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Part of
MATERIALS SCIENCE
& A Learner’s Guide
ENGINEERING
AN INTRODUCTORY E-BOOK
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur- 208016
Email: [email protected], URL: home.iitk.ac.in/~anandh
http://home.iitk.ac.in/~anandh/E-book.htm
UNIT CELLS (UC)
 A unit cell (also sometimes causally referred to as a cell) is a representative unit of the
structure.
 which when translationally repeated (by the basis vector(s)) gives the whole structure.
 The term unit should not be confused with ‘having one’ lattice point or motif
(The term primitive or sometimes simple is reserved for that).
 If the structure is a lattice, the unit cell will be unit of that (hence will have points* only).
 If the structure under considerations is a crystal, then the unit cell will also contain atoms
(or ions or molecules etc.).
 Note: Instead of full atoms (or other units) only a part of the entity may be
present in the unit cell (a single unit cell)
 The dimension of the unit cell will match the dimension of the structure**:
 If the lattice is 1D  the unit cell will be 1D,
if the crystal is 3D  then the unit cell will be 3D,
if the lattice is nD  the unit cell will be nD.
Unit cell
Lattice
Will contain lattice points only
Crystal
Will contain entities which decorate the lattice
of a
* Strangely in crystallography often we even ‘split a point’ (and say that 1/8th belongs to the UC).
** One can envisage other possibilities– e.g. a 2D motif may be repeated only along one direction (i.e. the crystal is 3D but the repeat direction
is along 1D)
Why Unit Cells?
Instead of drawing the whole structure I can draw a representative part and specify the repetition
pattern.

ADDITIONAL POINTS
 A cell is a finite representation of the infinite lattice/crystal
 A cell is a line segment (1D) or a parallelogram (2D) or a parallelopiped (3D) with lattice
points at their corners  This is the convention
 If the lattice points are only at the corners, the cell is primitive.
Hence, a primitive unit cell is one, wherein the lattice points are only at the corners of the
unit cell (or the ends of a line segment unit cell in 1D)
 If there are lattice points in the cell other than the corners, the cell is
non-primitive.
Consider an infinite pattern made of squares



This can be thought of as
a single square repeated
in x and y directions
This way the infinite information content of
a crystal can be reduced to the information
required to specify the contents of a unit
cell (along with the lattice translation
vectors).
In general the following types of unit cells can be defined:
 Primitive unit cell
 Non-primitive unit cells
 Voronoi cells
 Wigner-Seitz cells
1D
1D
 Unit cell of a 1D lattice is a line segment of length = the lattice parameter
 this is the PRIMITIVE UNIT CELL (i.e. has one lattice point per cell).
Each of these lattice points contributes half a lattice point to the unit cell
Primitive UC
Contributions to the unit cell: Left point = 0.5, Middle point = 1, Right point = 0.5. Total = 2
Doubly Nonprimitive UC
Triply Nonprimitive UC
 Unit cell of a 1D crystal will contain Motifs in addition to lattice points
 NOTE:
 The only kind of motifs possible in 1D are line segments
 Hence in ‘reality’ 1D crystals are not possible as Motifs typically have a finite
dimension (however we shall call them 1D crystals and use them for illustration
of concepts)
 We could have 2D or 3D motifs repeated along 1D (hence periodicity and
‘crystallinity’ is only along 1D)
Correct unit cell
Though the whole lattice point is
shown only half belongs to the UC
Each of these atoms contributes ‘half-atom’ to the unit cell
Though this is the correct unit cell
Often unit cells will be drawn like this
 Unit cell in 1D is described by 1 (one) lattice parameter: a
2D
2D
 Unit cell in 2D is described by 3 lattice parameters: a, b, 
 Special cases include: a = b;  = 90 or 120
 Unit Cell shapes in 2D

 Square

 Rectangle

 120 Rhombus

 Parallelogram (general) 
Lattice parameters
(a = b,  = 90)
(a, b,  = 90)
(a = b,  = 120)
(a, b, )
b 
a
2D
UC-1
b
90
Rectangular
lattice
Note: basis vectors
(& included angle)
will change based on
the ‘unit cell’ chosen
[which implies that
lattice parameters
will change as well
!]
a
Note: these are the
basis vectors (and
included angle) for
UC-1 above
90
Note: Symmetry of the Lattice or the crystal is not altered by our choice of unit cell!!
IMPORTANT
Symmetry (or the kind) of the Lattice or the
crystal is not altered by our choice of unit cell!!
You say this is obvious
 I agree!
How to choose a unit cell?
 When possible we chose a primitive unit cell
 The factors governing the choice of unit cell are:
 Symmetry of the Unit Cell  should be maximum (corresponding to lattice*)
 Size of the Unit Cell  should be minimum
 Convention  if above fails to resolve the issue we use some convention
(We will see later - using an example- that convention is not without common sense!)
How does convention come into play in the choice of unit cell for Orthorhombic lattices?
* The lattice may have higher symmetry than the crystal→ but in choosing the unit cell we focus on the
lattice.
E.g. if we decorate a square lattice with a triangle motif, we land up with a rectangle crystal. But we
prefer to chose a square unit cell for the crystal as well.
Centred square lattice = Simple square lattice
This is nothing but a
square lattice viewed
at 45!
Continued…
 In this case the primitive (square) and the non-primitive square cell both have the same
symmetry
 But the primitive square cell is chosen as it has the smaller size
 The primitive parallelogram cell is not chosen as it has a lower symmetry
 The lattice has 4-fold symmetries as shown
 The square cells also have 4-fold symmetry
 The parallelogram cell does NOT have 4-fold symmetry
(only 2-fold  lower symmetry)
Note these are symmetries of the UC
and not of the lattice!
Centred Rectangular Lattice
Unit Cell of
Lattice
Lattice parameters: a, b,  = 90
Note that the distribution of symmetry elements has not changed
(as compared to the Simple Rectangular Lattice)
Continued…
Simple rectangular Crystal
(Not a centred crystal)
Now the UC of the crystal will have a motif
Part of the structure
Unit Cell the
way it is
usually
shown
Though the whole
lattice point is
shown only one
fourth belongs to
the UC
True
Unit Cell of
Crystal
Correct unit cell
The centres of only the green circles are lattice points
(of course equivalently the centres of only the maroon circles)
Note that the UC has entities of the
motif in parts!
Click here
Solved
Example
Choice of Lattice, Motif, UC, Symmetry Elements etc are
illustrated in the example
(try and understand those concepts with which you are familiar at this juncture and
postpone the other concepts for a later discussion)
3D
Cells- 3D
 In order to define translations in 3-d space, we need 3 non-coplanar vectors.
 With the help of these three vectors, it is possible to construct a parallelepiped
called a UNIT CELL.
 Conventionally, the fundamental translation vector is taken from one lattice point
to the next in the chosen direction.
Unit Cell shapes in 3D:
Shape
Preferred unit cell for
_____ crystal system
Constraints on lattice
parameters (distances)
Constraints on lattice
parameters (angles)
Cube
Cubic
a=b=c
 =  =  = 90
Square Prism
Tetragonal
a=bc
 =  =  = 90
Rectangular Prism
Orthorhombic
abc
 =  =  = 90
120 Rhombic Prism
Hexagonal
a=bc
 =  = 90,  = 120
a=b=c
 =  =   90
Parallopiped
Trigonal (rhombohedral)
(Equilateral, Equiangular)
Paralleogramic Prism
Monoclinic
abc
 =  = 90  
Parallopiped (general)
Triclinic
abc

• The symbol “” implies → “need not be equal to”.
• Some common names of unit cells are given here → alternate names are also used for these cells.
• As we have noted in many places, these are conventional unit cells chosen and alternate unit cells
are also possible for the structure (for which this unit cell shape is chosen).
• Also, as we have noted elsewhere, these are unit cell shapes and not be confused with the
definition of crystal system (i.e. these unit cell shapes do not define the crystal system).
Different kinds of CELLS
Unit cell
A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional
space to describe the crystal.
Primitive unit cell
For each crystal structure there is a conventional unit cell, usually chosen to make
the resulting lattice as symmetric as possible. However, the conventional unit cell is
not always the smallest possible choice. A primitive unit cell of a particular crystal
structure is the smallest possible unit cell one can construct such that, when tiled, it
completely fills space.
Wigner-Seitz cell
A Wigner-Seitz cell is a particular kind of primitive cell, which has the same
symmetry as the lattice.
Q&A
How many ‘shapes’ of primitive unit cells are possible?
 1D → one.
 2D, 3D → Infinite (few examples in 2D given below).
1D
2D
Q &A
Funda Check
Should a Unit Cell have Lattice Points only at the Corners?
 The conventional unit cell chosen has lattice points at the ends of line segment/corners/vertices …
1D
Conventional UCs

But in principle any unit cell like the ones below (space
filling) should work fine! (all the illustrated cells fill space!)
2D
 We had earlier seen that conventional choice of unit cells can ‘cut into’ the lattice points
(and hence into entities of motif) (as below).
 Choices of some non-conventional cells (like the ones drawn before) can alleviate this
problem of ‘cutting into’ lattice points.
 The new unit cell may still (or may not as below) cut into parts of the motif.
New choice of “nonconventional” cell
Problem:
UC has entities of the motif in parts!
Corners of
the unit cell
still have to
be lattice
points
By convention
Funda Check
Define a unit cell.
 An unit cell unit of the structure [is a line segment in 1D, a parallelogram in 2D and a parallelepiped in 3D], such
that lattice points are at the ends of the line segment (1D) and at the vertices of the
parallelogram (2D) or parallelepiped (3D); which when repeated by the translational
symmetry vector(s) generates the whole structure (lattice or crystal).
 A primitive unit cell has only one lattice point per cell, which are at: the ends of the line
segment (1D) and at the vertices of the parallelogram (2D) or parallelepiped (3D).
(A primitive unit cell need not have 1 atom per cell!!)
Wigner-Seitz Cell
 Is a primitive unit cell with the symmetry of the lattice
 Created by Voronoi tessellation of space
 The region enclosed by the Wigner-Seitz cell is closer to a given lattice point than
to any other lattice point
Square lattice
Centred Rectangular lattice
Wigner-Seitz cells
BCC
Tetrakaidecahedron
The Tetrakaidecahedron is a space filling solid, which is semi-regular.
FCC
Rhombic Dodecahedron
• The rhombic dodecahedron has been considered as the least ‘photogenic’ solid!
• This is also a ‘semi-regular’ space filling solid.
• This is the coordination polyhedron for the BCC lattice.