56223895 chapter1 crystalline solids

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EP 364 SOLID STATE

PHYSICS

Course CoordinatorProf. Dr. Beşire Gönül



EP364 SOLID STATE PHYSICSINTRODUCTION

INTRODUCTION

AIM OF SOLID STATE PHYSICS

WHAT IS SOLID STATE PHYSICSAND WHY DO IT?

CONTENT

REFERENCES




Aim of Solid State Physics

Solid state physics (SSP) explains the properties of solid materials as found on earth.

The properties are expected to follow fromSchrödinger’s eqn. for a collection of atomic nucleiand electrons interacting with electrostatic forces.

The fundamental laws governing the behaviour of solids are known and well tested.




Crystalline Solids

We will deal with crystalline solids, that is solidswith an atomic structure based on a regularrepeated pattern.

Many important solids are crystalline.

More progress has been made in understanding thebehaviour of crystalline solids than that of non-crystalline materials since the calculation are easierin crystalline materials.




What is solid state physics?

Solid state physics, also known as condensed matterphysics, is the study of the behaviour of atoms when

they are placed in close proximity to one another.

In fact, condensed matter physics is a much bettername, since many of the concepts relevant to solids

are also applied to liquids, for example.




What is the point?

Understanding the electrical properties of solids isright at the heart of modern society and technology.

The entire computer and electronics industry relieson tuning of a special class of material, thesemiconductor, which lies right at the metal-

insulator boundary. Solid state physics provide abackground to understand what goes on insemiconductors.




Solid state physics (SSP)is the applied physics

New technology for the future will inevitably involvedeveloping and understanding new classes of

materials. By the end of this course we will see whythis is a non-trivial task.

So, SSP is the applied physics associated withtechnology rather than interesting fundamentals.




Electrical resistivity of threestates of solid matter

How can this be? After all, they each contain a system of atoms and especially electrons of similar density. And theplot thickens: graphite is a metal, diamond is an insulator andbuckminster-fullerene is a superconductor.

They are all just carbon!



Among our aims - understand why one is a metal andone an insulator, and then the physical origin of themarked features.

Also think about thermal properties etc. etc.




CONTENT

Chapter 1. Crystal Structure

Chapter 2. X-ray Crystallography

Chapter 3. Interatomic Forces

Chapter 4. Crystal Dynamics

Chapter 5. Free Electron Theory




CHAPTER 1.CRYSTAL STRUCTURE

Elementary Crystallography Solid materials (crystalline, polycrystalline,

amorphous)

Crystallography Crystal Lattice Crystal Structure Types of Lattices Unit Cell

Directions-Planes-Miller Indices in Cubic Unit Cell Typical Crystal Structures

(3D– 14 Bravais Lattices and the Seven Crystal System)

Elements of Symmetry




CHAPTER 2.X-RAY CRYSTALLOGRAPHY

X-ray

Diffraction

Bragg equation X-ray diffraction methods

Laue Method

Rotating Crystal Method

Powder Method Neutron & electron diffraction




CHAPTER 3.INTERATOMIC FORCES

Energies of Interactions Between Atoms

Ionic bonding

NaCl

Covalent bonding

Comparision of ionic and covalent bonding

Metallic bonding

Van der waals bonding

Hydrogen bonding




CHAPTER 4.CRYSTAL DYNAMICS

Sound wave

Lattice vibrations of 1D cystal

Chain of identical atoms

Chain of two types of atoms

Phonons

Heat Capacity

Density of States

Thermal Conduction

Energy of harmonic oscillator Thermal energy & Lattice Vibrations

Heat Capacity of Lattice vibrations




CHAPTER 5.FREE ELECTRON THEORY

Free electron model

Heat capacity of free electron gas

Fermi function, Fermi energy

Fermi dirac distribution function

Transport properties of conduction electrons




REFERENCES

Core book:Solid state physics, J .R.Hook and H.E.Hall,Second edition (Wiley)

Other books ata similar level:Solid state physics, Kittel (Wiley)Solid state physics, Blakemore (Cambridge)Fundamentals of solid state physics, Christman(Wiley)

More advanced: Solid state physics, Ashcroft andMermin



CHAPTER 1

CRYSTAL STRUCTURE

Elementary Crystallography

Typical Crystal StructuresElements Of Symmetry



Crystal Structure 18

Objectives

By the end of this section you should:

be able to identify a unit cell in a symmetricalpattern

know that there are 7 possible unit cell shapes

be able to define cubic, tetragonal,orthorhombic and hexagonal unit cell shapes




Matter

GASESLIQUIDS

and LIQUID

CRYSTALS

SOLIDS

matter




Gases

Gases have atoms or molecules that do notbond to one another in a range of pressure,temperature and volume.

These molecules haven’t any particular orderand move freely within a container.




Liquids and Liquid Crystals

Similar to gases, liquids haven’t any atomic/molecularorder and they assume the shape of the containers.

Applying low levels of thermal energy can easily

break the existing weak bonds.

Liquid crystals have mobilemolecules, but a type of longrange order can exist; themolecules have a permanentdipole. Applying an electric fieldrotates the dipole and establishesorder within the collection of

molecules.




Crytals

Solids consist of atoms or moleculesexecuting thermal motion about anequilibriumposition fixed at a point in space.

Solids can take the form of crystalline,polycrstalline, or amorphous materials.

Solids (at a given temperature, pressure, and

volume) have stronger bonds betweenmolecules and atoms than liquids.

Solids require more energy to break the

bonds.




SOLID MATERIALS

CRYSTALLINE POLYCRYSTALLINEAMORPHOUS

(Non-crystalline)

Single Crystal

ELEMENTARY CRYSTALLOGRAPHY

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




Types of Solids

Single crsytal, polycrystalline, and amorphous, are thethree general types of solids.

Each type is characterized by the size of orderedregion within the material.

An ordered region is a spatial volume in which atoms

or molecules have a regular geometric arrangementor periodicity.




Crystalline Solid Crystalline Solid is the solid form of a substance in

which the atoms or molecules are arranged ina definite, repeating pattern in three dimension.

Single crystals, ideally have a high degree of order, or

regular geometric periodicity, throughout the entirevolume of the material.




Crystalline Solid

Single Crystal

Single Pyrite

Crystal

Amorphous

Solid

Single crystal has an atomic structure that repeatsperiodically across its whole volume. Even at infinite lengthscales, each atom is related to every other equivalent

atomin the structure by translational symmetry

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




Polycrystalline Solid

Polycrystalline

Pyrite form

(Grain)

Polycrystal is a material made up of an aggregate of many small single crystals

(also calledcrystallites or grains).

Polycrystalline material have a high degree of order over many atomic or moleculardimensions.

These ordered regions, or single crytal regions, vary in size and orientation wrt

one another.

These regions are called as grains ( domain) and are separated fromone anotherby grain boun daries. The atomic order can vary from one domain to the next.

The grains are usually 100 nm - 100 micr ons in diameter . Polycrystals with grainsthatare <10 nmindiameterare called nanocrystalline

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




Amorphous Solid

Amorphous (Non-crystalline) Solid is composed of randomlyorientated atoms , ions, or molecules that do not form definedpatterns or lattice structures.

Amorphous materials have order only within a few atomic ormolecular dimensions.

Amorphous materials do not have any long-range order, but theyhave varying degrees of short-range order.

Examples to amorphous materials include amorphous silicon,plastics, and glasses.

Amorphous silicon can be used in solar cells and thin filmtransistors.




Departure From Perfect Crystal

Strictly speaking, one cannot prepare a perfect crystal. Forexample, even the surface of a crystal is a kind of imperfection because the periodicity is interrupted there.

Another example concerns the thermal vibrations of the atomsaround their equilibriumpositions for any temperature T>0°K.

As a third example, actualcrystal always contains someforeign atoms, i.e., impurities.

These impurities spoils theperfect crystal structure.




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 influenceon its properties.

Any crystal structure should be specified completely,concisely and unambiguously.

Structures should be classified into different typesaccording to the symmetries theypossess.

Crystallography




A basic knowledge of crystallography is essentialfor solid state 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 profoundinfluence on its properties.

We will concern in this course with solids withsimple structures.

ELEMENTARY

CRYSTALLOGRAPHY




CRYSTAL LATTICE

What is crystal (space) lattice?

In crystallography, only the geometrical properties of thecrystal are of interest, therefore one replaces each atomby

a geometrical point located at the equilibrium position of thatatom.

Platinum Platinum surface Crystal lattice andstructure of Platinum(scanning tunneling microscope)




An infinite array of points in space,

Each point hasidentical surroundingsto all others.

Arrays are arrangedexactly in a periodicmanner.

Crystal Lattice

α

a

b

CB ED

O A

y

x




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



A two-dimensional Bravais lattice with

different choices for the basis




E

HO A

CB

Fb G

D

x

y

a

α

a

b

CBED

O A

y

x

b) Crystal lattice obtained by

identifying all the atoms in (a)

a) Situation of atoms at the

corners of regular hexagons

Basis

A group of atoms which describe crystal structure




Crystal structure

Don't mix up atoms withlattice points

Lattice points are

infinitesimal points inspace

Lattice points do notnecessarily lie at thecentre of atoms

Crystal Structure = Crystal Lattice + Basis




Crystal Lattice

Bravais Lattice (BL) Non-Bravais Lattice (non-BL)

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

Atoms can be of differentkind Some lattice points are not

equivalent

A combination of two ormore BL




Types Of Crystal Lattices

1) Bravais lattice is an infinite array of discrete points withan arrangement and orientation that appears exactly thesame, 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




Types Of Crystal Lattices

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

Red (and blue) sides are equivalentand have the same appearance

Red and blue sides are notequivalent. Same appearance canbe obtained rotating blue side 180º.

2) Non-Bravais LatticeNot only the arrangement but also the orientation mustappear exactly the same from every point in a bravais lattice.

Honeycomb




Translational Lattice Vectors – 2D

A space lattice is a set of points suchthat a translation fromany point in thelattice by a vector;

Rn =n1 a +n2 b

locates an exactly equivalent point,i.e. a point with the same environmentas P . This is translational symmetry.

The vectors a, b are known as latticevectors and (n1, n2) is a pair of integers whose values depend on thelattice point.

P

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

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




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

The choice of lattice

vectors is not unique. Thus one could equally

well take the vectors aand b’ as a lattice vectors.

Lattice Vectors – 2D




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 vector

where u, v and w are arbitrary integers.

r ’ = r + u a + v b + w c (1)




Five Bravais Lattices in 2D




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




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 beatoms - 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 thesame - empty space is not allowed!




In 2D, this IS a unit cellIn 3D, it is NOT




Why can't the blue trianglebe a unit cell?




Unit Cell in 3D




Three common Unit Cell in 3D




UNIT CELL

Primitive Conventional & Non-primitive

Single lattice pointpercell Smallestarea in2D, orSmallestvolume in 3D

More thanone lattice pointpercell Integral multibles of the area of

primitive cell

Body centered cubic(bcc)

Conventional ≠ Primitive cell

Simple cubic(sc)

Conventional = Primitive cell




The Conventional Unit Cell

A unit cell just fills space whentranslated through a subset of Bravais lattice vectors.

The conventional unit cell ischosen to be larger than theprimitive cell, but with the fullsymmetryof the Bravais lattice.

The size of the conventional cellis given by the lattice constanta.




Primitive and conventional cells of FCC



1

2

3

1ˆ ˆ ˆ( )

21

ˆ 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:




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 ≠primitive cell

a

b cFractional coordinates of lattice points in

conventional cell:000,100, 010, 001, 110,101, 011, 111, ½ ½ ½




Body centered cubic (bcc):primitive (rombohedron) ≠conventional cell

a

b

c

Fractional coordinates:000, 100, 101, 110, 110,101, 011, 211, 200

Face centered cubic (fcc):

conventional ≠ primitive cell

a

b

c

Fractional coordinates:000,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ , ½ ½ 1

Primitive and conventional cells




Hexagonal close packed cell (hcp):

conventional =primitive cell

Fractional coordinates:100, 010, 110, 101,011, 111,000, 001

points of primitive cell

a

b

c

Primitive and conventional cells-hcp




The unit cell and, consequently,the entire lattice, is uniquelydetermined by the six latticeconstants: a, b, c, α, β and γ.

