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1013/Engg/FE/Pre Pap/2013/Phy_Soln 1

x

y

z −a/2

−c

(2 3 1)

c b/3 x

y

z (1 1 0) x

y

z

b/2

−c

(1 2 1) a c

Vidyalankar F.E. Sem. I

Applied Physics - I Prelim Question Paper Solution

(1 2 1) (2 3 1) (1 1 0)

SPACE LATTICE The lattice is purely a mathematical

abstraction. Space lattice is a regular, periodic, repeated three dimensional array of points.

Space lattice can also be defined as the set of those points selected in space such that irrespective of the reference, neighbourhood or environment remains the same.

This concept was introduced by Bravais therefore sometimes it is called Bravais lattice.

BASIS If on every lattice point, we consider an

unit or assembly of atoms which are identical in composition, arrangement and orientation then it is called basis.

Crystal structure = Lattice + Basis If basis is single atom, it is called

monoatomic, when there are two atoms, it is diatomic and so on.

UNIT CELL The smallest cell, which when repeated

throughout space is called primitive cell or unit cell.

A primitive cell is specified by axes a, b and c called primitive axes.

The magnitudes a, b and c are called lattice parameters.

1. (a)

Fig. 1 : Lattic points

Fig. 2 : Single crystal structure

Fig. 3 : Space lattice and unit cell in 2–dimension

1. (b)

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Vidyalankar : F.E. – Physics I


As shown in figure 3, a parallelogram ABCD formed by primitive vectors a and b , forms a primitive cell which is called unit cell.

The choice of unit cell is not unique. It may be ABCD or EFGH as shown in figure 3.

In 3-dimensional case, the unit cell is a parallelopipped formed by the basis vectors a, b and c and angle , and . Together they are called lattice parameters.

In general, a unit cell is defined as the smallest volume of the entire crystal constructed by translational repetitions in 3-dimension and which represents all the characteristics of the crystal.

Properties of Ultrasonic Waves are as follows

The speed of propagation of ultrasonic waves increases with increase in frequency. Since wavelength is very small exhibits negligible diffraction. They can travel over long distances as a highly directional beam without

appreciable loss of energy. Highly energetic owing to the high frequencies involved. The produce cavitation effect in liquids. All the above mentioned properties put them in a special category. Hence

ultrasonic waves are widely used in marine, metallurgy, medical, non destructive testing and so on. HOLES

If a trivalent impurity (Group III) is added to a pure semiconductor, it becomes ptype extrinsic semiconductor. The impurity added is called as acceptor impurity.

As a result a vacancy is left in the bonding. This vacancy is not a hole. The introduction of impurity atom does not disturb the environment and the

vacancy arising due to the non-formation of bond is not a hole. When an electron from a neighbouring bond acquires energy and jumps into

this vacancy, it leaves behind a positively charged environment in the broken bond. Therefore a hole is generated there.

According to Fermi-Dirac statistics, we make use of function f(E) which determines

the carrier occupancy of the energy. In otherwords f(E) governes the distribution of electrons among the energy levels as a function of temperature. It is given by

f(E) =

F

1E E

1 expKT

where f(E) = Probability that a particular energy level E is occupied by an electron EF = Fermi-energy K = Boltzman constant T = Temperature in Kelvin


1. (c)

1. (d)

1. (e) Vidyala

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Prelim Question Paper Solution


CONDUCTORS Conducting materials are those in which plenty

of free electrons are available for electrical conduction. In terms of energy bands.

i) Electric conductors are those which have overlapping valence band and conduction band.

ii) There is not physical distinctions between the two bands and hence a large number of conduction electrons are available.

iii) In the absence of forbidden energy gap in good conductors, there is no structure to establish holes. The total current in such conductors is simply a flow of electrons. The resistivity of the conductors is of the order of 106 -cm.

SEMICONDUCTORS A semiconductor material is one whose

electrical properties are between those of insulators and conductors e.g., Ge and Si in terms of energy bands.

i) Semiconductors are those materials which have an almost empty conduction band and a filled valence band with a narrow energy gap (Ge = 0.7 eV, Si = 1.1 eV)

iii) At D K there are no electrons in the conduction band and valence band is completely filled.

With rise in temperature width of the forbidden energy gap is decreased so that some of the electrons are liberated into the conduction band i.e., conductivity of semiconductor increases with temperature. The electrons departing into the conduction band leave behind positive holes in the valence band. Hence semiconductor current is the sum of electron and hole current flowing in the opposite direction.

iii) There resistivity varies from 1012 to 1010 -cm.

INSULATORS Insulators are those materials in which valance

electrons are bound very tightly to their parent atoms. Hence they require very large electric field to remove from the attraction of the nuclei. In terms of energy bands :

i) They have a full valance band. ii) They have an empty conduction band. iii) They have a large energy gap (Eg 5 10 eV). iv) At all ordinary temperatures the probability of electrons from full valance band

gaining sufficient energy so as to surmount energy gap and becoming available for conduction in the conduction band is slight.

v) Their resistivity is of the order of 109 -cm. In the case of materials like glass the valance band is full at 0 K and energy gap

is of the order of 10 eV. Even in the presence of electric field electrons, do not move from VB to CB. Increase in temperature enables some electrons to go to conduction band. This explains why certain materials which are insulators become conductors at high temperatures.

Ban

d E

nerg

y

V.B.

C.B.

Eg

Ban

d E

nerg

y

V.B.

C.B.

Eg

Ban

d E

nerg

y

C.B.

V.B.

2. (a)

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POINT DEFECTS All the atoms in a solid posses vibrational energy and at all temperature

above absolute zero, there will be a finite number of atoms which have sufficient energy to break the bonds which hold them in their equilibrium position.

Once the atoms become free from their lattice sites, they give rise to point defects.

Also due to the presence of impurity atoms point defects are likely to come in the crystal.

Types of point defect : (a) Vacancy (b) Interstitial (c) Substitutional impurities (d) Interstitial impurities (e) Shottky defect (f) Frankel defect (a) Vacancy : Vacancy is produced due to the removal of an atom from its

regular position in the lattice.

Fig. 1 : Vacancy

The removed atom does not vanish. It travels to the surface of the material.

For low concentration of vacancies, a relation is n = Nexp (–EV/kT) … (1)

Where n = Number of vacancies N = Total number of atoms T = Temperature in (°K) Ev = Average energy required to create a vacancy (b) Interstitial : An extra atom of the same

type is fitted into the void between the regularly occupied sites.

Since in general the size of atom is larger

than the void into which it is fitted, so the energy required for interstitial formation is higher than that of vacancy formation.

(c) Substitutional Impurities : In this, a

foreign atom is found occupying a regular site in a crystal lattice.

Fig. 2 : Interstitial

Fig. 3 : Substitutional impurity

2. (b)

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(d) Interstitial Impurities : Here a foreign atom is found at a non regular site.

(e) Shottky Defect : The point imperfection

in ionic crystals occurs when a negative ion vacancy is associated with a positive ion vacancy. It is therefore a localised vacancy pair of positive and negative ions. This type of defect maintains the crystal electrically neutral, it is called schottky defect.

For ionic crystal numbers of pair ion production is n = Ne–Ep/2kT …(2)

where N = Number of lattice site k = Boltzman constant

Ep = Energy required to create a pair of ion vacancy inside crystal lattice

T = Temperature in K The following essential features about the good acoustics are :

i) The sound heard must be sufficiently loud in every part of the hall and no echoes should be present.

ii) The total quality of the speech and music must be unchanged i.e., the relative intensities of the several components of a complex sound must be maintained.

iii) For the sake of clarity, the successive syllables spoken must be clear and distinct i.e., there must be no confusion due to overlapping of syllables.

iv) The reverberation should be quite proper i.e., neither too large nor too small. The reverberation time should be 1 to 2 seconds for music and 0.5 to 1 second for speech.

v) There should be no concentration of sound in any part of the hall. vi) The boundaries should be sufficiently sound proof to exclude extraneous

noise. vii) There should be no Echelon effect. viii) There should be resonance within the building. By an acoustically good hall we mean that in which every syllable or musical note

reaches an audible level of loudness at every point of the hall and then quickly dies away to make room for the next syllable or group of notes. The departure from this makes the hall defective acoustically. Following factors affect the architectural acoustics :

Fig. 5 : Schottky defect

Fig. 4 : Interstitial impurity

3. (a)

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Reverberation In a hall, when reverberation is large, there is overlapping of successive

sounds which results in loss of clarity in hearing. On the other hand, if the reverberation is very small, the loudness is inadequate. Thus the time of reverberation for a hall should neither be too large nor too small. It must have a definite value which may be satisfactory both to the speaker as well as to the audience. This preferred value of the time of reverberation is called the optimum reverberation time. A formula for standard time of reverberation was given by W.C. Sabine, which is

T = Σ

0.165V 0.165V

A aS

where A is the total absorption of the half V its volume in cubic metre and S is the surface are in square metre.

Experimentally it is observed that the time of reverberation depends upon the size of the hall, loudness of sound, and on the kind of the music for which the hall is used. For a frequency of 512 c.p.s., the best time of reverberation lie between 1 and 1.5 sec. for small hall and upto 2-3 seconds for large ones. The reverberation can be controlled by the following factors :

i) By providing windows and ventilators which can be opened and closed to make the value of the time of reverberation optimum.

ii) Decorating the walls by pictures and maps. iii) Using heavy curtains with folds. iv) The walls are lined with absorbent material such as felt, celotex, fibre board,

glass wool etc. v) Having full capacity of audience. vi) By covering the floor with carpets. vii) By providing acoustic tiles. Adequate loudness With great absorption, no doubt, the time of reverberation will be smaller

which will minimize the chances of confusion, between the different syllables but the intensity of sound is weakened and may go below the level of intelligibility of hearing Sufficient loudness in every portion of the hall is an important factor for satisfactory hearing. The loudness may be increased by :

i) Using large sounding boards behind the speaker and facing the audience. Large polished wooden reflecting surfaces immediately above the speaker are also helpful.

ii) Low ceilings are also of great help in reflecting the sound energy towards the audience.

iii) By providing additional sound energy with the help of equipments like loudspeakers.

Focusing due to walls and ceilings If there are focusing surfaces (viz, concave, spherical, cylindrical or

parabolic) on the walls or ceiling or the floor of the hall, they produce concentration of sound into particular regions while in some other parts no sound reaches at all. In this way there will be regions of silence or of poor

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audibility while there should be a uniform distribution of sound in the hall. If there are extensive reflecting surfaces in the hall, the reflected and direct sound waves may form stationary wave system thus making the sound intensity distribution bad and uneven. For uniform distribution of sound energy in the hall :

i) There should be no curved surfaces. If such surfaces are present, they should be covered with absorbent material.

ii) Ceiling should be low. iii) A paraboloidal reflected surface arranged with the speaker at the focus is

also helpful in sending a uniform reflected beam of sound in the hall. Absence of echoes An echo is heard when direct and reflected sound waves coming from the

same source reach the listener with a time interval of about 1/7 second. The reflected sound arriving earlier than this helps in raising the loudness while those arriving later produce echoes and cause confusion. This should be avoided or weakened as far as possible by absorption. Echoes may be avoided by covering the long distant walls and high ceiling with absorbent material.

Freedom from resonance Sometimes the window-panes, sections of the wooden portions and walls

lacking in rigidity are thrown in vibrations and they create other sounds. For some note of audio frequency, the frequencies of new sounds may be the same thus resulting in the resonance. Moreover if the frequency of the created sound is not equal to the original sound, at least certain tones of the original music will be reinforced. Due to the interference between original sound and created sound, the original sound is distorted. Thus the intensity of the note is entirely different from the original one. Enclosed air in the hall also causes resonance. Such resonant vibrations should be suitably damped.

Echelon effect A set of railings or any regular spacing of reflected surfaces may produce a

musical note due to the regular succession of echoes of the original sound to the listener. This makes the original sound confused or unintelligible. So this type of surface should be avoided.

Extraneous noise and Sound insulation In a good hall no noise should reach from outside. Noise may be defined as

unwanted sound. The noise may be due to high frequency of sound or intensity of sound or both. The noise produces jarring effect or displeasing effect on the ear. Generally, there are three types of noises which are very troublesome. These are :

(a) Air-borne noise, (b) Structure borne noise and (c) Inside noise. The prevention of the transmission of noise inside or outside the hall is known as

sound insulation. This is also known as sound proofing. The method of sound insulation will depend on the type of noise to be treated. Here we shall discuss the different types of noises and their sound insulation.

