Lecture 25 - Compton Effect - MODERN PHYSICS Notes | EduRev

: Lecture 25 - Compton Effect - MODERN PHYSICS Notes | EduRev

 Page 1


 Module 5 : MODERN PHYSICS
 Lecture 25 : Compton Effect
 
 
 
Objectives
 
In this course you will learn the following
Scattering of radiation from an electron (Compton effect).
Compton effect provides a direct confirmation of particle nature of radiation.
Why photoelectric effect cannot be exhited by a free electron.
 
Compton Effect
 
Photoelectric effect provides evidence that energy is quantized. In order to establish the particle nature of
radiation, it is necessary that photons must carry momentum. In 1922, Arthur Compton studied the
scattering of x-rays of known frequency from graphite and looked at the recoil electrons and the scattered
x-rays. 
According to wave theory, when an electromagnetic wave of frequency  is incident on an atom, it would
cause electrons to oscillate. The electrons would absorb energy from the wave and re-radiate
electromagnetic wave of a frequency . The frequency of scattered radiation would depend on the
amount of energy absorbed from the wave, i.e. on the intensity of incident radiation and the duration of
the exposure of electrons to the radiation and not on the frequency of the incident radiation.
 
Compton found that the wavelength of the scattered radiation does not depend on the intensity of incident
radiation but it depends on the angle of scattering and the wavelength of the incident beam. The
wavelength of the radiation scattered at an angle  is given by
 
 
.where  is the rest mass of the electron. The constant  is known as the Compton
wavelength of the electron and it has a value 0.0024 nm.
The spectrum of radiation at an angle  consists of two peaks, one at  and the other at . Compton
effect can be explained by assuming that the incoming radiation is a beam of particles with
 
Page 2


 Module 5 : MODERN PHYSICS
 Lecture 25 : Compton Effect
 
 
 
Objectives
 
In this course you will learn the following
Scattering of radiation from an electron (Compton effect).
Compton effect provides a direct confirmation of particle nature of radiation.
Why photoelectric effect cannot be exhited by a free electron.
 
Compton Effect
 
Photoelectric effect provides evidence that energy is quantized. In order to establish the particle nature of
radiation, it is necessary that photons must carry momentum. In 1922, Arthur Compton studied the
scattering of x-rays of known frequency from graphite and looked at the recoil electrons and the scattered
x-rays. 
According to wave theory, when an electromagnetic wave of frequency  is incident on an atom, it would
cause electrons to oscillate. The electrons would absorb energy from the wave and re-radiate
electromagnetic wave of a frequency . The frequency of scattered radiation would depend on the
amount of energy absorbed from the wave, i.e. on the intensity of incident radiation and the duration of
the exposure of electrons to the radiation and not on the frequency of the incident radiation.
 
Compton found that the wavelength of the scattered radiation does not depend on the intensity of incident
radiation but it depends on the angle of scattering and the wavelength of the incident beam. The
wavelength of the radiation scattered at an angle  is given by
 
 
.where  is the rest mass of the electron. The constant  is known as the Compton
wavelength of the electron and it has a value 0.0024 nm.
The spectrum of radiation at an angle  consists of two peaks, one at  and the other at . Compton
effect can be explained by assuming that the incoming radiation is a beam of particles with
 
Energy 
Momentum 
 
In arriving at the last relationship, we use the energy - momentum relation of the special theory of
relativity , according to which,
 
 
where  is the rest mass of a particle. Since photons are massless ( ), we get .
Compton's observation is consistent with what we expect if photons, considered as particles, collide with
electrons in an elastic collision.
Derivation of Compton's Formula
Consider a photon of energy  and momentum  colliding elastically with an electron at
rest. Let the direction of incoming photon be along the x-axis. After scattering, the photon moves along a
direction making an angle  with the x-axis while the scattered electron moves making an angle . Let
the magnitude of the momentum of the scattered electron be  while that of the scattered photon be 
.
 
