De Broglie Equation or Relationship

In this chapter, we will learn about the dual nature of particles: de Broglie equation or relationship. In 1905, Albert Einstein had suggested that the light has dual character, that is, sometimes it can behave as a wave as well as sometimes like a particle.

In 1924, the French Physicist, Louis de Broglie took the advantage of the above fact and proposed that all matter particles (such as electrons, protons, neutrons, atoms, molecules, etc.) have a dual character, i.e. wave as well as particle character. This proposal gave birth to a new theory known as wave mechanical theory of matter.

In Bohr’s theory, an electron is treated as a particle. But, according to de-Broglie’s theory, each and every moving particle, such as electrons, protons and even atoms, possesses dual character, both as a material particle and as a wave.

Louis De Broglie proposed the name, matter waves or de Broglie waves for the wave produced by the moving particles. The matter waves are fundamentally different from other types of waves, such as mechanical waves (e.g., sound waves) and electromagnetic waves (e.g., light waves).

Derivation of De Broglie Equation or Relationship

Louis de Broglie derived a relationship for the calculation of the wavelength (λ) associated with a particle of mass m, moving with the velocity v, as given below:

λ = h/mv = h/p (de Broglie equation)

Here, h is Planck’s constant and p is the momentum of the particle.

This relationship can be derived as follows:

According to Planck’s equation, E = hν – – – – – (1)

Energy of the photon on the basis of Einstein’s mass energy relationship, E = mc² — – – – – (2)

From equation (1) and (2), we get mc² = hν

mc² = hc/λ (∵ ν = c/λ )

Rearranging, we get λ = h/mc which is the same of de Broglie equation (λ = h/mv).

For a material particles, such as electron, proton, neutron, or any object such as ball, having mass m, and velocity v, the de Broglie equation is given by,

λ = h/mv = h/p

The wavelength associated with a material particle is commonly known as de Broglie wavelength. From the above de Broglie’s equation, it is clear that the momentum of a particle in motion is inversely proportional to wavelength, Planck’s constant ‘h’ being the constant of proportionality.

The below table gives the de Broglie wavelengths of some material particles.

Table: de Broglie Wavelengths of Some Material Particles

The result shown in the above table indicates that the de Broglie’s equation is true for all particles, but it is only with very small particles, such as electrons. Large particles in motion possess wavelength, but it is not measurable or observable. Let us, for instance consider de Broglie’s wavelengths associated with tennis ball which is 6.6 * 10^-34 meter. This value is too small to be measurable by any instrument and hence no significance.

Now consider the de Broglie wavelength of an electron which is 6.067 * 10-9 m. This value is quite comparable to the wavelength of X-rays and hence detectable.

Davison and Germers Experiment

De Broglie equation experimentally verified by Davisson and Germer by observing diffraction effects with an electron beam. In 1927, Davisson and Germer proved the physical reality of the wave nature of electrons by showing that a beam of electrons could also be diffracted by crystals just like light or X-rays.

In their experiment, Davison and Germer studied the scattering of slow moving electrons by reflection from the surface of nickel crystal, which provided experimental confirmation of de Broglie’s theory about the wave nature of matter.

They used a heated filament to generate electrons and passed the stream of electrons through charged plates kept at a potential difference V. This potential difference accelerated the electrons by exerting an electric force on each electron.

The kinetic energy acquired by an electron due to this electric field is mv²/2, where m is the mass of the electron and v is its velocity after acceleration.

This kinetic energy is equal to the work done by the electric field on the electron, which is eV, where e is the charge of the electron and V is the potential difference.

The kinetic energy acquired by an electron due to the electric field must be equal to the electrical force. Thus,

mv²/2 = eV

Multiplying m on both sides, we get

m²v² = 2eVm

mv = √2mVe – – – – – (1)

But according to de Broglie’s relationship, mv = h/λ – – – – – (2)

Comparing equation (1) and (2), we get

h/λ = √2mVe

λ = h/√2mVe

Substituting for h = 6.625 * 10-27 erg-sec, m = 9.1091 * 10-28 g, e = 4.803 * 10-10 esu, we will get the relation:

λ = √150/V volts Å

This is the relationship between the kinetic energy of the electrons and the potential difference accelerating them.

This relationship shows that by applying different potential differences, the wavelength of the electrons can be controlled. For instance, if a potential difference of 150 volts is applied, the wavelength of the electrons emerging is approximately 1 Å (angstrom). Similarly, if the potential difference is increased to 1500 volts, the wavelength of the electrons decreases to about 0.1 Å.

Thus, it is clear that electrons of different wavelengths can be obtained by changing the potential drop. Since these wavelengths are comparable to those of X-rays, electrons can undergo diffraction when directed at a crystalline surface. This phenomenon provides direct evidence of the wave-like properties of electrons, as predicted by de Broglie’s theory.

