Chapter I
Atomic theory of Matter
Introduction :- Engineering materials are used for different purposes like tools,
transportation, decoration, and jewellery. Properties of materials dictate their engineering
purpose. Properties of materials are in turn dependent on atomic structure of materials.
Other important components of Materials Science along with structure and properties are
processing, and performance. Most of the engineering materials in their raw form are not
useful, thus need to be processed. Usefulness of a material is dictated by its performance
under desired conditions.
• Materials Science is the combination of physics, chemistry, with the focus on the
relationship between the properties of a material and its microstructure.
• this science allowed designing materials and provided a knowledge base for the
engineering applications (Materials Engineering). Important components of the subject
Materials Science are structure, properties, processing, and performance.
• Materials which are used in the field of electrical engineering are called electrical
engineering materials. The engineer should select the right material with right
properties for particular application.
1.
2. Classification of properties of materials
Are
Density
Mechanical Properties
Physical
properties
Thermal Properties
ratio of mass of the material to its
volume
Elasticity, Plasticity, Strength, Hardness,
Brittleness, Toughness
Melting point, Boiling point, Thermal
expansion, thermal conductivity, thermal
capacity
Electrical Properties
Chemical
Properties
Technological
properties
Electrical conductivity, dielectric
constant, tracking resistance
Magnetic properties
Permeability, coercive field strength,
residual magnetism
Optical Properties
Colour, gloss, transparency
Resistance to corrosion, resistance to acid, resistance to alkali, scaling resistance
Castability, forgeability, weldability, machinability, cold workability, wears
resistance.
Properties of material are
• Elasticity: It is the property of a body of material of resuming its original form and
dimensions when the force acting upon it is removed. If the force is sufficiently large for
the deformation to cause a break in the molecular structure of the body or material, it
loses its elasticity and the elastic limit is said to have been reached.
• Plasticity: It is the property of a body or material of undergoing permanent
deformation under the effect of force, i..e of not returning to its original form when the
force is removed.
• Strength: It is the resistance of the material to breaking. Different types of strength are
tensile strength, compression strength, shearing strength, torsional strength, etc.,
• Hardness: It is the property of resisting penetration by another material.
• Brittleness: It is the property of breaking of a material without changing its shape
• Toughness: It is a property just opposite to brittleness. A tough material or work piece
will undergo considerable plastic deformation before breaking.
• Melting point: It is the temperature at which the solid material begins to melt. Melting
points are normally quoted for standard atmospheric pressure, i.e. 1bar or 760mm of
mercury
• Boiling point: The temperature at which a liquid boil freely and begins to turn into gas.
Boiling points are normally quoted for standard atmospheric pressure.
• Thermal expansion: It is the increase in volume of a material due to rise in temperature.
Coefficient of linear expansion is the measure of thermal expansion of solid material.
• Thermal conductivity: It is the ability of material to transmit heat
• Thermal capacity: It is the amount of heat required to raise the temperature of a
material.
• Electrical conductivity: It is the ability of material to allow current to flow through. It is
the reciprocal of specific resistance or resistivity of a conductor.
• Dielectric constant: It is the ratio of the capacitance of a capacitor with a specified
medium (dielectric material) between the plates, to the capacitance of the same capacitor
with free space between the plates. Thus, it is the property of the dielectric material that
causes the difference in the value of the capacitance, physical dimensions viz, plate area
and distance between the plates remaining the same.
• Tracking resistance: It is the resistance of an insulating material to the surface currents.
• Permeability: It is the ratio of the magnetic flux density in a medium to the external magnetic field strength
that induces it.
• Coercive field strength: It is that strength of the magnetizing force that is required to destroy or remove
residual magnetism from a magnetic material undergoing a hysteresis cycle.
• Residual magnetism: is the amount of the magnetism retained in a magnetic material when the magnetizing
force producing magnetism is reduced to zero or removed.
• Colour: The visual sensation resulting from the impact of light of a particular wave length on the cores of
the retina of the eye.
• Gloss: It represents the brightness of a material.
• Transparency: The quality of a material of permitting the passage of light in such a way that objects can be
seen clearly through the substance.
• Resistance of corrosion: It is the resistance of a material specifically of metals to getting damage by surface
chemical action due to action of moisture, air or chemicals.
• Castability: Ability of a material being given a particular shape by casting.
• Forgeability: Ability of a material being given a desired shape by forgeing.
• Weldability: Ability to join two metals by raising their temperature sufficiently to melt and fuse them
together.
