UNIT-III by Dr. Tahir Hussain - Aligarh Muslim University

Applied Physics − II

(Course Code: BPH-201)

UNIT-III

by

Dr. Tahir Hussain

Applied Sciences & Humanities Section

University Polytechnic, ZHCET

Aligarh Muslim University (AMU), Aligarh

Contents

1 Rutherford’s Alpha particle scattering experiment . . . . . . . . . . . . . . . . . . . . . . 1

2 Rutherford Atomic Model or Nuclear model of Atom . . . . . . . . . . . . . . . . . . . . 1

3 Shortcomings of Rutherford Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.1 Stability of atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.2 Line spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Bohr’s theory of Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4.1 Energy of a Hydrogen atom in different states . . . . . . . . . . . . . . . . . . . . 3

5 Excitation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

6 Excitation Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

7 Ionization Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

8 Ionization Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

9 Structure of solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

10 Idea of Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

11 Unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

12 Packing factor/efficiency/fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

12.1 Simple cubic (sc) structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

12.2 Body centered cubic (bcc) structure . . . . . . . . . . . . . . . . . . . . . . . . . 8

12.3 Face centered cubic (fcc) structure . . . . . . . . . . . . . . . . . . . . . . . . . . 8

13 Formation of bands in solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

14 Conductors, Insulators and Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . 10

15 Doped Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

15.1 n−type and p−type semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . 11

16 Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

17 X−rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

17.1 Production mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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BPH201:Unit−III Tahir Hussain

17.2 Continuous and characteristic X−rays . . . . . . . . . . . . . . . . . . . . . . . . 14

17.3 Duane and Hunt rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

17.4 Applications of X−rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

18 Problems for Unit-III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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Unit−III

1 Rutherford’s Alpha particle scattering experiment

Ernest Rutherford performed a series of experiments from 1906 to 1911 on alpha particle scattering. Inthese experiments, a beam of alpha particles was bombarded on a thin gold foil and their deflections werestudied. Most of the alpha articles passed through the gold foil either undeviated or with a small deviation.This was expected because an alpha particle is a heavy particle and will brush aside any tiny particle comingin its way. However, some of the alpha particles were deflected by large angles. Rutherford found that someof the alpha particles, about one in 8000, were deflected by more than 90°, i.e. they were turned back by thefoil.

This was interesting. When 8000 alpha particles could go through the gold atoms undeflected, why then onewas forced to turn back. The alpha particle itself is about 7350 times heavier than the electron. So neitheran electron, nor a similar positively charged particle could cause a large scale deflection of an alpha particle.The alpha particle must have encountered a very heavy particle in its path, a particle with mass of the orderof the mass of the atom itself. Also, thousands of alpha particles go undeviated or almost undeviated. Sothis heavy mass in the atom should occupy a very small volume so that the atom may contain lot of emptyspace.

From the pattern of the scattering of alpha particles, Rutherford made quantitative analysis. He found thatthe heavy particle from which an alpha particle suffered large deflection, had a positive charge and virtuallyall the mass of the atom was concentrated in it. Its size was also estimated from the same experiment. Thelinear size was found to be about 10 fermi (1 fermi = 1 femtometre = 10−15 m) which was about 10−5 ofthe size of the linear atom. As the volume is proportional to the cube of the linear size, the volume of thispositively charged particle was only about 10−15 of the volume of the atom.

2 Rutherford Atomic Model or Nuclear model of Atom

Based on the observations of the alpha particle scattering experiments, Rutherford proposed the model ofnuclear atom which remains accepted to a large extent even today. According to this model, the atomcontains a positively charged tiny particle at its center called the nucleus of the atom. This nucleus containsalmost all the mass of the atom. Outside this nucleus, there are electrons which move around it at someseparation. The space between the nucleus and the electrons is empty and determines the size of the atom.The amount of the positive charge on the nucleus is exactly equal to the total amount of negative charges onall the electrons of the atom.

So, Rutherford’s model explains the charge neutrality and the large empty space inside the atom. It alsoexplains why only negatively charged particles are ejected easily by an atom. It is so because the positivelycharged particle (nucleus) is so heavy that when an atoms gets energy from heating or otherwise, this particleis hardly affected.

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The movement of electrons around the nucleus was a necessary part of the Rutherford’s model. If theelectrons were at rest, they would fall into the nucleus because of Coulomb attraction. If the electrons movein circular orbits, the Coulomb force will only change the direction of velocity providing the necessarycentripetal force. This electronic motion, however, created difficulties for Rutherford’s model as we shallnow study.

