Constructing crystals in 1D, 2D & 3D

Understanding them using the language of: Lattices Symmetry

LET US MAKE SOME CRYSTALSLET US MAKE SOME CRYSTALS

http://cst-www.nrl.navy.mil/lattice/index.htmlhttp://cst-www.nrl.navy.mil/lattice/index.htmlAdditional consultations

MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: anandh@iitk.ac.in, 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

1D

Some of the concepts are best illustrated in lower dimensions hence we shall construct some 1D and 2D crystals before jumping into 3D

A strict 1D crystal = 1D lattice + 1D motif The only kind of 1D motif is a line segment(s) (though in principle a collection

of points can be included).

Making a 1D Crystal

Lattice

Motif

Crystal

=

+

Other ways of making the same crystal We had mentioned before that motifs need not sit on the lattice point- they are

merely associated with a lattice point Here is an example:

Note:For illustration purposes we will often relax this strict requirement of a 1D motif We will put 2D motifs on 1D lattice to get many of the useful concepts across

1D lattice +2D Motif*

*looks like 3D due to the shading! It has been shown that 1D crystals cannot be stable!!

Each of these atoms contributes ‘half-atom’ to the unit cell

Time to brush-up some symmetry concepts before going aheadTime to brush-up some symmetry concepts before going ahead

Lattices have the highest symmetry

(Which is allowed for it)

Crystals based on the lattice can have lower symmetry

In the coming slides we will understand this IMPORTANT point

If any of the coming 7 slides make you a little uncomfortable – you can skip them(however, they might look difficult – but they are actually easy)If any of the coming 7 slides make you a little uncomfortable – you can skip them(however, they might look difficult – but they are actually easy)

Progressive lowering of symmetry in an 1D lattice illustration using the frieze groups

Consider a 1D lattice with lattice parameter ‘a’

a

Unit cellAsymmetric

Unit

The unit cell is a line segment in 1D shown with a finite ‘y-direction’ extent for clarity and for understating some of the crystals which are coming-up

Asymmetric Unitis that part of the structure (region of space), which in combination with the symmetries (Space Group) of the lattice/crystal gives the complete structure (either the lattice or the crystal)

(though typically the concept is used for crystals only)The concept of the Asymmetric Unit will become clear in the coming slides

As we had pointed out we can understand some of the concepts of crystallography better by ‘putting’ 2D motifs on a 1D lattice. These kinds of patterns are called Frieze groups and there are 7 types of them (based on symmetry).

Three mirror planes The intersection points of the mirror planesgive rise to redundant inversion centres (i)

mmm

mirror

This 1D lattice has some symmetries apart from translation. The complete set is: Translation (t)

Horizontal Mirror (mh)

Vertical Mirror at Lattice Points (mv1)

Vertical Mirror between Lattice Points (mv2)

t mh mv1 mv2 mmmOr more concisely

Note:

The symmetry operators (t, mv1, mv2) are enough to generate the lattice

But, there are some redundant symmetry operators which develop due to their operation

In this example they are 2-fold axis or Inversion Centres (and for that matter mh)

mv1 mv2

mh

Note of Redundant Symmetry Operators

Three mirror planes Redundant inversion centres

mmm

mirror

Redundant 2-fold axes

It is true that some basic set of symmetry operators (set-1) can generate the structure (lattice or crystal) It is also true that some more symmetry operators can be identified which were not envisaged in the basic set

(called ‘redundant’) But then, we could have started with different set of operators (set-2) and call some of the operators used in set-1 as

redundant the lattice has some symmetries which we call basic and which we call redundant is up to us!

How do these symmetries create this lattice?How do these symmetries create this lattice?

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

t

Asymmetric Unit

We have already seen that Unit Cell is the least part of the structure which can be

used to construct the structure using translations (only).

Asymmetric Unit is that part of the structure (usually a region of space), which in

combination with the symmetries (Space Group) of the lattice/crystal gives the

complete structure (either the lattice or the crystal) (though typically the concept is used for crystals

only)

Simpler phrasing: It is the least part of the structure (region of space) which can be used to

build the structure using the symmetry elements in the structure (Space Group)

Asymmetric Unit

Lattice point

Unit cell

+ +

Which is theUnit Cell

t +Lattice

If we had started with the asymmetric unit of a crystal then we would have obtained a crystal instead of a lattice

mv2 mh

Decoration of the lattice with a motif may reduce the symmetry of the crystal

Decoration with a “sufficiently” symmetric motif does not reduce the symmetry of the lattice

Instead of the double headed arrow we could have used a circle (most symmetrical object possible in 2D)

1

2

mmm

mm

Decoration with a motif which is a ‘single headed arrow’ will lead to the loss of 1 mirror plane

mirror

t

t

Lattice points

Not a lattice point

g

Presence of 1 mirror plane and 1 glide reflection plane, with a redundant inversion centrethe translational symmetry has been reduced to ‘2a’

2 inversion centres

ii

mg3

4

glide reflectionmirror

t

t

g

1 mirror plane

m

g

1 glide reflection translational symmetry of ‘2a’

No symmetry except translation

5

6

7

glide reflectionmirror

t

t

t

2D

Video: Making 2D crystal using discsVideo: Making 2D crystal using discs

Some aspects we have already seen in 1D but 2D many more concepts can be clarified in 2D

2D crystal = 2D lattice + 2D motif As before we can relax this requirement and put 1D or 3D motifs!

