SPACE LATTICES MATERIALS SCIENCE &ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology,

SPACE LATTICESSPACE LATTICESMATERIALS SCIENCEMATERIALS SCIENCE

&&ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: [email protected], URL: home.iitk.ac.in/~anandh

AN INTRODUCTORY E-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

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’

Space Lattice

Translationally periodic arrangement of points in space is called a lattice*

or

A lattice is also called a Space Lattice (or even Bravais Lattice in some contexts)

Note: points are drawn with finite size for clarity in reality they are 0D (zero dimensional)

* this definition arises naturally from the first definition.

We can have 1D, 2D or 3D arrays (lattices)The motif associated with these lattices can themselves be 1D, 2D or 3D ‘entities’.

1D Lattices

Construction of a 1D lattice1D Lattices

Let us construct a 1D lattice starting with two points

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

In 1D spherical space a lattice can be finite!

The infinity on the sides would often be left out from schematics

These points are shown as ‘finite’ circles for better ‘visibility’!

1D Lattices

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.

a

Starting with a point the lattice translation vector (basis vector) can generate the lattice

Note: Basis vector should not be confused with the basis ( the motif)

Click here to see how symmetry operators generate the 1D latticeClick here to see how symmetry operators generate the 1D lattice

How can make some 1-D crystals out of the lattice we have constructed

How can make some 1-D crystals out of the lattice we have constructed

Click here

2D Lattices

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)

To simplify matters:In this set of slides we will NOT consider symmetries with translation built into them (e.g. glide reflection)

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.

This aspect can be quite confusing

a

b

2D Lattices

Two basis vectors generate the lattice

There are three lattice parameters which describe this latticeOne angle:

Two distances: a, b

= 90 in the current example

ba

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.

Square (p’ = 2, c = 2, t = 1)a = b

= 90º

Rectangle (p’ = 3, c = 1 , t = 2) a b = 90º

Parallelogram (p’ = 3, c = 0 , t = 3) a b

Progressive relaxation of the constraints on the lattice parameters amongst the FIVE 2D lattice shapes

Rhombus (p’ = 2, c = 2, t = 1)a = b

= 120º

Incr

easi

ng n

umbe

r t

Note how the Square and the Rhombus are in the same level

• 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

E.g. for Square: there are 3 parameters (p) and 1 “=“ amongst them (e) p’ = (p e) = (3 1) = 2

Now let us consider the 5 lattices one by one

Square Lattice

Why put rotational symmetry elements onto a lattice?(aren’t lattices built just out of translation?)

Why put rotational symmetry elements onto a lattice?(aren’t lattices built just out of translation?)

b a

Lattice parameters: a = b, = 90

Unit Cell with Symmetry elements (rotational) overlaid

Rotational + Mirrors

1

4mm

Symmetry

Note that these vectors are translational symmetry

operators (i.e. act repeatedly!)They are NOT ‘mere’ vectors!

4mhmd


4mm

Symmetry

A note on the symmetry

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)

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

Rectangle Lattice

The shortest lattice translation vector (a < b)

Lattice parameters: a, b, = 90



2

2mm

Centred Rectangle Lattice

Lattice parameters: a, b, = 90

Continued…


3


We will see the utility of the shortest lattice translation vector in the topic on dislocations

( )

2

a b

Centred Rectangular Lattice

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!!

2mm

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…

4

a b


120 Rhombus Lattice

6mm

The Hexagon shaped 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)

1/3 contribution to cell 1/3 6 = 2

1 (full) contribution to cell

Parallelogram Lattice

Lattice parameters: a, b, 90Lattice parameters: a, b, 90

Unit Cell with Symmetry elements

overlaid

There are no mirrors in parallelogram lattice

5

2

Lattice Symmetry Shape of UC Lattice Parameters

1. Square 4mm 1. Square (a = b , = 90) 2. Rectangle 2mm 2. Rectangle (a b, = 90)

3. Centred Rectangle 2mm " (a b, = 90)

4. 120 Rhombus 6mm 3. 120 Rhombus (a = b, = 120)

5. Parallelogram 2 4. Parallelogram (a b, general value)

Summary of 2D lattices

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

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.

This is nothing but a square lattice viewed at 45!

