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Sunday, July 1, 2018

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 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”.