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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids

http://folk.uio.no/ravi/CMP2013

Prof.P. Ravindran, Department of Physics,

Central University of Tamil Nadu, India

Crystal Structures of Solids

1


2


Table of metals, metalloids, and nonmetals3


Solids, Liquids and Gases

1A 8A

H 2A 3A 4A 5A 6A 7A He

Li Be B C N O F Ne

Na Mg 3B 4B 5B 6B 7B –– 8B –– 1B 2B Al Si P S Cl Ar

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

Fr Ra Ac Rf Db Sg Bh Hs Mt Ds

S L G Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

4


The Solid State1. Classification of Solid Structures

a. Crystalline Solids = regular arrangement of components in 3

dimensions

b. Amorphous Solids = disordered arrangement of components

2. Crystal Structure Basics

a. Crystal = a piece of a crystalline solid

b. Lattice = 3-dimensional system of designating where components are

c. Unit cell = smallest repeating unit of the lattice

d. Examples: simple cubic, body-centered cubic, face-centered cubic

5


6

Early Crystallography

René-Just Haüy (1781): cleavage of calcite

Common shape to all shards: rhombohedral

How to model this mathematically?

What is the maximum number of distinguishable shapes

that will fill three space?

Mathematically proved that there are only 7 distinct

space-filling volume elements


7

Christian Huygens - 1690

Studying calcite crystals made

drawings of atomic packing

and bulk shape.

http://upload.wikimedia.org/wikipedia/en/3/37/Calcite-HUGE.jpg

http://upload.wikimedia.org/wikipedia/en/3/37/Calcite-HUGE.jpg

P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids8

Beryl

Be3Al2(SiO3)6

http://upload.wikimedia.org/wikipedia/commons/f/f3/Beryl09.jpg

http://upload.wikimedia.org/wikipedia/commons/f/f3/Beryl09.jpg


9

Anisotropy

The physical properties of single crystals of some

substances depend on the crystallographic direction in

which measurements are taken.

For example, the elastic modulus, electrical conductivity,

and the index of refraction may have different values in

the [100] and [111] directions.

The directionality of the properties is termed anisotropy

and is associated with the atomic spacing.


10

Isotropic

If measured properties are independent of the direction

of measurement then they are isotropic.

For many polycrystalline materials, the crystallographic

orientations of the individual grains are totally random.

So, though, a specific grain may be anisotropic, when the

specimen is composed of many grains, the aggregate

behavior may be isotropic.


11

Single Crystals

For a crystalline solid, when the periodic and

repeated arrangement of atoms extends

throughout without interruption, the result is a

single crystal.

The crystal lattice of the entire sample is

continuous and unbroken with no grain

boundaries.

For a variety of reasons, including the distorting

effects of impurities, crystallographic defects

and dislocations, single crystals of meaningful

size are exceedingly rare in nature, and difficult

to produce in the laboratory under controlled

conditions.

Huge KDP (monopotassium phosphate)

crystal grown from a seed crystal in a

supersaturated aqueous solution at

LLNL. Below, silicon boule.

http://en.wikipedia.org/wiki/File:KDP_crystal.jpg

http://en.wikipedia.org/wiki/File:KDP_crystal.jpg

http://upload.wikimedia.org/wikipedia/commons/2/23/Monokristalines_Silizium_f%C3%BCr_die_Waferherstellung.jpg

http://upload.wikimedia.org/wikipedia/commons/2/23/Monokristalines_Silizium_f%C3%BCr_die_Waferherstellung.jpg


Single Crystal

The periodic and repeated arrangements of atoms is perfect or extends

throughout the entirety of the specimen without interruption.

All unit cells interlock in the same way and have the same orientation.

Single crystals exist in nature, but they may also produced artificially. They are

ordinarily difficult to grow, because the environment must be carefully

controlled.

Single crystals are needed for modern technologies today.

Electronic micro-chips uses single crystals of silicon and other semiconductors.

A garnet single crystal found in

Tongbei, Fujian Province, China.

If the extremities of a single crystal

are permitted to grow without any

external constraint, the crystal will

assume its geometric shape, with

flat surfaces as shown in the figure.

12

http://www.youtube.com/watch?v=PILlP_yDZ2o


Polycrystalline Materials

Composed of a collection of many small crystals or grains.

Stages in the solidification

of a polycrystalline

material:

a. Crystallite Nuclei

b. Growth of the

Crystallites

c. Formation of grains

d. Microscopic view

13


Anisotropy

Physical properties of single crystals of some substances depend

on the crystallographic direction in which measurements are

made.

This directionality of properties is termed anisotropy, and it is

associated with the variance of atomic or ionic spacing with

crystallographic direction.

Substances in which measured properties are independent of

the direction of measurement are isotropic.

14


15


Structures

The properties of some materials are

directly related to their crystal structures.

Significant property differences exist

between crystalline and noncrystalline

materials having the same composition.

16

P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids 17

• Some engineering applications require single crystals:

• Properties of crystalline materials

often related to crystal structure.

-- Ex: Quartz fractures more

easily

along some crystal planes than

others.

-- diamond

Single crystals for abrasives

-- turbine blades

Crystals as Building Blocks


Atomic Arrangement

Minerals must have a highly

ordered atomic arrangement

The crystal structure of

quartz is an example


Examples of silicate mineralsolivine

epidote

beryl

augite

hornblende

muscovite

quartz

Mineral pictures

from: mindat.org

19


Pink (Rose) : due to traces of iron, manganese or titanium.