Making a 2D Crystal

Continued

We shall make various crystals starting with a 2D lattice and putting motifs and we shall analyze the crystal which has thus been created

+

Square Lattice

Circle Motif

=

Square Crystal

Continued…

+

Square Lattice

Circle Motif

=

Square Crystal

Continued…

Symmetry of the lattice and crystal identical Square Crystal

Including mirrors4mm

Important Note

>Symmetry of the Motif Symmetry of the lattice

Hence Symmetry of the lattice and Crystal identical (symmetry of the lattice is preserved) Square Crystal

Any fold rotational axis allowed! (through the centre of the circle) Mirror in any orientation passing through the centre allowed! Centre of inversion at the centre of the circle

Symmetry of the Motif

Funda Check What do the ‘adjectives’ like square mean in

the context of the lattice, crystal etc?

Let us consider the square lattice and square crystal as before. In the case of the square lattice → the word square refers to the symmetry of the lattice

(and not the geometry of the unit cell!). In the case of the square crystal → the word square refers to the symmetry of the crystal

(and not the geometry of the unit cell!)

+Square Lattice Square Motif = Square Crystal

Continued…

Important Note

=Symmetry of the Motif Symmetry of the lattice

Hence Symmetry of the lattice and Crystal identical

Square Crystal

Continued…

4mm symmetry

Symmetry of the Motif

4mm

Important Rule

If the

Symmetry of the Motif Symmetry of the Lattice

The Symmetry of the lattice and the Crystal are identical

i.e. Symmetry of the lattice is NOT lowered but is preserved

Common surviving symmetry determines the crystal system

In a the above example we are assuming that the square is favourably orientedAnd that there are symmetry elements common to the lattice and the motif

+Square Lattice Square Motif = Square CrystalRotated

4

Funda Check How do we understand the crystal made out of

rotated squares?

Is the lattice square → YES (it has 4mm symmetry) Is the crystal square → YES (but it has 4 symmetry → since it has at least a 4-fold

rotation axis- we classify it under square crystal- we could have called it a square’ crystal or something else as well!)

Is the ‘preferred’ unit cell square → YES (it has square geometry) Is the motif a square → YES (just so happens in this example- though rotated wrt to the

lattice)

Infinite other choices of unit cells are possible → click here to know more

+

Square Lattice

Triangle Motif

=

Rectangle Crystal

Continued…

Symmetry of the lattice and crystal different NOT a Square Crystal

Square Crystal

Here the word square does not imply the shape in the usual sense

m

Symmetry of the structure

Only one set of parallel mirrors left

m

Important Note

<Symmetry of the Motif Symmetry of the lattice

The symmetry of the motif determines the symmetry of the crystal it is lowered to match the symmetry of the motif (common symmetry elements between the lattice and motif which survive) (i.e. the crystal structure has only the symmetry of the motif left: even though the lattice started of with a higher symmetry)

Rectangle Crystal (has no 4-folds but has mirror)

Mirror 3-fold

Symmetry of the Motif

Continued…

Note that the word ‘Rectangle’ denotes the symmetry of the crystal and NOT the shape of the UC

Important Rule

If the

Symmetry of the Motif < Symmetry of the Lattice

The Symmetry of the lattice and the Crystal are NOT identical

i.e. Symmetry of the lattice is lowered with only common symmetry elements

Funda Check How do we understand the crystal made out of

triangles?

Is the lattice square → YES (it has 4mm symmetry) Is the crystal square → NO (it has only m symmetry → hence it is a rectangle crystal) Is the unit cell square → YES (it has square geometry) (we have already noted that other shapes of unit cells are also possible)

Is the motif a square → NO (it is a triangle!)

+Square Lattice Triangle Motif = Parallelogram CrystalRotated

Crystal has No symmetry except translational symmetry as there are no symmetry elements common to the lattice and the motif (given its orientation)

Some more twists

+

Square Lattice

Random shaped Motif

=

Parallelogram Crystal

Symmetry of the lattice and crystal different NOT Square Crystal

Square Crystal

In Single Orientation

Except translation

+

Square Lattice

Random shaped Object

=

Amorphous Material(Glass)

Symmetry of the lattice and crystal different NOT even a Crystal

Square Crystal

Randomly oriented at each point

Funda Check Is there not some kind of order visible in the

amorphous structure considered before? How can understand this structure then?

YES, there is positional order but no orientational order. If we ignore the orientational order (e.g. if the entities are rotating constantly- and the

above picture is a time ‘snapshot’- then the time average of the motif is ‘like a circle’)

Hence, this structure can be considered to be a ‘crystal’ with positional order, but without orientational order!