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(a) Air-borne noise : The noise which commonly reaches the hall from outside through open windows, doors and ventilators is known as air-born noise. The noise is transmitted through the air. Sound insulation for the reduction of air-borne noise can be achieved by the following methods.

i) By avoiding openings for pipes and ventilators. ii) By allotting proper places for doors and windows. iii) Using double doors and windows with separate frames and having

insulating material between them. iv) By making arrangements for perfectly shutting doors and windows. v) Using heavy glass in doors, windows and ventilators. vi) By providing double wall construction, floating floor construction,

suspended ceiling construction, box type construction, etc. (b) Structure borne noise : The noises which are conveyed through the

structure of the building are known as structural noises. These noises may be caused due to structural vibration due to activity at, around, above or below the structure. The most common sources of this type of sound are footsteps, street traffic, hammering, drilling, operating machinery, moving of furniture, etc. Sound insulation for the reduction of structure-borne noise is done by the following ways i) Breaking the continuity by interposing layers of some acoustical

insulators. ii) Using double walls with air space between them. iii) Using anti-vibrations mounts iv) Soft floor finish (carpet, rubber, etc)

v) By insulating the machinery. Mechanical equipments such as refrigerators, lifts, fans etc. produce vibrations in the structure. These vibrations can be checked by insulating the equipments property.

(c) Inside noise : The noises which are produced inside the hall or rooms in

big offices are called as inside noises. They are produced due to machinery, type writers etc. in the hall. Following methods are used for sound insulation of inside noise : i) The machinery like type-writers etc. should be placed on absorbent

pads. ii) Any engine inside the hall should be fitted on the floor with a layer of

wood or felt between them. iii) The floor should be covered with carpet. iv) The walls, floors and ceilings should be provided with sound

absorbing materials. v) The sound absorbing materials should be mounted on the surfaces

near the source of noise. The acoustical treatment of the hall considerably reduces the noise level in the hall.

For noise reduction and sound proofing, the information regarding sounds

and their loudness is quite essential. Here we give some common sounds with their loudnesses (expressed in dB)

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i) Threshold of audibility 0 dB v) Ordinary Conversation 60 dB ii) Normal breathing 10 dB vi) Heavy traffic 80 dB iii) Whisper 20 dB vii) Thander 110 dB iv) Average house 40 dB viii) Threshold of feeling 120 dB

Given : n = 4 (FCC structure)

A = 63.55 = 8.9 gm/cc N = 6.023 1026 /kg−mole

Formula : a3 = nAN

a =

3nA 1N

Substitution in CGS, =

3

23

4 63.55 18.96.023 10

Lattice constant, a = 3.62 10−8 cm Calculation of diameter of Cu atom

Formula : r = 2 a4

for FCC

=

83.62 10

24

= 1.28 10−8 cm diameter = 2r = 2 1.28 10−8 cm diameter = 2.56 10−8 cm

SPACE LATTICE / BRAVAIS LATTICE The lattice is purely a mathematical abstraction.

Space lattice is a regular, periodic, repeated three dimensional array of points.

Space lattice can also be defined as the set of those points selected in space such that irrespective of the reference, neighbourhood or environment remains the same.

This concept was introduced by Bravais therefore sometimes it is called Bravais lattice.

BASIS If on every lattice point, we consider an

unit or assembly of atoms which are identical in composition, arrangement and orientation then it is called basis.

Crystal structure = Lattice + Basis

If basis is single atom, it is called monoatomic, when there are two atoms, it

is diatomic and so on.

Fig. 1 : Lattice points


3. (b)

4. (a)

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UNIT CELL The smallest cell, which when repeated throughout space is called primitive

cell or unit cell. A primitive cell is specified by axes a, b and c called primitive axes. The magnitudes a, b and c are called lattice parameters.

Fig. 3 : Space lattice and unit Fig. 4 : Unit cell in 3-dimension

cell in 2-dimension. As shown in figure 3, a parallelogram ABCD formed by primitive vectors a

and b , forms a primitive cell which is called unit cell. The choice of unit cell is not unique. It may be ABCD or EFGH as shown in

figure 3. In 3-dimensional case, the unit cell is a parallelopipped formed by the basis

vectors a , b and c and angles , and . Together they are called lattice parameters.