3 Shortcomings of Rutherford Atomic Model

The two important phenomena which could not be explained by the Rutherford atomic model are as follows:

3.1 Stability of atom

In Rutherford’s model, it is assumed that the electron revolves around the nucleus and the necessarycentripetal force is provided by the Coulomb force. But according to Maxwell’s theory of electromagnetism,an accelerated charged particle (in our case electron) must continuously emit electromagnetic radiation. Therevolving electron should, therefore, continuously lose energy by emitting electromagnetic radiation. So, ifthe electron will continuously lose energy in the form of radiation the radius of the circle should graduallydecrease and the electron would finally fall into the nucleus. Thus, atom cannot be stable in the Rutherfordatomic model.

3.2 Line spectrum

When a material body is heated, it emits electromagnetic radiation. The radiation may consist of variouscomponents having different wavelengths. When the filament of an electric bulb is heated, it gives whitelight and all wavelengths in the visible range are present in the emitted radiation. If the emitted light ispassed through a prism, components of different wavelengths deviate by different amounts and we get acontinuous spectrum.

If hydrogen gas enclosed in a sealed tube is heated to high temperatures, it emits radiation. If this radiationis passed through a prism, components of different wavelengths are deviated by different amounts and thuswe get the hydrogen spectrum. The most striking feature in this spectrum is that only some sharply defined,discrete wavelengths, i.e. line spectrum exist in the emitted radiation. For example, light of wavelength656.3 nm is observed and then the light of wavelength 486.1 nm is observed. Hydrogen atoms do not emitany radiation between 656.3 nm and 486.1 nm. A hydrogen sample also emits radiation with wavelengthsless than those in the visible range and also with wavelengths larger than those in the visible range.

However, according to Rutherford’s atomic model the frequency of revolution changes continuously as theelectron is losing its energy continuously. As a result, the electron should emit radiation of all wavelengths,that is, the spectrum of these waves will be continuous.

Thus, the Rutherford model was unable to explain the line spectrum. The sharply defined discrete wave-lengths in hydrogen spectra posed a serious puzzle before physicists.

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Tahir Hussain BPH201:Unit−III

4 Bohr’s theory of Hydrogen Atom

Bohr immediately recognized the importance of the Rutherford nuclear atom in understanding the structureof atoms. He was led to propose a model in which the electron circulates about the nucleus like a planet aboutthe Sun. However, he recognized that such a model would violate one of the predictions of classical physics,namely, that an accelerated electron (even centripetally accelerated) would emit a continuous spectrum ofradiation as it loses energy and spirals into the nucleus. Clearly this does not happen.

To remove the shortcomings of Rutherford’s model Bohr put forward the following postulates:

1. The electron can revolve around the nucleus only in those orbits in which their angular momentum(l) about the nucleus is an integral multiple of h/2π, where h is Planck’s universal constant. If themass of the electron be m and it is revolving with velocity v in an orbit of radius r , then its angularmomentum will be mvr . According to this Bohr’s postulate, we have

l = mvr = nh

2π, n = 1,2,3, ...

where n is an integer and is called the ‘principal quantum number’ of the orbit. This postulate is calledBohr’s quantization rule.Thus, according to Bohr’s atomic model, electrons can revolve only in certain discrete orbits of‘definite’ radii, not in all. These are called stable orbits.

2. While revolving in stable orbits, the electrons do not radiate energy as expected from Maxwell’s law.Hence, the atom remains stable and is said to exist in a stationary states.

3. The energy of the atom has a definite value in a given stationary state. Bohr assumed that the hydrogenatom can emit or absorb radiation only when the atom changes from one of its stationary states toanother. The energy of the emitted (or absorbed) photon is equal to the difference in energy betweenthese two states. When the atom receives energy from outside, then one (or more) of its outer electronleaves its orbit and goes to some higher orbit. This state of the atom is called ‘excited state’Theelectron in the higher orbit stays only for 10−8 second and returns back to a lower orbit. Whilereturning back, the electron radiates energy in the form of electromagnetic waves.If the electron jumps from an orbit of higher energy E2 to an orbit of lower energy E1, it emits aphoton of energy E2 − E1. The wavelength (or frequency) of the emitted radiation is given by theEinstein-Planck equation

E2 − E1 = hν =hcλ

This postulate ties together two new ideas (the photon hypothesis and energy quantization) with one familiarold idea (the conservation of energy).

4.1 Energy of a Hydrogen atom in different states

Let us now use the above postulates to find the allowed energies of the Hydrogen atom for different allowedorbits of the electron. Let the electron moves in a circle of radius r with velocity v around the nucleus. TheCoulomb attraction between the nucleus and the electron is given by

F =e2

4πϵ0r2

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If m is the mass of the electron, then from Newton’s law,

e2

4πϵ0r2 =mv2

r(1)

as the centripetal acceleration is provided by the Coulomb force. Also, from Bohr’s quantization rule, theangular momentum of the electron is

l = mvr = nh

2π(2)

where n is a positive integer.

