1. Anandh Subramaniam & Kantesh Balani
SPACE LATTICES
MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur
URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
A Learner’s Guide

2.  In Euclidean space lattices are infinite (infinite array)
A lattice is also called a Space Lattice (or even Bravais Lattice in some contexts)
An array of points in space such that every point has identical surroundings
 This automatically implies two properties of lattices
 In Euclidean space lattices are infinite (infinite array)
 Lattices ‘have translational periodicity’ or
Translationally periodic arrangement of points in space is called a lattice*
We can have 1D, 2D or 3D arrays (lattices)
The motif associated with these lattices can themselves be 1D, 2D or 3D ‘entities’.
* this definition arises naturally from the first definition.
Note: points are drawn with finite size for clarity  in reality they are 0D (zero dimensional)

3. 1D Lattices

4. 1D Lattices Construction of a 1D lattice
Let us construct a 1D lattice starting with two points
These points are shown as ‘finite’ circles for better ‘visibility’!
The point on the right has one to the left and hence by the requirement of identical surrounding the one of the left should have one more to the left
By a similar argument there should be one more to the left and one to the right
This would lead to an infinite number of points  
The infinity on the sides would often be left out from schematics
In 1D spherical space a lattice can be finite!

5. Click here to see how symmetry operators generate the 1D lattice
1D Lattices
Starting with a point the lattice translation vector (basis vector) can generate the lattice
In 1D there is only one kind of lattice.
This lattice can be described by a single lattice parameter (a).
In 1D Mirror  2-fold  Inversion. (The mirror and the 2-fold axis reduce to a points in 1D). (Shown below for a two line segment object).
To obtain a 1D crystal this lattice has to be decorated with a motif.
The unit cell for this lattice is a line segment of length a.  
Click here to see how symmetry operators generate the 1D lattice
Note: Basis vector should not be confused with the basis ( the motif)

6. How can make some 1-D crystals out of the lattice we have constructed
Click here

7. 2D Lattices

8. This aspect can be quite confusing
2D Lattices
2D lattices can be generated with two basis vectors
They are infinite in two dimensions
There are five distinct 2D lattices: 1 Square 2 Rectangle 3 Centered Rectangle 4 120 Rhombus 5 Parallelogram (general)
This aspect can be quite confusing
Note that in the classification of lattices, we are considering the shape of the unit cell and the disposition of lattice points with respect to that unit cell (i.e., “are there a lattice points only in the corners?”, “is there lattice point at the centre also?”).
However, at the heart of the classification is the symmetry of the lattice.
To simplify matters:
In this set of slides we will NOT consider symmetries with translation built into them (e.g. glide reflection)

9. 2D Lattices b  a Two distances: a, b One angle: 
There are three lattice parameters which describe this lattice
One angle: 
Two basis vectors generate the lattice
= 90 in the current example

10. Four (4) Unit Cell shapes in 2D can be used for 5 lattices as follows:  Square  (a = b,  = 90)  Rectangle  (a, b,  = 90)  120 Rhombus  (a = b,  = 120)  Parallelogram (general)  (a, b, )
It is clear some of them require more parameters to describe than others
Some of them have special constraints on the angle
Can we put them in some order?
The next slide defines a parameter called ‘terseness’ to order them.

11. c = number of constraints (positive  “= some number“)
Progressive relaxation of the constraints on the lattice parameters amongst the FIVE 2D lattice shapes
p’ = number of independent parameters = (p  e) (discounting the number of =)
c = number of constraints (positive  “= some number“)
t = terseness = (p  c) (is a measure of the ‘expenditure’ on the parameters
Increasing number t
Square (p’ = 2, c = 2, t = 1) a = b  = 90º
Rhombus (p’ = 2, c = 2, t = 1) a = b  = 120º
Note how the Square and the Rhombus are in the same level
Rectangle (p’ = 3, c = 1 , t = 2) a  b  = 90º
E.g. for Square: there are 3 parameters (p) and 1 “=“ amongst them (e)
 p’ = (p  e) = (3  1) = 2
Parallelogram (p’ = 3, c = 0 , t = 3) a  b 

12. Now let us consider the 5 lattices one by one

13. 1 Square Lattice 4mm 4mhmd Symmetry Lattice parameters: a = b,  = 90
Unit Cell with Symmetry elements (rotational) overlaid 1
Square Lattice
Rotational + Mirrors
Symmetry
4mm
Lattice parameters: a = b,  = 90
4mhmd
Note that these vectors are translational symmetry operators (i.e. act repeatedly!)
They are NOT ‘mere’ vectors!
Why put rotational symmetry elements onto a lattice?
(aren’t lattices built just out of translation?)

