Charactersitics of electron in one dimensional periodic potential

CRYSTAL MOMENTUM

For free electron, the quantity ℏk represents true momentum of electron as follows:

E= (ℏk)²/2m = (1/2m)*(ℏ)²*(k)² = (1/2m)*(h/2𝝅)²*(2𝝅/ƛ)² = (1/2m)*(h/ƛ)² = P²/2m

Therefore, the dynamical behavior of free electron can be represented by true momentum.

But when we consider an electron in periodic potential, ℏk doesn't represent true momentum. ℏk does not represent true momentum. The energy doesn't vary with 'k' as in previous case.

The true instantaneous momentum of an electron in presence of lattice potential is not a constant of motion and cannot be calculated by quantum mechanical method we take average value.
So in order to describe the dynamical behavior of electron in periodic potential we introduce a new type of momentum called as Crystal Momentum.

When we deal with interactions of electrons with lattice, we use conservation of crystal momentum and not of true momentum.

The crystal momentum is perfectly well defined constant for a state of given energy.


VELOCITY

The quantum mechanical part describes that the velocity of electron in a one dimensional lattice will be equal to Group Velocity of waves representing the electron.

v = (dw/dk) ...................................................................................................(1)

where 'w' is angular frequency of debroglie waves.

Eqn(1) depends on actual E-K curve.

(dE/dk) = ℏ (dw/dk) ; v=(1/ℏ)(dE/dk) ..........................................................(2)

for free electrons, substituting E= ℏk, v=p/m

giving linear variation of 'v' with 'k'.

In band theory of solids, however, 'E' is not proportional to k².

The variation of 'E' with 'k' is as sown in fig:

using this type of variation of 'E' with 'k' as shown in fig below.

We observe that at bottom (k=0) of energy band, the velocity is zero and as the value of 'k' increases ('E' increases) the velocity increases reaching its maximum at k=kₒ, where kₒ corresponds to "point of inflection" on E-K curve. Beyond this point the velocity begins to decrease and finally assumes zero value at k=𝝅/a, which is top of band. These are entirely new features which do not appear at all in behavior of free electrons.

EFFECTIVE MASS OF ELECTRON

It is known for long time that an electron has well defined mass and when accelerated by an electric field, it obeys Newtonian Mechanics. What happens when electron is to be accelerated inside a crystal? How will it react to electric field.

The mass of an electron inside the crystal appears, in general, different from free electron mass and is usually referred to as "effective mass".

The velocity of an electron in one dimensional lattice is given by

v = (2𝝅/ℏ)(dE/dk)..................................................................................................(3)

a= dv/dt = (2𝝅/h)(d²E/dk²)*(dk/dt) .......................................................................(4)

so we have to find value of dk/dt.

Let an electron is subjected to influence of an electric field of  strength 'E' for a time dt. If velocity of electron is v, the distance travelled in time dt=vdt

Therefore Work done, dE=(e𝜀)*v*dt

we know

v = (2𝝅/h)(dE/dk) ; therefore  dE= (e𝜀)*(2𝝅/h)*(dE/dk)*dt

(dk/dt) =  2𝝅e𝜀/h .....................................................................................................(5)

substituting (5) in (4), we get

a =  (2𝝅/h)² *(e𝜀)*(d²E/dk²)  ...................................................................................(6)

For free particle, a= m(dv/dt) = eE;

a=e*E/m  ..................................................................................................................(7)

comparing (6) & (7) , both forms are identical, we introduce a new mass known as effective mass given by

m* = (h/4𝝅) * (d²E/dk²)⁻¹  ......................................................................................(8)

For free electron,

m* = m

• Effective mass can also be determined using "Cyclotron Resonance Experiment".

From experimental values of effective mass, we can conclude that

•  Effective mass need not always be greater than 'm'. It can be smaller than 'm'.
• It can be negative.

Variation of m* with k:  

Physically speaking near the bottom of band the effective mass m* has a constant value which is positive because the quadratic eqn [E ∝k²] is  satisfied near the bottom of band.

But as 'k' increases m* is no longer a constant, being now a function of k, because quadratic relation is no longer valid.

The degree of freedom of an electron is defined by a factor

fk = (m/m*) = (m/ℏ2)*(d²E/dk²)

fk is measure of extent to which an electron in state 'k' is free.

Principles of Special Theory of Light

1. Does the speed of light depend on motion of source of light?

No, the motion of light is not affected by motion of source of light.

2. Is photon a particle?

The photon is a particle of light, but it doesn’t possess all essential properties we ascribe to a tiny ball i.e. photon doesn’t behave as a common sense particle but it has got some peculiar properties.

3. When we follow Albert Einstein in developing special theory of relatively, we are developing a theory of space and time.

4.  The principles of special theory of light.

Principle 1:

Colloquial statement: If we are in unaccelerated vehicle, its motion has no effect on the way things happen inside it.

Formal statement: The laws of physics are the same in all unaccelerated reference frames.

Principle 2: The motion of light is not affected by motion of source of light.

5. The special theory of relativity

      

          Special: The word special in name arises because we employ only unaccelerated reference frames, not all reference frames that one can think of. In other words, we special to the way things appear when observed from uniformly moving reference frames.

     

          Relativity:-The word relativity comes from a phrase coined by Henri Poincare, an eminent French physicist and mathematician.

In 1904, Poincare was invited to address the international congress of arts and science, held in st Louis to commemorate the 100^th anniversary of Louisiana Purchase. Poincare spoke of a principle of relativity.

If you are in plane on its way from Chicago to phoenix, another plane making the return flight, over wheat fields of Kansas. A farmer, looking up, notes that you are flying south west at 500 miles/hr relative to his wheat fields.

The pilot of return flight notes that the distance between the two planes is decreasing at about 1000 miles/hr. So far as the pilot is concerned, you are travelling at about 1000 miles/hr relative to his plane.

The essence is this:  statements about uniform motion relative to a specified reference frame wheat fields or another air plane are meaningful.

A quantitative statement about uniform motion without specification of a reference frame is not meaningful. Why? Because our principle 1 says we cannot discern uniform motion without recourse to some reference frame.

Take first the colloquial form of that principle if we are in an unaccelerated vehicle, its motion has no effect on the way things happen inside it. So by just doing experiments inside the vehicles, we have no way to assign a velocity to the vehicle. Only if we look out of window and thereby use wheat fields of Kansas as an outside reference frame. We can decide on velocity (velocity to that outside reference frame).

      

          Theory: It appears because principles 1 & 2 are generations from observation and experiment.

6. THE CONSTANCY OF SPEED OF LIGHT

•  Observes in all un accelerated reference frames measure the same speed for light ( in vacuum) from any given source.

• They all measure 3*10 ⁸m/sec   always for light in vacuum.

• This remarkable property is called “constancy of speed of light”.

Note:-Some factors other than light may be observed differently in unaccelerated frames.

7.  An “event” is anything that happens at some definite locations at some definite time. Proto typical examples are your birth, assassination of Abraham Lincoln etc. In contrast, a forest fire that sweeps across 10000 acres in 5 days does not constitute an “event” because the fire is spread out in space and time.

The adjective “definite” means   “distinct” or  “limited” for any one observing the happening.

8. THE RELATIVITY OF SIMULTANEITY:

•  Spatially separated events that are simultaneous in one frame are, in general, not simultaneous when viewed from other reference frame.

•  Simultaneity is a relative concept, but not an absolute one.

• The concept of simultaneity between two events in different space points has an exact meaning only in relation to a given inertial system i.e.   “Each frame of reference has its own particular time”.

• To measure the length of an object means to locate its end points simultaneously. As simultaneity    depends on frame of reference, the length measurements will also depend on frame of reference.

• Thus, “The length i.e.  Space is a relative concept, not an absolute one”.

• Thus there is no such thing as an absolute, global “now”.

Brewsters Law

When light is transferred from glass surface it may be polarized. It was explained by Brewster. According to Brewster the light reflected from a surface is completely polarized if the reflected beam and the beam refracted into material form a right angle. If the incident beam is polarized in plane of incidence there will be no reflection at all. Only if the incident beam is polarized normal to the plane of incidence it will be reflected.

Reason for above property

In the reflecting material, the light is polarized transversely and we know  that it is motion of charges in material which generates emergent beam which we call as reflected beam.The source of this so called reflected light is not simply that the incident beam is reflected; our deeper understanding of this phenomenon tells us that the incident beam drives an oscillation of charges in the material, which in turn generates reflected beam.

From below fig, it is clear that only oscillations normal to paper can radiate in direction of reflection and consequently reflected beam will be polarized normal to plane of incidence. If the incident beam is polarized in plane of incidence there will be no reflected light.

This phenomenon is readily demonstrated by reflecting a linear polarized beam from a flat piece of glass. If glass is turned to different angles of incidence to polarized beam, sharp attenuation of reflected intensity is observed when angle of incidence passes through Brewsters angle. This attenuation is observed only if plane of polarization lies in plane of incidence. If plane of polarization is normal to plane of incidence, the usual reflected intensity is observed at all angles.

BLOCH THEOREM

Bloch assumed that electrons move in a perfect periodic potential. He considered one dimensional array of lattice. The potential of electron at positive ion site is zero and is maximum in between. So long any line passing through the centers of positive ions, the potential variation must be as shown in below figure.

So Bloch gave a condition which is

𝚿(x+Na)=𝚿(x) .............................................................................................................(1)

It is considered as boundary condition.

Consider Schrodinger wave equation for one dimensional lattice.

(d²𝚿(x)/dx²) + (2m/ħ²)*[E-V(x)]*𝚿(x) = 0  .................................................................(2)

The Schrodinger equation for an electron in the potential at x+a is

[d²𝚿(x+a)/d(x+a)²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0  ..............................................(3)

Because of periodicity,

[d/d(x+a)] = d/dx  ; V(x+a) = V(x)

With  this, eqn (3) reduces to

[d²𝚿(x+a)/dx²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0  ...................................................(4)

This is Schrodinger  equation at x+a.

as 𝚿 at x+a is also obeying Schrodinger wave equation as 𝚿 at x there should exist a relation between 𝚿(x+a) & 𝚿(x).

Let    𝚿(x+a) = A𝚿(x)..................................................................................................(5)

𝚿(x+2a) = A²𝚿(x) [i.e. A𝚿(x+a) = A.A𝚿(x) = A²𝚿(x)]

𝚿(x+na) = Aⁿ𝚿(x)

from eqn(1),  Aⁿ =1 [i.e by using bloch condition]

Aⁿ =exp(2πij) [i.e. exp(2πij) =1 for j=01,2............]

or

A=exp(2πij/n)

Therefore,  𝚿(x+a) = exp(2πij/n)*𝚿(x) --------------------------------------------------(6) [ from eqn 5]

𝚿(x) can be written in terms of other function Uk(x )

𝚿(x) = exp(ikx)*Uk(x) where k=(2πj/n) ..................................................................(7)

From eqns (6) & (7),

exp[ik(x+a)]*Uk(x+a) = exp(2πij/n)*exp(ikx)*Uk(x)

exp[ika]*Uk(x+a) = exp(2πij/n)*Uk(x)

noting that  Ka = 2πj/n,

we can write that  Uk(x+a) = Uk(x) ..........................................................................(8)

Conclusion

Bloch Theorem is a mathematical theorem and it gives us the form of electron wave function in a periodic potential.

 𝚿(x) = exp(ikx)*Uk(x) represents Plane Wave

Thus, electron in a one dimensional lattice behaves a a plane wave.It only gives Wave nature of electron.

GIBBS CANONICAL ENSEMBLE

(SYSTEM IN CONTACT WITH A HEAT RESERVOIR)

Canonical ensemble describes those systems that are not isolated but are in contact with a heat reservoir.

The system under consideration together with a heat reservoir forms a closed system and then system of interest is taken as sub system of this closed system.

Any sub system of an isolated system in statistical equilibrium can be represented by a canonical ensemble.

The probability density of a canonical ensemble depends both on Energy 'E' and temperature 'T'

It is given by

ρ(E,T) = Ae-E/𝜏

Partition function for canonical ensemble

z=(1/h³ⁿ.n!)∫exp(-E(q,p)/KTdΓ

The statistical energy of a system in a canonical ensemble is given by

𝜎 = logZ  +(E/𝜏);  𝜎 = logZ  +T[∂(log Z)/∂T]

Thermodynamical entropy 'S' of a system in a canonical ensemble is given by

S=K𝜎

S=KlogZ + (E/T); U=KT2[∂(log Z)/∂T]

The entropy at absolute zero in a canonical ensemble can be expressed as

 S=K.log gₒ

gₒ is statistical weight of ground state

What is Chemical Potential?

For a chemical system, molar free energy is known as Chemical Potential.

A chemical substance that is free to move from one place to another place, will move spontaneously from a state of higher chemical potential to a state of lower chemical potential.

In the position of equilibrium, the chemical potential is constant through the entire system.

Let us consider a general heterogeneous system consisting of an independent components in several coexisting phases.

To start with, it is convenient to describe a given phase by its chemical composition, which is specified by the no. of mole 'Ni' of each species i, its volume V and its entropy 'S'.

If we consider internal energy (U)

U=U(S,V,N₁,N₂,.....Nᵢ,.....Nn)

μi=❴∂U/∂Nᵢ❵S,V,Nj ; j= except 'i'

'μi' is chemical potential of component 'i' in given phase.

dU=TdS-PdV+Σμᵢi.dNᵢ for i=1...n

We can also consider chemical potential 'μ' in terms of Helmoltz free energy 'F'.

F = F(T,V,N₁,N₂......Nn)

μ1=❴∂F/∂N1❵T,V,N₂,....

μ2=❴∂F/∂N2❵T,V,N₁,N₃,....

The chemical potentials are thus the rate of change of free energy per mole, at constant volume and temperature.

μ can also be expressed as

μi=❴∂G/∂Nᵢ❵T,P,Nj

A System in external field will be in equilibrium if the temperature and chemical potential of each component of the system is constant through out, i.e.

dT₁=0 and dμᵢ=0

Plot of Binding Energy per Nucleon against Mass Number - Important Conclusions

What is Binding Energy?

Binding Energy (BE) is the energy required to break a Nucleus into free neutrons and free protons.

According to Einstein's relative theory, mass of a system bound by energy 'B' is less than mass of its constituents by B/c².

BE/Nucleon(B/A) vs Mass Number (A) Plot:

Important Conclusions

a) Approximately for most of Nuclei B/A ~ Constant.
b) B/A falls off at small values of A

Reason: For very light Nuclei a large fraction of their nucleons resides on the surface rather than inside. This reduces the B/A value as a surface nucleon is surrounded by fewer nucleons compared to a nucleon residing in interior and consequently is not so strongly bound.

c) B/A falls off at large values of A. This is clearly a Coulomb effect. Between every pair of Protons, there is a Coulomb repulsion which increases as Z². Notice that for naturally occurring nuclei, Z² increases faster than A and so Coulomb effect cannot adequately compensated by an increase in A.

d) B/A against A plot is peaked about A~50.
 Binding Energy can be increased by either breaking a heavy nucleus into parts or fusing light nuclei together.  It is easy to see that when binding energy is increased, energy in other forms can be released , since a decrease in 'M' corresponds to conversion of mass into energy.

e) The peak of the plot corresponds to iron. This explains large abundance of Fe(iron) in nature.

f) The plot indicates that binding becomes strong for a grouping of four particles. This unit is 𝛂 particle (2 neutrons + 2 protons).

The peaks in figure at mass numbers 4,8,12,16,20 & 24 are clear evidence of this effect. This effect is due to a pairing  force which exists  between a pair of neutrons and pair of protons.

g)  On closer inspection, it is found that B/A against A plot shows discontinuities  at neutron or proton number values 2,4,8,20,50,82 & 126. At these values of neutron or proton numbers, the BE is found to be unusually large. Large BE means high stability.

What is a Nuclear Reactor?

A Nuclear Reactor is a systematic arrangement to convert Nuclear Energy into thermal energy and then to Electrical energy .  Nuclear Reactor uses fissile material, heavy atomic nuclei, called as Nuclear Fuel. Fissile material leads to nuclear fission when the nuclei are hit by suitable energy Neutrons.

  

Example for Fissile Material is Uranium oxide.

Fission reaction of Uranium is as follows:

The energy evolved is distributed as kinetic energy of fission fragments  and heat.

This heat energy transmitted to a coolant which leads to generation of steam that could drive turbine system for conversion of thermal energy into electrical energy.

There are different types of Nuclear Reactors operating across the world.

a)  Boiling Water Reactors

b)  Pressurized Water Reactors

c)  Pressurized heavy water Reactors

d)  Fast Breeder Reactors

How does Stern Gerlach Experiment proved spatial quantization and spin of electron

THE STERN-GERLACH EXPERIMENT

 Procedure implemented:

The silver atoms beam is produced by heating silver in a small electric oven. The beam is passed through an inhomogenous Magnetic Field.

Arrangement to produce inhomogenous Magnetic Field

We have one of pole pieces of the magnet flat with a cylindrical groove and the other in the form of a knife edge, parallel to groove.