Click here to know moreClick here to know more

CrystalHighest

Symmetry Possible

Other symmetries possible

Lattice Parameters(of conventional unit cell)

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

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

4. Parallelogram 2 1 (a b, general value)

Summary of 2D Crystals

Click here to see a summary of 2D lattices that these crystals are built onClick here to see a summary of 2D lattices that these crystals are built on

From the previous slides you must have seen that crystals have:

CRYSTALS

Orientational Order Positional Order

Later on we shall discuss that motifs can be:

MOTIFS

Geometrical entities Physical Property

In practice some of the strict conditions imposed might be relaxed and we might call a something a crystal even if

Orientational order is missing There is only average orientational or positional order Only the geometrical entity has been considered in the definition of the crystal and not

the physical property

3D

A strict 3D crystal = 3D lattice + 3D motif We have 14 3D Bravais lattices to chose from As an intellectual exercise we can put 1D or 2D motifs in a 3D lattice as well

(we could also try putting higher dimensional motifs like 4D motifs!!) We will illustrate some examples to understand some of the basic concepts

(most of which we have already explained in 1D and 2D)

Making a 3D Crystal

+Simple Cubic (SC) Lattice Sphere Motif

=

Simple Cubic Crystal

Graded Shading to give 3D effect

Unit cell of the SC lattice

If these spheres were ‘spherical atoms’ then the atoms would be touching each other The kind of model shown is known as the ‘Ball and Stick Model’

+Body Centred Cubic (BCC) Lattice Sphere Motif

=

Body Centred Cubic Crystal

Note: BCC is a lattice and not a crystalSo when one usually talks about a BCC crystal what is meant is a BCC lattice decorated with a mono-atomic motif

Unit cell of the BCC lattice

Atom at (½, ½, ½)

Atom at (0, 0, 0)

Space filling model

Central atom is coloured differently for better visibility

To know more about Close Packed Crystals

click here

To know more about Close Packed Crystals

click here

+Face Centred Cubic (FCC) Lattice Sphere Motif

=

Cubic Close Packed Crystal(Sometimes casually called the FCC crystal)

Note: FCC is a lattice and not a crystalSo when one talks about a FCC crystal what is meant is a FCC lattice decorated with a mono-atomic motif

Point at (½, ½, 0)

Point at (0, 0, 0)

Unit cell of the FCC lattice

Space filling model

Close Packed implies CLOSEST

PACKED

More views

All atoms are identical- coloured differently for better visibility

+Face Centred Cubic (FCC) Lattice Two Ion Motif

=

NaCl Crystal

Note: This is not a close packed crystal Has a packing fraction of 0.67

Na+ Ion at (½, 0, 0)Cl Ion at (0, 0, 0)

+Face Centred Cubic (FCC) Lattice Two Carbon atom Motif(0,0,0) & (¼, ¼, ¼)

=Diamond Cubic Crystal

Note: This is not a close packed crystal

It requires a little thinking to convince yourself that the two atom motif actually sits at all lattice points!

There are no close packed directions in this crystal either!

Tetrahedral bonding of C (sp3 hybridized)

+Face Centred Cubic (FCC) Lattice Two Ion Motif

=

NaCl Crystal

Note: This is not a close packed crystal Has a packing fraction of 0.67

The Na+ ions sit in the positions (but not inside) of the octahedral voids in an CCP crystal click here to know more

SolvedExample

Na+ Ion at (½, 0, 0)

Cl Ion at (0, 0, 0)

NaCl crystal: further points

This crystal can be considered as two interpenetrating FCC sublattices

decorated with Na+ and Cl respectively

Click here: Ordered CrystalsClick here: Ordered Crystals

Inter-penetration of just 2 UC are shown here

Coordination around Na+ and Cl ions

More views

The blue outline is NO longer a Unit Cell!!

Amorphous Material (Glass)(having no symmetry what so ever)

Triclinic Crystal(having only translational symmetry)

Now we present 3D analogues of the 2D cases considered before:those with only translational symmetry and those without any symmetry

We have seen that the symmetry (and positioning) of the motif plays an important role in the symmetry of the crystal.

Let us now consider some examples of Molecular Crystals to see practical examples of symmetry of the motif vis a vis the symmetry of the crystal.(click here to know more about molecular crystals → Molecular Crystals)

It is seen that there is no simple relationship between the symmetry of the molecule and the symmetry of the crystal structure. As noted before: Symmetry of the molecule may be retained in crystal packing (example of hexamethylenetetramine) or May be lowered (example of Benzene)

Making Some Molecular Crystals

6 12 4

1 12 4

1 1 16 6

60

Hexamethylenetetramine (C H N ) 43m I43m 43m

2 2 2 2 2 2 2Ethylene (C H ) P

m m m n n m m6 2 2 2 2 2

Benzene (C H ) P 1m m m b c a

2 4 2 4 2Fullerene (C ) 35 3 3F

m m m m m

Click here → connection between geometry and symmetryClick here → connection between geometry and symmetry

From reading some of the material presented in the chapter one might get a feeling that there is no connection between ‘geometry’ and ‘symmetry’. I.e. a crystal made out of lattice with square geometry can have any (given set) of symmetries.

In ‘atomic’ systems (crystals made of atomic entities) we expect that these two aspects are connected (and not arbitrary). The hyperlink below explains this aspect.

Funda Check