14. 4mm A note on the symmetry
Note that the peridicty of the lattice is a & b  but the periodicity of the mirrors along x, y are a/2 and b/2
Rotational + Mirrors
Symmetry
4mm
This (4mm) is the symmetry of the square lattice
Crystals based on the square lattice can have lower symmetry than the lattice itself
If the crystal based on the square lattice has 4mm or 4 symmetry then the crystal will be called a Square Crystal (else not)

15. Unit Cell with Symmetry elements (rotational) overlaid2
Rectangle Lattice
Rotational + Mirrors
2mm
Lattice parameters: a, b,  = 90
The shortest lattice translation vector (a < b)

16. Unit Cell with Symmetry elements (rotational) overlaid3
Centred Rectangle Lattice
Lattice parameters: a, b,  = 90
Continued…

17. Shape of Unit Cell does not determine the lattice or the crystal!!
Centred Rectangular Lattice
Rotational + Mirrors
2mm
We have chosen a different unit cell but this does not change the structure!
 It still remains a centred rectangular lattice
Shape of Unit Cell does not determine the lattice or the crystal!!
We will see the utility of the shortest lattice translation vector in the topic on dislocations

18. Unit Cell with Symmetry elements (rotational) overlaid4
120 Rhombus Lattice
Lattice parameters: a = b,  = 120
Q: I have seen a different representation of the same unit cell WITHOUT the 6-folds. How come?
Continued…

19. 120 Rhombus Lattice
Rotational + Mirrors
6mm

20. The Hexagon shaped cell
1/3 contribution to cell  1/3 6 = 2
1 (full) contribution to cell
Often one might see a cell in the form of a hexagon:
This is not a conventional cell (as it is not in the shape of a parallelogram)
This is actually a combination of 3 cells
This cell brings out the hexagonal symmetry of the lattice
It is triply non-primitive (3 lattice points per cell)

21. Unit Cell with Symmetry elements overlaid5
Parallelogram Lattice 2
Lattice parameters: a, b,   90
Lattice parameters: a, b,   90
There are no mirrors in parallelogram lattice

22. Summary of 2D lattices   (a = b ,  = 90) (a  b,  = 90)
Symmetry
Shape of UC
Lattice Parameters
1. Square
4mm
(a = b ,  = 90)
2. Rectangle
2mm
(a  b,  = 90)
3. Centred Rectangle
"
4. 120 Rhombus
6mm
3. 120 Rhombus
(a = b,  = 120)
5. Parallelogram 2
4. Parallelogram
(a  b,  general value)
Lattice
Simple
Centred
Square  
Rectangle
120 Rhombus
Parallelogram
Every lattice that you can construct is present somewhere in the list  the issue is where to put them!
Shows the equivalence

23. Why are some of the possible 2D lattices missing?
We had seen that there is a rectangle lattice and a centred rectangle lattice.
The natural question which comes to mind is that why are there no centred square, centred rhombus and centred parallelogram lattices?
We have already answered the question regarding the centred square lattice. (However, we will repeat the answer here again).
We will also answer the question for the other cases now.

24. This is nothing but a square lattice viewed at 45!
The case of the centred square lattice
Centred square lattice = Simple square lattice
4mm
Note that the symmetries of are that of the square lattice
Based on size the smaller blue cell (with half the area) is preferred
This is nothing but a square lattice viewed at 45!
Hence this is not a separate case

25. The case of the centred rhombus lattice
Centred rhombus lattice = Simple rectangle lattice
Note that the symmetries of the centred rhombus lattice are identical to the rectangle lattice (and not to the rhombus lattice)
Based on size the smaller green cell (with half the area) is preferred
Hence this is not a separate case

26. 2 The case of the centred parallelogram lattice
Centred parallelogram lattice = Simple parallelogram lattice 2
Note that the symmetries are that of the parallelogram lattice
Based on size the smaller green cell (with half the area) is preferred
Hence this is not a separate case

27. How can make some 2-D crystals out of the lattices we have constructed
Click here

28. 3D Lattices

29. 3D Lattices 3D lattices can be generated with three basis vectors
They are infinite in three dimensions
3 basis vectors generate a 3D lattice
The unit cell of a general 3D lattice is described by 6 numbers (in special cases all these numbers need not be independent)  6 lattice parameters  3 distances (a, b, c)  3 angles (, , )
A derivation of the 14 Bravais lattices or the existence of 7 crystal systems will not be shown in this introductory course

30. There are 14 distinct 3D lattices which come under 7 Crystal Systems  The BRAVAIS LATTICES (with shapes of unit cells as) :  Cube  (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)  Parallelepiped (Equilateral, Equiangular) (Trigonal)  (a = b = c,  =  =   90)  Parallelogram Prism (Monoclinic)  (a  b  c,  =  = 90  )  Parallelepiped (general) (Triclinic)  (a  b  c,     )
To restate: the 14 Bravais lattices have 7 different Symmetries (which correspond to the 7 Crystal Systems)

