Debroglie hypothesis

As the Newtonian classical approach to deal with problems involving particles at atomic level failed, the concept of wave mechanics was introduced by Louis Debroglie in 1924 and has been successfully applied to atomic and molecular structures.

Louis Debroglie, a French scientist extended the idea of dual nature of radiation to matter. 

According to Debroglie, a materialistic particle posessess both wave and particle characteristics.  

It is to be known that dual nature of light was explained by combining Planck's expression for energy of photon 

E = h𝜈  --------------------(1)

Einsteins Mass energy relation 

E = mc² ------------------(2)

h𝜈 = mc²------------------(3)

as we know,

𝜈 = C/ƛ  -------------------(4)

From (3) & (4)

ƛ = h/P   --------------------(5)

Just in analogy with equation 5, Debroglie proposed concept of matter waves according to which a material particles of mass M moving with a velocity 'v' should have an associated wave length  ƛ called Debroglie's wavelength given by 

ƛ = h/mv = h/P ----------(6)

'm' - according to theory of relativity is not an invariable quantity as in classical physics.

m = m₀/√ 1-(v²/c²) ------ (7)

If particles like electrons are accelerated to various velocities, we can produce waves of various wavelengths. The higher the velocity of electron, the smaller the debroglie's wavelength.

If electron is accelerated by applying potential V, then work done on electron is eV.

mv²/2 = eV ----------------(8)

mv = P = √ 2meV --------(9)

substituting (9) in (6), we have

ƛ = h/√ 2meV  ------------(10)

 

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.