Showing posts with label electron. Show all posts
Showing posts with label electron. Show all posts

The Davisson-Germer Experiment

The experiment gave the evidence of wave nature possession by materialistic particle(electron) for the first time.

The arrangement of equipment used for the experiment is as follows

                      Procedure:

An electron gun is used in order to produce electrons. The electrons so produced are accelerated by applying a high potential towards the target crystal, in this case the target crystal is nicker. The accelerated electron beam is made into fine beam by passing it through a Collimator ‘c’.

The crystal is mounted on an arrangement which could be rotated in different directions perpendicular to the plane of diagram.

The electrons are scattered in all directions by atomic planes of crystal.

The intensity of electron beam ( no. Of electrons) scattered in a particular direction is measured by electron collector which can be rotated about the same axis as target crystal.

The collector is connected to a sensitive Galvanometer whose deflection is proportional to intensity of electron beam entering collector . The electron collector is also called Faraday cylinder.

A retarding potential is applied to Faraday cylinder such that only fast electrons can reach it and secondary electrons emitted from crystals are stopped.

A graph is then plotted between galvanometer current against angle ‘θ' between incident beam and diffracted beam i.e, beam entering Faraday cylinder.



In the investigation , the electron beam accelerated by 54V and at an angle of 50 between incident and diffracted beam , a sharp maximum has occurred in electron distribution.


The incident beam and diffracted beam in this experiment make an angle of 65⁰   with Braggs plane. 




For a 54 V electron , the de-broglie wavelength associated with the electron is given by

ƛ = 12.25/√V = 12.25/√54 A⁰ = 1.66 A⁰ 

Now from Bragg’s equation for maxima in diffraction pattern for same energy electrons

2d sinθ' = nƛ; 2*0.91*10^-10*sin 65⁰ = 1*ƛ;

ƛ = 1.65 A⁰ 

Thus, both theoretical and experimental values are in excellent agreement.

Thus Davission - Germer experiment provides a direct verification of de-broglie hypothesis of wave nature of moving particles.

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.

Pair Annihilation

Positron and Electron coalesce to produce atleast two photons

e⁺ + e⁻→2𝛾

Annihilation into three or more Photons is possible but less likely. Each extra photon tends to supress the rate of annihilation by a factor of order of magnitude of fine structure constant 1/137.

A Positron moves thru matter and forms ion pairs giving up energy in the process. There is about 2% chance that a Positron will hit an electron and annihilate.

But more likely output is that Positron will stop and become attracted to an electron. The atom formed by these two particles is called Positronium.

The Positron-Electron system drops into successively lower energy states, emitting (low energy) photons, until it arrives in ground state.