physics 2 sengupta

8/2/2019 Physics 2 Sengupta

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Introduction

A solid composed of a closure collection of atoms. In solids atoms are very strongly held

together. Solids state maters may be crystalline or amorphous. A crystalline solid composed

of atoms, ions or molecule are arranged in long range of periodic order in three dimensions.

Examples of crystalline solid are metals, NaCl, KCl, diamond etc. A solid substance with its

atoms held apart at equilibrium spacing, but with no long-range periodicity in atom location

in its structure is an amorphous solid. Examples of amorphous solids are glass and some

types of plastic.

Types of bonding

A chemical bond is an attraction between atoms that allows the formation of chemical

substances that contain two or more atoms. The bond is caused by the electromagnetic force

of attraction between opposite charges, either between electrons and nuclei, or as the result of

a dipole attraction (dipole is an entity where two opposite kind of charges are separated by

some distance). The chemical bonds are classified into two groups based on the strength of

the bond i.e. primary bond and secondary bond. The Primary bonds are strong bonds andhave bond energies in the range of 0.1 to 10eV/bond. Ionic, covalent and metallic

bonding are the examples. Secondary bonding has energies in the range 0.01 to

0.5eV/bond. Hydrogen bonding and van der Waals bonding are the examples.

Ionic bonding

Ionic bond is the attractive force existing between a positive ion and a negative ion. These

ions are formed due to complete transfer of electron from one atom to another. Electro

positive elements readily give up electrons and are usually group I or II elements, e.g. Na, K

and Ba. Electro-negative elements readily take up electrons and are typically group VI andVII elements e.g. Cl, Br and O. This bond is non directional bond. To understand the


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formation of Ionic bonding, let us consider NaCl molecule. A sodium atom (Na) and a

chlorine atom (Cl) are shown below. A single circle represents the nucleus (protons and

neutrons) of the atoms. Dots represent the electrons. The sodium atom has a total of 11

electrons and one electron in its outer shell. Chlorine has a total of 17 electrons with seven in

its outer shell.

The outer valence electron of the sodium atom gets transferred to the chlorine atom to acquire

a stable electronic configuration fig.2. There exists an electrostatic attraction between

positively charged sodium cation and negatively charged chlorine anion and form NaCl

molecule.

Actually a positive charge attracts all negative charges in the neighborhood and vice versa.

Consequently in the crystalline solid, Na+

ions are surrounded by Cl-ions and Cl

-ions by Na

+

ions. The resulting NaCl structure is shown in Fig.3

Fig.1. Sodium and Chlorine atom

Fig.2. Sodium and Chlorine ions


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Properties of Ionic solids

1. Ionic solids are made up of metallic and nonmetallic elements

2. Ionic solids are hard and brittle in nature

3. Ionic solids shows high melting points

4. Ionic solids are soluble in polar solvents like water and ammonia. This solids are

insoluble in non polar solvents.

5. Ionic solids are poor conductor of electricity because they do not have free electrons.

However, Ionic solids dissolve in water and the solution conducts electricity.

6. Ionic solids are transparent in visible radiation and exhibits characteristic absorption

peaks in IR regions.

7. In an ionic crystal, a anion is surrounded by as many cations as possible and vice-

versa. Because of this ionic bonds are non directional bond.

Covalent Bond

A covalent bond is formed when two similar or dissimilar atoms achieve stability by equal

sharing of valence electrons between themselves. This kind of bonding is possible whenever

sharing results in lowering the potential energy of the system.

The shared electrons are under the influence of nuclei of both combining atoms. A stable

covalent bond is formed at the separation (distance between two nuclei) where the repulsive

force between electrons of two combining atoms and two nuclei is balanced by attractive

force between nucleus and electrons.

Hydrogen atoms only need two electrons in their outer level to reach the noble gas structure

of helium. Once again, the covalent bond holds the two atoms together because the pair of

electrons is attracted to both nuclei

Fig.3. NaCl structure


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In case of H2 molecule the attractive and repulsive forces balance at 0.740A and hence the

covalent bond exist with decrease in potential energy of the system. In order to break one H-

H bond 4.5eV of energy is required i.e.

H2+ 4.5ev H + H

Depending on the number of electrons shared, the covalent bonds may be single, double or

triple on sharing one two or three electrons per atom respectively.

Examples: H2 , HCl, HF, O2 , N2 , Diamond , Si, Ge etc.

The number covelent bonds that can be formed by an element is determined by number of

electrons that can be added to the valence shell to get octet state. Therefore the maximum

number of covalent bond is (8-N). where N is number of valence electron.

Single covalent bond: H2 ,HCl, CH4

Double covalent bond: O2 molecule

Triple covalent bond: N2 molecule

The important points about covalent bond formation are

1) Attainment of octet configuration is not essential

2) The essential feature of such bond is the fact that it involves the pairing of two

electrons with opposite spin

3) A covalent bond may be either polar or non polar depending on the fact that whether

the electrons pair is shared unequally (eg. HCl) or equally (O2 , H2 ) between the two

atoms.