Eliminating r from Equation (1) and Equation (2), we get

v =e2

2ϵ0hn(3)

Substituting this in Equation (2), we get the expression for the radius of the nth orbit

r =n2h2ϵ0

πme2 (4)

We see that the allowed radii are proportional to n2. For each value of n, we have an allowed radii. Forn = 1 and putting values of all the constants, we will get the radius of the first orbit (smallest orbit), i.e.r1 = 53 picometre (1 picometre = 10−12 m). This length is called the Bohr radius and is a convenientunit for measuring lengths in atomic physics. It is generally denoted by the symbol a0. We can rewriteEquation (4) in terms of a0 as

rn = n2a0

.

Using Equation (3), the kinetic energy of the electron in the nth orbit is given by

K =12

mv2 =me4

8ϵ02h2n2 (5)

The potential energy of the hydrogen atom is given by

V = − e2

4πϵ0r= − me4

4ϵ02h2n2 (6)

We have taken the potential energy to be zero when the nucleus and the electron are widely separated. Hence,the total energy of the atom is

E = K + V = − me4

8ϵ02h2n2 (7)

The energy is negative because the atom is in a bound state; that is, work must be done by some externalagent to pull it apart (The potential energy, which is zero at infinite separation of the proton and electron, is

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negative and larger in magnitude than the kinetic energy.) In the same way, the Earth-Sun system is a boundstate; work must be done by an external agent to tear this system apart against the gravitational force thatholds it together.

Equation (3) through Equation (7) give various parameters of the atom when the electron is in the nth orbit.Putting values of all the constants in Equation (7) and n = 1, E1 = −13.06 eV . This is the energy when theelectron revolves in the smallest allowed orbit r = a0, i.e. the one with radius 0.053 nm. Thus, Equation (7)can be written as

En =E1

n2 = −13.6 eVn2 (8)

The energy in the state n = 2 is E1 = −3.4 eV . In the state n = 3, it is E2 = −1.5 eV , etc. The lowest energycorresponds to the orbit of smallest radius. Note that the energy is negative and hence a larger magnitudemeans lower energy. The zero of energy corresponds to the state where the electron and the nucleus arewidely separated and the electron is no more bound to the nucleus. The state of an atom with the lowestenergy is called its ground state. The states with higher energies are called excited states. Thus, the energyof a hydrogen atom in the ground state is −13.6 eV and in the first excited state −3.4 eV .

Also, Equation (7) may be rewritten as

En = − me4

8ϵ02h2n2 = −(

me4

8ϵ02h3c

)hcn2 = −Rhc

n2 (9)

where R =(

me4

8ϵ02h3c

)is called the Rydberg constant. Putting the values of different constants, the Rydberg

constant R comes out to be 1.0973 × 107 m−1.

5 Excitation Energy

The energy needed to take the atom from its ground state to an excited state is called the excitation energy ofthat excited state. The hydrogen atom in ground state needs 10.2 eV to go into the first excited state. Thus,the excitation energy of hydrogen atom in the first excited state is 10.2 eV .

6 Excitation Potential

Generally, the atoms are excited or ionized by colliding them with electrons accelerated by high potentials.But the excitation or ionization of the atom is possible only when the energy of the colliding electron is atleast equal to that required for excitation or ionization of the atom.

The minimum accelerating potential required to energize an electron, which on collision, can excite an atomis called the excitation potential of that atom. Thus, the excitation potential of hydrogen atom in first excitedstate is 10.2 eV .

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7 Ionization Energy

The minimum energy needed to remove an electron from an atom in its ground state is called its ionizationenergy. Thus, in case of hydrogen atom, the ionization energy is 13.6 eV since the ground state of thehydrogen atom is −13.6 eV .

8 Ionization Potential

The minimum accelerating potential required to energize an electron, which on collision, can ionize an atomis called the ionization potential of that atom. Thus, the ionization potential of hydrogen atom is 13.6 V .

9 Structure of solids

The solids may be broadly classified as crystalline and non-crystalline (or amorphous) depending upon thearrangement of atoms or molecules.

The crystalline state of solids is characterized by regular or periodic arrangement of atoms or molecules.Most of the solids are crystalline in nature as the crystalline state is a low energy state and is, therefore,preferred by most of the solids. The crystalline solids may be further subdivided into single crystals andpolycrystalline solids. In single crystals, the periodicity of atoms extends throughout the material as thecase of diamond, quartz, mica, rock salt, calcite, etc. A polycrystalline material is an aggregate of a numberof small crystallites with random orientations separated by well-defined boundaries. The small crystallitesare known as grains and the boundaries are grain boundaries. Most of the metals and ceramics exhibitpolycrystalline structure.