• Machineability: Means whether the material can be cut on machines.
• Wear resistance: It is the resistance of a material to surface erosion e.g. due to friction.
• The other fators for material selections are cost, availability and easiness for use.
3. Atomic theory
 Matter is anything that has mass and occupies space. Matter is composed of atoms. An
atom is defined as a substance or particle that can neither be further broken up nor be
created by ordinary chemical means. Each atom of a matter consists of three different
particles; protons, neutrons and electrons. These are called the fundamental particles or sub
atomic particles. Understanding the behaviour of electrons in atoms is one of the key points
in understanding materials. Thus, the atomic theory is capable of explaining the electrical,
optical, thermal, and magnetic properties of materials. Here are some examples where such
important properties of materials are used.
• Electrical properties: transmission lines, conductors, electronic devices, batteries, cells, etc.
• Thermal properties: refrigeration, heating devices, etc.
• Optical properties: lasers, optical communication, lenses, solar collectors & reflectors, etc.
• Magnetic properties: electric generators & motors, loud speakers, transformers, tape
recorders, etc.
4. Atomic Structure
 All the materials available either in solid, liquid or gaseous form is made up of atoms, the
smallest indivisible particles. Atoms of the same element are identical to each other in
weight, size and all properties, whereas atoms of different elements differ in weight, size
and other characteristics. The size of the atoms is of the order of 1Å=(10-10 m). A material
which consists just one type of atoms is called element.e.g Nitrogen, carbon, hydrogen
 Group of atoms which tend to exist together in a stable form are called molecules, e.g. H2,
 Molecules containing one atom (known as monoatomic), two atoms (known as diatomic),
more atoms (known as polyatomic).
 J.J. Thomson, Rutherford, Niels Bohr and many other found atom, consists of smaller
particles called electrons, protons and neutrons.
 The electrons and protons are electrical in nature, where as neutrons are neutral.
 The atom is made up of a heavy nucleus, consisting of protons and neutrons, surrounded by
highly structured configurations of electrons, revolving around the nucleus.
 The size of the atomic nucleus is of the order of 10-14 m, whereas at its density is found to
be about 2x1017 kg/m3.
4.1 Electrons
 M. Faraday in 1983, in his experiments on the laws of electrolysis, found the existence of
electrical charges in discrete units.
 In 1897, J.J. Thomson, while studying the passage of electricity through gases at low
pressure, observed that the rays of light appear to travel in straight lines from the surface of
the cathode and move away from it in the discharge tube. These rays are called cathode
rays since they start from the cathode of the discharge tube.
 W. Crookes studied the properties of these cathode rays and showed that the rays,
• (i) travel in straight line and cast shadows
• (ii) carried negative charge and sufficient momentum
• (iii) possess high kinetic energy and can induce some chemical reactions, excite
fluorescence on certain substances.
 These properties of the cathode rays were best explained by J.J. Thomson by his hypothesis
that the cathode rays consist of a stream of particles, each of mass m and charge e (=
1.602X10–19 C), originating at the cathode of the discharge tube. These particles are called
electrons
4.2 Proton
 The nucleus of hydrogen atom is called the proton. A proton has a unit positive charge of same
magnitude as that of electron (= 1.602X10–19 C). The mass of a proton is 1.672X10–27 kg.
 The number of protons in a nucleus is called the charge Z of the given nucleus, or the charge
number.
4.3 Neutron
 These are electrically neutral particles and 1.008 times heavier than protons. The mass of each
neutron is 1.675X10–27 kg. Each neutron is composed of one proton and one electron, i.e.
Neutron = Proton + Electron
4.4 Atomic number (Z)
 The atomic number of an element is numerically equal to the number of protons present in the
nucleus
 A normal atom is electrically neutral and hence the number of protons and electrons are equal.
For example, an iron atom contains 26 protons (Z = 26) and hence the balancing electrons are
also 26. Atomic mass of element is the sum of protons and neutrons.
5. Rutherford’s nuclear atomic model
•
 Rutherford and co-workers in 1911 by performing differential scattering experiments
proposed a new atomic model.
 Rutherford directed α -particles emitted from a radioactive source on a thin gold foil 4X10–6
m thick.
 He observed that some α -particles passed through the film, while others were scattered all
around. Those α -particles passing through the thin gold film also scattered over a wide area,
and produced luminescence on a fluorescent screen of zinc sulphide. Very small number of α particles were deflected through large angles, i.e. about 1 in 10,000 particles suffered a
deflection of more than 90°. Only very rarely α -particle reversed back along its own path, i.e.
suffered a deflection through 180°.