 
See the animation
 
Conservation of Momentum
x-direction :
 
(1)
 
y-direction :
 
(2)
Page 3


 Module 5 : MODERN PHYSICS
 Lecture 25 : Compton Effect
 
 
 
Objectives
 
In this course you will learn the following
Scattering of radiation from an electron (Compton effect).
Compton effect provides a direct confirmation of particle nature of radiation.
Why photoelectric effect cannot be exhited by a free electron.
 
Compton Effect
 
Photoelectric effect provides evidence that energy is quantized. In order to establish the particle nature of
radiation, it is necessary that photons must carry momentum. In 1922, Arthur Compton studied the
scattering of x-rays of known frequency from graphite and looked at the recoil electrons and the scattered
x-rays. 
According to wave theory, when an electromagnetic wave of frequency  is incident on an atom, it would
cause electrons to oscillate. The electrons would absorb energy from the wave and re-radiate
electromagnetic wave of a frequency . The frequency of scattered radiation would depend on the
amount of energy absorbed from the wave, i.e. on the intensity of incident radiation and the duration of
the exposure of electrons to the radiation and not on the frequency of the incident radiation.
 
Compton found that the wavelength of the scattered radiation does not depend on the intensity of incident
radiation but it depends on the angle of scattering and the wavelength of the incident beam. The
wavelength of the radiation scattered at an angle  is given by
 
 
.where  is the rest mass of the electron. The constant  is known as the Compton
wavelength of the electron and it has a value 0.0024 nm.
The spectrum of radiation at an angle  consists of two peaks, one at  and the other at . Compton
effect can be explained by assuming that the incoming radiation is a beam of particles with
 
Energy 
Momentum 
 
In arriving at the last relationship, we use the energy - momentum relation of the special theory of
relativity , according to which,
 
 
where  is the rest mass of a particle. Since photons are massless ( ), we get .
Compton's observation is consistent with what we expect if photons, considered as particles, collide with
electrons in an elastic collision.
Derivation of Compton's Formula
Consider a photon of energy  and momentum  colliding elastically with an electron at
rest. Let the direction of incoming photon be along the x-axis. After scattering, the photon moves along a
direction making an angle  with the x-axis while the scattered electron moves making an angle . Let
the magnitude of the momentum of the scattered electron be  while that of the scattered photon be 
.
 
 
See the animation
 
Conservation of Momentum
x-direction :
 
(1)
 
y-direction :
 
(2)
 
From Eqns. (1) and (2), we get
 
(3)
 
Conservation of Energy : (relativistic effect)
 
If the rest mass of the electron is taken to be , the initial energy is  and the final energy is 
. Thus
 (4)
 
From Eqn. (4), we get, on squaring,
 
 
Thus,
 
 
On substituting expression (3) for  in the above equation, we get
 
 
Recalling  and  and on simplification, we get
 
 
Using , we get Compton's formula
 
 
 is known as the Compton Wavelength of an electron.
Page 4


 Module 5 : MODERN PHYSICS
 Lecture 25 : Compton Effect
 
 
 
Objectives
 
In this course you will learn the following
Scattering of radiation from an electron (Compton effect).
Compton effect provides a direct confirmation of particle nature of radiation.
Why photoelectric effect cannot be exhited by a free electron.
 
Compton Effect
 
Photoelectric effect provides evidence that energy is quantized. In order to establish the particle nature of
radiation, it is necessary that photons must carry momentum. In 1922, Arthur Compton studied the
scattering of x-rays of known frequency from graphite and looked at the recoil electrons and the scattered
x-rays. 
According to wave theory, when an electromagnetic wave of frequency  is incident on an atom, it would
cause electrons to oscillate. The electrons would absorb energy from the wave and re-radiate
electromagnetic wave of a frequency . The frequency of scattered radiation would depend on the
amount of energy absorbed from the wave, i.e. on the intensity of incident radiation and the duration of
the exposure of electrons to the radiation and not on the frequency of the incident radiation.
 