The non-crystalline or amorphous solids are characterized by the completely random arrangement of atomsor molecules. The periodicity, if at all present, extends up to a distance of a few atomic diameters only. Inother words, these solids exhibit short range order. The common examples are glass, plastics and rubbers.

10 Idea of Lattice

An arrangement/array of infinite number of imaginary points in three-dimensional space with each pointhaving identical surroundings is known as point lattice or space lattice. However, it is customary todescribe the crystal structure in terms of atoms rather than points. Thus, a crystal structure is formedonly when a group of atoms or molecules is attached identically to each lattice point, This group of atomsor molecules is called basis, the basis is identical in composition, arrangement and orientation which isrepeated periodically in space to form the crystal structure. Basis acts as a building unit for the completecrystal structure. Mathematically,

Space lattice + Basis = Crystal Structure

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11 Unit cell

It is defined as the smallest unit of the lattice which, on continuous repetition, generates the complete lattice.The choice of unit cell is not unique. According to the requirement of the case, either a primitive or anon-primitive cell can be selected as a unit cell. A primitive unit cell is a minimum/smallest volume celland has only one lattice point in it. In a primitive cell, lattice points are located only at the corners, whileeach corner of the cell is common to eight neighboring unit cells and the contribution towards the unit cellper corner is only one-eight. Since the cell has eight corners, therefore, the number of lattice point per unitcell is only one. A non-primitive cell may have the lattice points at the corners as well as at other locationsboth inside and on the surface of the cell. Hence, effective number of lattice points in a non-primitive cellis greater than one.

12 Packing factor/efficiency/fraction

It is defined as the ratio of the volume occupied by the atoms present in a unit cell to the total volume of theunit cell. Thus,

Packing f raction ( f ) = Volume o f the atoms (in the unit cell)Volume o f the unit cell

While calculating packing factors, atoms can be regarded as rigid spheres of definite radii. On the basisof packing crystalline solids may be classified as (a) Close-packed structures (b) Loose-packed structures.Hexagonal close-packed (hep) structure and Face-centered cubic (fcc) structure are examples of Close-packed structures. Simple cubic (sc) structures and Body-centered cubic (bcc) structure are examplesLoose-packed structures.

12.1 Simple cubic (sc) structure

The conventional unit cell is primitive. Coordination number of each atom is 6, i.e. each atom is surroundedby 6 similar and equal sized neighbors.

Figure 1: Unit cell of a Simple cubic structure

As the unit cell is primitive the atoms are located only atthe corners and touch one another along the cube edges. Eachcorner of the cell is common to eight neighboring unit cells andthe contribution towards the unit cell per corner is only one-eight. The effective number of atoms per unit cell is (8× 1

8 ) = 1.

Let the length of the side of the cube be a and radius of spherebe r . Hence, we can see from the Figure 1, a = 2r .

Volume of atoms in one unit cell = (8 × 18 ) ×

43π r3

Volume of the unit cell a3 = (2R)3 = 8r3

Packing fraction (f)= 8× 18×

43 π r3

8r3 = π6 = 0.52 or 52%

Only Polonium exhibits this type of structure at room temper-ature.

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12.2 Body centered cubic (bcc) structure

The conventional unit cell is non-primitive. Coordination number ofeach atom is 8. The atoms are located at 8 corners and at the centerof body. The atoms touch one another along the body diagonal. TheFigure 2 shows a bcc unit cell. The effective number of atoms per unitcell is (8 × 1

8 + 1) = 2.Let the length of the side of the cube be a and radius of sphere be R.Hence, we can see from the Figure 2,

√3a = 4R.

Figure 2: bcc unit cell

Volume of atoms within the unit cell = 2 × 43π R3

Volume of the unit cell a3 = ( 4R√3)3 = 64R3

3√

3

Packing fraction (f)= 2× 43 π R3

64R33√

3

=√

3π8 = 0.68 or 68%

The examples of bcc structure are Na, K, Mo, W, etc.

12.3 Face centered cubic (fcc) structure

The conventional unit cell is non-primitive. Coordination number of each atom is 12. The atoms are locatedat 8 corners and at the faces of the body. The effective number of atoms per unit cell is (8 × 1

8 + 6 × 12 ) = 4.

The atoms touch one another along the face diagonals as shown in Figure 3a and Figure 3b. From Figure 3bwe can see that √

2a = 4R

.

(a)(b)

Figure 3

Volume of atoms within the unit cell = 4 × 43π R3 Volume of the unit cell a3 = ( 4R√

2)3 = 64R3

2√

2

Packing fraction (f) = 4× 43 π R3

64R32√

2

= π

3√

2= 0.74 or 74%

The examples of fcc structure are Ag, Au, Al, etc.

