STUDY ON DE-BROGLIE WAVELENGTH OF AN ELECTRON

The wavelength of an electron has been calculated for a given accelerating voltage by using the Broglie relation between the momentum and the wavelength of an electron. Wavelength can be measured in SI units of meter and cgs units of cm. Nanometer or Angstrom is used for the unit of de Broglie wavelength of electrons. An electron’s de Broglie wavelength is inversely proportional to its momentum.     

3.1 The de Broglie wavelength of an electron

Louis De Broglie developed an equation that can express the wavelength of an electron. Wavelength of an electron is denoted by λ. It is the total distance from one crest to another crest of an electron. The mass of an electron is around 9.1*10-31 kg. Mass of electrons has been used to determine the wavelength of electrons. From the derivation of de Broglie equation, the wavelength of an electron was measured as 10-10 m. Electrons are one of the smallest particles in the world. As a reason, waves of electrons are only observed by electron microscopes. According to de Broglie, all particles have the property of waves. Wavelength of electrons is dependent on the momentum of electrons. Momentum of an electron is known as the multiplication of mass and velocity of electrons. Wavelength of electrons is shorter than the visible light. Wavelength is a property of electrons because electrons can carry energy and these small particles have motion properties. Wavelength of visible light is around 7 * 10-7 meters for red lights and 4 * 10-7 meters for violet light.        

3.2 Calculation of de Broglie wavelength of an electron

Planck’s constant is used to calculate the de Broglie wavelength of an electron. Planck’s constant is denoted as h. Value of Planck’s constant is 6.62607015 * 10-34 J.S. The relation between matter and energy can be established with the derivation of the de Broglie wavelength of electrons. Einstein established the energy equation of matters which is given below:

“E = mc2”

In this equation E denotes energy, m denotes mass and c denotes the velocity of light.

According to Planck’s theory, this equation can be written as:

“E = hν”

Here h denotes Planck’s constant and v is the velocity.The relation between the energy and frequency is E=hν

From the above two equations, one another equation can be developed as mc2=hν.

Now, no particle can travel at the speed of light. So, c can be written as v. Now this equation can be written as mv2= hv/λ. Now λ can be expressed as λ = hv/mv2or λ = h/p. This equation is known as the de Broglie wavelength equation of electrons. 

3.3 Application of de Broglie wavelength equation

From de Broglie wavelength equation wavelength of electrons can be determined in an easy way. For example, at 100 electron volts, the wavelength of an electron can be determined from de Broglie wavelength equation by substituting Planck’s constant, mass of electron and velocity of electrons. Final result of wavelength value will be released as 1.227 * 10-10 m. Wavelength of electrons with different acceleration voltages can be calculated by the implementation of Planck’s constant and mass of electrons. For example, wavelengths of electrons with acceleration voltages 10V, 100V, 1000V and 10000V will be around 3.87 * 10-10 m, 1.22 * 10-10 m, 3.87 * 10-11 m and 1.22 * 10-11 m respectively.

Small particles such as electrons only show wave properties. Wavelength of electrons can help to compare spacing between two electrons. For example, wavelength of electrons with the speed of 2* 106 meters per second can be determined by the de Broglie wavelength equation. Here velocity of electron is given as 2* 106 meters per second. Mass of an electron is around 9.1 * 10-31 kg. Value of Planck’s constant is 6.62607015 * 10-34 J.S. Now, the de Broglie wavelength equation is λ = h/mv. From above values and wavelength equation, the wavelength can be found as:

λ = 6.62607015 * 10-34 /(9.1 * 10-31)(2* 106)

Or, λ = 0.364 * 109 m. 

 Conclusion

It can be concluded that de Broglie’s wavelength of electrons is related to momentum and mass of electrons. It also can be concluded that mass of an electron is inversely proportional to the wavelength of the electron. As a result, if the mass of an electron is high then the wavelength of that electron will be below. Similarly, when the mass of an electron is lower, the wavelength of the electron will be higher. The de Broglie wavelength equation can help to determine different wavelengths of an electron with different accelerating voltages. It also can be concluded that the wavelength of electrons can be changed by changing the accelerating voltage of that electron.