Bravais Lattices in 2D

In 2D there are five ways to order atoms in a lattice

Primitive unit cell: contains only one atom (but 4 points?)Are the dotted lattices primitive?Non-primitive unit cells sometimes useful if orthogonal coordinate system can be used

Special case where

angles go to 90

a=b

Special case where point

halfway

a=b

2

α

a1

a2

O

y

x

b) Crystal lattice obtained by identifying all the atoms in (a)

a) Situation of atoms at the corners of regular hexagons

Defining Crystal Structure(In your groups) determine the lattice vectors and basis for the

smallest possible unit cell (more than one possible answer)If you finish, draw the lattice.

Structure of SolidsObjectives

By the end of this section you should be able to:

• Compare bcc, fcc and hcp crystal structures• Calculate atomic packing factors (HW)• Determine/understand coordination numbers• Identify primitive unit cell lattice parameters• Be able to build the Wigner-Seitz cell for a lattice

Solid Models: Close-Packed Spheres• Many atoms or ions forming solids have spherical

symmetry (e.g. noble gases and simple metals)• Considering the atoms or ions as solid spheres we

can imagine crystals as closely packed spheres• How can we pack them?

Simple cubic

• The simple cubic structure is a Bravais lattice.• How many atoms are in each unit cell?

a1a2

a3

R = n1 a1 + n2 a2 + n3 a3

Simple cubic• We can also simply count the atoms we see in

one unit cell.• But we have to keep track of how many unit cells

share these atoms.

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

APF for the simple cubic structure = 0.52

APF = a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolume

ATOMIC PACKING FACTOR (APF)

contains 8 x 1/8 = 1 atom/unit cell

Lattice constant

close-packed directions

a

R=0.5a

Simple cubic• A simple cubic structure is not efficient at packing

spheres (atoms occupy only 52% of the total volume). Marbles will not resemble.

• Only two single elements crystallize in the simple cubic structure (F and O).

Another Reason Simple Cubic Structure is Rare

Groups: Using the spheres (like atoms) and magnetic sticks (like bonds between atoms),

create a simple cubic lattice.

How does this compare to a

triangular pyramid structure?

Vertex(corner) atom shared by 8 cells Þ 1/8 atom per cell

Three Cubic Unit Cell Types in 3DOften drawn as different colors (for easy viewing)

but these are NOT different atoms!

# of Atoms/Unit Cell

For atoms in a cubic unit cell:

Atoms in corners are ⅛ within the cell

# of Atoms/Unit Cell

For atoms in a cubic unit cell:

Atoms on faces are ½ within the cell

Atoms on FacesFace Centered Cubic (FCC)

Vertex(corner) atom shared by 8 cells Þ 1/8 atom per cell

Edge atom shared by 4 cells Þ 1/4 atom per cell

Face atom shared by 2 cells Þ 1/2 atom per cell

Body unique to 1 cell Þ 1 atom per cell

Three Cubic Unit Cell Types in 3DAre these primitive unit cells?

Crystal Structure 14

UNIT CELLS

Primitive Conventional & Non-primitive

§ Single lattice point per cell§ Smallest area in 2D, or§Smallest volume in 3D

§ More than one lattice point per cell§ Integral multiples of the area of primitive cell

Body centered cubic(bcc)

Conventional ≠ Primitive cellSimple cubic(sc)

Conventional = Primitive cell

What is the close packed direction?

--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

BODY CENTERED CUBIC STRUCTURE (BCC)

How would we calculate the atomic packing factor?

Better packing than SC

• In the body-centred cubic (bcc) structure 68% of the total volume is occupied.

• Next-nearest neighbors relatively close by – make structure stable in some instances. Examples: Alkali metals, Ba, V, Nb, Ta, W, Mo, Cr, Fe

• Is this cube a primitive lattice?• No. The bcc structure is a Bravais lattice but the edges of the cube

are not the primitive lattice vectors. Not smallest Vol.

--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

FACE CENTERED CUBIC STRUCTURE (FCC)What is the close packed direction?

What are the lattice directions of the

primitive unit cell?

APF = 0.74

Simple Crystal FCC

Another view

Homework 4.1A. simple cubic with

additional points in the horizontal faces

C. simple cubic with additional points at the

midpoints of lines joining nearest

neighbors

Note: green and orange are same atoms (only different colors for clarity)

What does it mean to ask if it’s a Bravais lattice?

That orientation looks the same from any point.Is it the smallest possible unit cell?

If you think not a Bravais lattice, define the crystal structure.

Groups: Fill in this Table for Cubic Structures

SC BCC FCC

Volume of conventional cell a3 a3 a3

# of atoms per cubic cell 1 2 4

Volume, primitive cell a3 ½ a3 ¼ a3

# of nearest neighbors 6 8 12

Nearest-neighbor distance a ½ a 3 a/2

# of second neighbors 12 6 6

Second neighbor distance a2 a a

Wigner-Seitz Method for Defining a Primitive Unit Cell(points are closest to each other)

1. Pick a center atom (origin) within the lattice2. Draw perp. bisector to all neighbors of reciprocal lattice3. Draw smallest polyhedron enclosed by bisectors

Always a hexagon in

2D, unless the lattice is

rectangular.

Wigner-Seitz for BCC & FCC

Looks a little different in FCC. Why?

Is this BCC or FCC?

Close packed crystals

A plane

B plane

C plane

A plane

…ABCABCABC… packing[Face Centered Cubic (FCC)]

…ABABAB… packing[Hexagonal Close Packing (HCP)]

Close-packed structures: fcc and hcp

hcpABABAB...

fccABCABCABC...

If time allows:In groups, build these two differing crystal structures.

• ABAB... Stacking Sequence

APF = 0.74 (same as fcc) What is the packing direction?

• 3D Projection • 2D Projection

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)

For ideal packing, c/a ratio of 1.633However, in most metals, ratio

deviates from this value

Lattice Planes and Miller Indices

Hexagonal structure:

a-b plane (2D hexagon) can be defined by 3 vectors in plane (hkl)

3D structure can be defined by 4 miller indices (h k l m)

Third miller index not independent:h + k = -l

Have more on HCP planes in the Additional Materials tab of website

e

hk

l

m

Other non close-packed structures• In covalently bonded materials, bond direction is

more important than packing

diamond (only 34 % packing)graphite

Packing isn’t the only consideration when building a lattice.

Simple Crystal StructuresDiamond

• Crystal class Td (tetrahedral) - Each atom has 4 nearest-neighbors (nn).

• Can be interpreted as two combined fcc structures– One atom at origin– Other atom displaced along

diagonal (¼, ¼, ¼)

• Includes C, Si, Ge, a-Sn

Diamond & Zincblende Crystal Structure

• Basis set: 2 atoms. Lattice face centered cubic (fcc).• The fcc primitive lattice is generated by r = n1a1+n2a2+n3a3

with lattice vectors:a1 = a(0,1,0)/2, a2 = a(1,0,1)/2, a3 = a(1,1,0)/2

NOTE: The ai’s are NOT mutually orthogonal!

Diamond: 2 identical atoms in basis (e.g. 2 C) fcc latticeZincblende: 2 different atoms in basis and fcc latticeFor FCC 2 atom ABCABC stacking, it is called zinc blende

How would we calculate the atomic packing factor of diamond or zincblende?