Compton found that the wavelength of the scattered radiation does not depend on the intensity of incident
radiation but it depends on the angle of scattering and the wavelength of the incident beam. The
wavelength of the radiation scattered at an angle  is given by
 
 
.where  is the rest mass of the electron. The constant  is known as the Compton
wavelength of the electron and it has a value 0.0024 nm.
The spectrum of radiation at an angle  consists of two peaks, one at  and the other at . Compton
effect can be explained by assuming that the incoming radiation is a beam of particles with
 
Energy 
Momentum 
 
In arriving at the last relationship, we use the energy - momentum relation of the special theory of
relativity , according to which,
 
 
where  is the rest mass of a particle. Since photons are massless ( ), we get .
Compton's observation is consistent with what we expect if photons, considered as particles, collide with
electrons in an elastic collision.
Derivation of Compton's Formula
Consider a photon of energy  and momentum  colliding elastically with an electron at
rest. Let the direction of incoming photon be along the x-axis. After scattering, the photon moves along a
direction making an angle  with the x-axis while the scattered electron moves making an angle . Let
the magnitude of the momentum of the scattered electron be  while that of the scattered photon be 
.
 
 
See the animation
 
Conservation of Momentum
x-direction :
 
(1)
 
y-direction :
 
(2)
 
From Eqns. (1) and (2), we get
 
(3)
 
Conservation of Energy : (relativistic effect)
 
If the rest mass of the electron is taken to be , the initial energy is  and the final energy is 
. Thus
 (4)
 
From Eqn. (4), we get, on squaring,
 
 
Thus,
 
 
On substituting expression (3) for  in the above equation, we get
 
 
Recalling  and  and on simplification, we get
 
 
Using , we get Compton's formula
 
 
 is known as the Compton Wavelength of an electron.
 
                     
 
Exercise 1
 
Show that the angle  by which the electron is scattered is related to the scattering angle  of the
photon by
 
Exercise 2
 
Is Compton effect easier to observe with I.R., visible, UV or X-rays ? In a Compton scattering experiment
the scattered electron moves in the same direction as that of the incident photon. In which direction does
the photon scatter ?
(Answer : X-rays, .)
 
Exercise 3
 
A 200 MeV photon strikes a stationary proton (rest mass 931 MeV) and is back scattered. Find the kinetic
energy of the proton after the scattering.
(Ans. 60 MeV)
Page 5


 Module 5 : MODERN PHYSICS
 Lecture 25 : Compton Effect
 
 
 
Objectives
 
In this course you will learn the following
Scattering of radiation from an electron (Compton effect).
Compton effect provides a direct confirmation of particle nature of radiation.
Why photoelectric effect cannot be exhited by a free electron.
 
Compton Effect
 
Photoelectric effect provides evidence that energy is quantized. In order to establish the particle nature of
radiation, it is necessary that photons must carry momentum. In 1922, Arthur Compton studied the
scattering of x-rays of known frequency from graphite and looked at the recoil electrons and the scattered
x-rays. 
According to wave theory, when an electromagnetic wave of frequency  is incident on an atom, it would
cause electrons to oscillate. The electrons would absorb energy from the wave and re-radiate
electromagnetic wave of a frequency . The frequency of scattered radiation would depend on the
amount of energy absorbed from the wave, i.e. on the intensity of incident radiation and the duration of
the exposure of electrons to the radiation and not on the frequency of the incident radiation.
 