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Properties of covalent bonded solids

1) The covalent bond exist in all three states of matter

a) Gases O2 , H2 etc.

b) Liquids CCl4

c) Solids Diamond, Ge, Si and Sn

2) In solids they exist in crystalline state.

3) Covalent crystals are usually hard and brittle materials with high binding energies (eg.

Diamond =7.4eV)

4) The melting and boiling points of covalent solids are usually low as compared to

those of ionic solids (Diamond high melting point, tin low melting point)

5) Covalent bond is a directional bond

6) They generally dissolve in non polar solvents such as benzene, toluene and CCl 4.

However some covalent compounds like, HCl, NH4, sugar, urea are soluble in water)

7) Basically all covalent crystals are insulators because of no free electrons available forconduction. However Si, Ge are semiconductor and Sn is a conductor.

8) These crystals are transparent to long wavelength radiation and opaque to shorter

wavelength

9) Covalent bonds may be polar or non polar in nature, depending on the fact that

whether the electron pair is unequally shared (HCl) or equally shared ( O2, H2)

between two atoms.

Website : http://www.youtube.com/watch?NR=1&v=1wpDicW_MQQ

Metallic bond

Metallic bond is formed by partially showing of valance electrons by neighboring atoms,

sharing is not localized. Metallic bonds may also be considered as delocalized or unsaturated

covalent bond.

Free electrons

+ ve ion core

Fig.4. Atomic arrangement in metal crystal
http://www.youtube.com/watch?NR=1&v=1wpDicW_MQQhttp://www.youtube.com/watch?NR=1&v=1wpDicW_MQQhttp://www.youtube.com/watch?NR=1&v=1wpDicW_MQQ


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In case of metal, the valance electrons are mobile and are not bound to any particular atom.

These are free to move throughout the volume of the metal. These free and mobile electrons

form a kind of electron cloud or electron gas. Thus a metal may be regarded as an array of

closely packed or positive ion core is immersed in a sea of electron gas. There exist

electrostatic force of attraction between the electron gas and +ve ion core.

A metallic bond is formed when the force of attraction between the +ve metal ions and

electron gas exceeds the mutual repulsion of the electrons in metal.

Examples : Na, Al, Cu, Au, Ag etc.

The binding energies of Na is 1.13eV, Al is 3.23eV

Properties of metallic bonded crystals:

1. The metallic bond is weaker than ionic or covalent bonds

2. Metallic crystals have crystalline structure

3. They have high thermal and electrical conductivities since metals posses a large no of

free electrons

4. The melting and boiling points of metallic solids are lower than ionic and covalent

solids

5. Metallic bond is a non directional bond

6. The metallic crystals are opaque to light

7. Metallic solids are not soluble in polar and non polar solvents

8. Metallic solids are soft and malleable and ductile

Dipole bond

Dipole bond is formed between molecules having permanent dipoles. The permanent

dipoles are arising due to unequal sharing of electrons between two atoms. Let us consider

the formation of HF molecule by electron sharing. The hydrogen atoms have only on electron

in its outermost energy level and fluorine atom has 7 electrons in its outermost energy level.

The hydrogen atom requires one more electron to acquire stable configuration and fluorine

atom also requires one more electron to acquire stable configuration. Therefore both the

hydrogen and fluorine atoms share a pair of electron to acquire stable configuration. This

leads formation of covalent bond and the resulting molecule is called as hydrogen fluoride.


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In this covalent bond, the fluorine atom has high affinity than the hydrogen atom. Thus the

shared electron pair shifts towards the fluorine atom. This shifting of electron pair produces

an electric dipole. Similar dipoles are formed in adjacent molecule also. Adjacent HF

molecule therefore attract each other by means of electro static attraction between their

oppositely charged ends and the dipole bond is formed

Examples: HCl, SO2, NH3, H2O

Properties of dipole bonds:

1. Dipole bonds are much weaker than primary bonds but are stronger than Van dar

waals or dispersion bonds. Thus these are weakly bonded solids

2. These bonds have directional properties

3. Solids with dipole bonds are good insulators since there are no valance electrons

Hydrogen bond

A hydrogen bond is a particular type of dipole bond in which one atom is a hydrogen

atom. When a covalent bond is formed between a hydrogen atom and a highly

electronegative atom, such as an atom of oxygen, fluorine, chlorine etc. the shared electrons

pair gets attracted move towards the electronegative atom than the hydrogen atom. Thus theelectronegative atom acquires a slight negative charge and the hydrogen atom acquires an

equal amount of +ve charge. The molecule so formed is said to be polarized and behaves like

permanent dipole. A number of such dipoles get attracted to one another due to the columbic

force of attraction. This type of interaction between the oppositely charged ends of

permanently polarized molecules each containing a hydrogen atom is called the hydrogen

bond.

