Compton Effect
When a monochromatic beam of X-rays of wavelength ‘λ’ is allowed to fall on a block of
paraffin or carbon, the beam is scattered i8nto two components. One component has a
wavelength equal to the incident wavelength and the other has a higher wavelength compared to
the incident wavelength. This phenomenon of change in wavelength of scattered X-rays is called
Compton shift and this effect is known as Compton Effect.
Fig. 2.Compton Effect
Theory of Compton Effect
Consider a photon of energy hν which collides elastically with a free electron in the
scattering substance as shown in Fig.2. During the elastic collision, the incident photon transfers
some of its energy to the electron. Due to this energy transfer, the electron acquires kinetic
energy and recoils with the velocity v. The scattered photon comes out with a reduced energy hν ’
and increased wavelength. The energy and momentum components of before and after collision
process are find as follows.
Energy – Before collision
Initial energy of photon = hν
Initial energy of electron = m0c2
... Total energy = hν+ m0c2
Energy – After collision
The final energy of the scattered photon = hν’
The final energy of the recoil electron = mc2
(m0 - rest mass of the electron)
--------- (1)
= hν’+ mc2
... Total energy after collision
--------- (2)
According to the law of conservation of energy, the energy of the electron and photon before and
after the collisions are equal. Therefore,
hν+ m0c2 = hν’+ mc2
---------- (3)
Momentum – Before collision
X – Component
= hν/c
Momentum of photon
Momentum of electron
=0
... Total Momentum along x - axis
= hν/c
---------- (4)
Y – Component
Momentum of photon
=0
Momentum of electron
=0
... Total Momentum along y - axis
=0
---------- (5)
Momentum – After collision
X – Component
ℎ𝜈 ′ 𝑐𝑜𝑠𝜙
Momentum of photon
=
Momentum of electron
= mv cosθ
... Total Momentum along x – axis =
𝑐
ℎ𝜈 ′ 𝑐𝑜𝑠𝜙
𝑐
+ mv cosθ
---------- (6)
Y – Component
Momentum of photon
=
ℎ𝜈 ′ 𝑠𝑖𝑛𝜙
𝑐
= - mv sinθ
Momentum of electron
... Total Momentum along y - axis
=
ℎ𝜈 ′ 𝑠𝑖𝑛𝜙
𝑐
- mv sinθ
---------- (7)
According to law of conservation of momentum,
hν/c =
0=
ℎ𝜈 ′ 𝑐𝑜𝑠𝜙
𝑐
ℎ𝜈 ′ 𝑠𝑖𝑛𝜙
𝑐
Rearranging equations (8) and (9),
+ mv cosθ
- mv sinθ
---------- (8)
---------- (9)
mvc cosθ = h(ν – ν’ cosϕ)
---------- (10)
mvc sinθ = hν’ sinϕ
---------- (11)
Squaring equation (10) and (11) and adding,
(mvc cosθ)2+( mvc sinθ)2 = h2(ν – ν’ cosϕ)2 + h2 ν’2 sin2ϕ
m2 v2 c2 = h2 (ν2 +ν’2 cos2 ϕ - 2ν ν’ cosϕ) + h2 ν’2 sin2 ϕ
[ . .. sin2θ + cos2θ = 1]
m2 v2 c2 = h2 (ν2 +ν’2 - 2ν ν’ cosϕ)
--------- (12)
From equation (3),
mc2 = hν - hν’+ m0c2
mc2 = h (ν - ν’) + m0c2
--------- (13)
Squaring equation (13),
m2 c4 = h2 (ν2 - ν’2 - 2ν ν’) + m20c4 + 2h (ν - ν’) m0c2
---------- (14)
Subtracting equation (12) from (14),
m2c2 (c2 – v2) = -2h2νν’ (1- cosϕ) + 2h (ν - ν’) m0c2 + m20c4
---------- (15)
According to Einstein’s theory of relativity,
𝑚=
𝑚0
2
√1−𝑣2
---------- (16)
𝑐
Squaring and rearranging equation (16),
m2c2 (c2-v2) = m20c4
Substituting equation (17) in (15),
m20c4 = -2h2νν’ (1- cosϕ) + 2h (ν - ν’) m0c2 + m20c4
2h (ν - ν’) m0c2 = 2h2νν’ (1- cosϕ)
---------- (17)
𝜈−𝜈 ′
𝜈𝜈 ′
1
𝜈′
𝑐
𝜈′
=𝑚
ℎ
0𝑐
1
-𝜈=𝑚
2
ℎ
0𝑐
𝑐
(1 - cosϕ)
2
ℎ
-𝜈=𝑚
0𝑐
ℎ
λ’ – λ = 𝑚
0𝑐
(1 - cosϕ)
(1 – cosϕ)
(1 – cosϕ)
--------- (18)
Therefore, the change in wavelength (or)
Compton shift (dλ) =
ℎ
𝑚0 𝑐
(1 – cosϕ)
--------- (19)
Where, h is Planck constant = 6.626x10-34 Js
Mo is the rest mass of electron = 9.1x10-31kg
C is the velocity of light = 3x108ms-1
ϕ is the scattering angle.
From equation (19), it is concluded that the change in wavelength dλ is independent of the
wavelength of the incident radiation as well as the nature of the scattering substance. This
relation shows that dλ depends only on the angle of scattering (ϕ).
Case (i): ϕ = 0
When ϕ = 0, cosϕ = 1, therefore the change in wavelength
dλ = 0
Case (ii): ϕ = 450
When ϕ = 450, cos450 = (1/√2) = 0.7071, therefore the change in wavelength
h
(dλ) = m
0
(1 – 0.7071) =
c
6.626𝑋 10−34 𝑋0.2929
9.1𝑋10−31 𝑋3𝑋108
=
1.9419𝑋10−34
27.3𝑋10−23
dλ = 0.0071A0
Case (iii): ϕ = 900
When ϕ = 900, cos900 = 0, therefore the change in wavelength
h
(dλ) = m
0
6.626𝑋 10−34
=
c 9.1𝑋10−31 𝑋3𝑋108
=
6.626𝑋 10−34
27.3𝑋10−23
dλ = 0.024271A0
Case (iv): ϕ = 1350
When ϕ = 1350, cos1350 = - 0.7071, therefore the change in wavelength
h
(dλ) = m
0
6.626𝑋 10−34
(1 - 0.7071) = 9.1𝑋10−31 𝑋3𝑋108 (1+0.7071)
c
=
11.31124𝑋10−34
27.3𝑋10−23
dλ = 0.04143A0
Case (v): ϕ = 1800
When ϕ = 1800, cosϕ = -1, therefore the change in wavelength
𝟐𝒉
dλ = 𝒎
𝟎𝒄
= 0.04854A0
Therefore, the change in wavelength dλ is maximum value at an angle ϕ = 1800.
Experimental Verification of Compton Effect
The conclusions of Compton were experimentally verified using a carbon scatterer.
Monochromatic X-rays produced from a Coolidge bulb become narrow by the slits S 1 and S2 are
incident on a carbon block. The scattered beam is received in a Bragg’s crystal spectrometer and
the wavelengths of the scattered X-rays are measured. The slit S3 is used to fix the direction of
the scattered ray. The experiment was performed using different wavelengths and different
materials as scatterers. The experimental results agreed well with the theoretical values.