Question

Describe Compton’s experiment. Present and explain the formula for the Compton wavelength shift.

Answer 1

The copmpton effect occurs when the incident x-ray photon with relatively high energy ejects an electron from an atom and xray photon of lower energy is scattered from the atom.

Compton shift is the increase in X-ray or gamma-ray wavelength resulting from the transfer of energy that accompanies the scattering of photons in the Compton effect.

It is given in the following mathematical form:

λs−λ0=hm0c(1−cosΘ)

Where,

Ө: angle at which radiation scattered

m₀: rest mass of an electron

hm0c : Compton wavelength of the electron

λ_s and λ₀: radiation spectrum peaks.

Derivation of Compton effect equation

Considering the elastic collide between a photon and an electron, following is the derivation:

hν : energy of photon

pi=hνc :momentum of the photon

pi=pfcosΘ+pecosϕ(1) (conservation of momentum in x direction)

0=−pfsinΘ+pesinϕ(2) (conservation of momentum in y direction)

p2e=p2e(cos2ϕ+sin2ϕ)

=(pi−pfcosΘ)2+p2fsin2Θ

=p2i+p2f−2pipfcosΘ

hν0+m0c2=hν+(√m20c4+p2ec2)

m20c4+p2ec2=(hν0−hν+m0c2)2

=(hν0−hν)2+m20c4+2m0c2(hν0−hν)

p2ec2=(hν0−hν)2+2m0c2(hν0−hν)

p2ic2+p2fc2−2pipfcosΘc2=(hν0−hν)2+2m0c2(hν0−hν)

hνν0(1−cosΘ)=m0c2(ν0−ν)

∴λs−λ0=hm0c(1−cosΘ)

Therefore, above is the Compton effect equation andhm0c≡λc is Compton wavelength of an electron.

The compton wavelength shift depends on scattering angle.

the maximum increase in a photon’s wavelength (which is a decrease in its energy or frequency) is twice the Compton wavelength of the interacting electron. It’s because the maximum energy is lost only when a high-energy photon scores a direct hit and bounces straight back.

After a direct hit, the photon shoots straight back—an angle of 180 degrees, right?

The cosine of 180 degrees is -1, correct? 1-(-1) = 2. The rest of the equation for Compton scattering is simply the Compton wavelength of an electron, which is the inverse of the mass of an electron divided by the speed of light—all multiplied by Planck’s constant.

It’s important to understand that Compton scattering is a phenomenon of the high-energy light of x-rays and gamma rays. It’s not observed at lower frequencies (longer wavelengths), because at those levels all of a photon’s energy during an interaction is transferred to the electron and the photon disappears.

That said, it is important to point out that unless a lower-energy photon is in a kind of resonance with an electron, it won’t interact with it at all but will instead pass by, through, and around. It’s why glass is transparent to visible light. A sheet of iron is not.

When a photon is fired at an atom, the mass of the entire atom must be converted to a Compton wavelength. Since the mass of an atom is in the denominator and that mass will always be many thousands of times greater than a free electron, the larger mass-term in the denominator will drive the change in energy of the photon to nearly zero.

In the case of atoms, the wavelength of the incoming photon will remain essentially unchanged except in the most extreme photon energy states.