The above explanation is described by bottom figure.
Fig 1.1 Rutherford 𝛼-scattering experiment
fig 1.2 Deflection of 𝛼-particles in the vicinity
of the nucleus.
• 1 Rutherford observations
1) The entire positive charge and mass of an atom are concentrated at the centre of the atom in
a nucleus of very small dimensions.
2) The diameter of the nucleus is about 10–15 m whereas the diameter of an atom is about 10–10
m. obviously; the nucleus of an atom occupies a volume which is a million-millionth part of
an atom. Thus there is a lot of empty space in the atom.
3) An atom consists of a positive nucleus with the electrons moving very rapidly around
different orbits. The positive charge of the nucleus equals to total negative charge of
electrons, making atom electrically neutral.
5.2 Drawbacks of Rutherford’s atomic model
1) There is no rule for distribution of electrons in various orbits round the nucleus.
2) Due to the attractive force of the nucleus it will move in orbits of continually decreasing
radii and its path will be a spiral. Finally it will fall upon the nucleus itself and will
disappear.
6. Bohr’s atomic model
• He accepted the Rutherford model of the atom.
• Rutherford model assumed that there is only one electron revolving in the orbit of the
hydrogen atom round the nucleus, which is a single proton.
• In Bohr’s model of the atom, the electrons are assumed to be in a definite planetary system
(in circular orbits) of fixed energy. These stationary states are called as energy levels having a
definite value of potential energy associated with each orbit.
• He considered that more than one energy level to be possible for any electron of the atom.
• Bohr’s postulates
1.
In the case of hydrogen atom, there is single electron which can revolve round the nucleus
in certain definite orbits, known as stationary orbits. The electrons are permitted to have
only those orbits for which the angular moments of the planetary electron are integral
multiple of h/2π or ħ, where h is the Planck’s constant(ℎ = 6.626 𝑋 10−34 𝐽𝑠). Thus, angular
momentum of an electron is given by relation L= nh/2π=nħ. This is Bohr’s first postulate.
2. When the electron revolves in a stationary orbit, it does not emit electromagnetic radiation
as predicted by the electromagnetic theory of light.
 Radiation occurs only when an electron falls from a higher energy state to a lower energy
state. If the transition is from an orbit of higher energy E2 to an orbit of energy E1, then the
energy hυ of the emitted radiation, according to Planck’s law, will be
𝒉𝝊 = ħ𝝎 = 𝑬𝟐 − 𝑬𝟏
• Where υ is the frequency of the emitted radiation; ω =2πυ is the angular frequency. hυ, the
energy difference ejected from the atom in the form of light radiations is called a photon. This
reveals the origin of light waves from atom. Obviously, light is not emitted by an electron
when moving in one of its stationary orbits, but it ejects light only when it jumps from one
orbit to another. This is Bohr’s second postulate.
 Fig 1.3 Bohr's atomic model of
Hydrogen atom
 On the basis of the above two postulates, Bohr first developed the theory of hydrogen atom
(which is also applicable to hydrogen like atoms).
 A hydrogen atom (Z = 1) is the simplest of all atoms consists of a nucleus with one positive
charge e (proton) and a single electron of charge –e revolving around it in a circular orbit of
radius r.
 The centripetal force acting on the electron of mass m and moving with velocity v in a orbit
of radius r around the nucleus = mv2/r.
 The electrostatic force of attraction between the positively charged nucleus (proton) and the
electron is 𝑭 =
𝟏 𝒁𝒆𝟐
(𝑍
𝟒𝝅𝜺𝒐 𝒓𝒏 𝟐
= 1 𝑓𝑜𝑟 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛)……….(1.1)
 Where 𝜺𝒐 is permittivity of free space(𝜺𝒐 = 8.85 𝑥 10 − 12 𝐹/𝑚), Z = atomic number,
(𝑒 = 1.602𝑥10−19 𝑐) electron charge in coulombs.
 Since the electron revolves in a circular orbit, it experiences a centripetal force
 𝑭𝒄 =
𝒎𝒗𝒏 𝟐
𝒓𝒏
= 𝒎𝒓𝒏 𝝎𝒏 𝟐 … … … … … … … … . . 𝟏. 𝟐
Where m is mass of the electron, 𝑣 and 𝜔 are the linear and angular velocity of the electron
in the 𝑛𝑡ℎ orbit respectively.
 The necessary centripetal force is provided by the electrostatic force of
attraction. For equilibrium, from equations (1.1) and (1.2)
1
𝑍𝑒 2
.
