MSE 630

Introduction to Solid State PhysicsTopics:Structure of Crystals

classification of latticesreciprocal latticesbonding in crystals

Vibrations and Waves in Crystals - PhononsElectrons in Crystals

free electrons in metalselectrons in periodic potentialssemiconductors

•Conductivity?•Types?•Metals, Semiconductors, Insulators, and Superconductors

•Atoms and Electrons

•Electronic Structure of materials

•The Bohr Model•Quantum Mechanics

– Schrodinger Wave Eqn– Atomic structure and Periodic Table– Bonding forces in materials

Ionic Bond – electron transfer and electrostatic attraction

Non directional

Covalent Bond: electron sharing for stable outer shellsHighly directional

Metallic bond: electron sharing between charged ion coresNon directional

The bond type influences the mechanical properties

Both ionic and metallic bonds form close packed structures. Ions, though, have to maintain charge neutrality, so any deformation in ionic solids must be large enough to move the atoms back into registry. Metals do not have this restriction. Hence, ionic solids are brittle, while metals are ductile.

Other, weaker bonds

Hydrogen bond: hydrogen acts as infinitesimal cation attracting two anions

van der Waals bond: weak attraction between molecular dipoles

The structure of an ionic compound depends upon the ratio of the cation to the anion

The crystal lattice

A point lattice is made up of regular, repeating points in space. An atom or group of atoms are tied to each lattice point

14 different point lattices, called Bravais lattices, make up the crystal system. The lengths of the sides, a, b, and c, and the angles between them can vary for a particular unit cell.

Three simple lattices that describe metals are Face Centered Cubic (FCC) Body Centered Cubic (BCC) and Hexagonal Close Packed (HCP)

Whether a close packed crystal is FCC or HCP depends upon the stacking sequence of close packed planes

Diamond, BeO and GaAs are examples of FCC structures with two atoms per lattice point

Polymers are made up of repeating units “mers”, that make up a long chain. The chain may be cross-linked, or held together with van der Waals or

hydrogen bonds

Structures may be crystalline, having repeating structure, or amorphous, having local structure but no long-range structure

Directions in a crystal lattice – Miller Indices

Vectors described by multiples of lattice constants: ua+vb+wc

e.g., the vector in the illustration crosses the edges of the unit cell at u=1, v=1, c=1/2

Arrange these in brackets, and clear the fractions:

[1 1 ½] = [2 2 1]

Negative directions have a bar over the number

e.g., _

11

Families of crystallographically equivalent directions, e.g., [100], [010], [001] are written as <uvw>, or, in this example, <100>

Directions in HCP crystals

'

)''(

)''2(3

)''2(3

][]'''[

nww

vut

uvn

v

vun

u

uvtwwvu

a1, a2 and a3 axes are 120o apart, z axis is perpendicular to the a1,a2,a3 basal plane

Directions in this crystal system are derived by converting the [u′v′w′] directions to [uvtw] using the following convention:

n is a factor that reduces [uvtw] to smallest integers. For example, if

u′=1, v′=-1, w′=0, then

[uvtw]= ]0011[_

Crystallographic Planes

To find crystallographic planes are represented by (hkl). Identify where the plane intersects the a, b and c axes; in this case, a=1/2, b=1, c=∞

Write the reciprocals 1/a, 1/b, 1/c:

11

12

11

l

k

h

Clear fractions, and put into parentheses:

(hkl)=(210)

If the plane interesects the origin, simply translate the origin to an equivalent location.

Families of equivalent planes are denoted by braces:

e.g., the (100), (010), (001), etc. planes are denoted {100}

Planes in HCP crystals are numbered in the same way

e.g., the plane on the left intersects a1=1, a2=0, a3=-1, and z=1, thus the plane is )1110(

_

3

• tend to be densely packed.

