What is Compton effect derive an expression for determination of kinetic energy of recoil electron in Compton effect?

A. H. Compton observed that “when a monochromatic beam of high frequency (lower wavelength) radiation (e.g., X-rays and γ-ray) is scattered by a substance, the scattered radiation contains two type of wavelengths one having same wavelength as that of incident radiation while the other having the wavelength greater (or lower frequency) than that of incident radiations. This effect is known as Compton Effect.

Quantum Explanation: The explanation was given by Compton which was based on quantum theory of light. According to quantum theory when photon of energy hυ strikes with the substance some of the energy of photon is transferred to the electrons, therefore the energy (or frequency) of photon reduces and wavelength increases.

Various assumptions were made for explaining the effect these were:

(i)                 Compton Effect is the result of interaction of an individual particle and free electron of target.

(ii)               The collision is relativistic and elastic.

(iii)             The laws of conservation of energy and momentum hold good.

The energy of the system before collision

According to the principle of conservation of energy

..............(1.32)

According to the principle of conservation of linear momentum along and perpendicular to the direction of incident photon (i.e., along x and y axis), we have

and

Squaring (1.33) and (1.34) and then adding, we get

or
..(1.35)

Squaring equation (1.32), we get

..(.1.36)

Subtracting (1.35) from (1.36), we get

..(1.37)

According to the theory of relativity

or

or

Multiplying both sides by c2, we get

Using equation (1.38) equation (1.37) becomes

To find the relation in term of wavelength, let us substitute

Compton shift

From above equations (1.39) and (1.40) following conclusions can be drawn

1.  The wavelength of the scattered photon λ’ is greater than the wavelength of incident photon λ.
2. ∆ λ is independent of the incident wavelength.
3. ∆ λ have the same value for all substance containing free electron
4. ∆ λ only depend on the scattering angle
   
   .

1. when   = 0; cos   = 1

             ∆ λ = λ’ – λ = 0

λ’  =  λ, the scattered wavelength is  same as the incident wavelength in the direction of incidence.

1.  when  =90 o ; cos f  =0

= λc ;                                                              ……..(1.41)

Where λc is called the Compton wavelength  of the electron.

1.  when  =180 o ; cos = -1
   
   
   ............(1.42)

Why Compton Effect is not observed in visible spectrum

The maximum change in wavelength

For X-ray, the incident radiation is about 1Ao ,max is 0.05 Aotherefore the percentage of incident radiation is about 5%(detectable)

For Visible radiation, the incident radiation is about 5000 Ao, max is 0.05 Ao therefore the percentage of incident radiation is about 0.001% (undetectable)

Direction of Recoil electron

Dividing equation (1.33) by (1.34) direction of recoil electron is given by

where

Kinetic Energy of Recoil Electron

The kinetic energy gained by electron is equal to the energy loss by the scattered photon

where