4𝜋𝜀𝑜 𝑟𝑛 2
=
𝒎𝒗𝒏 𝟐
𝒓𝒏
= 𝒎𝒓𝒏 𝝎𝒏 𝟐 ……(1.3)
From equation 1.3
𝑍𝑒 2
𝝎𝒏 =
………………………..(1.4)
4𝜋𝑚𝜀𝑜 𝑟𝑛 3
an electron in 𝑛𝑡ℎ orbit is
𝟐
The angular momentum of
𝐿 = 𝑚𝑣𝑛 𝑟𝑛 = 𝑚𝑟𝑛 2 𝜔𝑛 ….……………(1.5)
By Bohr's first postulate, the angular momentum of the electron
𝐿=
𝑛ℎ
2𝜋
…………………………………………..(1.6)
Where n is an integer and is called as the principal quantum number.
 From equation (1.5) and (1.6)
𝑛ℎ
𝑛ℎ
2
𝑚𝑟𝑛 𝜔𝑛 =
,
𝑜𝑟 𝜔𝑛 =
2𝜋
2𝜋𝑚𝑟𝑛 2
 Squaring both sides
𝜔𝑛2
=
𝑛2 ℎ 2
4𝜋2 𝑚2 𝑟𝑛 4
From equation (1.4) and (1.7)
𝝎𝒏 𝟐 =
…………………………..(1.7)
𝑍𝑒 2
4𝜋𝑚𝜀𝑜 𝑟𝑛 3
𝑛2 ℎ2
= 2 2 4
4𝜋 𝑚 𝑟𝑛
𝜀𝑜 𝑛 2 ℎ 2
𝑟𝑛 =
………………………………….(1.8)
𝜋𝑚𝑍𝑒 2
seen that the radius of the 𝑛𝑡ℎ orbit is proportional
From equation(1.8), it is
to the square of
the principal quantum number. Therefore, the radii of the orbits are 1:4:9…..
 For hydrogen atom, z=1
• From equation (1.8)
𝑟𝑛 =
𝜀𝑜 𝑛 2 ℎ 2
………………………………(1.9)
𝜋𝑚𝑒 2
• Substituting the known values to the above equation, results
• 𝑟𝑛 = 𝑛2 𝑥 0.53 if n=1, 𝑟𝑛 = 0.53 . This is called the Bohr radius.
 Energy of an electron in the 𝒏𝒕𝒉 orbit (𝑬𝒏 )
 The total energy of the electron is the sum of its potential energy and kinetic energy in its
orbit (see fig 1.4)
 The potential energy of the electron in the 𝑛𝑡ℎ orbit is given by
𝐸𝑝 =
(𝑍𝑒)(−𝑒)
4𝜋𝜀𝑜 𝑟𝑛
The kinetic energy of the electron in the
1
2
−𝑍𝑒 2
=
…………………(1.10)
4𝜋𝜀𝑜 𝑟𝑛
𝑛𝑡ℎ orbit is
𝐸𝑘 = 𝑚𝑣𝑛 2 ………………(1.11)
Fig 1.4 energy of the electron
• Substituting equation
1
𝑍𝑒 2
(1.3)(i.e.
.
4𝜋𝜀𝑜 𝑟𝑛 2
𝐸𝑘 =
1
1
𝑍𝑒 2
.
2 4𝜋𝜀𝑜 𝑟𝑛
=
=
𝒎𝒗𝒏 𝟐
)
𝒓𝒏
into equation (1.11)
𝑍𝑒 2
………………………..(1.12)
8𝜋𝜀𝑜 𝑟𝑛
• The total energy of an electron in its 𝑛𝑡ℎ orbit is
𝐸𝑛 = 𝐸𝑝 + 𝐸𝑘 =
−𝑍𝑒 2
4𝜋𝜀𝑜 𝑟𝑛
+
𝑍𝑒 2
8𝜋𝜀𝑜 𝑟𝑛
=
−𝑍𝑒 2
………….(1.13)
8𝜋𝜀𝑜 𝑟𝑛
• Substituting the value of 𝑟𝑛 from equation (1.9) in equation (1.13)
𝐸𝑛 =
−𝑍 2 𝑚𝑒 4
8𝜀𝑜2 𝑛2 ℎ2
…………………………(1.14)
For hydrogen atom, Z=1
Therefore,
−𝑚𝑒 4
−31 kg
𝐸𝑛 =
,
m
=
9.1085x10
8𝜀𝑜2 𝑛2 ℎ2
Substituting the known values and calculating in electron-volt,
𝐸𝑛 =
−13.6
𝑒𝑉…………..(1.15)
𝑛2
[1𝑒𝑉 = 1.602𝑥10−19 𝐽]
• As there is a negative sign in equation (1.15), it is seen that the energy of the electron in its orbit
increases as n increases.