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13 Formation of bands in solids

While studying Bohr’s model of hydrogen atom we have dealt with single isolated atom in which an electroninteracts only with the nucleus. Also, in an isolated atom the electrons are fairly well localized to anindividual atom. As a result, an isolated hydrogen atom had sharply defined energy levels. However, in caseof atoms having more than one electron, the electrons not only interact with the nucleus but with the otherelectrons of their “home” atom as well. In a solid, the atoms are so close together that the electron energylevels in one atom are affected by the presence of the other atoms. This has the effect of changing the sharpenergy levels of an isolated atom into broad energy bands. The bands are in reality collections of energylevels that are so numerous and so close together that we can regard them continuous.

For instance, the distance between nearest neighbours in a copper lattice is 0.26 nm. Consider, however, twocopper atoms separated by a much greater distance − say, 50 nm − so that we may describe them “isolated”.

Now let us bring the two atoms closer together so that an outer electron in either atom can be influenced,however slightly, by forces exerted on it by the other atom. The individual energy levels of each atomwill split into two closely spaced energy levels. Because the overlap is greater for the outer electrons, theenergy splitting will be greater for them than for the inner electrons. If we have three interacting atoms, theindividual levels of the isolated atoms become three closely spaced levels. Similarly, if we bring N copperatoms together to form a copper solid, each level of the isolated atom becomes N closely spaced levels ofthe solid. Thus, the energy levels become energy bands in solids (Figure 4a).

(a) (b)

Figure 4: As atoms are brought together to form a solid, the levels of the isolated atoms split, eventually forming bandsof closely spaced energy levels (left). An idealized representation of the energy bands for metals (i.e. conductors)semiconductors and insulators (right).

In order to study any property (in our case it is electrical conductivity) of a solid, we have to deal withcollection of atoms. The electrons that contribute most to the electrical conductivity in a solid do not “belongto” any particular atom; they are shared among all the atoms in the material. As a consequence, we may bedealing with very large numbers of electrons.

An isolated copper atom has 29 electrons. In solid copper, 28 of these are held close to their lattice sitesby electromagnetic forces and are not free to move throughout the volume of the solid. The remainingelectron is free to so move and, if we apply an emf between the ends of a copper wire, it is these “conductionelectrons” (one per atom) that constitute the current that is set up in the wire.

The conductivity (σ) is a property of the material, not of any particular sample of the material. A largevalue of conductivity indicates that the material is a good conductor of electric current. It is more commonto find materials characterized by their resistivity (ρ), which is the inverse of the conductivity (ρ = 1/σ). A

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(a) (b)

Figure 5: Resistivity chart for conductors, semiconductors and insulators (left). Conductivity chart for conductors,semiconductors and insulators (right).

perfect conductor would have ρ = 0 (or σ = ∞).

14 Conductors, Insulators and Semiconductors

The energy band structure of conductors, semiconductors and insulators are shown in Figure 4b and Figure 6.At an absolute temperature T , electrons exchange energy of the order of kT , where k is the Boltzmannconstant, in collisions with other electrons or ions. This is known as thermal energy. At room temperature(300K), kT is about 0.026 eV. Suppose the band gaps are much larger than kT . An electron in a completelyfilled band does not find an empty state with a slightly higher or a slightly lower energy. It, therefore, cannotaccept or donate energy of the order of kT and hence does not take part in processes involving energyexchange. This is the case with inner bands which are completely filled. The outermost electrons whichare in the highest occupied energy band (valence band), may take up this energy ≈ kT if empty states areavailable in the same band or next higher band.

Figure 6: Comparison of band structures of metals, semiconductors, and insulators. Metallic behavior can ariseeither from the presence of a single partially filled band or from two overlapping bands (one full and one empty).

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For conductors, either the valence band is partially filled or the valence band (completely filled) andconduction band (completely empty) overlap with each other, therefore, the energy band gap (or forbiddenenergy gap) Eg ≤ 0 eV for them. Consequently, in both the cases, slightly higher energy empty states areavailable at room temperature and, therefore, large number of electrons are available in conduction bandwhich take part in processes involving energy exchange. However, for insulators, the energy band gap isEg ≥ 3 eV. Due to large value of energy band gap the electrons are not able to go to conduction band fromthe valence band as the thermal energy at room temperature is not sufficient to lift the electron into the emptyband. Carbon in its diamond form is an excellent insulator because its forbidden energy gap is 5.5 eV.

Semiconductors differs from an insulator in that its energy gap is small enough so that thermal excitation ofelectrons across it can occur to some extent at room temperature. This puts electrons into the conductionband and leaves an equal number of vacant states, or holes, in the valence band. In a band that is nearly full,it turns out to be more convenient to analyze its contribution to the electrical conduction in terms of themotion of holes, which behave like positively charged particles. Silicon is a typical semiconductor. It hasthe same crystal structure as diamond but its band gap (= 1.1 eV) is considerably smaller. It may be notedthat at the absolute zero of temperature, where thermal agitation is absent, all semiconductors behave likeinsulators. Another common semiconductor is Germanium for which the band gap is = 0.68 eV.