Dipole bond

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Covalent bond

-

+

-

+

-

-

+

+


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For example this bond is formed between ice or water molecules due to attraction between

the positively charged hydrogen end of a molecule and negatively charged oxygen of other

molecule as shown below.

Properties of hydrogen bonded solids

1. The hydrogen bonds are directional

2. The bonding is relatively strong as compared to other dipole-dipole interactions.

3. Hydrogen bonded solids have low melting points

4. Hydrogen bonded solids are good insulators since no valence electrons5. They are soluble in both polar and non polar solvents

6. They are transparent to visible light

7. Since elements of low atomic numbers for such solids, they have low densities.

Van dar Waals bond

A relatively weak and temporary (fluctuating) kind ofintermolecular bondthat forms when

one side of a molecule develops a slight negative charge because a number ofelectrons havetemporarily moved to that side of the molecule. This negative charge attracts the nuclei of

the atoms of a neighboring molecule. The side of the molecule with fewer electrons develops

a slight positive charge that attracts the electrons of the atoms of neighboring molecules.

Van dar waals bonding arise from the fluctuating dipole nature of an atom with all

occupied electron shell filled .This bond is non directional.

If the symmetrical distribution of electrons around the nucleus is disturbed, the centre of

positive and negative charges may not coincide at that moment giving rise to weak

fluctuating dipole.

Hydrogen bond in Water molecules


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A weak attractive force exists between the opposite ends of the dipoles in the neighboring

atoms. This force only allows inert gas atoms to condense at low temperatures. Van der

Waals bonding is typically an order of magnitude weaker than the hydrogen bonding.

Properties of solids with van der waals bonding

1. Van der waals bonds are non directional

2. Solids with van der waals bonds are insulators since no valence electrons are

available.

3. Solids are soluble in polar and nonpolar solvent

4. They are transparent to light

5. Van der waals bonded solids has low melting point

6. Van der waals bonding is weaker than hydrogen bond

Examples: solid Argon, solid neon.

Interatomic forces

The attractive forces between the atoms bring them close together until a strong repulsive

force arises due to overlap of electron shells. When two atoms approach each other, the

negatively charged electron shells come much closer than their positive nuclei. At a certain

+

+

-

-

+

-=0

(a) (b)

(c)

Van der

waals bond

(a) Symmetric distribution of electrons in a

nobel gas atom(b) Asymmetric distribution

(c) Formation of van der waals bond


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separation, called the equilibrium separation, r0 the attractive and repulsive forces are equal.

The two atoms come to a stable condition and have minimum potential energy. At

equilibrium the bonding force F is

MN NM rB

r

A

rF

Where r is the inter atomic distance, A, B, M, N are constant depend on the type of bond. The

first term represents the force of attraction and the second term represents force of repulsion.

The value of M is 2. The value of N is 7-10 for metallic bond and 10 to 12 for ionic and

covalent bond. At equilibrium spacing r0 the net force is zero . The value of r0 is of the

order of 10-10

m. i.e. 1A0.

MN

BAr

1

0

Estimation of cohesive energy

To calculate the cohesive energy, let us consider the general situation of two identical atoms.

As the atoms approach, the attractive forces increase. Since the atoms do the work during

attraction, the energy of attraction is Ve. Hence the potential energy decreases. When the

separation decreases to the order of few atomic diameters, repulsive forces begin to act. Sinceexternal work must be done to bring the two such atom close together, the repulsive force is

positive and hence the Potential Energy increase. At equilibrium position the potential energy

of the either atom is given by

U= decrease in Potential energy due to attraction + increase in potential energy due to

repulsion

Since the work done on the system is stored as potential energy it can be calculated by

integrating F(r)

The work done in moving through a small distance dr is

dU(r)=F(r)dr

The potential energy of the atom

drrFrdurU )()()(

drr

B

r

A

NM


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C

rN

B

rM

A

NM

11

1

1

1

1

C

r

b

r

a

nm

Where a and b are new constants related to A and B as1

M

Aa ,

1N

Bb

When r=, U=0

Hence C=0

Thereforenm

r

b

r

arU

)(

All the stable arrangements of atoms in solids are such that the potential energy U(r) is a

minimum. This happens at equilibrium position (r=r0)

mn

m

n

a

br

1

0

The energy corresponding to the equilibrium position is called the bonding energy or energy

of cohesion of the molecule.

Calculation of cohesive energy of ionic solids

Cohesive energy of ionic crystals is mainly due to electrostatic interaction. The cohesive

energy of an ionic crystal is the energy that would be liberated by the formation of the crystal

from individual neutral atoms. Cohesive energy is usually expressed in eV/atom or

eV/molecule or in kJ/kmol.

The binding energy of ionic crystals are about 5-10eV per molecule. This is the energy

required to dissociate the lattice into +ve and ve ions at infinite separation. The common

types of structures found in ionic crystals are NaCl-FCC and CsCl-BCC.