Only 1/8 of each lattice point in aunit cell can actually be assignedto thatcell.

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

Unit Cell




A primitive unit cell is made of primitivetranslation vectors a1 ,a2, and a3 suchthat there is no cell of smaller volumethat can be used as a building block forcrystal structures.

A primitive unit cell will fill space byrepetition of suitable crystal translationvectors. This defined by the parallelpipeda1, a2 and a3. The volume of a primitiveunit cell can be found by

V =a1.(a2 x a3) (vector products) Cubic cell volume = a3

Primitive Unit Cell and vectors




The primitive unit cell must have only one lattice point.

There can be different choices for lattice vectors , but thevolumes of these primitive cells are all the same.

P = Primitive Unit Cell

NP = Non-Primitive Unit Cell

Primitive Unit Cell

1a




Wigner-Seitz Method

A simplyway to find the primitivecell which is called Wigner-Seitzcell can be done as follows;

1. Choose a lattice point.2. Draw lines to connect these

lattice point to its neighbours.3. At the mid-point and normal

to these lines draw newlines.

The volume enclosed is called as a

Wigner-Seitz cell.




Wigner-Seitz Cell - 3D




Lattice Sites in Cubic Unit Cell




Crystal Directions

Fig. Shows

[111] direction

We choose one lattice point on the lineas an origin, say the point O. Choice of origin is completely arbitrary, sinceevery lattice point is identical.

Then we choose the lattice vector

joining O to any point on the line, saypoint T. This vector can be written as;

R = n1 a + n2 b + n3c

To distinguish a lattice direction from alattice point, the triple is enclosed insquare brackets [ ...] is used.[n1n2n3]

[n1n2n3] is the smallest integer of thesame relative ratios.




210

X = 1 , Y = ½ , Z = 0

[1 ½ 0] [2 1 0]

X = ½ , Y = ½ , Z = 1

[½ ½ 1] [1 1 2]

Examples




Negative directions

When we write the

direction [n1n2n3]

depend on the origin,

negative directionscan be written as

R = n1 a + n2 b + n3c

Direction must be

smallest integers.

Y direction

(origin) O

- Y direction

X direction

- X direction

Z direction

- Z direction

][ 321 nnn

][ 321 nnn




X = -1 , Y = -1 , Z = 0 [110]

Examples of crystal directions

X = 1 , Y = 0 , Z = 0 [1 0 0]




Examples

X =-1 , Y = 1 , Z = -1/6

[-1 1 -1/6] [6 6 1]

We can move vector to the origin.




Crystal Planes

Within a crystal lattice it is possible to identify sets of equallyspaced parallel planes. These are called lattice planes.

In the figure density of lattice points on each plane of a set isthe same and all lattice points are contained on each set of

planes.

b

a

b

a

The set of

planes in

2D lattice.




Miller Indices

Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographicaxes.

To determine Miller indices of a plane, take the following steps;

1) Determine the intercepts of the plane along each of the threecrystallographic directions

2) Take the reciprocals of the intercepts

3) If fractions result, multiply each by the denominator of the smallestfraction




Axis X Y Z

Intercept

points 1 ∞ ∞

Reciprocals 1/1 1/ ∞ 1/ ∞

Smallest

Ratio 1 0 0

Miller İndices (100)

Example-1

(1,0,0)




Axis X Y Z

Intercept

points 1 1 ∞

Reciprocals 1/1 1/ 1 1/ ∞

Smallest

Ratio 1 1 0


Example-2

(1,0,0)

(0,1,0)




Axis X Y Z

Intercept

points 1 1 1Reciprocals 1/1 1/ 1 1/ 1

Smallest

Ratio 1 1 1

Miller İndices (111)(1,0,0)

(0,1,0)

(0,0,1)

Example-3




Axis X Y Z

Intercept

points 1/2 1 ∞

Reciprocals 1/(½) 1/ 1 1/ ∞

Smallest

Ratio 2 1 0

Miller İndices (210)(1/2, 0, 0)

(0,1,0)