Shortcomings of Bohr’s theory
Bohr's theory was able to explain successfully a number of experimental observations and has
correctly predicted the spectral lines of hydrogen atom. However, the theory fails in the following
aspects.
 The theory could not account for the spectra of atoms more complex than hydrogen.
 The theory does not give any information regarding the distribution and arrangement of
electrons in an atom.
 When the spectral line of hydrogen atom is examined by spectrometers having high resolving
power, it is found that a single line is composed of two or more close components. This is
known as the fine structure of spectral lines. Bohr‟s theory could not account for the fine
structure of spectral lines.
 There was no satisfactory justification for the assumption that the electron can rotate only in
those orbits in which the angular momentum of the electron is a whole number multiple of h/2π
i.e. he could not give any explanation for using the principle of quantization of angular
momentum and it was introduced by him arbitrarily.
• It is found that when electric or magnetic field is applied to the atom, each of the spectral line
split into several lines. The former effect is called as Stark effect, while the latter is known as
Zeeman Effect. Bohr's theory could not explain the Stark effect and Zeeman Effect.
7. Quantum numbers
• (i) Principle of Quantum number (n)
• This quantum number of any electron in an atom determines the main energy level or shell to
which an electron belongs. This quantum number has integral positive value 1, 2, 3, 4,...... ∞, but
it is never be zero. This has been used in Bohr model as well as Somerfield model of the atom.
All the electrons that have the same value of n are at nearly the same distance from the nucleus
and have the same energy states. These electrons occupy the same energy level/shell. In
accordance with the value of n, the shells assigned a letter are given as under:
n
1 2 3 4 5
6
Shell designation letter K L M N O
Q
• With increase in distance of the shell from the nucleus, the energy in the energy level or shell
increases. Obviously, different shells possess different energies.
(ii) Orbital or Azimuthal Quantum number (l)
It specifies the number of units of angular momentum associated with an electron in a given
orbit and determines the shape of the orbit and the energy of the sublevel. This quantum
number is represented by a vector which is parallel to the axis of rotation of the electron and its
direction is given by the direction of advance of the right hand screw (Fig. 1.5). This
quantum number, l can have any integral value from 0 to n – 1, e.g.
For
n=1
l=0
n=2
l = 0, 1
n=3
l = 0, 1, 2
..........
.................
Fig 1.5 orbital quantum number
..........
.................
• Clearly, the electron in the smallest orbit n = 1 will have no angular momentum, i.e., l = 0. We
see that for n = 3 there are three possible orbital shapes or eccentrics: (i) a straight line
through the nucleus corresponding to l = 0, (ii) an ellipse with an angular momentum for l = 1
and (iii) a rounder ellipse with more angular momentum for l = 2.
• We have seen that n is the principal quantum number and gives principal shell. l gives the
possible orbital sub-shells. The sub-shells in the main shell are designated by s, p, d, f, g and h
with quantum number l = 0, 1, 2, 3, 4 and 5 respectively. We can represent it as follows
• For n = 1, l = 0, the electron is said to be in 1s sub-shell
n = 2, l = 1, the electron is said to be in 2p sub-shell
n = 2, l = 0, the electron is said to be in 2s sub-shell
n = 3, l = 1, the electron is said to be in 3p sub-shell
n = 3, l = 2, the electron is said to be in 3d sub-shell
 In designating an electron, the letter designating it is written after the number that indicates the
total quantum number n. For example a 3p electron is one for which n = 3 and l = 1.
 If an atom has more than one electron with particular values of n and l, this number is written
as a superscript of the letter, e.g. if an atom has 6 electrons for which n = 3 and l = 2, the atom
is said to have 3p6 electrons.
 We have stated above that for a given value of n the highest possible l value is n – 1. Hence we
can have 4d, 5f, 2p and 2s electrons whereas 1p, 2d, 3f electrons do not exist.