Amethyst : May be manganese but some believe it could be organic,

iron or even aluminum.

Citrine : iron

Aventurine : inclusion of green mica (fushite)

Tiger's eye : inclusion of fiber of silicified crocidolite (variety of asbestos)

Prasiolite : Iron or copper

Milk quartz : gas and liquid inclusions

Smoky : Radioactivity on quartz containing aluminium

Blue : pressure.

Chalcedony is a variety of quartz with micro-crystals. Agate is a multicolor

variety of chalcedony and onyx is a variety of agate with parallel strips of

various nuances of black.

Quartz Varieties20


Quartz Crystals

The external

appearance of the

crystal may reflect its

internal symmetry


Quartz Blob

Or the external

appearance may show

little or nothing of the

internal structure


Building Blocks

A cube may be used to

build a number of

forms


Fluorite

Fluorite may appear as

octahedron (upper photo)

Fluorite may appear as a

cube (lower photo), in this

case modified by

dodecahedral crystal faces


25

Crystal Growth

Ways in which a crystal can grow:

Dehydration of a solution

Growth from the molten state (magma or lava)

Direct growth from the vapor state


26

Types of Solids

Single crsytal, polycrystalline, and amorphous, are the

three general types of solids.

Each type is characterized by the size of ordered region

within the material.

An ordered region is a spatial volume in which atoms or

molecules have a regular geometric arrangement or

periodicity.


27

Crystalline Solid

Single Crystal

Single Pyrite

Crystal

Amorphous

Solid

Single crystal has an atomic structure that repeats

periodically across its whole volume. Even at infinite length

scales, each atom is related to every other equivalent atom in

the structure by translational symmetry

http://weblog.burningbird.net/fires/001936.htm

http://weblog.burningbird.net/fires/001936.htm


28

Ionic Crystals

Examples include sodium chloride, cupric sulfate.


29

Some Factors Affecting Crystalline Structure

Size of atoms or ions involved

Stoichiometry of salt

Materials involved

– Some substances do not form crystalline solids


30

Crystals

The periodic array of atoms, ions, or molecules that form the solids is called Crystal Structure

Crystal Structure = Space (Crystal) Lattice + Basis

– Space (Crystal) Lattice is a regular periodicarrangement of points in space, and is purely mathematical abstraction

– Crystal Structure is formed by “putting” the identical atoms (group of atoms) in the points of the space lattice

– This group of atoms is the Basis


The definition of crystals are based on symmetry and not on

the geometry of the unit cell

Our choice of unit cell cannot alter the crystal system a

crystal belongs to

Crystals based on a particular lattice can have symmetry

equal to or lower than that of the lattice

When all symmetry (including translation) is lost the

construct is called amorphous

31


Ideal crystals may have perfect positional and orientational

order with respect to geometrical entities and physical

properties

In (defining) real crystals some of these strict requirements

may be relaxed:

the order considered may be only with respect to the

geometrical entity

the positional order may be in the average sense

the orientational order may be in the average sense

In addition real crystals:

are finite

may contain other defects .

Ideal versus Real crystals32


When we look around us many of the organic materials are non-crystalline

But, many of the common ‘inorganic’ materials are ‘usually*’ crystalline:

◘ Metals: Cu, Zn, Fe, Cu-Zn alloys

◘ Semiconductors: Si, Ge, GaAs

◘Ceramics: Alumina (Al2O3), Zirconia (Zr O2), SiC, SrTiO3

Also, the usual form of crystalline materials (say a Cu wire or a piece of

alumina) is polycrystalline and special care has to be taken to produce

single crystals

Polymeric materials are usually not ‘fully’ crystalline

The crystal structure directly influences the properties of the material(as we have seen in the Introduction chapter many additional factors come in)

Why study crystal structures?

Gives a terse (concise) representation of a large assemblage of species

Gives the ‘first view’ towards understanding of the properties of the crystal

Why study crystallography?

* Many of the materials which are usually crystalline can also be obtained in an amorphous form

33


34

Crystallography

What is crystallography?

The branch of science that deals with the geometric

description of crystals and their internal arrangement.

Crystallography is essential for solid state physics

Symmetry of a crystal can have a profound influence on its

properties.

Any crystal structure should be specified completely, concisely and

unambiguously.

Structures should be classified into different types according to the

symmetries they possess.


35

A basic knowledge of crystallography is essential for solidstate physicists;

– to specify any crystal structure and

– to classify the solids into different types according tothe symmetries they possess.

Symmetry of a crystal can have a profound influence on itsproperties.

We will concern in this course with solids with simplestructures.

Elementary Crystallography


36

Crystal Lattice

What is crystal (space) lattice?

In crystallography, only the geometrical properties of the

crystal are of interest, therefore one replaces each atom by a

geometrical point located at the equilibrium position of that

atom.

Platinum Platinum surface Crystal lattice and

structure of Platinum(scanning tunneling microscope)


37

An infinite array of

points in space,

Each point has identical

surroundings to all

others.

Arrays are arranged

exactly in a periodic

manner.

Crystal Lattice

α

a

b

CB ED

O A

y

x


The “unit cell” is the basic

repeating unit of the

arrangement of atoms, ions or

molecules in a crystalline solid.

The “lattice” refers to the 3-D

array of particles in a

crystalline solid. One type of

atom occupies a “lattice point”

in the array.