Example-4




Axis a b c

Intercept

points 1 ∞ ½Reciprocals 1/1 1/ ∞ 1/(½)

Smallest

Ratio 1 0 2


Example-5




Axis a b c

Intercept

points -1 ∞ ½Reciprocals 1/-1 1/ ∞ 1/(½)

Smallest

Ratio -1 0 2


Example-6




Miller Indices

Reciprocal numbers are:2

1 ,

2

1 ,

3

1

Plane intercepts axes at cba 2,2,3

Indices of the plane (Miller): (2,3,3)

(100)

(200)

(110)(111)

(100)

Indices of the direction: [2,3,3]a

3

2

2

bc

[2,3,3]




Example-7




Indices of a Family or Form

Sometimes when the unit cell has rotational symmetry,several nonparallel planes may be equivalent by virtue of thissymmetry, in which case it is convenient to lump all these

planes in the same Miller Indices, but with curly brackets.

Thus indices {h,k,l}represent all the planes equivalent to theplane (hkl) through rotational symmetry.

)111(),111(),111(),111(),111(),111(),111(),111(}111{

)001(),100(),010(),001(),010(),100(}100{

≡

≡




There are only seven different shapes of unit cell whichcan be stacked together to completely fill all space (in 3dimensions) without overlapping. This gives the seven

crystal systems, in which all crystal structures can beclassified.

Cubic Crystal System(SC, BCC,FCC) Hexagonal Crystal System(S)

Triclinic Crystal System(S) Monoclinic Crystal System(S, Base-C) Orthorhombic Crystal System(S, Base-C, BC, FC) Tetragonal Crystal System(S, BC) Trigonal (Rhombohedral) Crystal System (S)

3D – 14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL SYSTEM

TYPICAL CRYSTAL STRUCTURES




C di t N b




Coordinatıon Number

Coordinatıon Number (CN) : The Bravais lattice pointsclosest to a given point are the nearest neighbours.

Because the Bravais lattice is periodic, all points havethe same number of nearest neighbours or coordinationnumber. It is a property of the lattice.

A simple cubic has coordination number 6; a body-centered cubic lattice, 8; and a face-centered cubiclattice,12.



Atomic Packing Factor

Atomic Packing Factor (APF) is defined asthe volume of atoms within the unit cell

divided by the volume of the unit cell.




1-CUBIC CRYSTAL SYSTEM

Simple Cubic has one lattice point so its primitive cell.

In the unit cell on the left, the atoms at the corners are cut

because only a portion (in this case 1/8) belongs to that cell. The restof the atombelongs to neighboring cells.

Coordinatination number of simple cubic is 6.

a- Simple Cubic (SC)

a

bc

http://www.kings.edu/~chemlab/vrml/simcubun.wrl

http://www.kings.edu/~chemlab/vrml/simcubun.wrl




a- Simple Cubic (SC)




Atomic Packing Factor of SC




b-Body Centered Cubic (BCC)

BCC has two lattice points so BCCis a non-primitive cell.

BCC has eight nearest neighbors.Each atom is in contact with itsneighbors only along the body-diagonal directions.

Many metals (Fe,Li,Na..etc),

including the alkalis and severaltransition elements choose theBCC structure.

a

b c




0 . 6 =V

V = A P F

c e l lu n i t

a t o m s B C C

2 (0,433a)

Atomic Packing Factor of BCC




c- Face Centered Cubic (FCC)

There are atoms at the corners of the unit cell and at thecenter of each face.

Face centered cubic has 4 atoms so its non primitive cell.

Many of common metals (Cu,Ni,Pb..etc) crystallize in FCC

structure.




3 - Face Centered Cubıc

Atoms are all same.




4 (0,353a)

0 . 6 =V

V = A P F

c e l lu n i t

a t o m s B C C FCC

0,74

Atomic Packing Factor of FCC




Atoms Shared Between: Each atom counts:

corner 8 cells 1/8face centre 2 cells 1/2body centre 1 cell 1edge centre 2cells 1/2

lattice type cell contents

P 1 [=8 x 1/8]I 2 [=(8 x 1/8) + (1 x 1)]F 4 [=(8 x 1/8) + (6 x 1/2)]C 2 [=(8 x 1/8) + (2 x 1/2)]

Unit cell contents

Counting the number of atoms within the unit cell




Example; Atomic Packing Factor




2 - HEXAGONAL SYSTEM

A crystal system in which three equal coplanar axesintersect at an angle of 60 , and a perpendicular to theothers, is of a different length.