III. Magnetic Quantum number (m or ml)
This quantum number determines the possible quantized space orientation of the electron`s
elliptical orbit without any effect on energy levels. When an atom is subjected to an external
strong magnetic field, B, each electronic orbit will be subjected to torque due to which vector l
starts rotating around B. The angle of rotation 𝜃 is called the angle of precession. Due to
restrictions of orbits of electrons 𝜃 has only discrete values. The permitted values of 𝜃 are
governed by lcos 𝜃, i.e. the projection of l along the direction of the magnetic field B.
Fig 1.6 angle of precession due to mag field
We note that l cos θ = m is also discrete (integer). m varies from +l to –l, including zero. (from
the Fig). Obviously, there are (2l + 1) values of m for a given l. Thus, we have
• For
• l=1
• l=2
• .........
l=0
m=0
m = –1, 0, 1
m = –2, –1, 0 + 1, + 2
................
Reading assignment
The magnetic spin quantum number (ms)
8. Atomic bonding
• (i) Ionic bonding
• Perhaps ionic or heteropolar bonding which is formed by the actual transfer of electrons
from one atom to the other so that each atom acquires a stable configuration similar to the
nearest inert gas atoms, is the simplest type of chemical bonding to visualize since it is almost
totally electrostatic in nature.
• Ionic bonding occurs between electropositive elements (metals, i.e., those elements on the
left side of the periodic table) and the electronegative elements (non-metals; i.e. on the right
hand of the periodic table).
• These bonds are formed mainly in inorganic compounds, e.g. sodium chloride (common
salt NaCl), MgO, CuO, CrO2, and MoF2. In MgO the ions are doubly ionized leading to a
stronger inter atomic bond and hence a higher melting point (~2800°C), compared to 800°C for
NaCl. Examination of the formation of ionic bonds, e.g., cupric oxide, chromous oxide and
molybdenum fluoride, reveals that the metallic element need not be from Group I or II but that
any metal may get ionized by losing its valence electrons.
Fig 1.7 ionic bond (sodium chloride)
• A good example of ionic crystals is the crystal of sodium chloride (NaCl). The electronic
configurations of white soft metal Na and Cl atoms are as follows
• Na : 1s2 2s2 2p6 3 s1 [K (2) L (8) M (1)]
• Cl : 1s2 2s2 2p6 3s2 3p5 [K (2) L (8) M (7)]
(ii) Covalent bonding
 is formed by an equal sharing of electrons between two neighbouring atoms each having
incomplete outermost shells.
 A covalent bond is formed between similar or dissimilar atoms each having a deficiency of
an equal number of electrons.
 The sharing is effective if the shared electrons have opposite spins.
 As the participating atoms in the bond have the same valence state, this bond is also called
the ‘valence bond’.
• Fig 1.8 covalent bond between two atoms of chlorine
(iii) Metallic bonding
• Metallic bond is formed by the partial sharing of valence electrons by the neighbouring atoms.
They are formed by the elements of all subgroups A and I-III, subgroups B.
• Unlike the case of covalent bond, the sharing in metallic bond is not localized.
• metallic bond may also be considered as delocalized or unsaturated covalent bond. Metallic
bonds are electropositive.
• When interacting with elements of other groups, atoms in a metallic crystal can easily give off
their valence electrons and change into positive ions.
• When interacting with one another, the valence energy zones of atoms overlap and form a
common zone with unoccupied sublevels. The valence electrons thus acquire the possibility to
move freely with the zone.
• Obviously, valence electrons are shared in the volume of a whole crystal. Thus, the valence
electrons in a metal cannot be considered lost or acquired by atoms. They are shared by atoms in
the volume of a crystal, unlike covalent crystals where sharing of electrons is limited to a single
pair of atoms.
• If potential difference is applied across such a material the electrons are easily attracted towards
the positive terminal due to the following two reasons:
(i) Electrons are far removed from the positive ions.
(ii)Electrons are subjected to force of repulsion from other electrons.
• Fig 1.9 metallic bonding
Comparison among ionic, covalent and metallic bonding
Co...d
Secondary Bonds
i. London dispersion forces /Vander Waals bonding
• Vander Waal`s bonding is due to Vander Waal's forces. These forces exist over a very short
range. The force decreases as the power of the distance of separation between the constituent
atoms or molecules is large when the ambient temperature is low enough. These forces lead to
condensation of gaseous to liquid state and even from liquid to solid state though no other
bonding mechanism exists.
Properties:
The bonding is weak because of which they have low melting points.
They are insulators and transparent for visible and UV light.
They are brittle.
They are non-directional
ii. Dipole–dipole interactions/Hydrogen bonding:
Covalently bonded atoms often produce an electric dipole configuration. With hydrogen atom
as the positive end of the dipole if bonds arise as a result of electrostatic attraction between
atoms, it is known as hydrogen bonding.