Compton found that the wavelength of the scattered radiation does not depend on the intensity of incident
radiation but it depends on the angle of scattering and the wavelength of the incident beam. The
wavelength of the radiation scattered at an angle  is given by
 
 
.where  is the rest mass of the electron. The constant  is known as the Compton
wavelength of the electron and it has a value 0.0024 nm.
The spectrum of radiation at an angle  consists of two peaks, one at  and the other at . Compton
effect can be explained by assuming that the incoming radiation is a beam of particles with
 
Energy 
Momentum 
 
In arriving at the last relationship, we use the energy - momentum relation of the special theory of
relativity , according to which,
 
 
where  is the rest mass of a particle. Since photons are massless ( ), we get .
Compton's observation is consistent with what we expect if photons, considered as particles, collide with
electrons in an elastic collision.
Derivation of Compton's Formula
Consider a photon of energy  and momentum  colliding elastically with an electron at
rest. Let the direction of incoming photon be along the x-axis. After scattering, the photon moves along a
direction making an angle  with the x-axis while the scattered electron moves making an angle . Let
the magnitude of the momentum of the scattered electron be  while that of the scattered photon be 
.
 
 
See the animation
 
Conservation of Momentum
x-direction :
 
(1)
 
y-direction :
 
(2)
 
From Eqns. (1) and (2), we get
 
(3)
 
Conservation of Energy : (relativistic effect)
 
If the rest mass of the electron is taken to be , the initial energy is  and the final energy is 
. Thus
 (4)
 
From Eqn. (4), we get, on squaring,
 
 
Thus,
 
 
On substituting expression (3) for  in the above equation, we get
 
 
Recalling  and  and on simplification, we get
 
 
Using , we get Compton's formula
 
 
 is known as the Compton Wavelength of an electron.
 
                     
 
Exercise 1
 
Show that the angle  by which the electron is scattered is related to the scattering angle  of the
photon by
 
Exercise 2
 
Is Compton effect easier to observe with I.R., visible, UV or X-rays ? In a Compton scattering experiment
the scattered electron moves in the same direction as that of the incident photon. In which direction does
the photon scatter ?
(Answer : X-rays, .)
 
Exercise 3
 
A 200 MeV photon strikes a stationary proton (rest mass 931 MeV) and is back scattered. Find the kinetic
energy of the proton after the scattering.
(Ans. 60 MeV)
 Reason for the unshifted peak in the spectrum
 
When a photon strikes an atom (say carbon atom in a graphite crystal), it may scatter from a loosely
bound electron, which is essentially free. In this case there is a measurable shift in the wavelength of the
scattered photon. It is also likely that the photon scatters from an electron that is tightly bound to an
atom. In such a case, the mass appearing in Compton's formula must be replaced by the mass of the
carbon atom itself, which is approximately 20,000 times heavier than an electron. The maximum
wavelength shift of the photon for scattering from a free electron is twice the Compton wavelength of an
electron, i.e.  nm. In case of scattering from the carbon atom, the maximum wavelength
shift is approximately  nm, which is very small. Thus we find an intensity maximum at an
wavelength which is essentially equal to that of the incident wavelength.
 
A free electron cannot absorb a photon and increase its energy as doing so would violate energy-
momentum conservation.
 
Consider a free electron at rest which absorbs a photon of energy  (and momentum ). The final
energy of the electron would be . According to relativistic principle, if the momentum of the
electron is , the total energy is given by . When the electron absorbs the incident
photon, the momentum of the photon would be transferred to the electron. Since the electron was initially
at rest (i.e. with zero momentum), its final momentum is . Thus we have
 
 
which simplifies to , which is not possible.
 
The reason why an electron bound to an atom can absorb a photon ( as in Compton effect) is that the
electron can share some of the resulting momentum with the ion which has a much larger mass
 
Example 11
A photon of wavelength 6000 nm collides with an electron at rest. After scattering, the wavelength of the
scattered photon is found to change by exactly one Compton wavelength. Calculate (i) the angle by which
the photon is scattered, (ii) the angle by which the electron is scattered and (iii) the change in the energy
of the electron due to scattering.
 
Solution :
Since the change in wavelength is one Compton wavelength, , i.e. . Thus
the photon is scattered at right angles to the incident direction.
 
 
Initial momentum of the photon is
 
 
The final momentum of the photon is
 
 
Thus the final momentum of the electron is . The angle that the
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