Compton Wavelength

What is Compton?

Compton is the scattering of photons by a charged particle (electron).

Do you know what is light and how does it interact with matter? After the success of electromagnetic radiation in the 19th century, the light was considered a wave. It means the light was a self-propagating ripple in the electromagnetic field. 

However, in 1905, Einstein introduced the Photoelectric effect that considered light as a bunch of photons instead of waves. Initially, people didn’t believe in the photon theory of light; therefore, to make people believe in this theory, in 1922, Arthur Compton, worked on X-ray scattering, which was called Compton scattering. So, what is Compton scattering?

Compton Scattering

In 1912, Max Von Laue said that X-rays perform diffraction but the diffraction occurs in light only. If this is the case, then X-rays should be considered light and they have a wave nature. 

However, if we go to the past, in 1905, Einstein considered light as a bunch of small packets or quanta called photons. Since X-rays and light have the same nature, so if the light has a particle nature, X-rays must have a particle nature too.

Now, a person arrives named Arthur Holly Compton. In his experiment, named Compton scattering (or the effect called Compton effect), he said that when X-rays and electrons interact with each other, it seems like two balls are colliding with each other. 

For his experiment in 1927, Arthur was conferred Nobel Prize. Now, let us understand the Compton effect in detail.

Compton Effect

In the experiment, Arthur Compton considered a molybdenum X-ray tube from which a beam of X-rays of sharp wavelength (0.71 Å) when passed onto the graphite target, it scatters in different directions and the wavelength varies according to the scattering angle.

To study the X-rays, Compton employed Bragg’s spectrometer that can rotate around the graphite target, so why did he consider this instrument? It’s because we have to determine the Compton wavelength value and the spectrometer helps us determine the wavelength of electromagnetic waves falling on it.

Since the wavelength varies with the scattering angle, so the spectrometer rotates around the target to determine the varying wavelengths.

The scattering of rays continues; however, a case comes when the maximum part of wavelength λ0 when reaches the target changes to λ1 and that change is because of the scattering angle Θ. Compton observed that the difference between λ1 and λ0 totally depends upon the angle Θ. So, we arrive at the equation as:

λ1 - λ0 = h/mc(1 - Cos Θ) 

Here,

λ1 - λ0 = difference in the wavelength

m =  rest mass

c = speed of light ,i.e., 3  x 10⁸ m/s

h = Planck’s constant = 6.626 x 10⁻³⁴ J/s

Now, let’s derive the above equation mathematically:

Compton Wavelength

Let’s suppose that a photon collides with an electron. Initially, its kinetic energy and momentum are E0’ and p0’, when it collides with an electron along the x-axis, its kinetic energy decreases and changes to E1’ momentum to p1’, and the wavelength becomes '. The angle made by the scattering photon with the x-axis is Θ.

However, an electron gains momentum ‘p’ and kinetic energy ‘k’, and the angle made by an electron with the x-axis is Ф.

[Image will be Uploaded Soon]

We know that momentum remains conserved at all the axes. Here, we will apply the law of conservation of momentum at x and the y-axis:

• Along the X-axis

p0 = p1 Cos Θ + p Cos Ф

⇒ p0  - p1 Cos Θ= p Cos Ф……(1)

Now, along the y-axis, p0  becomes zero and dividing the component of the momentum of the electron and the photon along the y-axis will be:

0 = p1 Sin Θ - p Sin Ф

p1 Sin Θ = p Sin Ф  …(2)

Now, squaring equation (1) & (2), we get:

From equation (1):

(p0 - p1 CosΘ)² = p² Cos²Ф

= p0² + p1²CosΘ + 2p0p1 CosΘ = p² Cos²Ф …….(3)

From equation (2):

p1Sin²Θ = p²Sin²Ф …..(4)

And, adding equations (3) and (4), we get:

po² + p1² CosΘ + 2p0p1 CosΘ + p1Sin²Θ = p²Cos²Ф + p²Sin²Ф

po² + p1² + 2p0p1CosΘ = p² ……(5)

Now, applying the law of conservation of energy:

            E0 = E1 + k

            E0 - E1 = k

            cp0 - cp1 = k

⇒ p0 - p1 = k/c…..(5)

The momentum of a photon is given by:

                p = h/λ = hf/λf = E/c

So, we represent the energy of a photon as;

                       E = pc

The relation among the relativistic energy, momentum, and the rest mass of the electron after the collision is:

(k + mc²)² = p²c² + m²c⁴

⇒ k² + m²c⁴ + 2kmc² = p²c² + m²c⁴

⇒ k² + 2kmc² = p²c²

⇒ k²/c² + 2km = p² ……(6)

From eq (5) in (6):

(p0 - p1)² + 2km = p² …..(7)

From eq (5) in (6):

(p0 - p1)² + 2km = p0² + p1² - 2p0p1 CosΘ

On solving, we get the equation as:

1/p1 - 1/p0 = 1/mc (1 - CosΘ)

Now, putting p = λ/h 

⇒ λ0 - λ1 = h/mc (1 - Cos Θ)

The Compton wavelength value of an electron is 0.243 Å.