31. (a = b = c,  =  =  = 90) (a = b  c,  =  =  = 90)
Shape of UC
Used as UC for crystal:
Lattice Parameters
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)
Parallelepiped (Equilateral, Equiangular)
Trigonal
(a = b = c,  =  =   90)
Parallelogram Prism
Monoclinic
(a  b  c,  =  = 90  )
Parallelepiped (general)
Triclinic
(a  b  c,     )

32. Important Note: do NOT confuse the shape of the unit cell with the crystal systems
(as we have already seen we can always choose a different unit cell for a given crystal)

33. Click here to visualize a step by step construction
Building a 3D cubic lattice
Click here to visualize a step by step construction
a = b = c,  =  =  = 90
Each vertex of the cube is a lattice point (no points are shown for clarity)
Actually this is a part of the cubic lattice remember lattices are infinite!

34. 6 lattice parameters  3 distances (a, b, c)  3 angles (, , )
A General Lattice in 3D
6 lattice parameters  3 distances (a, b, c)  3 angles (, , )
a  b  c,     
In special cases some of these numbers may be equal to each other (e.g. a = b) or equal to a special number (e.g.  = 90)
(hence we may not require 6 independent numbers to describe a lattice)
Any general parallelepiped is space filling
Click here to know more about

35. An important property of a lattice
Bravais Lattice: various viewpoints
A lattice is a set of points constructed by translating a single point in discrete steps by a set of basis vectors. In three dimensions, there are 14 unique Bravais lattices (distinct from one another in that they have different space groups) in three dimensions. All crystalline materials recognized till now fit in one of these arrangements.
In geometry and crystallography, a Bravais lattice is an infinite set of points generated by a set of discrete translation operations.
A Bravais lattice looks exactly the same no matter from which point in the lattice one views it.
Bravais concluded that there are only 14 possible Space Lattices (with Unit Cells to represent them). These belong to 7 Crystal systems.
There are 14 Bravais Lattices which are the Space Group symmetries of lattices
An important property of a lattice
A derivation of the 14 Bravais lattices or the existence of 7 crystal systems will not be shown in this introductory course

36. Time to fasten you seat-belts the next few slides will take you on a 10 g-force dive

37. IMPORTANT Crystals and Crystal Systems are defined based on Symmetry
& NOT
Based on the Geometry of the Unit Cell
Example
Cubic Crystal  Does NOT imply a = b = c &  =  =   It implies the existence of two 3-fold axis in the structure
Intrigued!
Want to Know More?

38. If lattices are based on just translation (Translational Symmetry (t))
IMPORTANT
If lattices are based on just translation (Translational Symmetry (t))
then how come other Symmetries (especially rotational) come into the picture while choosing the Crystal System & Unit Cell for a lattice?
Example
Why do we say that End Centred Cubic Lattice does not exist?  Isn’t it sufficient that a = b = c &  =  =  to call something cubic? (why do we put End Centred Cubic in Simple Tetragonal?)
Answer
The issue comes because we want to put 14 Bravais lattices into 7 boxes (the 7 Crystal Systems; the Bravais lattices have 7 distinct symmetries) and further assign Unit Cells to them
The Crystal Systems are defined based on Symmetries (Rotational, Mirror, Inversion etc.  forming the Point Groups) and NOT on the geometry of the Unit Cell
The Choice of Unit Cell is based on Symmetry & Size (& Convention) (in practice the choice of unit cell is left to us!  but what we call the crystal is not!!)
Continued…

39.  hang on!  some up-coming examples will make things CRYSTAL clear
ONCE MORE:
 When we say End Centred Cubic  End Centred is a type of Lattice (based on translation) & Cubic is a type of Crystal (based on other symmetries)
&
Cubic also refers to a shape of Unit Cell (based on lattice parameters)
AND:
 To confuse things further  Cubic crystals can have lower symmetry than the cubic lattice (e.g. Cubic lattices always have 4-fold axis while Cubic Crystals may not have 4-fold axes)
Feeling lost!?!
 hang on!  some up-coming examples will make things CRYSTAL clear

40. A shape of Unit Cell (based on lattice parameters)
To emphasize:
 The word Cubic (e.g. in a cubic crystal) refers to 3 things  A type of Lattice (based on translation) & A type of Crystal (based on other symmetries)
&
A shape of Unit Cell (based on lattice parameters)
Hence the confusion!!