38


Examples of Unit Cells39


Crystal Systems

7 crystal systems

14 crystal lattices

Unit cell: smallest repetitive volume which contains the

complete lattice pattern of a crystal.

a, b, and c are the lattice constants

40


Unit Cell Concept

The unit cell is the smallest structural unit or building block that uniquely can describe the crystal structure. Repetition of the unit cell generates the entire crystal. By simple translation, it defines a lattice .

Lattice: The periodic arrangement of atoms in a crystal.

Lattice Parameter : Repeat

distance in the unit cell, one

for in each dimension

b

a

41


The Unit Cell Concept

The simplest repeating unit in a crystal is called a unit

cell.

Opposite faces of a unit cell are parallel.

The edge of the unit cell connects equivalent points.

Not unique. There can be several unit cells of a crystal.

The smallest possible unit cell is called primitive unit

cell of a particular crystal structure.

A primitive unit cell whose symmetry matches the

lattice symmetry is called Wigner-Seitz cell.

42


Each unit cell is defined in terms of lattice points.

Lattice point not necessarily at an atomic site.

For each crystal structure, a conventional unit cell,

is chosen to make the lattice as symmetric as

possible. However, the conventional unit cell is not

always the primitive unit cell.

A crystal's structure and symmetry play a role in

determining many of its properties, such as cleavage

(tendency to split along certain planes with smooth

surfaces), electronic band structure and optical

properties.

43


Crystal Systems

Units cells and lattices in 3-D:

– When translated in each lattice parameter direction, MUST fill

3-D space such that no gaps, empty spaces left.

Lattice Parameter : Repeat

distance in the unit cell, one

for in each dimension

a

bc

44


45

The Importance of the Unit Cell

One can analyze the crystal as a whole by investigating a

representative volume.

Ex: from unit cell we can

– Find the distances between nearest atoms for

calculations of the forces holding the lattice together

– Look at the fraction of the unit cell volume filled by

atoms and relate the density of solid to the atomic

arrangement

– The properties of the periodic Xtal lattice determine the

allowed energies of electrons that participate in the

conduction process.


Unit cell

46


47

The unit cell and, consequently, theentire lattice, is uniquely determinedby the six lattice constants: a, b, c,α, β and γ.

Only 1/8 of each lattice point in aunit cell can actually be assigned tothat cell.

Each unit cell in the figure can beassociated with 8 x 1/8 = 1 latticepoint.

Unit Cell


Unit cell: Simplest portion of the structure which is repeated

and shows its full symmetry.

Basis vectors a and b defines relationship between a unit cell

and (Bravais) lattice points of a crystal.

Equivalent points of the lattice is defined by translation vector.

r = ha + kb where h and k are integers. This constructs the entire

lattice.

a

b

48


By repeated duplication, a unit cell should

reproduce the whole crystal.

A Bravais lattice (unit cells) - a set of points

constructed by translating a single point in

discrete steps by a set of basis vectors.

In 3-D, there are 14 unique Bravais lattices.

All crystalline materials fit in one of these

arrangements.

In 3-D, the translation vector is

r = ha + kb + lc

49


50

Unit Cell in 2D

The smallest component of the crystal (group of atoms,ions or molecules), which when stacked together withpure translational repetition reproduces the wholecrystal.

S

a

b

S

S

S

S

S

S

S

S

S

S

S

S

S

S


51

Unit Cell in 2D

The smallest component of the crystal (group of atoms,ions or molecules), which when stacked together withpure translational repetition reproduces the wholecrystal.

S

S

The choice of

unit cell

is not unique.

a

Sb

S


2D Unit Cell example -(NaCl)

We define lattice points ; these are points with identical

environments


Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.


This is also a unit cell -

it doesn’t matter if you start from Na or Cl


- or if you don’t start from an atom


This is NOT a unit cell even though they are all the

same - empty space is not allowed!


In 2D, this IS a unit cell

In 3D, it is NOT


58

Unit Cell in 3D


Unit Cell in 3D

59


Crystal Structure

Crystal structure can be obtained by attaching atoms,groups of atoms or molecules which are called basis(motif) to the lattice sides of the lattice point.

Crystal Structure = Crystal Lattice + Basis


61

Lattice Sites in Cubic Unit Cell


A two-dimensional Bravais lattice with

different choices for the basis

62


63

Crystal structure

Don't mix up atoms withlattice points

Lattice points areinfinitesimal points in space

Lattice points do notnecessarily lie at the centreof atoms

Crystal Structure = Crystal Lattice + Basis


Crystal Lattice

Bravais Lattice (BL)

All atoms are of the same kind All lattice points are equivalent

Non-Bravais Lattice (non-

BL)

Atoms can be of different kind Some lattice points are not

equivalentA combination of two or more BL

64


65

Types Of Crystal Lattices

1) Bravais lattice is an infinite array of discrete points with an

arrangement and orientation that appears exactly the same,

from whichever of the points the array is viewed. Lattice is

invariant under a translation.

Nb film

http://www.sciencemag.org/cgi/content/full/274/5290/1167/F1

http://www.sciencemag.org/cgi/content/full/274/5290/1167/F1


66

Types Of Crystal Lattices

The red side has a neighbour to itsimmediate left, the blue one instead hasa neighbour to its right.

Red (and blue) sides are equivalent andhave the same appearance

Red and blue sides are not equivalent.Same appearance can be obtainedrotating blue side 180º.