15 Doped Semiconductors

Pure semiconductors (i.e. without any impurity) are called intrinsic semiconductors. In an intrinsicsemiconductor, only a small fraction of the valence electrons are able to reach the conduction band. Whenevera valence electron is shifted to conduction band, a hole is created (in the valence band, obviously). Thus,in a pure semiconductor, the number of conduction electrons equals the number of holes. The conductionproperties of a semiconductor can be drastically changed by diffusing a small amount of impurity in it. Theprocess of diffusing an impurity is known as doping. The doped semiconductors are also called extrinsicsemiconductors.

15.1 n−type and p−type semiconductors

n-type semiconductors are formed when group V elements like phosphorus, antimony, bismuth etc. aredoped to a pure semiconductor crystal. These are called so because doping these elements will cause thepresence of an additional unbounded electron in the valence band of the atom which is free to move into theconduction band. Figure 7a represents the doping of phosphorus element into a pure silicon material. Letus discuss it in more detail.

Suppose a small amount of phosphorous (Z=15) is diffused into a silicon crystal. Each phosphorous atomhas five outer electrons in the valence band. Some of the phosphorous atoms displace the silicon atoms andoccupy their place. A silicon atom has four electrons locked in covalent bonds with neighbouring four siliconatoms. Phosphorous comes in with five valence electrons. Four electrons are shared with the neighbouringfour silicon atoms. The fifth one moves around the phosphorous ion. The energy of this extra electron ismuch higher than the valence electrons locked in covalent bonds. In fact, the energy levels for these extraelectrons − known as impurity levels (donor levels in this case) − are only slightly below the conductionband. This small gap is easily covered by the electrons during thermal collisions and hence a large fractionof them are found in the conduction band. Figure 7a and Figure 8 a shows qualitatively the situation in sucha doped semiconductor. When a phosphorous atom is substituted for a silicon atom, an extra electron ismade available for conduction. Thus, the number of conduction electron increases due to the introductionof a pentavalent impurity in silicon. The conduction properties are, therefore, very sensitive to the amount

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(a) (b)

Figure 7: Silicon doped with pentavalent impurity, Phosphorous, forms n-type semiconductor (left). Silicon dopedwith trivalent impurity, Aluminium, forms p-type semiconductor (right).

Figure 8: In n-type semiconductor donor level is created near the conduction band (left). In p-type semiconductoracceptor level is created near the valence band (right).

of the impurity. Such impurities, which donate electrons for conduction, are called donor impurities. Asthe number of negative charge carriers is much larger than the number of positive charge carriers, thesesemiconductors are called n-type semiconductors.

What happens if a trivalent impurity − such as Aluminium − is doped into Silicon? p-type semiconductorsare formed when a pure or intrinsic semiconductor is doped with group III elements of periodic table such asboron, gallium, aluminium etc. These are called so because doping these elements will lead to the creationof a vacancy of an electron (or hole) in the valence band of the atom. It is the movement of the holes whichis mainly responsible for the conduction in the p-type semiconductor to take place. Hence, in this case, themajority charge carriers are holes rather than electrons. Figure 7b represents doping of aluminium to a puresilicon material.

Silicon atom with four valence electrons is substituted by an aluminium atom with three valence electrons.These three electrons are used to form covalent bonds with the neighbouring three silicon atoms but thebond with the fourth neighbour is not complete. A hole is then created in the valence band. Impurity levels(acceptor levels in this case) are created a little above the valence band. The valence electrons can crossover to these levels leaving behind the holes, which are responsible for conduction. As the energy gapbetween the valence band and the impurity levels is comparable to kT , large number of holes are created.

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The number of holes in such a doped semiconductor is much larger than the number of conduction electrons.As the majority charge carriers are holes, i.e. positive charges, these semiconductors are called p-typesemiconductors. The impurity of this kind creates new levels which can accept valence electrons, hencethese impurities are called acceptor impurities.

16 Superconductors

Even the best conductors (copper, silver, and gold) show a small but definitely nonzero resistance to the flowof electricity. As we reduce the temperature of a conductor, the resistivity grows smaller. What happensas we approach the absolute zero of the temperature scale? Certain metals and alloys exhibit almost zeroelectrical resistivity (i.e. infinite conductivity) when they are cooled to sufficiently low temperatures, whichmeans electric charge can flow through these materials with no resistance at all. Thus, the phenomenon ofdisappearance of electrical resistance of material below a certain temperature is called superconductivity,and the materials under these conditions are called superconductors.

