Transcript Making Crystals

LET US MAKE SOME CRYSTALS
 Constructing crystals in 1D, 2D & 3D
 Understanding them using the language of:
 Lattices
 Symmetry
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
Additional consultations
http://cst-www.nrl.navy.mil/lattice/index.html
1D
Making a 1D Crystal
 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).
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 ahead
In the coming slides we
will understand this
IMPORTANT point
Lattices have the highest symmetry
(Which is allowed for it)
 Crystals based on the lattice
can have lower symmetry
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)
 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).
Progressive lowering of symmetry in an 1D lattice illustration using the frieze groups
Consider a 1D lattice with lattice parameter ‘a’
Unit cell
Asymmetric
Unit
a
 Asymmetric Unit
is 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)
The concept of the
(though typically the concept is used for crystals only)
Asymmetric Unit will
become clear in the
coming slides
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
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)
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)
t mh mv1 mv2
Or more concisely
mmm
mh
mmm
Three mirror planes
mirror
mv1
mv2
The intersection points of the mirror planes
give rise to redundant inversion centres (i)
Note of Redundant Symmetry Operators
t
mmm
Three mirror planes




Redundant 2-fold axes
Redundant inversion centres
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?
mirror
Click here to see how symmetry operators generate the 1D lattice
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
+
mv2
+
mh
Lattice point
Which is the
Unit Cell
Unit cell
If we had started with the asymmetric unit of a crystal
then we would have obtained a crystal instead of a lattice
Lattice
t
+
Decoration of the lattice with a motif  may reduce the symmetry of the crystal
t
1
mmm
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)
t
2
mm
Decoration with a motif which is a ‘single headed arrow’ will lead to the loss of 1 mirror plane
mirror
Lattice points
t
3
g
mg
Not a lattice point
Presence of 1 mirror plane and 1 glide reflection plane, with a redundant inversion centre
the translational symmetry has been reduced to ‘2a’
t
ii
4
2 inversion centres
mirror
glide reflection
t
5
m
1 mirror plane
t
g
6
g
1 glide reflection translational symmetry of ‘2a’
t
7
No symmetry except translation
mirror
glide reflection
2D
Video: Making 2D crystal using discs
Making a 2D Crystal
 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!
 We shall make various crystals starting with a 2D lattice and putting motifs and
we shall analyze the crystal which has thus been created
Continued
Square Lattice
+
Circle Motif
=
Square Crystal
Continued…
Square Lattice
+
Circle Motif
=
Square Crystal
Symmetry of the lattice and
crystal identical
 Square Crystal
Including mirrors
4mm
Continued…
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
Symmetry of the Motif
 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
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
Hence Symmetry of the lattice
and Crystal identical
 Square Crystal
=
Symmetry of the lattice
4mm
Symmetry of the Motif
 4mm symmetry
Continued…
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 oriented
And that there are symmetry elements common to the lattice and the motif
Square Lattice
+
Square Motif
=
Square Crystal
Rotated
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
=
Square Crystal
Rectangle Crystal
Symmetry of the lattice and
crystal different
 NOT a Square Crystal
m
Here the word square does not imply the shape in the usual sense
Continued…
Only one set of parallel mirrors left
Symmetry of the structure
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)
Symmetry of the Motif
 Mirror
 3-fold
Note that the word ‘Rectangle’ denotes the symmetry of the
crystal and NOT the shape of the UC
Continued…
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 Crystal
Rotated
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
In Single Orientation
=
Square Crystal
Parallelogram Crystal
Symmetry of the lattice and
crystal different
 NOT Square Crystal
Except translation
Square Lattice
+
Random shaped Object
Randomly oriented at each point
=
Square Crystal
Amorphous Material
(Glass)
Symmetry of the lattice and
crystal different
 NOT even a Crystal
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 more
Summary of 2D Crystals
Highest
Symmetry
Possible
Other symmetries
possible
Lattice Parameters
(of conventional unit cell)
1. Square
4mm
4
2. Rectangle
2mm
m
(a = b ,  = 90)
(a  b,  = 90)
3. 120 Rhombus
6mm
6, 3m, 3
(a = b,  = 120)
4. Parallelogram
2
1
(a  b,  general value)
Crystal
Click 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
Making a 3D Crystal
 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)
Simple Cubic (SC) Lattice
+
Sphere Motif
Graded Shading to give 3D effect
Simple Cubic Crystal
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
To know more about
Close Packed Crystals
click here
Atom at (½, ½, ½)
Body Centred Cubic Crystal
Atom at (0, 0, 0)
Unit cell of the BCC lattice
=
Space filling model
Central atom is coloured differently for better visibility
So when one usually talks about a BCC crystal what is
Note: BCC is a lattice and not a crystal meant is a BCC lattice decorated with a mono-atomic motif
Face Centred Cubic (FCC) Lattice +
Sphere Motif
Close Packed
implies CLOSEST
PACKED
Point at (½, ½, 0)
Point at (0, 0, 0)
Cubic Close Packed Crystal
(Sometimes casually called the FCC crystal)
Unit cell of the FCC lattice
=
Space filling model
Note: FCC is a lattice and not a crystal
So when one talks about a FCC crystal what is meant
is a FCC lattice decorated with a mono-atomic motif
More views
All atoms are identical- coloured differently for better visibility
Face Centred Cubic (FCC) Lattice +
Two Ion Motif
NaCl Crystal
=
Cl Ion at (0, 0, 0)
Na+ Ion at (½, 0, 0)
Note: This is not a close packed crystal Has a packing fraction of 0.67
Face Centred Cubic (FCC) Lattice +
Two Carbon atom Motif
(0,0,0) & (¼, ¼, ¼)
=
Tetrahedral bonding of C
(sp3 hybridized)
It requires a little thinking to convince yourself that the two atom motif
actually sits at all lattice points!
Note: This is not a close packed crystal
There are no close packed directions in this crystal either!
Diamond Cubic Crystal
Face Centred Cubic (FCC) Lattice +
Two Ion Motif
NaCl Crystal
Cl Ion at (0, 0, 0)
Na+ Ion at (½, 0, 0)
=
The Na+ ions sit in the positions (but not inside) of the
octahedral voids in an CCP crystal  click here to know more
Note: This is not a close packed crystal Has a packing fraction of 0.67
Solved
Example
NaCl crystal: further points
Click here: Ordered Crystals
This crystal can be considered as two
interpenetrating FCC sublattices decorated
with Na+ and Cl respectively
Inter-penetration of just 2 UC are shown here
More views
Coordination around Na+ and Cl ions
 Now we present 3D analogues of the 2D cases considered before:
those with only translational symmetry and those without any symmetry
The blue outline is NO longer a
Unit Cell!!
Triclinic Crystal
(having only translational symmetry)
Amorphous Material (Glass)
(having no symmetry what so ever)
Making Some Molecular Crystals
 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)
Hexamethylenetetramine (C6 H12 N 4 )
Ethylene (C 2 H 4 )
Benzene (C6 H 6 )
Fullerene (C60 )
43m
2 2 2
mmm
6 2 2
mmm
2
35
m
I43m
2 2 2
P 1 1
n n m
2 2 2
P 1 1 1
b c a
4 2
F 3
m m
43m
2
m
1
4 2
3
m m
Funda Check
 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.
Click here → connection between geometry and symmetry