In classical Born-Madelung theory it is assumed that the e-s are transferred from electro +ve

atoms(Na, K, Mg) to electronegative atoms (O,F,Cl). The stability of the ionic crystal

depends on the balancing of at least three forces.

1) Coulombs forces between the ions2

1

r

2) Van der waals forces7

1

r

3) Inter ionic repulsive forces falling off still more rapidly with distance.


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The resultant of attractive and repulsive forces is to lead an equilibrium position of

minimum potential energy. i.e. of greatest stability

Consider two ions of charge Z1e and Z2e are separated by a distance r

r

eZZ

0

221

4energyattractiveThe

For the whole crystal the coulomb potential energy may be written asr

eZAZ

0

2

21

4

This term represent the net coulomb potential energy of any one ion due to the present of

all other similar and dissimilar ions present in the crystal

The minus sign shows that net coulomb energy is attractive

The Constant A is known as Madelung constant, it differs with crystal structures. It is

1.748 for NaCl crystal.

The repulsive energy of this ion, due to all the other ions present in the crystal isn

r

B

where n is called born repulsive exponent.

Therefore the total energy of one ion due to the presence of all other ions is given by

nr

B

r

eZAZrU

0

2

21

4

The total energy per kmol of the crystal is

nA

r

B

r

eZAZNrU

0

2

21

4

Where NA is the Avogadro number

If Z1 = Z2 = 1 then

nA

r

B

r

AeNrU

0

2

4

The potential energy will be minimum at equilibrium spacing r0

0

4

1

0

2

00

2

0

nA

rrr

Bn

r

AeN

dr

dU


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n

rAeB

n

0

1

0

2

4

Substituting this value of B along with r = r0 we get total equilibrium energy per kmol of

the crystal.

n

n

Arr

nr

rAe

r

AeNUU

00

1

0

2

00

2

0440

Rearranging the above equation for lattice energy per kmol of ionic crystal

nr

AeNU

A1

14

00

2

0

Then the expression for the lattice energy that is released in the process when the

constituent ions are placed in their respective positions in the crystal lattice

nr

AeU

11

400

2

0

The cohesive energy and lattice are equal and opposite in sign.

The expression for cohesive energy of a molecule

Bond formation energy isve and bond breaking energy is +ve.

Note: the cohesive energy is the energy required to separate the crystal into positive

and negative ions

nr

AeU

11

400

2

0


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Problems

1) Calculate the cohesive energy of NaCl from the following data

Equilibrium separation between the ion pair = 0.281nm

Born repulsive exponent = 9

Madelung constant = 1.748

Solution:

Cohesive energy per molecule is

e = 1.602x10-19

C

0 =8.854x10-12

F/m

Then cohesive energy per molecule of NaCl is

=-0.12755x10-17

Jouls

=-7.96eV

2) The madelung constant of KCL is 1.75. Its neighbor separation is 0.314nm. Find

the cohesive energy per atom ( given that the repulsive exponent value = 5.77;

Ionisation energy of potassium = 4.1eV; Electron affinity of chlorine = 3.6eV).

Solution:

Potential energy per pair of ions is

e = 1.602x10-19

C

0 =8.854x10-12

F/m

Then bond energy per molecule of NaCl is

=- 0.12755x10-17

Jouls

=- 6.63eV

Potential energy per ions is(6.63/2) = - 3.315eV

nr

AeU 114

00

2

0

00

2

04 r

e

U


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The average energy needed to create an ion( K+

or Cl-) from pair of K and Cl atom is

(4.1-3.6)/2=0.25eV

The cohesive energy per atom is = -3.315+0.25 = -3.065eV

3) Estimate the bond energy for the NaCl molecule as formed from sodium and

chlorine atoms. The inter ionic equilibrium distance is 236pm. The ionization

energy(IP) of sodium is 5.14eV and electron affinity(EA) of Cl is 3.65eV.

Solution:

Potential energy

= - 6.1eV

The bond energy is given by

= (U) + (IP - EA )

= -6.1+ (5.14-3.65)

= -4.61eV

00

2

4 r

eU


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Crystal structure

Matter consisting of one or more elements or their chemical compounds, exists in nature in

the solid, liquid and gases. As the atom or molecules in solids are attached to one another

with strong forces of attraction, the solid maintains a definite volume and shape.

On the basis of arrangement of constituent particles solids are divided into

1) Crystalline solids

2) Amorphous ( non crystalline solids)

Crystalline solids

The crystalline state of solid is characterized by regular and periodic arrangement of atoms or

molecules. Most of the solids are crystalline in nature since the energy released during the

formation of an ordered structure is more, than that released during the formation of

disordered structure, i.e. the crystalline state is a low energy state.

The crystalline solid may be further divided into

1) Single crystal (eg. Diamond, quartz, mica etc.)

2) Poly crystalline solids ( eg. Metals, ceramics etc.)

In single crystal, the periodicity of atoms extends throughout the crystal (Diamond, Quartz)

A poly crystalline material is an aggregate of number of small crystallites with random

orientations separated by well defined boundaries. The smallest crystallites are known as

grains and the boundaries are grain boundaries.

