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----

Class 11

Crystallography and Crystal Structures continued

Suggested Reading Ch. 3 M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge (2007) 69-78.

Ch. 1 C. Kittel, Introduction to Solid State Physics, 3rd Edition, Wiley (1956). Excerpt from ASM Metals Handbook.

Chs. 1 and 3 S.M. Allen and E.L. Thomas, The Structure of Materials, Wiley (1999).

Chs. 3 and 4 R. Tilley, Crystals and Crystal Structures, Wiley (2006).

34

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Types of LatticesTypes of LatticesTypes of LatticesTypes of Lattices2D/3D

Primitive (P)

One lattice point per unit cell

Termed a simple or primitive

-

.

Termed a XXX-centered

35XXX = body, face, or base

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Lattice Points Per Cell in 2DLattice Points Per Cell in 2DLattice Points Per Cell in 2DLattice Points Per Cell in 2D

edge corner

N NN N

cornerEdges are notallowed lattice sites

in 2D.

edge

interior

corner

positions are

the only

a owe a cepoint locations.

36WHY THESE?

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Consider primitive or body centered squares

37

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This lattice can be re-defined as one of the other allowed latticesNow consider edge placement of a lattice point.

This rectangular lattice is symmetric and allowed!

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Dont confuse lattice oint lacement with atom lacement.

Atoms can sit on edge sites.

Youll learn why a little later

39

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Lattice Points Per Cell in 3DLattice Points Per Cell in 3DLattice Points Per Cell in 3DLattice Points Per Cell in 3D

face cornerdgee

NNNN

N

Edges are not

allowed in 3D either.

We often seeexamples of crystals edge

face

interior

wit atoms sitting onedges.

corner

40

internal point or space symmetry

operations.

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Primitive vs. NonPrimitive vs. Non--primitive latticesprimitive latticesPrimitive vs. NonPrimitive vs. Non--primitive latticesprimitive lattices

There are 4 crystal systems in 2D. Thus we can define 4

primitive lattices in 2D.

4 primitive Bravais nets (aka. lattices)

Are there more? OF COURSE!

There are 7 crystal systems in 3D. Thus we can define 7

primitive lattices in 3D. 7 primitive Bravais lattices

Are there more? OF COURSE!

Can we add additional lattice points to a primitive lattice

and still have a lattice with the same shape?

41ABSOLUTELY!

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Primitive vs. NonPrimitive vs. Non--primitive latticesprimitive latticesPrimitive vs. NonPrimitive vs. Non--primitive latticesprimitive lattices

Answer: YES, if we maintain symmetry.

.

b

90

1 2

1 3

1 2

1 2

2D

rectangular

lattices

IMPOSSIBLE POSSIBLE

42

or equivalent to cornersn erna po n s symme r c

and equivalent to corners

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a

b

90

1 2

We could define a different lattice {e.g., primitive oblique lattice (shaded)}.

The primitive cell obscures the true fact that the lattice has higher symmetry.

A mirror image of the primitive unit cell is not identical to the original.

A mirror image of the rectangular cell with a lattice point in the center IS

identical to the original. It has higher symmetry!

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General things about latticesGeneral things about latticesGeneral things about latticesGeneral things about lattices

RECALL: You can always define a primitive

lattice/unit cell.

-,

also describes symmetry of the lattice, it should be

used instead.

Since all lattice oints must be identical new lattice

points can only be placed on positions centered

between primitive lattice points.

44Dont confuse lattice points with atoms

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22--D SynopsisD Synopsis

The Five 2The Five 2--DD BravaisBravais LatticesLattices

From the previous definitions of the four 2-D and seven 3-D crystal systems, we know that there are four

and sevenprimitive unit cells, respectively.

We can then ask: can we add additional lattice points to the primitive lattices (or nets), in such a way that we

First illustrate this for 2-D nets, where we know that the surroundings of each lattice point must be identical.

We can come up with centered rectangular net in (c) where A, B and C points have identical surroundings.

If we try to do same thing with other 2-D nets, we find that there are no new nets to be found

a non-primitive cell can be found that describes the symmetry of the net (lattice), then that cell should be

used to describe the net (lattice). Since the surroundings of every lattice point must be identical , we can only

add new lattice points at centered positions.

The 5 Bravais lattices of 2-D crystals: (a) square, (b) rectangular,

(c) centered rectangular, (d) hexagonal and (e) oblique: AB

These are

the only 5possible 2-D

Bravais

lattices

Prof. M.L. Weaver

45 45

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BravaisBravais Lattice or Not?Lattice or Not?

This is a 2-D Bravais Lattice:

This is not a 2-D Bravais Lattice: This is a 2-D Bravais Lattice:

Prof. M.L. Weaver

From point 1 to 2: environment changes by reflection (mirror plane half way in between), if tie vertical pairs of

points together you have 2-D Bravais lattice with 6 identical neighbors.

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Alternative ways to define unit cellAlternative ways to define unit cellAlternative ways to define unit cellAlternative ways to define unit cell

nce t s a ways poss e to escr e a att ce w t a pr m t ve un t ce , a

Bravais lattices can be described by primitive cells, even when they are centered

(i.e., non-primitive).

, -

If we select shorter vectors a1, a2, and a3, we can define a primitiverhombohedral lattice with angle =60.

a

There is also the Wigner-Seitz (WS) cell, which is used to describe the first

Brillouin zone of the reciprocal lattice.

r ou n zones are use n an eory o represen n rec proca space e

solutions of the wave equations for the propagation of phonons or electrons in

solids.

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Alternative ways to define unit cellAlternative ways to define unit cellAlternative ways to define unit cellAlternative ways to define unit cell

Figure 2.9 Wigner-Seitz cells: (a) the body-centered cubic lattice; (b)

the Wigner-Seitz cell of (a); (c) the face-centered cubic lattice; (d) the

Wigner-Seitz cell of (c). The fcc cubic lattice point marked * forms the

48

. -

cell or Dirichlet region: (a) draw a line from

each lattice point to its nearest neighbors; (b)

draw a set of lines normal to the first, throughtheir mid-points; (c) the polygon formed

shaded is the cell re uired.

central lattice point in the Wigner-Seitz cell.

[Figures from Tilley]

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Brillouin ZonesBrillouin ZonesBrillouin ZonesBrillouin Zones

-defined in reciprocal space.

points on the reciprocal spacelattice.

Importance stems from Bloch

wave description of waves in a.

Higher order zones are useful inhttp://en.wikipedia.org/wiki/File:Brillouin_zone.svg

- .

From Naumann, p. 128.