In general, a unit cell is defined as the smallest volume of the entire crystal constructed by translational repetitions in 3-dimension and which represents all the characteristics of the crystal.

The density primarily depends upon the nature of packing, co-ordination number and number of atoms / unit cell.

Now, Density = Mass

Volume =

(Number of atoms/unit cell) (Mass of atom)Volume of unit cell

= 3

n(M / N)

a

where = Density n = Number of atoms/unit cell

MN

= Mass of one atom with M as molecular weight

N = Avogardro's number = 6.023 1026 in SI, if CGS take N = 6.023 1023

a3 = Volume of unit cell

= 3

n(M / N)

a

or a3 = M

nN

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MILLER INDICES The crystal structure may be regarded as made up of an aggregate of a set

of parallel equidistant planes passing through at least one lattice point or a number of lattice points. In a given crystal a plane may be selected in number of ways.

The position of crystal plane can be specified in terms of three integers called 'Miller Indices'. Miller developed a method by which one can find three integers (hkl). This method is universally employed.

The procedure is as follows : i) Find the intercepts of the plane with the crystal axes along the basic

vectors

a, b and c . Let the intercepts be m, n and p respectively. ii) Express m, n and p in terms of the respective basic vectors, as fractional

multiples, we get

m n p

, ,a b c

iii) Take the reciprocals of the three fractions :

i.e., a b c

, ,m n p

iv) Find the LCM of the denominator, by which the above three ratios are multiplied. This operation reduces them to a set of three integers h, k and l. The resultant three integers are called Miller indices of the given plane, denoted by (hkl).

While finding Miller indices of a plane, following points should be kept in mind : i) When a plane is parallel to one of the axes, it is said to have intercept at

and its reciprocal is zero. ii) When the intercept of a plane is on the negative part, the corresponding

Miller index is distinguished by a bar over it. iii) Parallel plane have same Miller indices. iv) Miller indices, in practice do not define a particular plane but a set of

parallel planes. v) A plane passing through the origin is defined in terms of a parallel plane

having non-zero intercept. Conduction in Semiconductors

Intrinsic semiconductors :In this case, current flow is due to the movement of electrons and holes in opposite directions. Even if the number of electrons is equal to the number of holes, hole mobility n is half of electron mobility e.

The total current flow is due to the sum of electrons and hole flow the relation is given by

I = Ie + Ih

= nieVeA + peeVhA

where ni density of free electrons in an intrinsic semiconductor pi density of holes in an intrinsic semiconductor. e charge on an electron A Area of crosssection of a semiconductor ve drift velocity of electrons vh drift velocity of holes

4. (b)

4. (c)

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Since in case of intrinsic semiconductors No. of electrons = No. of holes n = p I = nie (eE + hE) A = nie E A (e + h)

= nie (e + h) V

. A (E = V

. is length of intrinsic semiconductor)

= nie (e + h). AV

R = VI

=

i e hne( )A

R = A

where = i e h

1ne ( )

The electrical conductivity which is the reciprocal of resistivity is given by = nie (e + h)

Current density J = IA

J = i e hne ( ) EA

A = nie (e + h) E

J = E

= JE

Thus, conductivity of semiconductors depends upon two factors i) number of current carriers present per unit volume and ii) the mobility of current carriers.

HALL EFFECT

VH +

+

I I

t

B

V

EH

Ex E

H

F

F

5. (a)

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If a metal or a semiconductor carrying a current ‘I’ is placed in a transverse magnetic field ‘B’, a potential difference is produced in the direction normal to both the current and magnetic field directions. The phenomenon is called Hall effect in honour of E.H. Hall, the physicist who discovered it.

Hall effect measurements showed that it is the negative charge carriers namely electrons that are responsible for electrical conduction in metals. It is the Hall effect measurements again which showed that there exists two types of charge carriers in semiconductors. The importance of Hall effect is that it helps

i) determine the sign of charge carriers, ii) determine the charge carrier concentration and

iii) determine the mobility of charge carriers if conductivity of the material is known.