It may be noted that although the periodicity of individual crystallites in interrupted at grain

boundaries. Yet the polycrystalline form of the material may be more stable compared to the

crystal form. Most of metals and ceramics exhibit poly crystalline structure.

Amorphous solids:

Amorphous solids are characterized by the complete random arrangement of atoms or

molecules. The periodicity if at all present, extends upto a distance of a few atomic diameters

only. In other words, these solids exhibit short range order.

Amorphous solids are formed when the atoms do not get sufficient time to undergo a periodic

arrangement. Eg. Glass, plastics, rubber, wood etc.


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Difference between crystalline and amorphous solids

Crystalline Amorphous

The constituent particles(atoms/molecules/ions) are arranged in

regular fashion containing short range as well

as long range order.

The constituent particles are not arranged inany regular fashion, short range order.

They are anisotropic They are isotropic

They show elastic behavior They do not show elastic behavior

They are ductile in nature They are brittle in nature

They have sharp melting point i.e. all the

bonds are broken at particular temperature as

they are of equal strength

They melt over a range of temperature as all

bonds are of not equal strength.

Crystallography

The branch of science, which deals with the study of geometrical form and physical

properties of crystalline solids, is called crystallography.

Lattice points and space lattice

A crystal is a 3 dimensional body, which is made up of regular and periodic arrangement of

atoms or molecules in space. The periodicity in the arrangement generally varies in different

directions. It is very convenient to imagine points in space about which these atoms are

located. Such points in space are called lattice points and totality of such points form a

crystal lattice or space lattice. If all the atoms at the lattice points are identical, the lattice is

called Bravias lattice.

A space lattice may be defined as an infinite array of points in three-dimensions in which

every point has surroundings or environments identical to that of every other point in

the array.


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Let us consider the case of a two dimensional square array of points as shown in figure. By

repeated translation of the two vectors a and b on the plane of the paper, we can generate the

square array.

The magnitudes of a and b are equal and can be taken to be unity. The angle between them is

900. A and b are called the fundamental translation vectors that generate the square array. Let

us choose any arbitrary point O as origin. If we choose a lattice point P at position r it can be

represented by translation vectors as

bmalr

Where l and m are integers. In the figure l=2 and m=1

Similarly for three dimensional lattice

cnbmalr

Thus a three dimensional space lattice is generated by repeated translation of three

noncoplaner vectors a, b and c.

A crystal lattice refers to the geometry of a set of points in space, where as the structure of

crystal refers to actual ordering of its constituent atom, ions, or molecules in space.

Lattice points: - Lattice points denote the position of atoms or molecules in the crystals.

It is an Infinite array of points in space, in which each point has identical surroundings to allothers

a

O

P

b

r


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Space lattice:- An infinite three dimensional array of points showing how atoms are

molecules are arranged in a crystal is known as space lattice or simply lattice.

The Basis and Crystal structure

In order to obtain a crystal structure, an atom or group of atoms must be placed on eachlattice point in a regular fashion such an atom or group of atoms is called the basis or the

pattern. When the basis is repeated with correct periodicity in all directions it gives the actual

crystal structure. The crystal structure is real, while the lattice is imaginary. Mathematically it

is expressed as

Space lattice + basis ----------- Crystal structure

-The number of atoms in a basis varies from one to several thousands

- In crystalline solids like Cu and Na the basis is Single atom, In NaCl & CsCl the basis in

diatomic, where as in crystals, like CaF2 the basis in tri-atomic.

Complex basis are found in biological materials.

Unit cell and lattice parameters

The unit cell is the smallest block or geometrical figure from which the entire crystal is built

up by repetition in three dimensions.

The unit cell may also be defined as the fundamental elementary pattern of minimum number

of atoms, molecules which represents fully all the characteristics of the crystal

Unit cell may be defined as that volume of a solid from which entire crystal may beconstructed by translation repetitions in three dimensions.

It should be noted that the choice of a unit cell is not unique but can be constructed in number

of ways. The unit cell should be chosen in such a way that it conveys the symmetry of the

crystal lattice and makes the mathematical calculations easy.

Space Lattice + Basis = Crystal Structure

=


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Let us consider a unit cell of a three dimensional crystal lattice. Unit cell can be completely

described by three vectorsa , b , c when the length of the vectors and the angle between

them , , or specified.

The lines drawn parallel to the lines of intersection of any three faces of the unit cell which

do not lie in the same plane are called crystallographic axes. An arbitrary arrangement of

crystallographic axes marked x,y and z defining the unit cell is shown in fig below.

The angles between the three crystallographic axes are known as interfacial angles or

interaxial angles

The angle between the axes y and z is

The angle between the axes z and x is

The angle between the axes x and y is

Thus the above angles , and are the interfacial angles.