Centred square lattice = Simple square lattice

The case of the centred square lattice

Hence this is not a separate case

Based on size the smaller blue cell (with half the area) is preferred

Note that the symmetries of are that of the square lattice

4mm

The case of the centred rhombus lattice

Centred rhombus lattice = Simple rectangle lattice

Based on size the smaller green cell (with half the area) is preferred


Note that the symmetries of the centred rhombus lattice are identical to the rectangle lattice (and not to the rhombus lattice)

The case of the centred parallelogram lattice

Centred parallelogram lattice = Simple parallelogram lattice

Based on size the smaller green cell (with half the area) is preferred


Note that the symmetries are that of the parallelogram lattice

2

How can make some 2-D crystals out of the lattices we have constructed

How can make some 2-D crystals out of the lattices we have constructed

Click here

3D Lattices

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

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)

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, )

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)

Building a 3D cubic lattice Click here to visualize a step by step construction

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!

a = b = c, = = = 90

A General Lattice in 3D

6 lattice parameters 3 distances (a, b, c) 3 angles (, , )

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

a b c,

Click here to know more

about

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

A derivation of the 14 Bravais lattices or the existence of 7 crystal systems will not be shown in this introductory course

Bravais Lattice: various viewpoints

An important property of a lattice

Time to fasten you seat-belts the

next few slides will take you on a 10 g-force dive

IMPORTANT

Crystals and Crystal Systems are defined

based on Symmetry

& NOT

Based on the Geometry of the Unit Cell

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?

Example

IMPORTANTIf 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?

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?)

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!!)

Answer

Example

Continued…

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)

hang on! some up-coming examples will make things CRYSTAL clear

Feeling lost!?!

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!!

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 moreClick here to know more

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)

Why are some of the entries missing? Why is there no C-centred cubic lattice? Why is the F-centred tetagonal lattice missing? ….?

14 Bravais Lattices divided into 7 Crystal Systems

Continued…

P Primitive

I Body Centred

F Face Centred

C A/B/C- Centred

A Symmetry based concept

We will take up these cases one by one(hence do not worry!)

‘Translation’ based conceptSome guidelines apply

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 I8 Corners + 1 body centre

= [1 (for corners)] + [1 (BC)] = 2

3 F8 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

1 Cubic Cube P I F C

Lattice point

PPII

FF

a b c 90

4 23

m m

Symmetry of Cubic latticesSymmetry of Cubic lattices

P I F C

2 Tetragonal Square Prism (general height)

II

PP

a b c

90

Symmetry of Tetragonal latticesSymmetry of Tetragonal lattices

4 2 2

m m m

P I F C

3 Orthorhombic Rectangular Prism (general height)

PPII

FFCC

a b c

90

Symmetry of Orthorhombic latticesSymmetry of Orthorhombic lattices

2 2 2

m m m

Note the position of ‘a’ and ‘b’

a b c One convention

Why is Orthorhombic called Ortho-’Rhombic’?Why is Orthorhombic called Ortho-’Rhombic’?Is there a alternate possible set of unit cells for OR?Is there a alternate possible set of unit cells for OR?

P I F C

4 Hexagonal 120 Rhombic Prism

What about the HCP?(Does it not have an additional atom somewhere in the middle?)

What about the HCP?(Does it not have an additional atom somewhere in the middle?)

A single unit cell (marked in blue) along with a 3-unit cells forming a

hexagonal prism

Note: there is only one type of hexagonal lattice (the primitive one)

a b c

90 , 120

Symmetry of Hexagonal latticesSymmetry of Hexagonal lattices

6 2 2

m m m

P I F C

5 Trigonal Parallelepiped (Equilateral, Equiangular)

90

a b c

Symmetry of Trigonal latticesSymmetry of Trigonal lattices

Rhombohedral

23

m

Note the position of the origin and of ‘a’, ‘b’ & ‘c’

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

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 UCVideo: Cubic to Trigonal UC

P I F C

6 Monoclinic Parallogramic Prism

90

a b c

Symmetry of Monoclinic latticesSymmetry of Monoclinic lattices

2

m

a b c

Note the position of ‘a’, ‘b’ & ‘c’

One convention

P I F C

7 Triclinic Parallelepiped (general)

a b c

Symmetry of Triclinic latticesSymmetry of Triclinic lattices

1

Let us make some 3-D crystalsLet us make some 3-D crystals Click here

An important property of a lattices

If one sits at any lattice point the space around looks identical to the person

This aspect might seem trivial here but is very useful to remember!

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

2D

3D

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

SolvedExample

The Graphene Crystal



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)

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

Note the loss in a mirror as well

BackBackClick here Example of a 3D analogue of this

Documents

SPACE LATTICES MATERIALS SCIENCE &ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology,