Introduction: Do you know that not only photons of light but also ‘particles’ of matter such as electrons and atoms possess a dual character? Yes, sometimes it behaves like a particle and sometimes as a wave. Davisson and Germer experiment proved the wave nature of electrons and verified the de Broglie equation. The dual nature of matter was discussed back by De Broglie in \(1924,\) but later Davisson and Germer’s experiment verified the results.

The results of this experiment established the first experimental proof of quantum mechanics. In this article, we will study the scattering of electrons by a Ni crystal in the Davisson-Germer experiment. We will also discuss the de Broglie hypothesis and its relation to the Davisson-Germer experiment.

Davisson-Germer Experiment

Davisson and Germer set up an experiment to test de Broglie’s hypothesis that matter behaved like waves very similar to what might be used to examine the interference pattern from \(x-\)rays scattering from a crystal surface. The basic idea is that the planar nature of crystal structure provides scattering surfaces at regular intervals. Thus, waves that scatter from one surface can constructively or destructively interfere with waves that scatter from the next crystal plane deeper into the crystal. Their experimental apparatus is given below.

Experiment Setup

The deflection and scattering of electrons by the medium are prevented because all the experimental setup of Davisson and Germer’s experiment is enclosed within a vacuum chamber. The experimental parts are given below:
1. Electron gun: An electron gun emits electrons via thermionic emission produced by the Tungsten filament used in it, i.e. when heated to a particular temperature, it emits electrons.
2. Electrostatic particle accelerator: Two opposite charged plates (positive and negative plates) accelerate the electrons at a known potential.
3. Collimator: The accelerator is enclosed within a cylinder with a narrow passage for the electrons along its axis. Collimators are used to render a narrow and straight beam of electrons ready for acceleration.
4. Target: The target is a Nickel crystal. On the nickel crystal, the electron beam is fired normally.
5. Detector:  When the electrons are scattered from the Ni crystal, it is captured by the detector. It can be moved in a semicircular arc.

Observations of the Davisson and Germer Experiment

As in the figure below, the experimental arrangement used by Davisson and Germer is schematically shown in fig (1). By applying a suitable potential/voltage from a high voltage power, electrons emitted by the filaments of the electron gun are accelerated to the desired velocity. They are made to pass through a cylinder with fine holes along its axis, producing a finely collimated beam. Now, this collimated beam is made to fall on the surface of a nickel crystal.

These electrons are scattered in all directions due to the crystal of the atom. The electron detector is used to measure the intensity of the electron beam, scattered in a given direction. The current is recorded by the sensitive galvanometer in which the detector is connected. The deflection of the galvanometer is proportional to the intensity of the electron beam entering the collector. Then, the intensity of the scattered electron beam is measured for different values of angle of scattering \(\left( \theta \right)\) by moving the detector on the circular scale at different positions, where \(\theta \) is the angle between the incident and the scattered electron beams.

The variation of the intensity \(\left( I \right)\) of the scattered electrons with the scattering angle \(\left( \theta \right)\) is obtained for different accelerating voltages, as shown in the figure below. Now by varying the accelerating voltage from \(44\,\rm{V}\) to \(68\,\rm{V}\) the experiment was performed. It was noticed that a strong peak appeared in the intensity \(\left( I \right)\) of the scattered electron for an accelerating voltage of \(54\,\rm{V}\) at a scattering angle \( = {50^{\rm{o}}}.\)

Bragg’s Equation

Since the value of \(d\) was already known from the \(X-\)ray diffraction experiments. Hence for various values of \(\theta ,\) we can find the wavelength of the waves producing a diffraction pattern. Due to the constructive interference of electrons scattered from different layers of the regularly spaced atoms of the crystals, we get the appearance of the peak in a particular direction. And the wavelength of matter waves was found to be \(0.165\,{\rm{nm}}\) from the electron diffraction measurements by the given equation.
\(n\lambda = 2d\,\sin \left( {90 – \frac{\theta }{2}} \right)\)