The intensity of magnetic field increases as we go towards upper knife edge pole from center and it decreases as we go below towards lower pole.

A photographic plate is arranged to record the configuration of beam after its passage through the field.

The whole arrangement is placed in a vacuum. In absence of magnetic field, a trace of form of a narrow strip is obtained as shown in fig (a).

In presence of inhomogenous magnetic field the strip splits up into two components as shown in fig(b). 

The splitting of silver beam into two components in inhomogenous field verifies existence of electro spin and postulate of space quantization as shown below: 

Silver has an atomic number 47. According to Pauli's exclusion principle, all inner shells and sub shells are completely filled except outer most electron in 5S state.  Thus, it is a monovalent element.
The 5S electron is responsible for magnetic moment of atom.

When all silver atoms possessing a magnetic moment 'μᴊ' pass through inhomogenous magnetic field, they experience different amount of force in vertical direction depending on their orientation and alignment with magnetic field.

If magnetic moment  'μᴊ' can have all possible orientations then beam of Silver atoms consisting of Millions of atoms having all possible orientations will spread out into a broad continuous band on emerging from magnetic field. So a broad continuous patch should be observed on photographic plate.

Experimentally only two narrow strips are obtained on photographic plate. Therefore, predictions of classical physics are not correct in this case.

The two narrow strips show that 'μᴊ' cannot have all possible orientations, but only two possible orientations as shown in below figure

We know that 'μᴊ' is proportional to angular momentum 'J' and hence direction of 'J' relative to a well defined direction is given by

J = [√j(j+1)   ]* h/2Π

There are (2j+1) possible orientations of J. The Stern-Gerlach experiment shows that (2j+1)=2 or j=1/2.

Thus, J=(√3 /2)* (h/2Π)

It is known that angular momentum 'J' of Silver atoms entirely due to spin of its valence electrons. Thus, we conclude that the electron has a spin angular momentum

S = [√s(s+1)]*(h/2Π); where s=1/2.

Thus, Stern and Gerlach found that intial beam split into two distinct parts, corresponding to two opposite spin orientations in magnetic field that are permitted by space quantization.

THE RAINBOW - EXPLANATION

Of all the optical phenomenon in every day life, the rainbow is loveliest.

Reflection of sun light by the rain drops is certainly an essential  element of an explanation but refraction-plays a role, too.

The following figure shows crucial path of light

The circle represents the cross section of a spherical rain drop. For the light ray, the sequence is Refraction, Reflection, and Refraction.

The angle (less than 90o) between the incident direction and the emergent direction is called as "Return Angle".

 A Ray from the sun strikes the spherical rain drop and some light is refracted into the water. Here we may ignore the portion that is reflected by the drops' surface. Next, the Ray proceeds to the far side of drop and is reflected there. Now we may ignore the portion that is refracted. Finally, the ray strikes the underside of drop and is refracted out into the air.

In precisely which direction does the emergent ray travel?

Two rules - i) Snell's Law for Refraction & ii) Equality of the angles of incidence and reflection

 suffice for answering that question. Once the initial point of contact between the rain drop and the ray from the sun has been specified. Also because the index of refraction depends on color, we must specify the color of light.

Lets start considering Red. Working out the complete path for many rays-i.e. for many different initial points of contact-reveals a surprising geometric property: The return angle for red light never exceeds 42.5 deg C, and most rays have a return angle 42 deg.

So return angle will decide the color of light reaching our vision of sight. Hence, different colors emerge from different sets of rain drops and produce a colored Rainbow.

What are Light Pipes

Sending light in a staright line is easy. Getting it around corners -with out losing intensity, is a different story.

Even when a light beam reflects from a good silvered mirror, not all of the incident light bounces off; some is absorbed by metal, as much as 20%. A sequence of reflections, in which 20% is lost at each bounce, will quickly reduce a powerful beam to a faint trace of its former itself.

A light pipe is made of glass, designed in such a fashion that there is no refraction of Light out of glass, and hence, there is no loss of intensity.

The Glass just "pipes" the light up and around the curve.

In commercial use, light pipes are drawn out as very long, fine fibers of glass. The diameter is about 0.1mm and field of applications itself is called fiber optics.

SPEED OF LIGHT IN VACUUM

Newton composed his "Opticks", which he published in 1704, he addressed the question of speed and wrote as follows:

[Light is propogated from luminous bodies in Time, and speeds about seven or eight minutes of an hour in passing from Sun to Earth.