(a) (b)

Figure 9: Comparison of the dependence of resistivity on temperature for a normal conductor and a superconductor.The resistivity of a normal conductor falls gradually with decreasing temperature. In superconductors, the resistivitydrops suddenly to zero at the critical temperature, Tc (left). On the contrary, for semiconductors, the resistivitydecreases with the increasing temperature (right).

In 1911, the Dutch physicist Kammerlingh Onnes discovered that at the temperature of 4.2K, the resistanceof mercury suddenly vanishes. In other words, it loses its resistivity and became a perfect conductor,called a superconductor. This was not a gradual change, but a sudden transition, as indicated by Figure 9a.The resistivity of a superconductor is not only very small; it is zero! If a current is established in asuperconducting material, it should persist forever, even with no electric field present. The temperatureat which a material becomes superconducting is called its critical temperature Tc. Superconductivity hasbeen observed in 27 elements [such as Al (Tc = 1.19K), Pb (Tc = 7.22K)] and in numerous compounds[such as MgB2 (Tc = 39K)]. Some materials can be relatively poor conductors at room temperature but canbe superconductors at low temperatures. However, superconductivity has not been observed for the bestmetallic conductors (copper, silver, and gold).

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17 X−rays

X−rays was discovered by the German physicist Wilhelm Roentgen in 1895 for which he was awarded thefirst Nobel Prize in Physics in 1901.

17.1 Production mechanism

When highly energetic electrons are made to strike a metal target, electromagnetic radiations are produced.A large part of this radiation has wavelength in the range of about 0.01−10 nm and is known as X−rays.The important characteristics of X−rays are that they travel in straight lines, unaffected by electric andmagnetic fields, pass readily through opaque materials, and expose photographic plates. The faster theoriginal electrons, the more penetrating the resulting X−rays, and the greater the number of electrons, thegreater the intensity of the X−ray beam.

X−ray tube is shown in Figure 10a. A heated cathode supplies electrons by thermionic emission. The highpotential difference V maintained between the cathode and a metallic target accelerates the electrons towardthe latter. The face of the target is at an angle relative to the electron beam, and the X−rays that leave thetarget pass through the side of the tube. The tube is evacuated to permit the electrons to get to the targetunimpeded.

(a) (b)

Figure 10: An X−ray tube or Coolidge tube. The higher the accelerating voltage V, the faster the electrons and theshorter the wavelengths of the X−rays (left). X−ray spectrum of molybdenum at 35 keV accelerating potential (right).

17.2 Continuous and characteristic X−rays

If the X−rays coming from a Coolidge tube are examined for the wavelengths present, and the intensity ofdifferent wavelength components are measured, we obtain a plot of the nature shown in Figure 10b. We seethat there is a minimum wavelength below which no X−ray is emitted. This is called the cutoff wavelengthor the threshold wavelength. The X−rays emitted can be clearly divided in two categories. Intensitypeaks occur that indicate the enhanced production of X−rays at certain wavelengths marked by Kα, Kβ in

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Figure 10b. These X−rays are known as characteristic X−rays. These peaks occur at specific wavelengthsfor each target material and originate in rearrangements of the electron structures of the target atoms afterhaving been disturbed by the bombarding electrons. At other wavelengths the intensity varies gradually andthese X−rays are called continuous X−rays. Let us examine the origin of these two types of X−rays.

Suppose, the potential difference applied between the target and cathode is V and the electrons are emittedwith negligible speed. The electrons are accelerated in their journey from the cathode to the target. Thekinetic energy of an electron when it hits the target is

K = eV (10)

As the electron enters into the target material, it readily loses its kinetic energy and is brought to rest insidethe metal. The electron before finally being stopped, makes several collisions with the atoms in the target.At each collision, one of the following two processes may occur:

1. The kinetic energy of the electron is reduced. A part of this lost kinetic energy is converted intoa photon of electromagnetic radiation and the remaining part increases the kinetic energy of thecolliding particle of the target. The energy received by the colliding particle goes into heating thetarget. The electron makes another collision with its reduced energy.

2. The electron knocks out an inner electron of the atom with which it collides.

The fraction of kinetic energy appearing as the energy of a photon varies from collision to collision. In acertain collision, the electron may lose its entire kinetic energy to bring out a photon or it may not createa photon at all. Thus, the energy of the photon created can be anything between 0 and eV depending onhow much energy has already been lost to the target and what fraction of the available energy is convertedinto the photon. The maximum energy of such a photon can be E = eV when the electron converts all itskinetic energy into a photon in the first encounter itself. Most of the electrons that strike the target undergonumerous glancing collisions, with their energy going simply into heat (This is why the targets in X−raytubes are made from high-melting point metals such as tungsten, and a means of cooling the target is usuallyemployed). A few electrons, though, lose most or all of their energy in single collisions with target atoms.This is the energy that becomes X−rays.