2) Non-Bravais Lattice

Not only the arrangement but also the orientation must

appear exactly the same from every point in a bravais lattice.

Honeycomb


67

Translational Lattice Vectors – 2D

A space lattice is a set of points such thata translation from any point in thelattice by a vector;

Rn = n1 a + n2 b

locates an exactly equivalent point, i.e. apoint with the same environment as P .This is translational symmetry. Thevectors a, b are known as lattice vectorsand (n1, n2) is a pair of integers whosevalues depend on the lattice point.

P

Point D(n1, n2) = (0,2)

Point F (n1, n2) = (0,-1)


68

The two vectors a and bform a set of latticevectors for the lattice.

The choice of latticevectors is not unique.Thus one could equallywell take the vectors a andb’ as a lattice vectors.

Lattice Vectors – 2D


69

Five Bravais Lattices in 2D


70

Lattice Vectors – 3D

An ideal three dimensional crystal is described by 3fundamental translation vectors a, b and c. If there is alattice point represented by the position vector r, there isthen also a lattice point represented by the position vectorwhere u, v and w are arbitrary integers.

r’ = r + u a + v b + w c (1)


Gold (Why one naturally get Gold in pure form?

71


Hydrated Crystals - Water

molecules become

chemically bonded to ions in

the crystal.

Anhydrous Crystals Crystal

without water

CuSO4 • 5 H2O

a hydrated crystal 72

Hydrated and Unhydrated Crystals


Hydrated Crystals

Ex:

CuSO4

• 5H2O + heat CuSO

4+ 5 H

2O

(Blue) (White)

Hydrate

Anhydrous

Hydrate

73


Ionic solids

Group 1A (alkali metals) contains lithium (Li), sodium (Na), potassium(K),..and these combine easily with group 7A (halogens) of fluorine (F),chlorine (Cl), bromine (Br),.. and produce ionic solids of NaCl, KCl, KBr,etc.

Rare (noble) gases

Group 8A elements of noble gases of helium(He), neon (Ne), argon(Ar),… have a full complement of valence electrons and so do not combineeasily with other elements.

Elemental semiconductors

Silicon(Si) and germanium (Ge) belong to group 4A.

Compound semiconductors

1) III-V compound s/c’s; GaP, InAs, AlGaAs (group 3A-5A)

2) II-VI compound s/c’s; ZnS, CdS, etc. (group 2B-6A)

The periodic table74


Molecular Solids

H2O, S8, P4

Molecules occupy positions in crystal lattice

Melting points increase with size and polarity

75


SOLID MATERIALS

CRYSTALLINE POLYCRYSTALLINEAMORPHOUS(Non-crystalline)

Single Crystal

Classification of solids

http://www.alaskanessences.com/gembig/Pyrite.jpg



77

Crystalline Solid is the solid form of a substance in

which the atoms or molecules are arranged in a

definite, repeating pattern in three dimension.

Crystalline Solid


Atomic order present in sections (grains) of the solid. Different order of arrangement from grain to grain.

Grain sizes = hundreds of m. An aggregate of a large number of small crystals or

grains in which the structure is regular, but the crystals

or grains are arranged in a random fashion.

Polycrystalline Solids78


Polycrystal

Polycrystalline

Pyrite form

(Grain)

• Polycrystal is a material made up of an

aggregate of many small single crystals

(also called crystallites or grains).

• The grains are usually 100 nm - 100

microns in diameter. Polycrystals with

grains that are <10 nm in diameter are

called nanocrystalline

Polycrystalline Solid




No regular long range order of arrangement in the atoms.

Eg. Polymers, cotton candy, common window glass, ceramic.

Can be prepared by rapidly cooling molten material.

Rapid – minimizes time for atoms to pack into a more

thermodynamically favorable crystalline state.

Two sub-states of amorphous solids: Rubbery and Glassy

states. Glass transition temperature Tg = temperature above

which the solid transforms from glassy to rubbery state,

becoming more viscous.

Amorphous Solids80


81

Amorphous (non-crystalline) Solid is composed ofrandomly orientated atoms, ions, or molecules thatdo not form defined patterns or lattice structures.

Amorphous Solid


Properties of single crystalline materials vary with direction,

ie anisotropic.

Properties of polycrystalline materials may or may not vary

with direction.

If the polycrystal grains are randomly oriented, properties

will not vary with direction i.e isotropic.

If the polycrystal grains are textured, properties will vary

with direction i.e anisotropic

Single- Vs Poly- Crystal82


-Properties may/may not

vary with direction.

-If grains are randomly

oriented: isotropic.

(Epoly iron = 210 GPa)

-If grains are textured,

anisotropic.

200 m



85

Physical Properties

Related to Solid Structure

Density

Luster

Hardness

Electrical Properties

Melting Point

Magnetic Properties


Tetrakaidecahedron

(Truncated Octahedron)

Octahedron

Cube

Crystal Shape

Tetrahedron

Examples of few crystal shapes (cubic crystal system)

86


The shape of the crystal (Eumorphic-

well formed) will ‘reflect’ the point

group symmetry of the crystal

87


88

Early ideas

Crystals are solid - but solids are not necessarily

crystalline

Crystals have symmetry (Kepler) and long range

order

Spheres and small shapes can be packed to

produce regular shapes (Hooke, Hauy)

?


Kepler wondered why snowflakes have 6 corners,

never 5 or 7. By considering the packing of

polygons in 2 dimensions, it can be shown why

pentagons and heptagons shouldn’t occur.

