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1/17

Primitive cells, Wigner-Seitz cells,

and 2D lattices

4P70, Solid StatePhysics

Chris Wiebe


2/17

Choice of primitive cells

! Which unit cell is a goodchoice?

! A, B, and C are primitive

unit cells. Why?! D, E, and F are not. Why?

! Notice: the volumes of A,B, and C are the same.Also, the choice of origin isdifferent, but it doesntmatter

! Also: There is only onelattice point in the primitive

unit cells.


3/17

Wigner-Seitz cells

! How can we chooseprimitive cells?

! One algorithm is the Wigner-Seitz cell

! Steps: (1) Draw lines toconnect a given lattice pointto all nearby lattice points

! (2) At the midpoint andnormal to these lines, drawnew lines (or planes in 3D)

! (3) The smallest volumeenclosed in this way is theWigner-Seitz primitive cell

Wigner-Seitz cell


4/17

Finding Wigner-Seitz cells

Phopholipid bilayers

(P. C. Mason et al, PRE,

63, 030902 (2001))(M. C. Escher)

Homework: What are the Wigner-Seitz cells for these lattices?


5/17

Fundamental types of lattices

! Crystal lattices can be mapped intothemselves by the lattice translations T,and by other symmetry operations

! Physicists use the symmetry of the unitcells to classify crystal structures and

how they fill space.! In this course, you will not have to know

how this is done (this involves the use ofpoint operations and translationoperators)

! However, you do have to understand

how we classify lattices! In 2D, there are only 5 distinct lattices.

These are defined by how you canrotate the cell contents (and get thesame cell back), and if there are anymirror planes within the cell.

! From now on, we will call these distinctlattice types Bravais lattices.

! Unit cells made of these 5 types in 2Dcan fill space. All other ones cannot.

/3

We can fill

space with a

rectangular

lattice by

180 orotations

(not 90o

why?)

We can fill

space with a

hexagonal

lattice by 60o

rotations

Note: this is the primitive cell of a hexagonal

lattice (why? See Kittel, fig 9b)


6/17

Fivefold rotations and quasicrystals

! It turns out that mathematiciansdiscovered that you can only fillspace by using rotations of unit cellsby 2, 2/2, 2/3, 2/4, and 2/6radians (or, by 360o, 180o, 120o, 90o,

and 600)! But, rotations of the kind 2/5 or

2/7 do not fill space!

! A quasicrystal is a quasiperiodicnonrandom assembly of two types

of figures (since we need two types,it is not a Bravais lattice youcannot fill space with just onerepeating unit cell)

! We will discuss these in Chapter

TwoPenrose tiling in 2D


7/17

The 2D Bravais lattices

Note that this is the

proper primitive cell

for the centered

rectangular latticetype (why? It contains

only one lattice point)

(this is called a rhombus)


8/17

The 3D Bravais Lattices

! In 3D, there are 7lattice systems,which give rise to14 bravais lattices

! The general latticeis triclinic, and allothers are derivedfrom putting

restraints on thetriclinic lattice

! Know theconditions of eachlattice (excellent

test question )


9/17

The 14 Bravais lattices in 3D

! Notice that weknow havedifferent latticetypes:

! P = primitive(1 lattice point)

! I = Body-centred(2 lattice pts)

! F = Face- centred

(4 lattice pts)! C = Side-Centred

(2 lattice pts)

! In this course, wewill concentrate

upon only a few ofthese types


10/17

Cubic lattices: SC, BCC and FCC

! For cubic systems, we havethree bravais lattices: thesimple cubic (SC), body-centred cubic (BCC) and face-

centred cubic (FCC).! The simple cubic has 1 latticepoint per unit cell, with a totalarea of a3

! Number of nearest neighbours:

6! Nearest neighbour distance: a

! Number of next-nearestneighbours: 12

! Next-nearest neighbour

distance: 2a (prove this!)

Simple

cubic

lattice

(4/3)(a/2)3

(a3)

Packing fraction =

(atomic volume)/

(bravais unit cell

volume)= / 6


11/17

The Body-Centred Cubic Lattice(Li (at room temp.), Na, K, V, Cr, Fe, Rb, Nb, Mo, Cs, Ba, Eu, Ta)

! The body-centred cubic latticehas 2 lattice points per unitcell, with a total primitive cellvolume of a3/2

! Number of nearestneighbours: 8

! Nearest neighbour distance:3 a/2 = 0.866 a (prove this!)

! Number of next-nearestneighbours: 6 (where arethese?)

! Next-nearest neighbour

distance: a (prove this!)

Packing fraction =

(atomic volume)/

( bravais unit cell

volume)

(4/3)(3a/4)3

(a3/2)

=3

/ 8

Body-

Centred

Cubic

Lattice(BCC)


12/17

The Body-Centred Cubic Lattice

! The primitive cell of the BCC

lattice is defined by the

translation vectors:

a1

a2

a3

x

y

z

a1 = a (x + y - z)a2 = a (-x+y + z)

a3 = a (x - y + z)a

where x, y, and z are the Cartesian unitvectors. These translation vectors

connect the lattice pt at the origin to the

points at the body centres (and make a

rhombohedron)

See figure 11 in Kittel for a picture

of the primitive cell in more detail

The angle between adjacent edges is 109.3


13/17

The Face-Centred Cubic Lattice(Al, Cu, Ni, Sr, Rh, Pd, Ag, Ce, Tb, Ir, Pt, Au, Pb, Th)

! The face-centred cubic latticehas 4 lattice points per unitcell, with a total primitive cellvolume of a3/4

! Number of nearestneighbours: 12

! Nearest neighbour distance:a/2 = 0.707 a (prove this!)

! Number of next-nearestneighbours: 6 (where arethese?)

! Next-nearest neighbourdistance: a (prove this!)

= 2 / 6Packing fraction =

(atomic volume)/

(bravais unit cellvolume) (do this on

your own)


14/17

The Face-Centred Cubic Lattice

! The primitive cell of the

FCC lattice is defined

by the translation

vectors:

a1 = a (x + y)

a2 = a (y + z)

a3 = a (z + x)

where x, y, and z are the Cartesian unit

vectors. These translation vectors

connect the lattice pt at the origin to the

points at the face centres.

The angles between the axes are 60


15/17

Wigner-Seitz cells for BCC, FCC lattices

! Note: You can also constructWigner-Seitz cells for the FCCand BCC lattices. These looka bit strange, but they will beuseful when we look atreciprocal space in the nextchapter.

! These are made by taking thelines to the nearest and next-

nearest neighbour points, andbisecting them with planes.The resulting figure is theWigner-Seitz cell (which havethe same volumes as the

primitive cells we made above)


16/17

Wigner-Seitz cells

BCC lattice Simple cubic lattice


17/17

Hexagonal unit cell

a1

a2

a3

120

! The primitive cell in ahexagonal system is a rightprism based on a rhombus

with an included angle of120

! Note here that a1 = a2 = a3

! Later, we will look at thehexagonal close-packedstructure, which is thisstructure with a basis (and is

related to the fcc structure).(primitive cell is in bold)

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