considering
•Maximum Symmetry, and
•Minimum Size
Bravais concluded that there are only
14 possible Space Lattices (or
Unit Cells to represent them).
These belong to 7 Crystal Classes

29. Arrangement of lattice points in the unit cell
•8 Corners (P)
•8 Corners and 1
body centre (I)
•8 Corners and 6
face centres (F)
•8 corners and 2
centres of opposite
faces (A/B/C)
Effective number of l.p.

30. •Cubic Crystals
•Simple Cubic (P)
•Body Centred
Cubic (I) – BCC
•Face Centred
Cubic (F) - FCC

31. •Tetragonal Crystals
•Simple Tetragonal
•Body Centred
Tetragonal

32. 1.Orthorhombic
Crystals
•Simple
Orthorhombic
•Body Centred
Orthorhombic
•Face Centred
Orthorhombic
•End Centred
Orthorhombic

33. 1.Hexagonal Crystals
•Simple Hexagonal
or most commonly
HEXAGONAL
5.Rhombohedral
Crystals
•Rhombohedral
(simple)

34. •Monoclinic Crystals
•Simple Monoclinic
•End Centred
Monoclinic (A/B)
6.Triclinic Crystals
•Triclinic (simple)

35. Crystal Structure
Space Lattice + Basis (or Motif)
Basis consists of a group of atoms located
at every lattice point in an identical fashion
To define it, we need to specify
•Number of atoms and their kind
•Internuclear spacings
•Orientation in space

36. Atoms are assumed to be hard spheres