The intercepts a, b, and c defines the dimensions of a unit cell and are known as its primitives

or characteristics intercept on the axes or fundamental translational vectors.


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The primitives a, b and c and interfacial angles , , and are the basic lattice parameter and

they determine the form and actual size of the unit cell and hence the space lattice. But if we

do not know actual values of primitives but only their ratio and the value of interfacial

angles, then we can only determine the form of the unit cell but not its actual size.

The vectors a , b , c may or may not be equal. Also the angles , and may or may not be

right angles. Based in these conditions, there are seven crystal systems. If the existing of

atoms is only at the corners of the unit cell the seven crystal systems yield seven types of

lattices.

We have seen that a three dimensional space lattice is generated by repeated translation of

three non coplanar vectors a, b and c. there are only fourteen distinguishable ways of

arranging points in three dimensional space. These 14 space lattices are known as Bravais

lattices.

Primitive cell

Primitive cell in the smallest volume cell, all the lattice points belong to a primitive cell lies

at its corners. Therefore effective number of lattice points in a primitive cell is one.

Non primitive cell

A non primitive cell many have the lattice points at the corner as well as other locations both

inside and on the surface of the cell. Therefore the effective number of lattice points in a non-

primitive cell in greater than one.


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Crystal systems

On the basis of the length and directions of the axes of the symmetry all the crystals may be

classified into following seven systems


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.


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Important crystal structure terms

1. Coordination number (N): The coordination number is defined as the number of equidistant

neighbours that an atom has in the given lattice.

2. Nearest neighbour distance (2r): The distance between the centers of two nearest

neighbouring atoms is called the nearest neighbour distance.

3. Atomic radius (r) : Atomic radius is defined as half the distance between nearest neibours in

a crystal of pure element.

4. Unit cell volume (V): The volume of unit cell is given by

1/2222 coscoscoscoscoscos1abcV

If a = b = c and = = =90o

then

V= a3 (cubic unit cell)

5. Atomic packing factor(APF): The fraction of space occupied by atoms in a unit cell is known

as atomic packing factor or packing factor or density of packing. It is the ratio of volume

occupied by the atoms in unit cell (v) to the volume of the unit cell relating to that structure.

V

v

cellunittheofvolume

cellunitatomsbyoccupiedvolumefactorpackingAtomic

cellunittheofvolume

atomeachofvolumecell)atoms/unitof(number(APF)factorpackingAtomic

6. Void space: The void space in the unit cell is the vacant space left unutilized in the cell. It is

equal to (1-APF). It is often expressed in percentage.

Void space = (1-APF) 100

The void space is most commonly known as Interstitial space.

7. Density : As the unit cell posses all the structural properties of a bulk material. The density

of unit cell must be equal to that of the bulk crystal. Then,

AVN

zM

V

zw

volume

massDensity

Where w is mass of each atom =M/NA, z is number of atoms per unit cell, M is molecular

weight and NA is Avagadro number.

Simple cubic (BCC) structure :


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The unit cell of a simple cubic lattice, there is one lattice point (atom) at each of the eight

corners of unit cell. Only polonium at certain temperature region exhibits this type of structure.

Coordination number:

In this structure for every one corner atom there are four nearest neighbours in the same

plane plus two nearest neighbours, one exactly above and the other exactly below that corner atom.

Thus, the coordination number is (4+1+1) six.

Number of atoms per unit cell:

This unit cell as eight corner atoms. Each corner atom is shared by eight adjacent unit cells.

The total number of atoms per unit cell 18

18

Atomic radius:

In this structure the atoms touches each other along cube edges. Hence the nearest

neighbour distance

a2r

Atomic radius2

ar

Atomic packing factor:

Volume of all atoms in unit cell, v =3

r3

41

v3

3

a6

1

2

a

3

41

Volume of the unit cell, V= a3


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V

v

cellunittheofvolume


6

a

a

6

1

V

vAPF

3

3

%250.526

APF

Void space :

Void space = (1-APF) 100 = (1-0.52) 100

=48%

Body centered cubic (BCC) structure :

In this structure in a unit cell there are eight atoms at the eight corners and one atom at the

body center. The unit cell of the body center cubic structure is shown in fig.

Iron (at room temperature), chromium, tungsten, sodium etc. have BCC structure.


27/38


In body centered cubic structure the corner atoms do not touch each other but each corner

atom touches the body center atom along body diagonal. Hence the coordination number this

structure is eight.


This unit cell as eight corner atoms and one body centered atom. As body centered atom is

contained with in the unit cell and it is not shared by and surrounding unit cell.