This was observed first by Roemer, and then by others, by means of eclipses of satellites of Jupiter. For these eclipses, when the Earth is between the Sun and Jupiter, happen about seven or eight minutes sooner than they ought to do by the tables, and when the Earth is beyond the Sun they happen seven or eight minutes later than they ought  to do; The reason being, that the lights of satellites has farther to go in the latter case than in the former by diameter of earths orbit. ]

Newton is referring to Olaus Roemers calculations of 1676.

The Astronomers of seventeenth century had studied 'IO' (satellie of Jupiter) ever since Galileo discovered the Moons of Jupiter in 1610. They noted the eclipses and sought to predict their recurrence. But the eclipses -as

observed on Earth - did not recur perfectly periodical. If one used observations made when the Earth was nearest to Jupiter to predict when eclipses will terminate when the earth is farthest from Jupiter (about half a year

later), then the eclipses actually seem to terminate later than caluclated. The delay reasoned Roemer, is due to travel time for light to cross Earth's Orbit.

Newton gives the Time for Light to travel from Sun to Earth as 7 or 8 minutes. Let us average and take 15 minutes as the time for light to cross diameter of Earth's Orbit.

Already in Newton's day, the diameter of Earth's orbit was known to be 3 x 1011 meters.

Speed of Light in Vacuum = Diameter of Earth's Orbit / Apparent delay in eclipse termination
                                           = 3x10¹¹/900 Sec = 3x10⁸ m/sec

Properties of Longitudinal Progressive Waves

Longitudinal wave motion refers to wave motion in which particles of medium vibrate along the direction of propagation of wave.

Properties:

1. All the particles have same Amplitude, Frequency and Time Period

2. There is a gradual Phase difference between successive particles

3. All the particles vibrating in Phase will be at a distance equal to nƛ. Here n=1,2,3etc. It means the   minimum distance between two particles vibrating in Phase is equal to wave length.

4. When the particle moves in same distance as that of wave, it is in a region of compression.

5. When the particle moves in opposite direction as that of wave it is in a region of Refraction.

6. When the particle is at mean position, it is a region of maximum Compression or Refraction.

7. When the particle is at extreme position, the medium around particles has its normal density, with compression on one side and rare fraction on other side.

Ionic Conductivity - Detailed Explanation

Thermal Radiation - Important points

The process of heat transfer from a body by virtue of its temperature with out involvement of intervening medium is called Radiation. The radiant energy is transported by electromagnetic waves because these waves can travel through vacuum.

The Radiation emitted by a body by virtue of its temperature is called Thermal Radiation. It is an inherent property of all bodies.

According to Prevost theory of heat exchanger, every body emits and absorbs radiant energy continuously as long as its temperature is above 0 K.

At low temperature, the emission rate is small while at higher temperatures it increases rapidly as 4th power of absolute temperature. 

At ordinary and moderate high temperature, mostly longer waves(infrared) are emitted but at very high temperatures shorter waves are also emitted.

Properties of Thermal Radiation:

i)  It travels through empty space with the velocity of light.

ii) It undergoes Reflection, Refraction and total internal reflection obeying the same law as light.

iii) It exhibits the phenomenon of interference, diffraction and polarisation.

iv) It exerts a small, but finite pressure on the surface on which it is incident. This is called as pressure of thermal radiation.

v)  It obeys inverse square law

Some important terms related to Thermal Radiation are  Spectral Energy Density, Total Energy Density, Emmisive Power & absorptive power

Spectral Energy Density

Spectral Energy Density for a particular wavelength is the energy per unit volume per unit range of wavelength.

Total Energy Density

Total energy density of thermal radiation at any point is the total radiant energy per unit volume around that point due to all wavelengths.

Emmissive Power

The emissive power of a body at a given temperature and for a given wavelength, is defined as the ratio radiant energy absorbed for a second by unit surface area of the body per unit wavelength range.

Absorptive power

The absorptive power of a body at a given temperature and for a given wavelength is defined as the ratio of radiant energy absorbed per second by unit surface area of the body to the total energy falling per second on the same area. 

Black body and its Radiation

What is a Black body?

Energy distribution in black body radiation

Laws of Black body Radiation

Kirchoff's Law

Stefan's-Boltzmann law

Wien's Law

Rayleigh-Jeans Law

Planck's Law

MILLER INDICES