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Chapter One Crystal Structure

Drusy Quartz in Geode Tabular Orthoclase Feldspar Encrusting Smithsonite

Peruvian Pyrite

http://www.rockhounds.com/rockshop/xtal

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Snow crystals

the Beltsville Agricultural Research Center

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Solid : Crystal vs. Amorphous (glassy)

Ordered array of atoms Disordered arrangement

Competition between attractive (binding) force and repulsive force.

Regular array lowers system energy.

Complicated ! -- difficult to predict the structure of materials

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☺ Importance : structure plays a major role indetermining physical properties of solids

☺ Determination : X-ray and neutron scattering are key toolsfor determining crystal structures. (bulk)

Also microscopic techniques such as SEM, TEM,STM, AFM… (surface)

☺ Deviations : There is no perfect crystal. Many key properties depend on deviation more. Defects – imperfection in crystal Phonons – lattice vibrations

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Calcite(CaCO3) crystal is made from spherical particles.

Christiaan Huygen, Leiden 1690

A crystal is made from spherical particles.

Robert Hooke, London 1745

depicted by René Haüy, Paris, 1822

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Crystalperiodic array of atoms : point lattice + basis

Point lattice – mathematical points in space

vectorslatticea,a,a

integeru,u,uauauaurr

321

321

332211

=

∈+++=′

r

2ar

1ar

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Primitive lattice cell

2 Dimensions

• A cell will fill all space by the repetition of suitable crystaltranslation operations. ---- A minimum volume cell.⊗.One lattice point per primitive cell. ⊗.The basis associated w/. a primitive cell -- a primitive basis⊗.Not unique.

2ar

1ar

φsinaa

aaA

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21cellrr

rr

=

×=

Same for all primitive cells

Uniform mass density

Wigner-Seitz Primitive cell -- lattice point is at its center

the highest symmetry cell possible

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Wigner-Seitz Primitive cell in 2D (or 3D)

⊗ Draw lines to connect a given lattice point to all nearby lattice points.⊗. Draw bisecting lines (or planes) to the previous lines.⊗. The smallest area (or volume) enclosed.

2 D Oblique Lattice

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Fundamental types of Bravais lattices

Based on symmetries :

1) Translational – same if translate by a vector

2) Rotational – same if lattice is rotated by an angle about a point2-fold by 180o 4-fold by 90o

3-fold by 120o 6-fold by 60o

3) Mirror symmetry – same if reflected about a plane

4) Inversion symmetry – same if reflected through a point (equivalent to rotation 180o and mirror ⊥ rotational axis r → -r

332211 auauauT ++=Auguste Bravais

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Five Bravais lattices in two dimensions

Square lattice a1=a2, φ=90o

Unit cellSymmetry element

1ar2ar φ

Rectangular lattice a1≠a2, φ=90oSymmetry elementUnit cell

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Oblique lattice a1≠ a2, φ≠60o,90oSymmetry elementUnit cell

a1= 2a2cosφCentered Rectangular latticeSymmetry element

φ1ar

2arUnit cell

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a1=a2, φ=60oHexagonal lattice

Symmetry element

Unit cell twofold axis (dia)

threefold axis (triad)

fourfold axis (tetrad)

sixfold axis (hexad)

mirror line

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A Bravais lattice is a lattice in which every lattice point has exactly the same environment.

A honeycomb crystal = A hexagonal lattice + two-pint basis

How about a honeycomb lattice ?

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The seven crystal systems divided into fourteen Bravais lattices

One 3-foldrotation axis

a1 = a2≠a3

α = β = 90o, γ =120o1: SimpleHexagonal

One 3-fold rotation axis

a1 = a2 = a3

α = β = γ<120o≠ 90o1: SimpleTrigonal

Four 3-fold rotation axes

a1 = a2 = a3

α = β = γ = 90o3: Simple, BCC, FCCCubic

One 4-fold rotation axis

a1 = a2≠a3

α = β = γ = 90o2: Simple, BCCTetragonal

Three mutually orthogonal 2-fold

rotation axes

a1≠a2≠a3

α = β = γ = 90o4: BCC, FCCSimple, Base-Centered

Orthorhombic

One 2-fold rotation axis

a1≠a2≠a3

α = β = 90o ≠ γ2: Simple, Base-Centered

Monoclinic

Nonea1≠a2≠a3

α≠ β≠ γ1: SimpleTriclinic

Characteristic symmetry elements

Unit cell characteristics

Number of latticesSystem

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a1≠a2≠a3TRICLINIC (α≠β≠γ)