Let us consider a rectangular plate of a ptype semiconductor. When a potential difference is applied across its ends, a current of ‘I’ flows through it along the xdirection. If holes are the charge carriers in the ptype semiconductors the current is given by

I = p A e vd … (1)

where p is the concentration of holes e is the charge on a hole A is the area of crosssection of the end face vd is the average drift velocity of holes

The current density

Jx = IA

= p e vd … (2)

Any plane perpendicular to the current flow direction is an equipotential surface. Therefore, the potential difference between the front and rear faces F and F is zero.

If a magnetic field B is applied normal to the crystal surface and also to the current flow, a transverse potential difference is produced between faces F and F. It is called Hall voltage, VH. The origin of Hall voltage is as follows :

Before the application of magnetic field holes move in an orderly way parallel to

faces F and F. Upon the application of the magnetic field B, the holes experience a sideway deflection due to the Lorentz force FL (fig. a). The magnitude of the magnetic force is given by

FL = B e vd

Because of this force, holes are deflected towards the front face F and pile up there. Initially the material is electrically neutral everywhere. However as holes

I I

ptype

EH FL I

I

ntype

EH FL

(a) (b) Vidyala

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pile up on the front side, a corresponding equivalent negative charge is left on the rear face F. As a result an electric field is produced across the two faces F and F. The direction of electric field will be from the front to rear face. It is such that it opposes the further filling up of holes on the front face F. A condition at equilibrium will be reached when the force FE due to transverse electric field EH balances the Lorentz force FL. The transverse electric field EH is known as Hall field. Equilibrium state is usually attained in about 1014 sec and after that, the holes flow once again along xdirection parallel to the faces F and F. In the equilibrium condition

FE = FL

FE = e EH = ω

HVe … (3)

where ‘’ is the width of the semiconductor plate. FL = e B vd

From equation (2)

vd = xJ

p e

FL = xBJ

p … (4)

From (3) and (4) we get

ω

HeV = xBJ

p

HV = ω xBJ

pe =

ωBIpe A

If ‘t’ is the thickness of the semiconductor plate, A = wt. and the above equation reduces to

HV = BI

pe t … (5)

Hall field unit current density per unit magnetic induction is called Hall coefficient, RH.

Thus, RH = H

x

E

J B =

ωH

x

V /

J B =

ωω

x

x

BJ

peJ B

RH = 1

pe … (6)

Using (6) in (5) we obtain

HV = H

BIR

t … (7)

RH = HV t

BI … (8)

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Fig. 1 : Rotation of an electron

ORIGIN OF MAGNETIZATION USING ATOMIC THEORY Permanent magnetic moment can arise from the following : i) The orbital magnetic moment of the electron. ii) The spin magnetic moment of the electron. iii) The spin magnetic moment of the nucleus. Origin of magnetization A current loop behaves as a magnetic dipole having magnetic dipole

moment. M = iA Where A is the area of current loop and i is the current flowing through the

current loop. In an atom, electrons revolve around the nucleus. So the circular orbits of

electrons may be considered as the small current loops. Because of this, an atom possesses magnetic dipole moment and therefore behaves like a magnetic dipole.

One can calculate the magnetic dipole moment of an atom due to orbital motion of the electron. Consider an electron of mass me and charge e revolves in a circular orbit of radius r around the positive nucleus in anticlockwise direction as shown in figure 1.

The angular momentum of the electron due to its orbital motion is given by,

L = me vr

The direction of L is normal to the plane of the electron orbit and in upward

direction. If the period of orbital motion of the electron is T, Orbital motion of electron is equivalent to a current

i =

1e

T

The time period of revolution of the electron

T = 2 rv

i =

1e

(2 r / v) =

ev2 r

The area of the electron orbit A = r2 Magnetic dipole moment of the atom

M = i A =

2evr

2 r =

evr2

Using equation L , we get

M = e

e2m

L

In vector form

M =

e

eL

2m (1)

Negative sign is due to the charge of electron, which is negative. Magnetic dipole moment vector is opposite to angular momentum vector.

5. (b)

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According to Bohr's theory, angular momentum of electron in a stationary

orbit can have only those values which are multiple of

h2

i.e. L =

hn

2 n = 1, 2, 3,

M = e

e nh2m 2

= e

ehn

4 m

Eq. (1) represents the magnetic moment of an electron due to its orbital

motion and e

eh4 m

is the least value of the magnetic moment.

