Symmetry & Crystal Symmetry & Crystal StructuresStructures

Solid State Physics 355Solid State Physics 355

Topic 1Topic 1

CrystalsCrystals

Atoms that are bound together, do so in a Atoms that are bound together, do so in a way that minimizes their energy.way that minimizes their energy.

This most often leads to a periodic This most often leads to a periodic arrangement of the atoms in space.arrangement of the atoms in space.

If the arrangement is purely periodic we If the arrangement is purely periodic we say that it is say that it is crystalline.crystalline.

Crystal = Lattice + BasisCrystal = Lattice + Basis

Unit CellUnit Cell

Face-centered Cubic LatticeFace-centered Cubic LatticeIn the Face Centered Cubic (FCC) unit cell there is one basis atom at each corner and one basis atom in each face. 

There are 4 atoms per unit cell:(1/8)8 + (1/2)6 = 4.

The simplest primitive cell is actually rhombohedral. Its volume is ¼ that of the cubic, as we might expect.1

2

3

2

2

2

ˆ ˆa (x y)

ˆ ˆa (y z)

ˆ ˆa (z x)

a

a

a

The three interior angles formed between unit cell edges are called: alpha, between edges a2 & a3 (beta, between edges a1 & a3) (gamma, between edges a1 & a2)In the FCC rhombohedral standard reduced cell, it can be shown that = = = 60.  Note that a cube is a just a special rhombohedron, with = = = 90.

0 1 2 3 a a a

BaTiO3

Body-centered Cubic LatticeBody-centered Cubic Lattice

In the Body Centered Cubic (BCC) unit cell there is one host atom (lattice point) at each corner of the cube and one host atom in the center of the cube: Z = 2

Symmetry OperationsSymmetry Operations

TranslationalTranslational Reflection at a planeReflection at a plane Rotation about an axisRotation about an axis

– Inversion through a pointInversion through a point Glide (=reflection + translation)Glide (=reflection + translation) Screw (=rotation + translation)Screw (=rotation + translation)

RotationsRotations

Mirror SymmetryMirror Symmetry

InversionInversion

2D - Point Groups2D - Point Groups

Point and Space GroupsPoint and Space Groups

7 crystal systems7 crystal systems 14 Bravais lattices14 Bravais lattices 230 non-Bravais lattices 230 non-Bravais lattices

32 point symmetries32 point symmetries– 2 triclinic2 triclinic– 3 monoclinic3 monoclinic– 3 orthorhombic3 orthorhombic– 7 tetragonal7 tetragonal– 5 cubic5 cubic– 5 trigonal5 trigonal– 7 hexagonal 7 hexagonal

Space GroupsSpace Groups

Crystal DirectionsCrystal Directions

]100[],010[],001[],001[],010[],100[ 100

Crystal PlanesCrystal Planes

Crystal Planes: Miller IndicesCrystal Planes: Miller Indices

Procedure for finding Miller indices in four-index notation:

1. Find the intersections, r and s, of the plane with any two of the basal plane axes.

2. Find the intersection t of the plane with the c axis. 3. Evaluate the reciprocals 1/r, 1/s, and 1/t. 4. Convert the reciprocals to smallest set of integers that are in the same ratio. 5. Use the relation i = -(h+ k), where h is associated with a1

, k is associated

with a2, and i is associated with a3

.

6. Enclose all four indices in parentheses: (h k i l)

Example: What is the designation, using four Miller indices, of the lattice plane shaded pink in the figure?    

• The plane intercepts the a1, a3

, and c axes

at r = 1, u = 1, and t = ∞, respectively. • The reciprocals are

•1/r = 1 •1/u = 1 •1/t = 0

• These are already in integer form. • We can write down the Miller indices as

(h k i l) = (1 k 1 0), where the k index is undetermined so far.

• Use i = (h + k). That is, k = 2. • The designation of this plane is (h k i l) =  

)1021(

Why are planes in a lattice important?

(A) Determining crystal structureDiffraction methods directly measure the distance between parallel planes of lattice points. This information is used to determine the lattice parameters in a crystal and measure the angles between lattice planes.

(B) Plastic deformationPlastic (permanent) deformation in metals occurs by the slip of atoms past each other in the crystal. This slip tends to occur preferentially along specific lattice planes in the crystal. Which planes slip depends on the crystal structure of the material.

     (C) Transport Properties

In certain materials, the atomic structure in certain planes causes the transport of electrons and/or heat to be particularly rapid in that plane, and relatively slow away from the plane.Example: Graphite

Conduction of heat is more rapid in the sp2 covalently bonded lattice planes than in the direction perpendicular to those planes.

Example: YBa2Cu3O7 superconductors

Some lattice planes contain only Cu and O. These planes conduct pairs of electrons (called Cooper pairs) that are responsible for superconductivity. These superconductors are electrically insulating in directions perpendicular to the Cu-O lattice planes.

(GPa)

d spacingd spacing

2

2

2

2

2

2

c

l

b

k

a

h

ndhkl

ExampleThe lattice constant for aluminum is 4.041 angstroms. What is d220?

AnswerAluminum has an fcc structure, so a = b = 4.041 Å

angstroms 43.1

041.4

22

11

2

22

2

22

a

khdhkl

InterstitialsInterstitials

BCCFCC

What types/sizes of atoms or ions can fit in a given interstitial site?

Answer:

1. Calculate the effective radius ρ of the void space using trigonometry AND the hard-sphere model with the radius r for the host atom (for metals) or ions (for ionic crystals).