Properties:
The bonding is weak because of which they have low melting points.
They are insulators and transparent for visible and UV light.
They are brittle.
The hydrogen bonds are directional.
9. Energy Bands
One of the most important reasons for the initial success of the band theory of
solids was that it offered a simple explanation of markedly different electrical
behaviour of solids (i.e., conductors, semiconductors and insulators).
9.1 Conductors
 A crystalline solid is called a conductor or metal if the uppermost energy band is partly filled
or the uppermost filled band and the next unoccupied band overlap in energy.
 the electrons in the uppermost band find neighbouring vacant states to move in, and thus
behave as free particles.
 When an electric field is applied, these electrons gain energy from the field and produce an
electric current, so that a metal is a good conductor of electricity.
 The partially filled band is called the conduction band. The electrons in the conduction are
known as free electrons or conduction electrons. Silver, copper, aluminium, etc. are few
examples of good conductors.
 Fig 1.10 conductors feature
9.2 Insulators
• In some crystalline solids, the forbidden energy gap between the uppermost filled band,
called the valence band, and the lowermost empty band, called the conduction band, is very
large .In such solids at ordinary temperatures only a few electrons can acquire enough
thermal energy to move from the valence band into the conduction band.
• Such solids are known as insulators. Since only a few free electrons are available in the
conduction band, an insulator is a bad conductor of electricity. It has been observed that if the
temperature of an insulator is increased, some electrons do jump to the conduction band.
Therefore, a small electrical conduction may take place. Most covalent solids, which are
composed of atoms having an even number of valence electrons, are insulators. Rubber,
Bakelite, mica etc. are good examples of insulators.
• Fig 1.11 insulator feature
9.3 Semiconductors
• A material for which the width of the forbidden energy gap between the valence and the
conduction band is relatively small (~1 eV) is referred as semiconductor.
• Germanium and Silicon having forbidden energy gaps of 0.78 and 1.2 eV respectively at 0K
are typical semiconductors. As the forbidden gap is not very wide, some of the valence
electrons acquire enough thermal energy to go into the conduction band. These electrons then
become free and can move about under the action of an applied electric field. The absence of
an electron in the valence band is referred to as a hole. The holes also serve as carriers of
electricity. The electrical conductivity of a semiconductor is less than that of a metal but greater
than that of an insulator.
• Fig 1.12 semiconductor conduction feature
De Broglie wave nature of particle
• In 1924 de Broglie suggested that particles in motion should exhibit properties characteristic
of waves. He further suggested that certain basic formulae should apply both to waves and
particles. The wavelength of such particles, e.g., electron, proton, neutron, etc. is given by
the relation.
𝜆=
ℎ
𝑚𝑣
……………………(16)
where h is Planck’s constant, m is mass of the particle and v is the velocity of the particle. de
Broglie called these waves as matter waves. From de Broglie’s hypothesis , wave mechanics,
which deals about the details of atomic and molecular spectra, electron diffraction and
reflection, nuclear properties, etc. has emerged.
from Bohr theory We have
𝑛ℎ
𝑚𝑣𝑟 =
………………………………(17)
2𝜋
From de Broglie relation
𝑛ℎ
𝑛ℎ
2𝜋𝑟 =
= = 𝑛𝜆 ………………(18)
𝑚𝑣
𝑝
• This provides an ‘explanation of the quantum condition on the basis of de Broglie’s
theory.
10. Schrodinger’s Wave Equation
 The Schrodinger’s wave equation like Newton’s laws of motion is a fundamental relationship
showing logical coherence to a vast amount of the experimental observation.
 Newton’s laws of motion can be applied only to macroscopic systems and events. But
Schrodinger’s wave equation can be applied both to macroscopic and microscopic systems
and events.
 The classical wave equation
Let a function given by 𝒚 𝒙, 𝒕 = 𝑨𝒄𝒐𝒔 𝒌𝒙 − 𝝎𝒕 has a classical wave function
𝜕2 y(x,t)
𝜕x2
=
1 𝜕2 y(x,t)
v2 𝜕t2
………………(19)
Lets differentiate to the second order the function y(x, t) with respect to x and y .