Empty space not allowed

89


Crystal Types

Three types of solids, classified according to atomic

arrangement:

(a) crystalline and (b) amorphous materials are illustrated by

microscopic views of the atoms, whereas (c) polycrystalline

structure is illustrated by a more macroscopic view of adjacent

single-crystalline regions, such as (a).

90


quartz

Crystal structure

Amorphous structure

91


Crystals

The periodic array of atoms, ions, or molecules that form the solids is called Crystal.

Crystal Structure = Space (Crystal) Lattice + Basis

– Space (Crystal) Lattice is a regular periodicarrangement of points in space, and is purely mathematical abstraction

– Crystal Structure is formed by “putting” the identical atoms (group of atoms) in the points of the space lattice

– This group of atoms is the Basis

92


The crystal system: Set of rotation and reflection

symmetries which leave a lattice point fixed.

There are seven unique crystal systems: the cubic

(isometric), hexagonal, tetragonal, rhombohedral

(trigonal), orthorhombic, monoclinic and triclinic.

Crystal System93


Bravais Lattice and Crystal System

Crystal structure: contains atoms at every lattice point.

The symmetry of the crystal can be more complicated than

the symmetry of the lattice.

Bravais lattice points do not necessarily correspond to real

atomic sites in a crystal. A Bravais lattice point may be

used to represent a group of many atoms of a real crystal.

This means more ways of arranging atoms in a crystal

lattice.

94


What is a lattice?

A lattice is a 3-D system of points designating the positions of

the components (atoms, ions, or molecules) that make up the

substance

Unit Cell: The smallest repeating unit of the lattice.

Eg:

simple cubic

body-centered cubic

face-centered cubic

95


We can pick out the smallest repeating unit…..

Lattice Example96


We can pick out the smallest repeating unit…called Unit Cell...

UNIT CELL97


98

Definitions

1. The unit cell: “The smallest repeat unit of a

crystal structure, in 3D, which shows the full

symmetry of the structure”

The unit cell is a box

with:

• 3 sides - a, b, c

• 3 angles - , ,

14 possible crystal structures (Bravais lattices)


99

Unit Cell

Simplest (smallest) parallel piped outlined

by a lattice

Lattice: a two or three (space lattice)

dimensional array of points

Environment about all lattice points must

be identical

Unit cell must fill all space, with no “holes”


100

Physical Properties & Structure

Hardness and Structure

– Hardness depends on how easily structural units can be

moved relative to one another

– Molecular solids with weak intermolecular attractions are

rather soft compared with ionic compounds, where forces

are much stronger

– Covalent network solids are quite hard because of the

rigidity of the covalent network structure

– Molecular and ionic crystals are generally brittle because

they fracture easily along crystal plane

– Metallic solids, by contrast, are malleable


101

Physical Properties

Electrical Conductivity and Structure

– Molecular and ionic solids are generally considered

nonconductors

– Ionic compounds conduct in their molten state, as ions are

then free to move

– Metals are all considered conductors

– Of the covalent network solids, only graphite conducts

electricity

This is due to the delocalization of the resonant p

electrons in graphite’s sp2 hybridization


102

UNIT CELL

Primitive

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

Conventional & Non-primitive

More than one lattice point per cell Integral multiples of the volume of

primitive cell

Body centered cubic(bcc)

Conventional ≠ Primitive cell

Simple cubic(sc)

Conventional = Primitive cell


103

The Conventional Unit Cell

A unit cell just fills space when

translated through a subset of

Bravais lattice vectors.

The conventional unit cell is chosen

to be larger than the primitive cell,

but with the full symmetry of the

Bravais lattice.

The size of the conventional cell is

given by the lattice constant a.


Primitive and conventional cells of FCC

104


1

2

3

1ˆ ˆ ˆ( )

2

1ˆ ˆ ˆ( )

2

1ˆ ˆ ˆ( )

2

a x y z

a x y z

a x y z

Primitive and conventional cells of BCC

Primitive Translation Vectors:

105


a

b c

Simple cubic (sc):

primitive cell=conventional cell

Fractional coordinates of lattice

points:

000, 100, 010, 001, 110,101, 011, 111

Primitive and conventional cells

Body centered cubic (bcc):

conventional cell

a

b c Fractional coordinates of lattice points in

conventional cell:

000,100, 010, 001, 110,101, 011, 111, ½ ½ ½

106


Face centered cubic (fcc):

primitive (rombohedron) cell

a

b

c

Fractional coordinates:

000, 100, 101, 110, 110,101, 011, 211, 200

Face centered cubic (fcc):

conventional cell

a

bc


000,100, 010, 001, 110,101, 011,111,

½ ½ 0, ½ 0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ ,

½ ½ 1

Primitive and conventional cells

107


Hexagonal close packed cell

(hcp): conventional =primitive

cell


100, 010, 110, 101,011, 111,000,

001

points of primitive cell

a

b

c

Primitive and conventional cells-hcp

108


A primitive unit cell is made ofprimitive translation vectors a1 ,a2, anda3 such that there is no cell of smallervolume that can be used as a buildingblock for crystal structures.

A primitive unit cell will fill space byrepetition of suitable crystal translationvectors. This defined by theparallelpiped a1, a2 and a3. The volumeof a primitive unit cell can be found by

V = a1.(a2 x a3) (vector products)

Cubic cell volume = a3

Primitive Unit Cell and vectors


110

The primitive unit cell may have only one lattice point.