The natural unit of magnetic moment is called Bohr magneton (B). Its value

is calculated as follows:

B =

19 34

31

(1.6 10 C) (6.63 10 J)

4 (9.1 10 kg)

= 9.27 1024 Am2 In an atom, in addition to orbital motion, an electron has got spin motion also.

The electron possesses magnetic moment due to its spin motion also. The total magnetic moment of the electron is the vector sum of its magnetic moments due to orbital and spin motion.

Fig. 2

Even the nucleus is supposed to be possessing spin called nuclear spin and

hence a magnetic moment is associated with it. Due to its heavy mass compare to electron, nuclear magnetic moment is found nearly three orders smaller of magnitude smaller than the moment associated with electron.

Nuclear magnetron is the unit used to express the nuclear magnetic moment. It is defined as

m = eh

4 mp = 5.05 1029 A-m2

where mp = mass of proton

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Difference between Diamagnetic, Paramagnetic and Ferromagnetic material

Diamagnetic Substance

Paramagnetic Substance

Ferromagnetic Substance

(i) Diamagnetic substances get weakly magnetized in the direction opposite to that of the external magnetic field.

Paramagnetic substances get weakly magnetized in the direction of external magnetic field.

Ferromagnetic substances get strongly magnetized even in the presence of a weak magnetic field.

(ii) When a bar of diamagnetic substance is placed in a strong external magnetic field, the total number of lines of induction through the substance is less than that in free space and magnetism is induced in the opposite direction of external field.

When a bar of paramagnetic substance is placed in a strong external magnetic field, the total number of lines of induction through the substance is less than that in free space and magnetism is induced in the direction of the external field.

Ferromagnetic substances show all the properties of paramagnetic substances to a great extent. When a bar of ferromagnetic substance is placed in a magnetic field, the total number of lines of induction in the substance is much greater than that of in free space. Therefore, intensity of magnetization, magnetic susceptibility is very large and the permeability may be order of hundreds or several thousands.

(iii) The intensity of magnetization and magnetic susceptibility are negative and relative permeability is less than 1.

The intensity of magnetization and magnetic susceptibility are positive and relative permeability is slightly greater than 1.

The relative permeabilities of ferromagnetic materials are of the order of hundreds and thousands (r >> 1). The susceptibilities of ferro-magnetic materials are independent of temperature and have very large and positive values.

(iv) A diamagnetic liquid shows a depression in the limb of a Utube when it is placed in between the poles of strong magnet.

A paramagnetic liquid shows a rise in the limb of a Utube when it is placed in between the poles of a strong magnet.

(v) A diamagnetic substance tends to move from stronger to weaker part in a nonuniform magnetic field.

A paramagnetic substance tends to move from weaker to stronger part in a non-uniform magnetic field.

Ferromagnetic substance tends to move strongly from weaker to stronger part in a non-uniform magnetic field.

(vi) A diamagnetic gas in between the poles of magnet spreads across the field.

A paramagnetic gas in between the poles of magnet spreads along the field.

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(vii) The magnetic susceptibility of diamagnetic substance is independent of temperature.

The magnetic susceptibility of paramagnetic substance is inversely proportional to temperature m 1/T i.e. Curie law

The magnetic susceptibility is inversely proportional to temperature. Above a certain temperature the ferromagnetic substance changes to paramagnetic. This temperature is called Curie temperature.

FERMI-ENERGY OR FERMI LEVEL

When the filling up of electrons is undertaken, the universal rule is that the lowest energy level gets filled first. However there will be many more allowed energy levels left vacant as shown in figure 1 below

Fig. 1 : Fermi energy.

Here we define Fermi energy or Fermi level as : The energy of the highest occupied level at zero degree absolute is called

the Fermi-energy and the level is referred to as the Fermi level EF. All the energy levels above the Fermi level at T = 0 °K are empty and those

lying below are completely filled. EF may or may not be an allowed state. It provides a reference with which other energy levels can be compared.

Fermi Level in Conductor As mentioned in classification the conductors are having many free electrons. Let us see how Fermi function helps us understand their distribution. (a) At T = 0 °K

At 0 °K electrons occupy the lower energy levels in the conduction band leaving upper energy levels vacant.

