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