There can be different choices for lattice vectors , but the

volumes of these primitive cells are all the same.

P = Primitive Unit Cell

NP = Non-Primitive Unit Cell

Primitive Unit Cell

1a


111

Wigner-Seitz Method

A simply way to find theprimitive cell which is calledWigner-Seitz cell can be doneas follows;

1. Choose a lattice point.

2. Draw lines to connect theselattice point to its neighbours.

3. At the mid-point and normalto these lines draw new lines.

The volume enclosed is called as a

Wigner-Seitz cell.


112

Wigner-Seitz Cell - 3D


Wigner-Seitz primitive unit cell and first Brillouin zone

The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are

closer to that lattice point than to any of the other lattice points.

The cell may be chosen by first picking a lattice point. Then, lines are drawn to all nearby

(closest) lattice points. At the midpoint of each line, another line (or a plane, in 3D) is drawn

normal to each of the first set of lines.

1D case

2D case

3D case: BCCImportant

113


The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice

1D

2D

Real space Reciprocal space

114


3D: Recall that the reciprocal lattice of FCC is BCC.

44

4

4/a

Why is FCC so important?

X = ???

115


The Unit Cell Concept summary

The simplest repeating unit in a crystal is called a unit cell.

Opposite faces of a unit cell are parallel.

The edge of the unit cell connects equivalent points.

Not unique. There can be several unit cells of a crystal.

The smallest possible unit cell is called primitive unit cell of a

particular crystal structure.

A primitive unit cell whose symmetry matches the lattice

symmetry is called Wigner-Seitz cell.

116


Each unit cell is defined in terms of lattice points.

Lattice point not necessarily at an atomic site.

For each crystal structure, a conventional unit cell,

is chosen to make the lattice as symmetric as

possible. However, the conventional unit cell is not

always the primitive unit cell.

A crystal's structure and symmetry play a role in

determining many of its properties, such as cleavage

(tendency to split along certain planes with smooth

surfaces), electronic band structure and optical

properties.

117


The crystal system: Set of rotation and reflection

symmetries which leave a lattice point fixed.

There are seven unique crystal systems: the cubic

(isometric), hexagonal, tetragonal, rhombohedral

(trigonal), orthorhombic, monoclinic and triclinic.

Crystal System118


Bravais Lattice and Crystal System

Crystal structure: contains atoms at every lattice point.

The symmetry of the crystal can be more complicated than

the symmetry of the lattice.

Bravais lattice points do not necessarily correspond to real

atomic sites in a crystal. A Bravais lattice point may be used

to represent a group of many atoms of a real crystal. This

means more ways of arranging atoms in a crystal lattice.

119


Geometry Of Crystals

Space Lattices

Motifs

Crystal Systems

Elementary Crystallography

M.J. Buerger

John Wiley & Sons Inc., New York (1956)The Structure of Materials

Samuel M. Allen, Edwin L. Thomas

John Wiley & Sons Inc., New York (1999)

Advanced Reading


The language of crystallography is one shortness

We shall consider two definitions of a crystal:

1) Crystal = Lattice + Motif

2) Crystal = Space Group + Asymmetric unit

The second definition is the more advanced one (the language of

crystallographers) and we shall only briefly consider it in this

introductory text

The second definition becomes important as the classification of crystals

(7 crystal systems) is made based on symmetry and the first definition

does not bring out this aspect

Note: Since we have this precise definition of a crystal, loose definitions

should be avoided (Though often we may live with definitions like: a 3D translationally

periodic arrangement of atoms in space is called a crystal)

Initially we shall start with ideal mathematical crystals and then slowly

we shall relax various conditions to get into practical crystals

121


Lattice the underlying periodicity of the crystal

Basis Entity associated with each lattice points

Lattice how to repeat

Motif what to repeat

Crystal = Lattice + Motif

Motif or Basis:

typically an atom or a group of atoms associated with each lattice point

Definition 1

Translationally periodic

arrangement of motifs

Crystal

Translationally periodic

arrangement of points

Lattice

122

lattice.ppt

Motifs.ppt

Motifs.ppt


Symmetry is perhaps the most important principle of

nature:

though often you will have to dig deeper to find this

statement

The analogous terms to symmetry are:

Symmetry Conservation Invariance

The kind of symmetry of relevance to crystallography is

geometrical symmetry

The kind of symmetry we encountered in the definition of a

lattice is TRANSLATIONAL SYMMETRY (t)

As mentioned before crystals are understood based on the language of symmetry 123

Symmetry

symmetry.ppt


Auguste Bravais

Found fourteen unique lattices which satisfy the

requirements

Published Études Crystallographiques in 1849


Isometric (cubic) Lattices

P = primitive I = body-centered (I for German innenzentriate) F = face centered a = b = c, α = β = γ = 90 ̊


Tetragonal Lattices

a = b ≠c

α = β = γ = 90 ̊


Tetragonal Axes

The tetragonal unit cell vectors

differ from the cubic one by

either stretching the vertical

axis, so that c > a (upper image)

or compressing the vertical

axis, so that c < a (lower image)


Orthorhombic Lattice

a ≠ b ≠c

α = β = γ = 90 ̊ C - Centered: additional point in the center of each

end of two parallel faces


Orthorhombic Axes

The axes system is

orthogonal

Common practice is to assign

the axes so the the magnitude

of the vectors is c > a > b


Monoclinic Lattice

a ≠ b ≠c

α = γ = 90 ̊ (β ≠ 90 ̊)