2 - HEXAGONAL SYSTEM

Atoms are all same.

http://www.kings.edu/~chemlab/vrml/hcpun.wrl




3 - TRICLINIC




3 TRICLINIC4 - MONOCLINIC CRYSTAL SYSTEM

Triclinic minerals are the least symmetrical. Their three axesare all different lengths and none of them are perpendicularto each other. These minerals are the most difficult torecognize.

Triclinic (Simple)

α ≠ ß ≠ γ ≠ 90oa ≠ b ≠ c

Monoclinic (Simple)

α = γ = 90o, ß ≠ 90o

a ≠ b ≠c

Monoclinic (Base Centered)

α = γ = 90o, ß ≠ 90o

a ≠ b ≠ c,

http://www.kings.edu/~chemlab/vrml/naclun.wrl

http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bondd.html

http://hyperphysics.phy-astr.gsu.edu/hbase/chemical/bond.html




5 - ORTHORHOMBIC SYSTEM

Orthorhombic (Simple)α = ß = γ = 90o

a ≠ b ≠ c

Orthorhombic (Base-centred)

α = ß = γ = 90o

a ≠ b ≠ c

Orthorhombic (BC)α = ß = γ = 90o

a ≠ b ≠ c

Orthorhombic (FC)

α = ß = γ = 90o

a ≠ b ≠ c

http://www.kings.edu/~chemlab/vrml/cesun.wrl







6 – TETRAGONAL SYSTEM

Tetragonal (P)

α = ß = γ = 90o

a = b ≠ c

Tetragonal (BC)

α = ß = γ = 90o

a = b ≠ c






7 - Rhombohedral (R) or Trigonal

Rhombohedral (R) or Trigonal (S)

a = b = c, α = ß = γ ≠ 90o

THE MOST IMPORTANT




CRYSTAL STRUCTURES

Sodium Chloride Structure Na+Cl-

Cesium Chloride Structure Cs+

Cl-

Hexagonal Closed-Packed Structure

Diamond Structure

Zinc Blende




1 – Sodium Chloride Structure

Sodium chloride also

crystallizes in a cubic lattice,

but with a different unit cel l.

Sodium chloride structure

consists of equal numbers of

sodium and chlorine ions

placed at alternate points of a

simple cubic lattice.

Each ion has six of the other kind of ions as its nearest

neighbours.




Sodium Chloride Structure

If we take the NaCl unit cell and remove all the red Cl ions, we

are left with only the blue Na. If we compare this with the fcc /

ccp unit cell, it is clear that they are identical. Thus, the Na is

in a fcc sublattice.




Sodium Chloride Structure

This structure can beconsidered as a face-centered-cubic Bravais latticewith a basis consisting of a

sodium ion at 0 and a chlorineion at the center of theconventional cell,

LiF,NaBr,KCl,LiI,etc

The lattice constants are inthe order of 4-7 angstroms.

)(2/→→→

++ z y x a

2-Cesium Chloride Structure




Cs+Cl-

Cesium chloride crystallizes in a

cubic lattice. The unit cell may be

depicted as shown. (Cs+ is teal,

Cl- is gold).

Cesium chloride consists of equal

numbers of cesium and chlorine

ions, placed at the points of abody-centered cubic lattice so that

each ion has eight of the other kind

as its nearest neighbors.

Cesium Chloride Structure







Cs+Cl-

The translational symmetry of thisstructure is that of the simple cubicBravais lattice, and is described as a

simple cubic lattice with a basisconsisting of a cesium ion at the origin 0

and a chlorine ion at the cube center

CsBr,CsI crystallize in this structure.Thelattice constants are in the order of 4

angstroms

)(2/

→→→

++ z y xa



8 cell

Cesium Chloride Cs+Cl-




3–Hexagonal Close-Packed Str.