𝜕𝑦(𝑥, 𝑡)
= −𝑘𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡)
𝜕𝑥
𝜕2 𝑦(𝑥,𝑡)
𝜕𝑥 2
= −𝑘 2 𝐴𝑐𝑜𝑠 𝑘𝑥 − 𝜔𝑡 = −𝑘 2 𝑦(𝑥, 𝑡)………(20)
𝜕𝑦(𝑥, 𝑡)
= −𝜔𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡)
𝜕𝑡
𝜕2 𝑦(𝑥,𝑡)
𝜕𝑡 2
= −𝜔2 𝐴𝑐𝑜𝑠 𝑘𝑥 − 𝜔𝑡 = −𝜔2 𝑦(𝑥, 𝑡)………..(21)
• where ‘y’ is the displacement of the particle, which is moving in ‘x’ direction at any instant
‘t’. This equation can be applied to waves in stretched string, sound waves in air and light
waves in vacuum. Equating equation (20) and (21) with equation (19) the solution of this
equation will be
1 2
𝑥, 𝑡 = − 2 𝜔 𝑦(𝑥, 𝑡)
v
1
𝜔
𝑘 2 = 2 𝜔2 ≫≫≫ 𝑣 =
……………(22)
−𝑘 2 𝑦
v
𝑘
• According to Schrodinger, for atomic particle like electron, one must take the whole solution
of ’y’, since one cannot determine the momentum and position of it simultaneously as
Heisenberg uncertainty principle. He called this complex displacement as wave function ‘ψ’.
Ψ 𝑥, 𝑡 = 𝐴𝑐𝑜𝑠(𝑘𝑥 − 𝜔𝑡)
• Let`s assume
𝟐𝝅
𝝎 = 𝟐𝝅𝒇 𝑎𝑛𝑑 𝒌 =
………….(23)
𝝀
• Take the following plausible argument
1. Taking de Broglie-Einstein (energy and momentum) postulate assumption
ℎ𝑓 = ℎ
𝜆
2𝜋
= ℏ𝜆 = 𝑚𝑐 2 = 𝐸……..(24)
From equation (16) and (24), we have 𝜆 =
ℎ
𝑝
and 𝑓 =
𝐸
ℎ
2. The total energy of a particle is the sum of its kinetic energy and potential energy
𝐸 = 𝐸𝑝 + 𝐸𝑘 …………………(25)
The kinetic energy is given by
1
𝑚𝑣 2
2
𝐸𝑘 =
=
𝑚𝑣 2
2𝑚
=
𝑝2
2𝑚
=
ℏ2 𝑘 2
2𝑚
Considering equation (16) and (23)
𝐸𝑘 =
𝑝2
2𝑚
=
ℎ2
2𝑚𝜆2
=
ℎ2
2𝑚
2𝜋 2
𝑘
=
ℎ2 𝑘 2
2𝑚 2𝜋 2
…………..(26)
Re-writing the energy equation using equation(24), (25) and (26)
ℏ2 𝑘 2
+ 𝑤(𝑥, 𝑡) = ℎ𝑓
2𝑚
Using equation (23), 𝑓
𝜔
=
2𝜋
ℏ2 𝑘 2
+
2𝑚
𝑤 𝑥, 𝑡 = ℎ𝑓 =
h𝜔
2π
= 𝜔ℏ …………..(27)
The term (𝑘 2 ) and (𝜔) in equation (27) gives us to relate this equation with equation(20) and
first derivative of equation (21).
𝜕2 Ψ(𝑥,𝑡)
𝛼
𝜕𝑥 2
+ 𝑤 𝑥, 𝑡 𝜓 𝑥, 𝑡 =
𝜕𝜓(𝑥,𝑡)
𝛽
𝜕𝑡
Solving for 𝛼 and 𝛽
ℏ2
𝛼=−
, 𝛽 = iℏ
2𝑚
…………..(28)
Then equation(28) becomes
ℏ2 𝜕2 Ψ(𝑥,𝑡)
−
2𝑚 𝜕𝑥 2
+ 𝑤 𝑥, 𝑡 𝜓 𝑥, 𝑡 =
𝜕𝜓(𝑥,𝑡)
iℏ
𝜕𝑡
…………………..….(29)
• This can be written in three dimensions as
ℏ2 𝜕2 Ψ(𝑥,𝑦,𝑧,𝑡)
−
2𝑚 𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2
+ 𝑤 𝑥, 𝑦, 𝑧, 𝑡 𝜓 𝑥, 𝑦, 𝑧, 𝑡 =
𝜕𝜓(𝑥,𝑦,𝑧,𝑡)
iℏ
𝜕𝑡
…….(30)
• Equation (29) and (30) are the famous time dependent Schrodinger wave
equations in one dimension and three dimensions respectively.