Monoclinic Axes

The monoclinic axes

system is not orthogonal


Triclinic Lattice

a ≠ b ≠c

α ≠ β ≠ γ ≠ 90 ̊


Triclinic Axes

None of the axes are at

right angles to the

others

Relationship of angles

and axes is as shown


134

Hexagonal

Some crystallographers call the hexagonal group a

single crystal system, with two divisions

Rhombohedral division

Hexagonal division

Others divide it into two systems, but this practice

is discouraged


Hexagonal Lattice

a = b ≠ c

α = γ = 90 ̊

β = 120 ̊


Rhombohedral Lattice

a = b = c

α = β = γ ≠ 90 ̊


What are the symmetries of the 7 crystal systems?

Characteristic symmetry

Cubic Four 3-fold rotation axes

(two will generate the other two)

Hexagonal One 6-fold rotation axis

(or roto-inversion axis)

Tetragonal (Only) One 4-fold rotation axis


Trigonal (Only) One 3-fold rotation axis


Orthorhombic (Only) Three 2-fold rotation axes


Monoclinic (Only) One 2-fold rotation axis


Triclinic None

Note: translational symmetry is always present in crystals (i.e. even in triclinic crystal)

We have stated that basis of definition

of crystals is ‘symmetry’ and hence

the classification of crystals is also

based on symmetry

The essence of the required symmetry

is listed in the table

more symmetries may be part of the

point group in an actual crystal

137


Crystal Structures - Cubic

a

aa

Simple Face-Centered Body-Centered


Crystal Structures - Monoclinic

c

ab

Simple End Face-Centered


Crystal Structures - Tetragonal

c

a

a

Simple Body-Centered


Crystal Structures - Orthorhombic

c

ab

SimpleEnd

Face-Centered

Body

Centered

Face

Centered


For a well grown crystal (eumorphic crystal) the external shape

‘reflects’ the point group symmetry of the crystal

the confluence of the mathematical concept of point groups

and practical crystals occurs here!

The unit cell shapes indicated are the conventional/preferred

ones and alternate unit cells may be chosen based on need

It is to be noted that some crystals can be based on all possible

lattices (Orthorhombic crystals can be based on P, I, F, C

lattices); while others have a limited set (only P triclinic lattice)

Emphasis

142


Lattices can be constructed using translation alone

The definition (& classification) of Crystals is based on symmetry and NOT

on the geometry of the unit cell (as often one might feel after reading some

books!)

Crystals based on a particular lattice can have symmetry:

equal to that of the lattice or

lower than that of the lattice

Based on symmetry crystals are classified into seven

types/categories/systems known as the SEVEN CRYSTAL SYSTEMS

We can put all possible crystals into 7 boxes based on symmetry

Crystal system

Symmetry operators acting at a point can combine in 32 distinct ways to

give the 32 point groups

Lattices have 7 distinct point group symmetries which correspond to the

SEVEN CRYSTAL SYSTEMS

Alternate view

143

lattice.ppt

symmetry.ppt

symmetry.ppt


Crystal System Shape of Unit Cell Bravais Lattices

P I F C

1 Cubic Cube

2 Tetragonal Square Prism (general height)

3 Orthorhombic Rectangular Prism (general height)

4 Hexagonal 120 Rhombic Prism

5 Trigonal Parallopiped (Equilateral, Equiangular)

6 Monoclinic Parallogramic Prism

7 Triclinic Parallopiped (general)

Why are some of the entries missing?

Why is there no C-centred cubic lattice?

Why is the F-centred tetagonal lattice missing?

….?

14 Bravais Lattices divided into 7 Crystal Systems

P Primitive

I Body Centred

F Face Centred

C A/B/C- Centred

A Symmetry based concept ‘Translation’ based concept

144

mystry_missing_lattices.ppt

mystry_missing_lattices.ppt


Crystal System Bravais Lattices

1. Cubic P I F

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

?

Why so many empty boxes?

E.g. Why cubic C is absent?

P: Simple; I: body-centred;F: Face-centred; C: End-centred

145


End-centred cubic not in the Bravais list ?

End-centred cubic = Simple Tetragonal

2

a

2

a

146


14 Bravais lattices divided into seven crystal systems

Crystal system Bravais lattices

1. Cubic P I F C

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

147


THE 7 CRYSTAL SYSTEMS

148


1. Name of crystal system

lattice parameters and

relationship amongst them (preferred Unit Cell)

Possible Bravais lattices

Point groups belonging to the crystal system

Diagram of preferred Unit Cell

149


1. Cubic Crystals

a = b= c = = = 90º

m

23

m

4 432, ,3m 3m,4 23, groupsPoint

Note the 3s are in the second position

SC, BCC, FCC are lattices

while HCP & DC are crystals!

• Simple Cubic (P) - SC

• Body Centred Cubic (I) – BCC

• Face Centred Cubic (F) - FCC

Elements with Cubic structure → SC: F, O, Po ||

BCC: Cr, Fe, Nb, K, W, V||

FCC: Al, Ar, Pb, Ni, Pd, Pt, Ge

150


Examples of elements with Cubic Crystal Structure

Po

n = 1n = 2 n = 4

Fe Cu

BCC FCC/CCPSC

C (diamond)

n = 8 DC

151


Note that cubic crystals can have the shape of a cube, an octahedron, a

truncated octahedron etc.