This is another structure that iscommon, particular ly in metals.In addition to the two layers of atoms which form the base and

the upper face of the hexagon,there is also an intervening layer of atoms arranged such thateach of these atoms rest over adepression between three atomsin the base.





Bravais Lattice : Hexagonal LatticeHe, Be, Mg, Hf, Re (Group II elements)ABABAB Type of Stacking

Hexagonal Close-packed Structure

a=b a=120, c=1.633a,basis : (0,0,0) (2/3a ,1/3a,1/2c)




A A

AA

AA

A

AAA

AA

AAA

AAA

B B

B

B

B B

B

B

B

BB

C C C

CC

C

C

C C C

Sequence ABABAB..-hexagonal close pack

Sequence ABCABCAB..-face centered cubic close pack

Close pack

B

AA

AA

A

A

A

A A

B

B B

Sequence AAAA…

- simple cubic

Sequence ABAB…- body centered cubic

Packing




4 - Diamond Structure

The diamond lattice is consist of two interpenetrating

face centered bravais lattices.

There are eight atom in the structure of diamond.

Each atom bonds covalently to 4 others equally spread

about atom in 3d.

d S



4 - Diamond Structure

The coordination number of diamondstructure is 4.

The diamond lattice is not a Bravaislattice.

Si, Ge and C crystallizes in diamond

structure.




5 Zi Bl d



5- Zinc Blende

Zincblende has equal numbers of zinc andsulfur ions distributed on a diamond latticeso that each has four of the opposite kind as

nearest neighbors. This structure is anexample of a lattice with a basis, which mustso described both because of thegeometrical position of the ions andbecause two types of ions occur.

AgI,GaAs,GaSb,InAs,

5 Zi Bl d



5- Zinc Blende




5- Zinc Blende

Zinc Blende is the name given to the mineral ZnS. It has a cubicclose packed (face centred) array of S and the Zn(II) sit intetrahedral (1/2 occupied) sites in the lattice.

http://euch3i.chem.emory.edu/proposal/www.ncl.ox.ac.uk/icl/heyes/structure_of_solids/Coords/ZnBlende.cmdf




Each of the unit cells of the 14 Bravais lattices has oneor more types of symmetry properties, such asinversion, reflection or rotation,etc.

SYMMETRY

INVERSION REFLECTION ROTATION

ELEMENTS OF SYMMETRY

Lattice goes into itself throughS t ith t t l ti




Symmetry without translation

Operation Element

Inversion Point

Reflection Plane

Rotation Axis

Rotoinversion Axes

I i C t




Inversion Center

A center of symmetry: A pointat the center of the molecule.(x,y,z) -->(-x,-y,-z)

Center of inversion can only be in a molecule. It is notnecessary to have an atom in the center (benzene, ethane). Tetrahedral, triangles, pentagons don't have a center of inversion symmetry. All Bravais lattices are inversionsymmetric.

Mo(CO)6

R fl ti Pl




Reflection Plane

A plane in a cell such that, when a mirror reflectionin this plane is performed, the cell remains invariant.

E l




Examples

Triclinic has no reflection plane.

Monoclinic has one plane midway between andparallel to the bases, and so forth.

Rotation Symmetry




We can not find a lattice that goesinto itself under other rotations

• A single molecule can have any degreeof rotational symmetry, but an infinite

periodic lattice – can not.

Rotation Symmetry

Rotation Axis




Rotation Axis

This is an axis such that, if the cell is rotated around it

through some angles, the cell remains invariant.

The axis is called n-fold if the angle of rotation is 2π/n.

90

°

120° 180°

A i f R t ti




Axis of Rotation

Axis of Rotation




Axis of Rotation

5 fold symmetry




Can not be combined with translational periodicity!

5-fold symmetry

Group discussion




Group discussion

Kepler wondered why snowflakes have 6 corners,never 5 or 7.Byconsidering the packing of polygonsin 2 dimensions, demonstrate why pentagons andheptagons shouldn’t occur.

Empty space

not allowed

Examples




90°

Examples

Triclinic has no axis of rotation.

Monoclinic has 2-fold axis (θ= 2π/2 =π) normal to thebase.

56223895 chapter1 crystalline solids

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