Davisson Germer Experiment and de Broglie Relation

Let us consider an electron of mass \(\left( m \right),\) charge \(\left( e \right)\) accelerated from rest through a potential \(V.\) Then, the kinetic energy \(K\) of the electron equals the work done \(\left( {eV} \right)\) on it by the electric field:
\(K = eV\)
Now,
\(K = \frac{1}{2}m{v^2} = \frac{{{p^2}}}{{2m}}\)
So that.
\(p = \sqrt {2\,m\,K} = \sqrt {2\,m\,eV} \)…..(i)
Then, the de-Broglie wavelength \(\lambda \) of the electron is given as:
\(\lambda = \frac{h}{p} = \frac{h}{{\sqrt {2\,m\,K} }} = \frac{h}{{\sqrt {2\,m\,eV} }}\)….(ii)
Again, substituting the numerical values \(h = 6.626 \times {10^{ – 34}}{\rm{J}}.{\rm{s}},\,m = 9.11 \times {10^{ – 31}}\,{\rm{kg}},\,e = 1.6 \times {10^{ – 19}}\,{\rm{C,}}\) we get
\(\lambda = \frac{{1.227}}{{\sqrt V }}{\rm{nm}}\)…..(iii)
Where \(V = \)The magnitude of accelerating potential in volts.
Now, de Broglie wavelength \(\lambda \) associated with electrons for \(V = 54\,\rm{V},\) by using equation (iii), we have
\(\lambda = \frac{{1.227}}{{\sqrt {54} }}\;{\rm{nm}}{\rm{.}}\)
\(\lambda = 0.167\;{\rm{nm}}{\rm{.}}\)

Results of the Davisson and Germer Experiment

Thus, from the above result of the experimental and theoretical data, we can say that there is a similarity between the theoretical and experimental values obtained from the De-Broglie wavelength and the Davisson- Germer experiment. Thus we can say that this experiment confirms the wave nature of electrons and the de Broglie relation.

Also, the wave nature of a beam of electrons was experimentally demonstrated in a double-slit experiment in \(1989,\) similar to the wave nature of light. For the development of modern quantum mechanics, we can use the basics of the de Broglie hypothesis. It is used in the field of electron optics. In the electron microscope design, the wave properties of electrons have been utilized, which is a great improvement, with higher resolution, over the optical microscope.

Summary

Like photons, particles of matter also have a dual nature, one is of particle, and the other is of the wave. de Broglie gave a formula connecting their mass, velocity, momentum (particle characteristics) with their wavelength and frequency (wave characteristics). In \(1927\) Thomson, and Davisson and Germer, in separate experiments, showed that electrons did behave like waves with a wavelength that agreed with that given by de Broglie’s formula.

Their experiment was on the diffraction of electrons through crystalline solids, in which the regular arrangement of atoms acted like a grating. Very soon, diffraction experiments with other ‘particles such as neutrons and protons were performed, confirmed with de Broglie’s formula. These experiments confirmed the wave-particle duality.

Sample Problem on Davisson-Germer Experiment

Q.1. What is the de Broglie wavelength associated with an electron, accelerated through a potential difference of \(100\) volts?
Ans: Accelerating potential \(V = 100\,\rm{V}.\) The de Broglie wavelength \(\lambda \) is
\(\lambda = \frac{h}{p} = \frac{{1.227}}{{\sqrt V }}\,{\rm{nm}}\)
\( \Rightarrow \lambda = \frac{{1.227}}{{\sqrt {100} }}\,{\rm{nm}}\)
\( \Rightarrow \lambda = 0.123\;{\rm{nm}}.\)
In this case, the de Broglie wavelength associated with an electron is of the order of X-ray wavelengths.

Study Equation of Photoelectric Effect

FAQs

Q.1. Davisson – Germer confirmed which theory in their experiment?
Ans: Davisson and Germer’s experiment proves the concept of the wave nature of matter particles. This experiment confirms that particles, such as electrons, are of dual nature as given in the de-Broglie hypothesis.

Q.2. Which crystal is used in the Davisson – Germer experiment?
Ans: The crystal used in the Davisson – Germer experiment is nickel. The electrons are scattered in all directions by the crystal atoms when a fine beam of electrons is made to fall on the surface of the nickel crystal.

Q.3. State the importance of Davisson- Germer experiment.
Ans: Some of the importance of Davisson-Germer experiment are given below:
(i) This experiment demonstrated the wave nature of the electron, confirming the hypothesis of de-Broglie.
(ii) It confirmed wave-particle duality.
(iii) It is an apparatus built to measure the energies of e- scattered from the surface of the metal.

Q.4. How does Davisson-Germer explain the wave nature of electrons?
Ans: Davisson-Germer experimented with verifying the de-Broglie hypothesis that a material particle possesses a wave nature. They use an electron beam to pass through a hole and strike the nickel crystal normally. After observing the results, they find that the maximum intensity was found due to constructive interference of two waves, and hence the wave nature of electrons was experimentally proved.

Q.5. Which phenomena supports that matter has a wave nature?
Ans: The wave nature of de Broglie was proved when accelerated electrons showed diffraction by metal foil in the same manner as \(X-\)ray diffraction.