• have several reasons for dense packing:-Typically, only one element is present, so all atomic radii are the same.-Metallic bonding is not directional.-Nearest neighbor distances tend to be small in order to lower bond energy.

• have the simplest crystal structures.

We will look at three such structures...

METALLIC CRYSTALS

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Example: Copper

n AVcNA

# atoms/unit cell Atomic weight (g/mol)

Volume/unit cell

(cm3/unit cell)Avogadro's number (6.023 x 1023 atoms/mol)

Data from Table inside front cover of Callister (see next slide):• crystal structure = FCC: 4 atoms/unit cell• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)• atomic radius R = 0.128 nm (1 nm = 10 cm)-7

Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3

Compare to actual: Cu = 8.94 g/cm3Result: theoretical Cu = 8.89 g/cm3

THEORETICAL DENSITY,

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• Charge Neutrality: --Net charge in the structure should be zero.

--General form: AmXp

m, p determined by charge neutrality• Stable structures: --maximize the # of nearest oppositely charged neighbors.

- -

- -+

unstable

- -

- -+

stable

- -

- -+

stable

CaF2: Ca2+cation

F-

F-

anions+

IONIC BONDING & STRUCTURE

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• Coordination # increases with Issue: How many anions can you arrange around a cation?

rcationranion

rcationranion

Coord #

< .155 .155-.225 .225-.414 .414-.732 .732-1.0

ZnS (zincblende)

NaCl (sodium chloride)

CsCl (cesium chloride)

2 3 4 6 8

COORDINATION # AND IONIC RADII

The Reciprocal Lattice and Waves in Crystals

We use the reciprocal lattice to calculate wave behavior in crystals because sound, optical and electrical properties pass through the crystal as waves

Because crystals are periodic, properties throughout the crystal will be the same as those surrounding any lattice point, contained in a volume known as a “Brillion Zone”

X-ray diffraction and crystal structure

• X-rays have a wave length, Å.• This is on the size scale of the structures

we wish to study X-rays interfere constructively when the interplanar spacing is related to an integer number of wavelengths in accordance with Bragg’s law:

sin2dn

Because of the numbering system, atomic planes are perpendicular to their corresponding vector,

e.g., (111) is perpendicular to [111]

The interplanar spacing for a cubic crystal is:

222 lkh

adhkl

Because the intensity of the diffracted beam varies depending upon the diffraction angle, knowing the angle and using Bragg’s law we can obtain the crystal structure and lattice parameter

The Reciprocal Lattice

To analyze the atomic structure and resulting properties of crystals, we introduce the concept of the “reciprocal lattice”

A reciprocal lattice vector is defined as

G = 1b1 + 2b2 + 3b3

Where 1, 2, 3 are integers and b1, b2 and b3 are primitive vectors in the reciprocal lattice

Why do we need reciprocal space?

Because waves or vibrations can be made up of a series of waves of other frequencies, i.e., expressed as a Fourier Series:

)sin()cos()(

)exp()(

rGirGnrn

or

riGnrn

g

G

We want G∙r to equal 2

at boundaries. Therefore ijji ab

ora

b

2

2

Wave reflection in reciprocal space

The reciprocal lattice vector, G = k

Calculating the reciprocal lattice

We construct the axis vectors b1, b2 and b3 of the reciprocal lattice using the following formulas: )(

2

)(2

)(2

321

213

321

132

321

321

aaa

aab

aaa

aab

aaa

aab

First brillion zone in a 2-D lattice

zyxaa

zyxaa

zyxaa

ˆˆˆ2

1

ˆˆˆ2

1

ˆˆˆ2

1

3

2

1

BCC primitive lattice vectors: FCC primitive lattice vectors:

xzaa

zyaa

yxaa

ˆˆ2

1

ˆˆ2

1

ˆˆ2

1

3

2

1

Brillion Zones

BCC Brillion Zone FCC Brillion Zone

Brillion Zones are the same as Wigner-Seitz cells in reciprocal space