2. Any element with atomic/ionic radius less than or equal to ρ can occupy that interstitial site.

3. If the element has radius larger than ρ, then it will cause some distortion to the crystal structure

4. The increased energy due to distortion will limit the number of interstitial sites that can be occupied (example: carbon in iron).

Examples of common structures: Examples of common structures: (1) The Sodium Chloride (NaCl) Structure (1) The Sodium Chloride (NaCl) Structure

(LiH, MgO, MnO, AgBr, PbS, KCl, KBr)(LiH, MgO, MnO, AgBr, PbS, KCl, KBr)

The NaCl structure is FCCThe NaCl structure is FCC The basis consists of one Na The basis consists of one Na

atom and one Cl atom, atom and one Cl atom, separated by one-half of the separated by one-half of the body diagonal of a unit cubebody diagonal of a unit cube

There are four units of NaCl There are four units of NaCl in each unit cubein each unit cube

Atom positions: Atom positions: Cl : 000 ; ½½0; ½0½; 0½½Cl : 000 ; ½½0; ½0½; 0½½ Na: ½½½; 00½; 0½0; ½00Na: ½½½; 00½; 0½0; ½00 Each atom has 6 nearest Each atom has 6 nearest

neighbours of the opposite neighbours of the opposite kindkind

Often described as 2 interpenetrating

FCC lattices

NaCl structureNaCl structure

Crystal a

LiH 4.08 Å

MgO 4.20

MnO 4.43

NaCl 5.63

AgBr 5.77

PbS 5.92

KCl 6.29

KBr 6.59

a

(2) The Cesium Chloride (CsCl) structure (2) The Cesium Chloride (CsCl) structure

The CsCl structure is BCCThe CsCl structure is BCC The basis consists of one Cs The basis consists of one Cs

atom and one Cl atom, with atom and one Cl atom, with each atom at the center of a each atom at the center of a cube of atoms of the cube of atoms of the opposite kindopposite kind

There is on unit of CsCl in There is on unit of CsCl in each unit cubeeach unit cube

Atom positions: Atom positions: Cs : 000Cs : 000 Cl : ½½½ (or vice-versa)Cl : ½½½ (or vice-versa) Each atom has 8 nearest Each atom has 8 nearest

neighbours of the opposite neighbours of the opposite kindkind

(CsBr, CsI, RbCl, AlCo, AgZn, BeCu, MgCe, RuAl, SrTl)

CsCl structureCsCl structure

Crystal a

BeCu 2.70 Å

AlNi 2.88

CuZn 2.94

CuPd 2.99

AgMg 3.28

LiHg 3.29

NH4Cl 3.87

TlBr 3.97

CsCl 4.11

TlI 4.20

a

Why are the a values smaller for the CsCl structures

than for the NaCl (in general)?

Closed-packed structuresClosed-packed structures

(or, what does stacking fruit have to do with solid state physics?)

Closed-packed structuresClosed-packed structures

There are an infinite number of ways to There are an infinite number of ways to organize spheres to organize spheres to maximizemaximize the packing the packing fraction.fraction.

There are different ways you can pack spheres together. This shows two ways,

one by putting the spheres in an ABAB… arrangement, the other with ACAC…. (or

any combination of the two works)

The centres of

spheres at A, B,

and C positions

(from Kittel)

(3) The Hexagonal Closed-packed (HCP) structure(3) The Hexagonal Closed-packed (HCP) structure

The HCP structure is made up of stacking spheres in a ABABAB… The HCP structure is made up of stacking spheres in a ABABAB… configurationconfiguration

The HCP structure has the primitive cell of the hexagonal lattice, with a The HCP structure has the primitive cell of the hexagonal lattice, with a basis of two identical atomsbasis of two identical atoms

Atom positions: 000, 2/3 1/3 ½ (remember, the unit axes are Atom positions: 000, 2/3 1/3 ½ (remember, the unit axes are notnot all all perpendicular)perpendicular)

The number of nearest-neighbours is 12.The number of nearest-neighbours is 12.

Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl

Conventional HCP unit cell

The FCC and hexagonal closed-packed structures (HCP) are The FCC and hexagonal closed-packed structures (HCP) are formed from packing in different ways. FCC (sometimes called formed from packing in different ways. FCC (sometimes called the cubic closed-packed structure, or CCP) has the stacking the cubic closed-packed structure, or CCP) has the stacking arrangement of ABCABCABC… HCP has the arrangement arrangement of ABCABCABC… HCP has the arrangement ABABAB….ABABAB….

HCP

ABAB

sequence

FCC

(CCP)

(looking along [111]

direction

ABCABC

sequence

[1 1 1][0 0 1]

HCP and FCC structuresHCP and FCC structures

The hexagonal-closed The hexagonal-closed packed (HCP) and FCC packed (HCP) and FCC structures both have the structures both have the ideal packing fraction of ideal packing fraction of 0.74 (Kepler figured this 0.74 (Kepler figured this out hundreds of years ago)out hundreds of years ago)

The ideal ratio of c/a for The ideal ratio of c/a for this packing is this packing is (8/3)(8/3)1/21/2 = 1.633 = 1.633

Crystal c/a

He 1.633

Be 1.581

Mg 1.623

Ti 1.586

Zn 1.861

Cd 1.886

Co 1.622

Y 1.570

Zr 1.594

Gd 1.592

Lu 1.586

Amorphous MaterialsAmorphous Materials

GlassThe continuous random network structureof amorphous silicon dioxide, notice thateach Si atom (gold spheres) has 4 bonds, and each oxygen atom (red spheres) has 2 bonds.