三斜晶系

a1

a2

a3

α

γ

β

a1

a3

a2

MONOCLINIC (β=γ=90o ≠ α)單斜晶系

Simple

Simple Base-centered

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a1≠a2≠a3Orthorhombic (α=β=γ=90o)正交晶系

Base-centered BCC

a1

a2

a3

α

γβ

Simple

FCC

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a1≠aTetragonal (α=β=γ=90o)

正方晶系

a

a1

a

α

γ

β

BCCSimple

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Cubic (α=β=γ=90o)

立方晶系

a

a

a

α

γ

β

FCCSimple BCC

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Hexagonal (α=β=90o, γ=120o)

Trigonal(α=β=γ<120o , ≠90o )

a

aa3

γ=120o

a3≠a六角晶系三角（菱面）晶系

aa

a

Simple

Simple

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Characteristics of cubic lattices

Packing fraction

Nearest- neighbor distance

1286Number of nearest neighbors

421Lattice points/cell

Face-centeredBody-centeredSimple

2a

a23a

524.06=

π680.0

83

=π 740.062

=π

Maximum, same as hexagonal

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Crystal – Periodic arrangement of atoms

Bravais lattice + Basis of atom(Array of point in space) (Arrangement of atoms within unit cell)

2 Dimensions → 5 types

3 Dimensions → 14 typesBy symmetry

Crystal may have same or less symmetry than original Bravais lattice.

Basis

Symmetry “4mm”

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Introducing new symmetries for basis with multiple atoms :

• Point group symmetries : combinations of rotation, inversion, reflection that hold one point fixed and return original structure.

2 Dimensions , 103 Dimensions , 32

• Space group symmetries : point group operations + translations that return original structure.2 Dimensions , 173 Dimensions , 230

There are many possible types by symmetry but most are never observed.

Real crystals form a few types due to energies of crystal formation.

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Directions and planes in crystalsUseful to develop a notation for describing directions and

identifying planes of atoms in crystals

Consider lattice defined by

Vector direction is described as [u1 u2 u3]

where u1, u2, and u3 are the lowest reduced integers.

321 a ,a ,a

332211 auauaur ++=r

Note :

• [8 6 0] same as [4 3 0]

• use ū instead of -u

• [u1 u2 u3] not as the same as Cartesian coordinate direction except for simple cubic crystal

] 1 2 1 [

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[ 1 1 0 ]

[ 1 1 1 ]

[ 1 0 0 ]

[ 0 0 1 ]

[ 0 1 0 ]

Cubic has highest symmetric directions

By symmetry, [ 1 0 0 ], [ 0 1 0 ], [ 0 0 1 ] are equivalent

] 1 0 0 [ ], 0 1 0 [ ], 0 0 1 [Denote { 1 0 0 } ≡ set of equivalent directions

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Crystal planes – atoms fall on planes labeled by “Miller indices”

Note :

• Find the intercepts on the axes in terms of a1, a2, and a3 → h’ k’ l’

• Take reciprocals and make integers

• Reduce to smallest three integers

• If the plane does not intersect one of the crystal axes, that intercept is taken to be infinitely far from the origin and the corresponding index is zero.

''kh') '

1 k'1

h'1 ( l

l×

( h k l ) is called Miller index of the plane

Miller indices specify a vector normal to the plane, and not a specific plane : all parallel planes have the same indices

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(25)

2ar

1ar

Schematic illustrations of lattice planesLines in two dimensional crystals

(01)

(11)

Low index plane : more dense and more widely spaced

High index plane : Less dense and more closely spaced

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Most common crystal structures :

Simple Cubic lattice : Po conventional cell : 1 atom/cube

Body Centered Cubic lattice :

Conventional cell : 2 atoms/ cube

(0,0,0)

(1/2,1/2,1/2)

Not a primitive lattice

8 nearest neighbors

Alkali metals : Li, Na, K, Rb, Cs

Ferromagnetic metals : Cr, Fe

Transition metals : Nb, V, Ta, Mo, W

• BCC lattice + single atom basis

• SC lattice + basis of 2 atoms at (0,0,0) and (1/2,1/2,1/2)

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Body Centered Cubic lattice

x

z

ya1

a2a3

)zyx(2aa

)zyx(2aa

)zyx(2aa

3

2

1

+−=

++−=

−+=

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Body-centered Cubic lattice

Primitive cell :

Rhombohedron

w/.