The band is filled up to a certain energy level EF therefore Fermi level may be regarded as the uppermost filled energy level in conductor at 0 °K. Let us see some important conclusions

at T = 0 °K, levels below EF have E < EF

f(E) = F(E E )/KT

1

1 e =

1

1 e =

1

1 0 = 1

f(E) = 1 means all the levels below EF are occupied by electrons.

6. (a)

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At T = 0 K, level above EF have E > EF

f(E) = F(E E )/KT

1

1 e =

1

1 e =

1

1 = 0

f(E) = 0 mean all the levels above EF are vacant. At T = 0 K for E = EF

f(E) = F(E E )/KT

1

1 e =

%

1

1 e

f(E) is indeterminable This is summarized in figure 2.

Fig. 2 : Fermi-Dirac distribution at T = 0 K

(b) At T > 0 K

At temperature above 0° K, few electrons are excited to vacant levels above EF. This happens to those electrons which are close to EF hence probability to find an electron at E > EF will become greater than unity which was zero at T = 0 °K.

Similarly, due to excitation of electrons, few levels just below EF will be become vacant and f(E) will be slightly reduced which was unity at T = 0 °K.

In simple way one can understand that what increase in f(E) at T > 0 °K above E = EF we get is equal to reduction in f(E) below E = EF. This is shown as below in figure 3.

Fig. 3 :Electron occupancy at T > 0 K

At E = EF for T > 0 K

f(E) = o

1

1 e =

1

1 1 =

12

= 0.5

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Fermi Level in Semiconductor: Once the concept of Fermi level is understood properly by considering

conductors, it is proper to go to semiconductors. A semiconductor has conduction band and valence band separated by a

small energy gap. At normal temperature, a significant number of electrons are excited to

conduction band (CB) and from Valence band (VB) leaving behind same number of holes.

Therefore f(E) has non zero probability above Fermi level and f(E) reduces by same amount below EF as shown in figure 4.

Fermi level is half way between CB and VB if it is intrinsic.

Fig. 4 : Fermi-Dirac distribution of semi-conductor at T > 0 K

Sodium Chloride Structure

The sodium chloride structure is shown in figure below. This may be regarded as interpenetrating two fcc lattices one with Na+ ions having their origin at (0, 0, 0) and one with Cl ions with their origin mid way along a cube edge, that is, say at

a,0,0 .

2

The space lattice is fcc with a basis of Na+ and Cl separated by one half the body diagonal of a unit cube. There are four molecules of NaCl in a unit cell with ions in the positions.

Na+ : 0, 0, 0;12

, 12

, 0;12

, 0,12

; 0, 12

,12

.

Cl : 12

, 12

,12

; 0, 0,12

; 0, 12

, 0;12

, 0, 0.

In the crystal, each ion is surrounded by six nearest neighbours of the opposite kind (Hence co-ordination number is six.) and twelve next nearest neighbours of the same kind as the reference ion. In the formation of NaCl crystal, sodium atom loses its outermost electron and acquires an excess of positive charge while the chlorine atom accepts one electron from sodium, acquiring negative charge. There are electrostatic forces of attraction between the opposite charges and the

NaCl Crystal

6. (b)

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strong repulsive forces between the electronic shells of the two ions when they come closer. When attractive and repulsive forces balance, equilibrium is reached, resulting in stable structure.

Representative crystals with NaCl structure are LiH, KBr, KCl, PbI, PbS, AgBr, MnO, MgO. Data : (i) Frequency = 50 Hz (ii) Velocity of ultrasound in atmosphere = 348 m/s (iii) Velocity of ultrasound in sea water = 1392 m/s (iv) Time difference = 2 sec. Let ships be anchored at two points A and B at distance d apart.

Let a signal going from A to B take t1 seconds to come back in atmosphere and t2 seconds in sea water. For atmosphere d = v1 t1 (1) and for sea water d = v2 t2 (2) v1 t1 = v2 t2

1

2

t

t = 2

1

v

v =

1392348

= 4 (3)

t1 = 4 t2 but t1 t2 = 2 sec (given) 4t2 t2 = 2

t2 = 23

sec = 0.667 sec

using eqn. (2) d = v2 t2 = 1392 0.667

Distance between two ships = 928.46

6. (c)

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