The total number of atoms per unit cell of BCC 218

18

Atomic radius:

For calculation of r in case of BCC, consider the fig

2222 2aaaAC

22222 a2aCDACAD

22 3AD a

Body diagonal AD = r+ 2r + r = 4r

22

a3)4( r

Lattice parameter3

4ra

Nearest neighbor distance2

3a2r

Atomic radius4

3ar



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r

3

42

v3

3

a8

3a

4

3

3

42


V

v

cellunittheofvolume


8

3

a

a

8

3

V

vAPF

3

3

68%0.688

3APF

Void space :

Void space = (1-APF) 100 = (1-0.68) 100

= 32%

Face centered cubic (FCC) structure:

In the case of FCC lattice, there are eight atoms at eight corners and six atoms at the six face

centers of the unit cell. Copper, aluminum, argon, gold, platinum, lead, thorium etc. are of FCC

structure.


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In this case the nearest neighbors of any corner atom are the four face centered atoms of

surrounding unit cells. The corner atom will have four face centered nearest neighbor atoms in it

own plane, four in a plane above it and four in a plane below it. Thus the coordination number is

(4+4+4) twelve.


This unit cell as eight corner atoms and six face center atoms. Each of these face centered

atom is shared by the two adjacent unit cells

The total number of atoms per unit cell of FCC 42

16

8

18

Atomic radius:

For calculation of r in case of FCC, consider the fig

2222 2aaaAC

Body diagonal AC = r+ 2r + r = 4r

22 a2)4( r

Lattice parameter2

4ra

Nearest neighbor distance2

2a2r

Atomic radius4

2ar



r

3

44

v3

3

a6

2a

4

2

3

44



30/38

V

v

cellunittheofvolume


6

2

a

a

6

2

V

vAPF

3

3

%470.746

2APF

Void space :

Void space = (1-APF) 100 = (1-0.74) 100

= 26%

Hexagonal close packed structure:

The lattice parameter of simple hexagonal structure are a=bc and == 90o; =120

o. It contains

one lattice point at each corner of hexagonal face and one in the center of hexagonal faces. Each

corner atom is shared by six other unit cells and each central atom is shared by two unit cells.

Hence number of atoms per unit cell 3

2

16

6

12

The total number of atoms in a unit cell is only 3, its packing fraction is low hence metals do not

crystallize in this simple hexagonal structure. The atoms attain a lower energy and more stable

configuration only by forming hexagonal close packed (HCP) structure. The unit cell of HCP is shown

in the fig.


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HCP structure belongs to hexagonal Bravias lattice with a basis of two atoms associated with

each lattice point. In HCP there are three atoms at interstices between two hexagonal faces.

Example of metals that crystallize in HCP structure are Zn, Co, Mg, Cd, Ti etc.


The total number of atoms per unit cell in HCP structure is six calculated as below.

The total number of atoms per unit cell 632

16

6

12


Consider the bottom layer, the central atom has 6 nearest neighboring atoms in the same

plane. Further at a distance of c/2 from this plane there are two layers one above and another below

it containing three atoms in each layer. Thus there are 12 nearest neighboring atoms.

The coordination number = 6+3+3= 12

Atomic radius:

The atoms are in contact along the edges of hexagon. The nearest neighbor distance

a2r

Atomic radius

2

ar


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c/a ratio:

In HCP structure the three body atoms lie in a horizontal plane at a height c/2 from the orthocenter

of alternate equilateral triangles in the base or at top of the hexagonal cell. From the fig.

2

22

2

cx2r

But

2

3a

3

2cos30a

3

2AN

3

2x o

3

ax

and 2r =a

4

c

3

a

2

c

3

aa

2222

2

32a

3aa

4c

222

2

3

8

a

c


Volume occupied by atoms in a unit cell v =3

r

3

46

v3

3

a2

a

3

46

Volume of the unit cell may be determined by calculating the area of the base of the unit cell and

then multiplying it by the height of the unit cell.


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Area of the base of the unit cell = 6 (area of triangle ABO)

2o a2

33

2

3aa

2

16asin60a

2

16

the volume of the unit cell = ca2

33 2

Substituting the value of a3

8c we get,

volume of the unit cell =32 a23a

3

8a

2

33

Vv

cellunittheofvolumecellunitatomsbyoccupiedvolumefactorpackingAtomic

6

2

23

a23

a

V

vAPF

3

3

%7474.06

2APF

Void space :

Void space = (1-APF) 100 = (1-0.74) 100

= 26%

Density of unit cell:

The density of HCP cell is given by

3A3AA aNM2

.a23N

6M

VN

zM


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Diamond structure:

In crystalline form diamond is one of the most important precious stone, it crystallizes in

cubic as well as in hexagonal structures. The diamond cubic (DC) structure is very important

structure as besides diamond, the semiconductors Ge and Si crystallizes in this structure. The unit of

diamond cubic is shown in fig. The diamond structure is a FCC structure with basis of two carbon

atoms one located at (0,0,0) and other at (1/4,1/4,1/4) associated with each lattice point.

The diamond lattice can be considered to be formed by the interpenetrating two FCC sub-

lattices along the body diagonal by 1/4th cube edge. If one sub-lattice has its origin at a point (0,0,0)

and the other at point quarter of the way along the body diagonal i.e. at (a/4,a/4,a/4).