1. edge

2. the angle between two adjacent edges is 109o28’

a23

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Face Centered Cubic lattice :

Conventional cell : 4 atoms/ cube

(0,0,0)

(1/2,1/2,0)

(1/2,0,1/2)(0,1/2,1/2)

Not a primitive lattice

12 nearest neighbors

Noble metals : Cu, Ag, Au

Transition metals : Ni, Pd, Pt,

Inert gas solids : Ne, Ar, Kr, Xe

• FCC lattice + single atom basis

• SC lattice + basis of 4 atoms at (0,0,0), (1/2,1/2,0) (1/2,0,1/2), and (0,1/2,1/2)

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Face Centered Cubic lattice

xy

z

a1

a2

a3

)zx(2aa

)zy(2aa

)yx(2aa

3

2

1

+=

+=

+=

The angle between two adjacent edges : 60o

Edge a22Rhombohedral Primitive cell

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Hexagonal Close-Packed lattice

a1, a2, and c do not construct a primitive lattice

a1, a2, and a3 construct a primitive lattice

a633.1c

120 angle includedan with , aa

1

o21

=

=

a aa 321 ==

a1

a2

c

Basal Plane

a3 12 nearest neighbors

Transition metals : Sc, Y, Ti, Zr, Co…

IIA metals : Be, Mg

Hexgonal lattice + basis of 2 atoms at (0,0,0) and (2/3,1/3,1/2)

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HCP lattice A B A B A B …

x

y

z

FCC lattice A B C A B C A B C …

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Diamond structure : two FCC displaced from each other by ¼of a body diagonal

0

0

0

0

0

1/2

1/2

1/2 1/2

1/4

1/43/4

3/4

FCC lattice + basis of 2 atoms at (0,0,0) and (1/4,1/4,1/4)

Tetrahedral bonding : 4 nearest neighbors12 next nearest neighbors

The maximum packing fraction = 0.34Si, Ge, Sn, C, ZnS, GaAs, …

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Some atoms form multiple stable structures:for example, C→ diamond or graphite (hexagonal)

An STM image of a graphite surfaceclearly shows the interconnected 6-membered rings of graphite

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graphite diamond

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Many crystals undergo structural changes with T, P:for example,

TemperatureFe 910oC 1400oC 2100oC

LiquidBCCFCCBCC

δ-ferrite α-ferrite

TemperatureNa 36K 371K

LiquidFCCHCP

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Compounds

NaCl structures NaCl, NaF, KCl, AgCl, MgO, MnO, …

• FCC lattice + basis of two atoms

• Cl (0,0,0), Na (1/2,1/2,1/2)

SC w/. alternating atoms

CsCl structures CsCl, BeCu, ZnCu(brass), AlNi, AgMg, …

• SC lattice + basis of two atoms

• Cs (0,0,0), Cl (1/2,1/2,1/2)

BCC w/. alternating atoms

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ZnS, CuF, CuCl, … compounds

CdS Photoconductor

GaAs, GaP, InSb, … III-V semiconducting compounds

ZnS structures

Znicblende

• FCC lattice + basis of two atoms• Zn (0,0,0), S (1/4,1/4,1/4)• Ga(0,0,0), As(1/4,1/4,1/4)

diamond w/. alternating atoms

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High Tc superconductors

a b

c

Orthorhombic O(I) (Pmmm)

O(3)

Cu(2)

Cu(1)

O(1) Ba/Pr

PrO(4)

O(2)

O(5)

Tetragonal T (P4/mmm)

Ba/Pr

Pr

O(2)

O(1)

O(3)

Cu(2)

Cu(1)

Ba/Pr

Pr

O(2)

O(1)

Cu(2)

Cu(1)O(3)

Orthorhombic O(II) (Cmmm)

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C-60

“Bucky balls”

BuckminsterFullurene