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Wednesday, August 22, 2018

Crystal Growth Methods - Brief Explanation

Various types of crystal growth methods are

Growth from Water Solution
Growth from Flux
Hydro Thermal Growth
Electrode Deposition
Gel Growth

Growth from Water Solution


This technique is used for soluble crystals like sugar, salt crystals for example NaCl, KCl, KBr are used. Their growth rates are very small. They have 5 mole percent solubility.

Nucleation is one such process. Liquid containing crystal solution solution having low viscosity is taken into a beaker. Crystal which has to be grown is taken in very small size which is called as seed crystal. We have to hang this crystal in the liquid in beaker. The molecules join crystal to form the crystal big in size.


Growth from Flux

This method is used for crystals which are not dissoluble. This technique uses oxide crystals/metal crystals. Crystals like quartz having high melting point of 1400 oC which is attained at higher energy are grown using this technique. For this they are combined with other crystals called as flux whose benefit is to reduce the melting point of crystals to form.

Advantages of this technique

a) Growth is at temperature well below the melting point
b) High quality crystals can be obtained
c) Doping with suitable materials could be done
d) Solid solution can be grown easily

Hydro thermal growth

This method is used for crystals whose melting point is very high. For instance, Al2O3 cannot be soluble in water. Normally Al2O3 dissolves in water at critical temperature of 353 ⁰C.

So when pressure is exerted on crystal then melting point of material decreases (of about 50000 pounds per sq. inch). So special devices such as autoclaves are used for this purpose.

conditions of growth in hydro thermal process:

i) The Temperature
ii) The Pressure
iii) The temperature difference between top and bottom ends of autoclave



Sunday, July 1, 2018

Charactersitics of electron in one dimensional periodic potential

CRYSTAL MOMENTUM

For free electron, the quantity โ„k represents true momentum of electron as follows:

E= (โ„k)²/2m = (1/2m)*(โ„)²*(k)² = (1/2m)*(h/2๐…)²*(2๐…/ฦ›)² = (1/2m)*(h/ฦ›)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, โ„k doesn't represent true momentum. โ„k 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) = โ„ (dw/dk) ; v=(1/โ„)(dE/dk) ..........................................................(2)

for free electrons, substituting E= โ„k, 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=kโ‚’, where kโ‚’ corresponds to "point of inflection" on E-K curve. Beyond this point the velocity begins to decrease and finally assumes zero value at k=๐…/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 = (2๐…/โ„)(dE/dk)..................................................................................................(3)

a= dv/dt = (2๐…/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=(e๐œ€)*v*dt

we know

v = (2๐…/h)(dE/dk) ; therefore  dE= (e๐œ€)*(2๐…/h)*(dE/dk)*dt

(dk/dt) =  2๐…e๐œ€/h .....................................................................................................(5)

substituting (5) in (4), we get

a =  (2๐…/h)² *(e๐œ€)*(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/4๐…) * (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/โ„2)*(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 100th 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”.

Saturday, June 30, 2018

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

๐šฟ(x+Na)=๐šฟ(x) .............................................................................................................(1)

It is considered as boundary condition.

Consider Schrodinger wave equation for one dimensional lattice.

(d²๐šฟ(x)/dx²) + (2m/ฤง²)*[E-V(x)]*๐šฟ(x) = 0  .................................................................(2)

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

[d²๐šฟ(x+a)/d(x+a)²] + (2m/ฤง²)*[E-V(x+a)]*๐šฟ(x+a) = 0  ..............................................(3)

Because of periodicity,

[d/d(x+a)] = d/dx  ; V(x+a) = V(x)

With  this, eqn (3) reduces to

[d²๐šฟ(x+a)/dx²] + (2m/ฤง²)*[E-V(x+a)]*๐šฟ(x+a) = 0  ...................................................(4)

This is Schrodinger  equation at x+a.

as ๐šฟ at x+a is also obeying Schrodinger wave equation as ๐šฟ at x there should exist a relation between ๐šฟ(x+a) & ๐šฟ(x).

Let    ๐šฟ(x+a) = A๐šฟ(x)..................................................................................................(5)

๐šฟ(x+2a) = A²๐šฟ(x) [i.e. A๐šฟ(x+a) = A.A๐šฟ(x) = A²๐šฟ(x)]

๐šฟ(x+na) = Aโฟ๐šฟ(x)

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

Aโฟ =exp(2ฯ€ij) [i.e. exp(2ฯ€ij) =1 for j=01,2............]

or

A=exp(2ฯ€ij/n)

Therefore,  ๐šฟ(x+a) = exp(2ฯ€ij/n)*๐šฟ(x) --------------------------------------------------(6) [ from eqn 5]

๐šฟ(x) can be written in terms of other function Uk(x )

๐šฟ(x) = exp(ikx)*Uk(x) where k=(2ฯ€j/n) ..................................................................(7)

From eqns (6) & (7),

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

exp[ika]*Uk(x+a) = exp(2ฯ€ij/n)*Uk(x)

noting that  Ka = 2ฯ€j/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.

 ๐šฟ(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.

Sunday, February 18, 2018

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

ฯ(E,T) = Ae-E/๐œ

Partition function for canonical ensemble

z=(1/h³โฟ.n!)∫exp(-E(q,p)/KTdฮ“

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


๐œŽ = logZ  +(E/๐œ);  ๐œŽ = logZ  +T[∂(log Z)/∂T]

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

S=K๐œŽ

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 g

gโ‚’ is statistical weight of ground state

Saturday, February 17, 2018

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,.....Nแตข,.....Nn)
ฮผi=❴∂U/∂Nแตข❵S,V,Nj ; j= except 'i'

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

dU=TdS-PdV+ฮฃฮผแตขi.dNแตข for i=1...n

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

F = F(T,V,N₁,N₂......Nn)

ฮผ1=❴∂F/∂N1❵T,V,N₂,....

ฮผ2=❴∂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.

ฮผ can also be expressed as

ฮผi=❴∂G/∂Nแตข❵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 dฮผแตข=0