Unit cell of diamond structure contains 18 atoms. Of these atoms 14 carbon atoms are

situated at 8 corners and 6 face centers of the cube, apart from those occupying the FCC lattice

points the other four carbon atoms are situated along the two body diagonals at a height of 1/4 of

the lattice parameter for the bottom and 3/4 of the lattice parameter from the bottom on the other

two body diagonals which are opposite to each other. The coordinates of four body diagonal atoms

in the unit cell are

4

3,

4

3,

4

3and

4

1,

4

3,

4

3,

4

3,

4

3,

4

1,

4

1,

4

1,

4

1.


Each carbon atom has four nearest neighbors in the tetrahedral coordination with a bond

angle 109.5o. Hence the coordination number in diamond cubic structure is four.


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This unit cell consists of eight corner atoms, six face centered atoms and four atoms inside

the cube along body diagonal

Hence the number of atoms per unit cell = 8462

18

8

1

Atomic radius:

From the figure, the nearest neighbour distance,

222

4

a0

4

a0

4

a02r

222

4

a

4

a

4

a2r

therefore4

a32r

Atomic radius8

a3r


Volume occupied by atoms in a unit cell v =3

r3

48

v 3

3

a16

3a83

348


V

v

cellunittheofvolume


16

3

a

a

16

3

V

vAPF

3

3


36/38

%430.3416

3APF

Void space :

Void space = (1-APF) 100 = (1-0.34) 100

= 76%

Diamond cubic structure is loosely packed structure because the atomic packing factor is

only 34%.

If the two atoms in the basis are identical, the structure is called diamond. Semiconductors

such as Ge, Si fall in this category. If the two atoms are different, the structure is called zincblende (ZnS) and examples are GaAs, AlAs, CdS, InSb etc.

Zinc Blende or cubic zinc sulphide structure:

Zinc Blende structure belongs to FCC Bravias lattice with a basis of two atoms (Zn and S). The

structure of ZnS is similar to diamond cubic structure except that the two atoms belonging to the

basis are not same. This structure can be considered to be formed by the two interpenetrating FCC

lattices (one sublattice with Zn atoms other with S atoms), which are displaced by one quarter of thecube edge, along the body diagonal.

There are four molecules of ZnS per unit cell. Each atom there are four equidistant nearest

neighbouring atoms of opposite king arranged in a regular tetrahedral configuration. The

coordination number, APF and other parameters in this structure would be same as in the case of

diamond structure. The materials that crystallizes in zinc blende structure are InSb, Cds,CuF, AgI, SiC,

BeO, ZnTe, CdTe, GaAs etc.
http://www.geocities.jp/ohba_lab_ob_page/Structure/ZnS.jpg


37/38

The coordination number, APF and other parameters in this structure would be same as in the

case of diamond structure.

Because of covalent nature of the bonds the material which crystallize with these structures

(i.e. Diamond and ZnS) have very high hardness and are used as abrasives.

NaCl structure

NaCl and many other ionic solids crystallize in the rock salt structure which is also known as

sodium chloride structure. It belongs to the FCC Bravias lattice with two ions, one anion and

one cation forming a basis. The two ions are separated by a distance of a/2. Where a is the

lattice parameter.

The larger anions are in FCC packing with all octahedral interstitial positions filled with the

small cations. The octahedral interstitial(voids) are at the body center and at midpoint of cube

edges as shown in fig.

The structure can be thought of as two interpenetrating mono-atomic FCC lattices along the

cube edge by of the cube edges. If the corner of one FCC is located at (0,0,0), the other

corner of the other is located at (1/2,0,0)

Each ion in NaCl lattice has six nearest neighbor ions at a distance of a/2 i.e. its coordination

number is six. The ionic radius of Cl is about 1.810A and for sodium it is about 0.98

0A.

Number of NaCl molecules per unit cell

1) Number of Cl- ions per unit cell

(1/8) x8+(1/2)x6 = 4

2) Number of Na+ ions per unit cell

(1/4) x12+1 = 4

Thus there are 4 molecules in each unit cell

The crystal having NaCl structure are KCl, KBr, AgBr, LiH


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Cesium chloride

CsCl structure belongs to the simple cubic Bravias lattice with two atoms forming a basis.

The two atoms of a basis which are unidirectional are at positions (0,0,0) and (1/2,1/2,1/2)

In CsCl structure the Cs and Cl ions have the same size approximately. Thestructure can be viewed as a BCC lattice with atoms of on type at the corners andthe atoms of the other type at the body centers. This structure is considered to be

formed by interpenetrations of two simple cubic lattices such that the corner of one

sub lattice is at the center of other sub lattice.

Each Cs ions is surrounded by eight Cl ions. Thus its coordination number is 8

Number of Cl ions per unit cell is (1/8)x8 = 1Number of Cs ions per unit cell is =1

Number of CsCl molecules per unit cell is = 1The crystals having CsCl structure are RbCl, LiHg, TiBr etc.

physics 2 sengupta

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