(some of these polyhedra have the same rotational symmetry axes; noting that

cube and octahedron are regular solids (Platonic) while truncated octahedron

with two kinds of faces is not a regular solid)

The external shape is a ‘reflection’ of the symmetry at the atomic level

Point groups have be included for completeness and can be ignored by

beginners

Cubic crystals can be based on Simple Cubic (SC), Body Centred Cubic

(BCC) and Face Centred Cubic Lattices (FCC)

by putting motifs on these lattices

After the crystal is constructed based on the SC, BCC or FCC lattice, it

should have four 3-fold symmetry axes (along the body diagonals)

which crystals built out of atomic entities will usually have

if the crystal does not have this feature it will not be a cubic crystal (even

though it is based on a cubic lattice)

152

../../../books/e-book/symmetry_of_solids.ppt


2. Tetragonal Crystals

a = b c = = = 90º

Simple Tetragonal

Body Centred Tetragonal -BCT

m

2

m

2

m

42m,4 4mm, 422, ,

m

4 ,4 4, groupsPoint

Note the 4 in the first place

Elements with Tetragonal structure → In, Sn

153


Example of an element with Body Centred Tetragonal Crystal Structure Indium

InLattice parameter(s) a = 3.25 Å, c = 4.95 Å

Space Group I4/mmm (139)

Strukturbericht notation A6

Pearson symbol tI2

Other examples with this structure Pa

Wyckoff

position

Site

Symmetryx y z Occupancy

In 2a 4/mmm 0 0 0 1

In

All atoms are In → coloured

for better visibility

[001] view

[100] views

BCT

Note: All atoms are identical (coloured differently for easy visualization)

154


3. Orthorhombic Crystalsa b c = = = 90º

Simple Orthorhombic

Body Centred Orthorhombic

Face Centred Orthorhombic

End Centred Orthorhombic

m

2

m

2

m

2 2mm, 222, groupsPoint

a b c

One convention

Elements with Orthorhombic structure → Br, Cl, Ga, I, S

155


Example of an element with Orthorhombic Crystal Structure Ga

GaLattice parameter(s) a = 2.9 Å, b = 8.13, c = 3.17 Å

Space Group Cmcm (63)

Strukturbericht notation

Pearson symbol oC4

Wyckoff

position

Site


Ga 4c m2m 0 0.133 0.25 1

[010] view

[001] view

Note: All atoms are identical (coloured differently for easy visualization)

156


4. Hexagonal Crystals

a = b c = = 90º = 120º

Simple Hexagonal

m

2

m

2

m

6 m2,6 6mm, 622, ,

m

6 ,6 6, groupsPoint

Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn

157


Example of an element with Hexagonal Crystal Structure Mg

Note: All atoms are identical (coloured differently for easy

visualization)

158


5. Trigonal/Rhombohedral

Crystals

a = b = c

= = 90º

• Rhombohedral (simple)

m

23 3m, 32, ,3 3, groupsPoint

Note the 3 s are in the first position

Elements with Trigonal structure → As, B, Bi, Hg, Sb, Sm

159


Example of an element with Simple Trigonal

Crystal Structure -Hg

[111] view

Wyckoff

position

Site


Hg 1a -3m 0 0 0 1

-HgLattice parameter(s) a = 3.005 Å

Space Group R-3m (166)

Strukturbericht notation A10

Pearson symbol hR1

Other examples with this structure -Po

160


6. Monoclinic Crystalsa b c

= = 90º

Simple Monoclinic

End Centred (base centered) Monoclinic (A/C)

m

2 ,2 2, groupsPoint

Elements with Monoclinic structure → P, Pu, Po

161


7. Triclinic Crystalsa b c

• Simple Triclinic

1 1, groupsPoint

162


163


164


165


Cubic(48)

Tetragonal(16)

Triclinic(2)

Monoclinic(4)

Orthorhombic(8)

Progressive lowering of symmetry amongst the 7 crystal systems

Hexagonal(24)

Trigonal(12)

Incr

easi

ng s

ym

met

ry

Superscript to the crystal system is the order of the lattice point group

Arrow marks lead from supergroups to subgroups

Ordering the 7 Crystal Systems: Symmetry

Cubic

Hexagonal

Orthorhombic

Trigonal

Tetragonal

Monoclinic

Triclinic

Order

166


Minimum symmetry requirement for the 7 crystal systems

Crystal

system

Characteric symmetry Point groups Comment

Cubic Four 3-fold rotation axes

m

23

m

4 432, ,3m 3m,4 23,

3 or 3 in the second place

Two 3-fold axes will generate the other

two 3-fold axes

Hexagonal One 6-fold rotation axis

(or roto-inversion axis) m

2

m

2

m

6 m2,6 6mm, 622, ,

m

6 ,6 6,

6 in the first place

Tetragonal (Only) One 4-fold

rotation axis


2

m

2

m

42m,4 4mm, 422, ,

m

4 ,4 4,

4 in first place but no 3 in second place

Trigonal (Only) One 3-fold

rotation axis


23 3m, 32, ,3 3,

3 or 3 in the first place

Orthorhombic (Only) Three 2-fold

rotation axes


2

m

2

m

2 2mm, 222,

Monoclinic (Only) One 2-fold

rotation axis


2 ,2 2,

Triclinic None 1 1, 1 could be present

167


168

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