Apparatus Used by Davison and Germer

A schematic representation of the apparatus used by Davison and Germer is shown in the below figure.

When the electrons fall upon the surface of a nickel crystal, they get diffracted, similar to how light or X-rays behave when interacting with a crystal lattice. Electrons of a specific wavelength get diffracted at specific angles. An electron detector measures the angle of diffraction, denoted as θ, on the graduated circular scale.

According to Bragg’s law of diffraction, the wavelength λ of the diffracted electrons is given by:

λ = d sin θ, where d is a constant (= 2.15 for Ni crystal) and θ is the angle of diffraction.

By substituting the experimentally determined value of θ into Bragg’s equation (λ = d sin θ), the wavelength λ of the electrons can be calculated. This experimentally determined wavelength matches the wavelength predicted by the de Broglie equation, confirming the wave like nature of electrons.

Since diffraction is a phenomenon associated exclusively with wave motion, the Davisson and Germer experiment provided conclusive evidence of the wave properties or nature of electrons. Thus, electrons not only behave like particles in motion but also have wave properties associated with them. However, it is not easy at this stage to get an idea of this new conception of the motion of an electron.

Bohr Theory versus De Broglie Equation

One postulate of the Bohr’s atomic theory states that the angular momentum of an electron in orbit is quantized, meaning that it is an integral multiple of h/2π, where h is Planck’s constant. This postulate can be derived using de Broglie’s concept of the wave nature of electrons.

Let us consider an electron moving in a circular orbit around the nucleus. According to de Broglie, a wave is associated with a moving electron. For the electron wave to be in phase and form a stable standing wave, the circumference of the Bohr orbit must be an integral multiple of the wavelength λ of the electron. This condition is expressed as:

2πr = nλ

where r is the radius of the orbit, n is a positive integer (1, 2, 3, etc.), and λ is the de Broglie wavelength of the electron.

Substituting de Broglie’s relation λ = h/mv in the above equation, we get:

Rearranging the terms,

mvr = nh/2π

Here, v is the velocity of the electron and r is the radius of the orbit.

Thus, angular momentum = nh/2π

This relation proves that de Broglie and Bohr concepts are in perfect agreement with each other.

Solved Examples on De Broglie Equation

Let’s take some solved examples based on de broglie equation.

Example 1: Calculate the de Broglie wavelength of a body of mass 1 mg moving with a velocity of 10 msec^-1.

Solution: According to de Broglie equation, λ = h/mv

Mass of the body m = 1 mg = 10^-6 kg

Velocity v = 10 msec^-1

Wavelength λ = 6.625 * 10^-34 / (10^-6 * 10) = 6.625 * 10^-29 m (Ans.)

Example 2: Calculate the momentum of a moving particle having de Broglie wavelength of 200 pm.

Solution: We know from the de Broglie equation, λ = h/mv = h/p

Momentum p = h/λ = 6.625 * 10^-34 / 2 * 10^-10 = 3.31 * 10^-24 kgms^-1 (Ans.)

Example 3: Calculate the de Broglie if a moving electron has 5 * 10^-25 Joules of kinetic energy.

Solution: Mass of electron m = 9.1 * 10^-31 kg

Kinetic Energy KE = mv²/2 = 5 * 10^-25 J

9.1 * 10^-31 * v² = 2 * 5 * 10^-25

v = √(2 * 5 * 10^-25) / 9.1 * 10^-31 = 1.048 * 10³ ms^-1

h = 6.62 * 10^-34 Js = 6.62 * 10^-34 kg m s^-1

De Broglie wavelength λ = h/mv = 6.62 * 10^-34 kg m s^-1 / (9.1 * 10^-31 kg * 1.048 * 10³ ms^-1)

De Broglie wavelength λ = 6.84 * 10^-7 m (Ans.)

Example 4: Calculate the wavelength of the wave associated with the electron beam if an electron beam is accelerated by a potential difference of 10000 volts.

Solution: Mass of electron m = 9.1 * 10^-31 kg

Kinetic energy of the electron = 10000 eV = 10000 * 1.602 * 10^-19 J = 1.602 * 10^-15 J

Velocity of the electron can be calculated from kinetic energy as follows:

mv²/2 = 1.602 * 10^-15 J

v = √(2 * 1.602 * 10^-15)/9.1 * 10^-31 = 5.9 * 10⁷ ms^-1

Now we know from de Broglie equation, λ = h/mv

λ = h/mv = 6.62 * 10^-34 / (9.1 * 10^-31 * 5.9 * 10⁷) = 0.123 * 10^-10 m = 0.123 Å (Ans.)

FAQ on De Broglie Equation or Relationship