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1. LET US MAKE SOME CRYSTALS

O Constructing crystals in ID, 2D & 3D

Q Understanding them using the language of:
> Lattices
> Symmetry MATERIALS SCIENCE

Part of & |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.titk.ac.in/~anandh

hup:Vhome.iitk.ac. in/-anandh/E-book.him

http://cst-www.nrl.navy.mil/lattice/index.html

3. Making a 1D Crystal

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

Q A strict 1D crystal = 1D lattice + 1D motif

Q The only kind of 1D motif is a line segment(s) (though in principle a collection
of points can be included).

Lattice ¡A Unit Cell

— ——— = —- -—- —_— —— -
Crystal
Unit Cell

4. 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:

ul u es a e +

a Unit Cell

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 + 900090000000

2D Motif” Unit Cell

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

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

29 ee 9.0.0.0.0.6

a

6. € “Time to brush-up some symmetry concepts before going ahead)

Lattices have the highest symmetry
(Which is allowed for it)

In the coming slides we
will understand this
IMPORTANT point

> Crystals based on the lattice

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)

7. Q 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

O 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) This consent art
(though typically the concept is used for crystals only) Asymm

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

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

= Horizontal Mirror (my)

= Vertical Mirror at Lattice Points (m,;)

= Vertical Mirror between Lattice Points (m,;)

Note:

= The symmetry operators (t, m,,, m,) 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 m,)

tm, m,, m,, - Or more concisely +
mmm

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

Three mirror planes m

9. Note of Redundant Symmetry Operators

amm

Three mirror planes Redundant inversion centres
Redundant 2-fold axes

= Itis true that some basic set of symmetry operators (set-1) can generate the structure (/attice or crystal)
= Itis 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?|

—E TH a pcr pe pee Te

10. Asymmetric Unit

QO) 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).

Q) 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 wpically ihe concept is used for orystals
only)

Q 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)

mmetric Unit

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

11. Decoration of the lattice with a motif — may reduce the symmetry of the crystal

oH RRA ma

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)

mH 0

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

—— Binet:

12. Nota lattice point

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

|
a ttm

2 inversion centres

— Siete Slide reflection

13. eo ee ee ma ar m aa
= - m

1 mirror plane

1 glide reflection translational symmetry of ‘2a’

a {++++4-++++5 I

No symmetry except translation

— Siete Slide reflection

15. Making a 2D Crystal

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

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

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

16. Square Lattice

e . . . . . .
. . > . . . .
. ° . . . . .

+ Circle Motif ©
Ld . . . . . .
Unit Cell 4-fold axes =

A u Ñ 2-fold axis
i
{] un @ à Square Crystal

For sake of clarity all axes have not been marked © O O O
0000
00

O —

17. Square Lattice

+ Circle Motif ©

Unit Cell

Square Crystal

For sake of clarity all axes hayénot been marked. © O © O
Symmetry of the lattice and
crystal identical ko o O O O O
= Square Crystal e

mr

18. [Im portant Note

Symmetry of the Motif Symmetry of the lattice

O AT

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

Symmetry of the Motif

O = 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

19. 000
Fu E Q What do the ‘adjectives’ like square mean in ‘5 * 0000
the context of the lattice, crystal etc? ss... oo

00

Q Let us consider the square lattice and square crystal as before.
Q 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!).

Q 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!)

20. Square Lattice + Square Motif = Square Crystal

on TEE

21. [Important Note]
Symmetry of the Motif E Symmetry of the lattice

Hence Symmetry of the lattice

and Crystal identical
> Square Crystal

Symmetry of the Motif

O = 4mm symmetry

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

Common surviving symmetry determines the crystal system

23. U Ina 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

sooo
8 sooo
on!
oo

No Mirrors

24. vu
oa

Fu et Q How do we understand the crystal made out of
rotated squares?

0 9 9
Bone

33

Q Is the lattice square > YES (it has 4mm symmetry)

Q 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!)

Q Is the ‘preferred’ unit cell square — YES (it has square geometry)

Q 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

25. Square Lattice

. . . . . . .
. . . . . . .
. . . . . . .
+ Triangle Motif
. . . .
Unit Cell = À aes
. .
E i } Rectangle Crystal
. . ty
Ford tatin me tel À A A A
Symmetry of the lattice and
crystal different A A
=> NOT a Square Crystal

A À

Here ihe word square does not imply the shape in the usual sense WIN | Continued...