17.3 Duane and Hunt rule

Continuous X−rays production represents an inverse photoelectric effect. Instead of photon energy beingtransformed into electron kinetic energy, electron kinetic energy is being transformed into photon energy.Suppose the bombarding electron loses all its kinetic energy K = eV in its first collision and produces acorresponding single X−ray photon of energy hνmax . Hence,

eV = hνmax =hcλmin

λmin =hceV=

1.24 × 10−6

VV .m (11)

Equation (11) is known as the Duane-Hunt formula. We see that the cutoff wavelength λmin depends onlyon the accelerating voltage V applied between the target and the cathode. It does not depend on the materialof the target. This explains the origin of continuous X−rays and the cutoff wavelength.

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We shall now discuss what happens if the electron knocks out an inner electron from the atom with which itcollides. As we know that the energy of an electron in any atom primarily depends on the principal quantumnumber n. The two electrons corresponding to n = 1 are said to be in K shell, those corresponding to n = 2are in L shell, etc. Suppose, the incident electron knocks out an electron from the K shell. This will createa vacancy in the K shell in the sense that now there is only one electron with n = 1, whereas two could beaccommodated by Pauli exclusion principle. An electron from a higher energy state may make a transitionto this vacant state. When such a transition takes place, the difference of energy ∆E is converted into anX−ray photon of wavelength λ = hc/∆E . X−rays emitted due to electronic transition from a higher energystate to a vacancy created in the K shell are called K X−rays. If an electron from the L shell (i.e. with n = 2)makes transition to the vacant state in the K shell, the X-ray emitted is called Kα X−ray. If an electron fromthe M shell (i.e. with n = 3) makes transition to the K shell, a Kβ X−ray is emitted. It should be noted thattransitions that involve the inner electrons in an atom are what give rise to characteristic X−rays because ofthe high photon energies involved (Inner electrons are more tightly bound to the nucleus as compared to theouter electrons).

17.4 Applications of X−rays

• Medical: Exploiting its penetrating power it is used extensively to detect diseases inside the body.It passes quite freely through the flesh but is stopped by the bones. So it can photograph the bonesinside the body on a photographic film. Such a photograph is called a radiograph. This is used todetect bone fractures. Chest radiographs are used to study diseases in lungs. Dentists also use X−raysto study teeth-decay. It is also used in cancer therapy.

• Industry and Research: X−rays are used in industry and material science research. It can detectstructural defects, fault of joints, welding, etc. Atomic and molecular structures of crystals are studiedby researchers by diffracting X−rays. This scientific technique is known as X−ray Crystallography.

• Security: X−ray machines are used at airports, railway stations and other public places to inspectsuitcases, boxes, etc. without opening them.

• Apart from numerous benefits of X−rays, it also has its flip side. Random and excess exposure toX−rays may induce diseases. It has a damaging effect on the living cells of a body which may lead tocell-death. High exposure for a long period (say for years) may lead to cancer or genetic defects.

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18 Problems for Unit-III

1. The energy of hydrogen atom in its nth orbit is En = −(13.6/n2) eV. Calculate the energy required forsending electron from first orbit to second orbit. (Ans: 10.2 eV)

2. In Bohr’s model of hydrogen atom, the radius of the first orbit of the electron is 0.53 . What will bethe radius of the third orbit ? (Ans: 4.77 )

3. Find the maximum wavelength λ0 of light which can ionize a hydrogen atom in its ground state.(Hint: The ground state energy of hydrogen atom is −13.6 eV.) (Ans: 91.3 nm)

4. What is the wavelength of the radiation emitted when the electron in a hydrogen atom jumps fromn = ∞ to n = 2 ? (Hint: The energy of n = ∞ state is zero.) (Ans: 365 nm)

5. Calculate the highest wavelength of the radiation emitted when hydrogen atom make a transition froma higher state to n = 2 state. (Hint: The highest wavelength corresponds to the lowest energy oftransition. This will be the case for the transition n = 3 to n = 2.) (Ans: 654 nm)

6. The energy of an electron in an excited hydrogen atom is -3.4 eV. Calculate the angular momentum ofthe electron according to Bohr’s theory. (Hint: The given energy corresponds to n = 2.) (Ans: h/π)

7. Find the maximum frequency of the X-rays emitted by an X-ray tube operating at 30 kV. (Ans: 7.2 ×1018 Hz)

8. An X-ray tube is operated at 50 kV. Calculate the energy of the electron striking the anode and theminimum wavelength of the emitted X-rays. (Ans: 8 × 10−15 joule, 0.25 )

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UNIT-III by Dr. Tahir Hussain - Aligarh Muslim University

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Transcript of UNIT-III by Dr. Tahir Hussain - Aligarh Muslim University