41. Lattices have the highest symmetry
(Which is allowed for it)  Crystals based on the lattice can have lower symmetry
Another IMPORTANT point
Click here to know more

42. 14 Bravais Lattices divided into 7 Crystal Systems
We will take up these cases one by one (hence do not worry!)
A Symmetry based concept
Some guidelines apply
‘Translation’ based concept
Crystal System
Shape of UC
Bravais Lattices P I F C 1
Cubic
Cube  2
Tetragonal
Square Prism (general height) 3
Orthorhombic
Rectangular Prism (general height) 4
Hexagonal
120 Rhombic Prism 5
Trigonal
Parallopiped (Equilateral, Equiangular) 6
Monoclinic
Parallogramic Prism 7
Triclinic
Parallelepiped (general) P
Primitive I
Body Centred F
Face Centred C
A/B/C- Centred
Why are some of the entries missing?  Why is there no C-centred cubic lattice?  Why is the F-centred tetagonal lattice missing?
 ….?
Continued…

43. Arrangement of lattice points in the Unit Cell. & No
Arrangement of lattice points in the Unit Cell & No. of Lattice points / Cell
Position of lattice points
Effective number of Lattice points / cell 1 P
8 Corners
= [8  (1/8)] = 1 2 I
8 Corners body centre
= [1 (for corners)] + [1 (BC)] = 2 3 F
8 Corners +
6 face centres
= [1 (for corners)] + [6  (1/2)] = 4 4 A/ B/ C
8 corners + 2 centres of opposite faces
= [1 (for corners)] + [2  (1/2)] = 2

44. P I F C 1
Cubic
Cube  I P
Symmetry of Cubic lattices F
Lattice point

45. I P P I F C 2 Tetragonal Square Prism (general height) 
Symmetry of Tetragonal lattices

46. I P F C P I F C 3 Orthorhombic Rectangular Prism (general height) 
One convention I P
Note the position of ‘a’ and ‘b’ F
Symmetry of Orthorhombic lattices C
Is there a alternate possible set of unit cells for OR?
Why is Orthorhombic called Ortho-’Rhombic’?

47. Symmetry of Hexagonal latticesP I F C 4
Hexagonal
120 Rhombic Prism 
A single unit cell (marked in blue) along with a 3-unit cells forming a hexagonal prism
Symmetry of Hexagonal lattices
What about the HCP?
(Does it not have an additional atom somewhere in the middle?)
Note: there is only one type of hexagonal lattice (the primitive one)

48. Note the position of the origin and of ‘a’, ‘b’ & ‘c’5
Trigonal
Parallelepiped (Equilateral, Equiangular) 
Rhombohedral
Note the position of the origin and of ‘a’, ‘b’ & ‘c’
Symmetry of Trigonal lattices

49. Some times an alternate hexagonal cell is used instead of the Trigonal Cell

50. (keeping the edge length of the cube constant)
A trigonal cell can be produced from a cubic cell by pulling along [111] (the body diagonal)
(keeping the edge length of the cube constant)
Video: Cubic to Trigonal UC

51. Note the position of ‘a’, ‘b’ & ‘c’6
Monoclinic
Parallogramic Prism 
One convention
Note the position of ‘a’, ‘b’ & ‘c’
Symmetry of Monoclinic lattices

52. P I F C 7
Triclinic
Parallelepiped (general) 
Symmetry of Triclinic lattices

53. Let us make some 3-D crystals
Click here
Let us make some 3-D crystals

54. The Xs themselves form an equivalent lattice
An important property of a lattices
This aspect might seem trivial here but is very useful to remember!
If one sits at any lattice point the space around looks identical to the person
Hence we can chart out a set of equivalent points in space (Which may or may not coincide with the lattice points) 1D
The Xs themselves form an equivalent lattice

55. 2D 3D Solved Example The Graphene Crystal
Hence, if for a given crystal (say with FCC lattice decorated with a single atom motif), the edge centre is a position of an octahedral void then the set of octahedral void positions will form a FCC lattice

56. Symmetries of the Crystal
Q: I have seen a different representation of the same unit cell WITHOUT the 6-folds. How come?
As we know lattices have the highest symmetry and hence a 120 rhombus lattice (noting that this is actually the shape of the UC) always has 6-fold symmetries
However crystals based on the lattice can have lower symmetry which includes only 3-fold symmetries
The list of crystals in 2D are (with shapes of UC):  Square  Rectangle  120 Rhombus  Parallelogram (general)
Unfortunately this does not include a crystal with 3-fold symmetry alone (which could be called TRIANGULAR  analogous to Trigonal in 3D)
Note the loss
in a mirror as well
Crystal
Symmetries of the Crystal
Hence the 120 Rhombus lattice always has 6-fold axes while crystals based on the lattice may have only 3-folds
Back
Click here
Example of a 3D analogue of this