26. El
:
2
|
E

27. [Im portant 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

| Note that the word ‘Rectangle ‘denotes the symmetry of ve |

= Mirror crystal and NOT the shape of the UC

= 3-fold

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

29. Fu et Q How do we understand the crystal made out of
triangles?

Q Is the lattice square — YES (it has 4mm symmetry)
Q Is the crystal square — NO (it has only m symmetry — hence it is a rectangle crystal)
Q Is the unit cell square YES (it has square geometry) (weve creat noted ha aer shapes of anit cls are ais poste)

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

30. Square Lattice + ‘Triangle Motif = Parallelogram Crystal

naam DD DD
an DD DD
ver > > »

>>

I

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

31. nO)

Some more
twists

32. Square Lattice

. . . . . . .
. . . . . . .
+ Random shaped Motif
. . . . In Single Orientation
Unit Cell = a
. . .
4 Parallelogram Crystal
For sake of clarity all axes havı
Symmetry of the lattice and
crystal different fe
= NOT Square Crystal |

Except translation

33. Square Lattice

né

+ Random shaped Object

Randomly oriented at each point
= Square Erystal—

Unit Cell ¿laos 2

Amorphous Material
(Glass)

RE SEE FE
Me HE Tk HE
He Me Me St

No Unit Cell

34. Fu et Q Is there not some kind of order visible in the

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

Q YES, there is positional order but no orientational order.

Q 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’)

=

Q Hence, this structure can be considered to be a ‘crystal’ wit
without orientational order!

h positional order, but

Click here to know more

35. Summary of 2D Crystals

Crystal

Highest
Symmetry
Possible

Other symmetries
possible

Lattice Parameters
(of conventional unit cell)

4mm

(a=b,a=90")

2mm

(a#b, a = 90°)

3. 120° Rhombus

6mm

(a = b, a = 120°)

4. Parallelogram

2

(a #b, @ general value)

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

36. 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

38. Making a 3D Crystal

O A strict 3D crystal = 3D lattice + 3D motif

Q We have 14 3D Bravais lattices to chose from

Q As an intellectual exercise we can put ID or 2D motifs in a 3D lattice as well
(we could also try putting higher dimensional motifs like 4D motifs!!)

Q We will illustrate some examples to understand some of the basic concepts
(most of which we have already explained in 1D and 2D)

39. Simple Cubic (SC) Lattice | +4 Sphere Motif

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’

40. To know more about |

Body Centred Cubic (BCC) Lattice | + Sphere Motif (Clots Packed Cisco

click here

O

Body Centred Cubic Crystal

Atom at (0, 0,0)

Unit cell of the BCC lattice

<_ Space filling model

Central atom is coloured differently for beter 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

41. Face Centred Cubic (FCC) Lattice MP Sphere Motif

Close Packed
implies CLOSEST
PACKED:

_ ETA Cubic Close Pácked Crystal
mi (Sometimes casually called the FCC crystal)

Unit cell of the FCC lattice

< Space filling model
So when one talks about a FCC crystal what is meant

[Note: FCC is a lattice and not a crystal] is a FCC lattice decorated with a mono-atomic motif

42. More views

43. Face Centred Cubic (FCC) Lattice | Two Ion Motif

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

44. Face Centred Cubic (FCC) Lattice |~ Two Carbon atom Motif
(0,0,0) & (14, Ya, 14)

A —
©

Diamond Cubic Crystal

It requires a little thinking to convince yourself that the two atom motif

actually sits at all lattice points! All Atoms are Carbon

(coloured differently for better visibility)

[Note: This is not a close packed esta]

There are no close packed directions in this crystal either!

45. Face Centred Cubic (FCC) Lattice | Two Ion Motif

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

" Solved |
\Example 2

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

46. NaCl crystal: further points

(Click here: Ordered Crystals )

| This crystal can be considered as two |
si ing ECC +

Inter-penetration of just 2 UC are shown here

47. More views

le
PA

Coordination around Na* and CF ions

48. Q 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) Amor phous Material (Glass)

(having no symmetry what so ever)

49. Making Some Molecular Crystals

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

U 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)

Q Itis 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 (C,H,,N,) 43m 143m 43m
Ethylene (C,H,) 222 P 2, 2, 2 2
mmm nom m
Benzene (C,H,) £ 2 2 P 2 2 2 7
mmm be a
Fullerene (C,,) 2 35 E 4 3 2 4 3 2
m m m m m

50. ria

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

Q 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 J