De-Broglie’s Hypothesis - Civil Engineering (CE) PDF Download

De-Broglie’s hypothesis
We know that the phenomenon such as interference, diffraction, polarisation etc. can be explained only with the help of wave theory of light. While phenomenon such as photoelectric effect, Compton effect, spectrum of blackbody radiation can be explained only with the help of Quantum theory of radiation. Thus radiation is assumed to exhibit dual nature. i.e., both particle and wave nature.

In 1924, Louis de-Broglie made a bold hypothesis, which can be stated as follows
“If radiation which is basically a wave can exhibit particle nature under certain circumstances, and since nature likes symmetry, then entities which exhibit particle nature ordinarily, should also exhibit wave nature under suitable circumstances.

” Thus according to De-Broglie’s hypothesis, there is wave associated with the moving particle. Such waves are called Matter waves and wavelength of the wave associated with the particle is called De-Broglie wavelength.

Expression for De-Broglie wavelength (from analogy)
We know that radiation consists of stream of particles called Photon. Energy of each photon is given by
E = hν = hc/λ

Where h = Plancks constant, ν is the frequency and λ is wavelength of the radiation. But according to Einsteins relativistic formula for energy of a particle

 Where p is momentum of the particle and m₀ is rest mass of the particle. For photon rest mass m₀ is zero. Therefore

E = pc = hc/λ

or  p = h/λ
is the momentum of the photon.
De-Broglie proposed that the same equation is applicable for matter waves also. Therefore, wavelength of the waves associated with the moving particle of mass m , moving with the velocity v is given by

where the momentum is given by p = mv and λ is the de Broglie wavelength.

De-Broglie wavelength of an electron accelerated by a potential difference of V volts: 
Consider an electron accelerated by a potential difference of V volts Kinetic energy gained by the electron is given by

substituting for h, m, and e we get

thus for V = 100 volts

λ = 1.226/√ 100 = 0.1226 nm

Note:
For a particle of mass M and charge q accelerated through a potential difference of V volts, expression for De-Broglie wavelength is given by

∴ λ = h/√ 2MqV

Experimental verification of De-Broglie’s hypothesis (Davisson and Germer experiment 
The apparatus used by Davisson and Germer is shown in the Fig. 1.12. Apparatus consists of an electron gun E, which produces a narrow and collimated beam of electrons accelerated to known potential V. This narrow beam of electron is made to incident normally on a Nickel crystal C mounted on turntable T. Electrons get scattered from the crystal in all directions. The turntable T can be rotated about an axis perpendicular to both incident beam and scat-tered beam. The scattered electron beam is received by an ionization chamber I mounted on an arm R capable of rotating about an axis same as that of turntable T. The angle of rotation can be measured with a help of circular scale S. Electrons entering the ionization chamber produces a small current, which can be measured with the help of galvanometer G connected to the ionization chamber. Keeping the accelerating potential constant at E,

electron beam is made to strike the nickel crystal C. Electrons scattered from the crystal is collected by the ionization chamber at various scattering angles φ and the corresponding value of ionization current I is noted. Experiment is repeated for various accelerating potentials V. A polar graph of representing φ and I for different values of V is plotted. The graphs obtained are shown in the fig 1.4.

When accelerating potential was 40 V , a smooth curve was obtained. When accelerating potential was increased to 44 V a small bump appeared in the curve. Bump became more and more pronounced as the voltage was increased, to reach maximum for the accelerating potential of 54 V . Beyond 54 V , bump started diminishing and vanished after 68 V . The scattering angle corresponding the accelerating voltage V = 54 V when bump became maximum was found to be φ = 50⁰ .

Davisson and Germer interpreted the result as follows: Electrons in the incident beam behave like waves. Thus when electrons strike the crystal they undergo Bragg’s diffraction

from the different planes of the crystal. The bump in the curve corresponds to constructive interference caused by the scattered electrons. According to Bragg’s law the condition for constructive interference is given by

2dSinθ = nλ

where d = Interplanar spacing for the crystal, θ = glancing angle made by incident beam with the crystal plane, n=order and λ is wavelength of the wave. Thus, when bump in the curve is maximum, θ = 65⁰ (see fig 1.5), n=1 and for nickel crystal d = 0.091 nm. Thus, wavelength of the waves associated with the electrons is given by

Figure 1.5: Angle between crystal plane and incident electrons

According to De-Broglie’s hypothesis wavelength of the wave associated with the electrons accelerated by a potential difference of V volts is given by

λ = h/√ 2meV

Since bump becomes maximum for the accelerating voltage of 54 V ,

Thus there is an excellent agreement between experimental results and theoretical prediction. Thus Davisson-Germer experiment provides direct experimental evidence for the existence of matter waves.