The document discusses Compton scattering and the Compton effect. It defines the Compton effect as an interaction between a photon and loosely bound electron that results in the photon being scattered at a lower energy and the electron ejected. Key equations presented include: - The Compton shift equation relating the wavelength shift to scattering angle and Compton wavelength. - Relationships between scattering angle and recoil angle, scattered photon energy as a function of incident energy and angle. - Expressions for the differential Klein-Nishina cross section and distributions of scattered photon energy and recoil electron kinetic energy.

1. COMPTON EFFECT
INCOHERENT SCATTERING

2. COMPTON EFFECT
• WHAT IS COMPTON EFFECT ?
• COMPTON SHIFT .
• RELATION BETWEEN SCATTERING ANGLE AND RECOIL ANGLE.
• SCATTERED PHOTON ENERGY AS A FUNCTION OF INCIDENT PHOTON ENERGY.
• ENERGY TRANSFER TO RECOIL ELECTRON.
• DIFFERENTIAL ELECTRONIC CROSS SECTION FOR COMPTON SCATTERING, SCATTERING ANGLE AND RECOIL ANGLE.
• DIFFERENTIAL KLEIN–NISHINA ENERGY TRANSFER CROSS SECTION.
• ENERGY DISTRIBUTION OF RECOIL ELECTRONS.
• TOTAL ELECTRONIC KLEIN–NISHINA CROSS SECTION FOR COMPTON SCATTERING.
• ELECTRONIC ENERGY TRANSFER CROSS SECTION FOR COMPTON EFFECT.
• BINDING ENERGY EFFECTS AND CORRECTIONS.
• COMPTON ATOMIC CROSS SECTION, MASS ATTENUATION COEFFICIENT AND MASS ENERGY TRANSFER COEFFICIENT.

3. WHAT IS COMPTON EFFECT
• An interaction of a photon of energy
ℎ𝜈 with a loosely bound orbital
electron of an absorber is called
Compton effect.(see
fig 1). This phenomenon was
discovered by ARTHUR COMPTON
• A photon, referred to as a scattered
photon with energy ℎ𝑣′ that is smaller
than the incident photon energy ℎ𝜈, is
produced in Compton effect and an
electron, referred to as a Compton
(recoil) electron, is ejected from the
atom with kinetic energy 𝐸𝑘.
Fig. 1) Diagrammatic representation of Compton Effect

4. WHAT IS COMPTON EFFECT
• A typical Compton effect interaction is
shown in Fig. 2. Here;
• 𝜃 = Scattering angle of the Scattered
Photon
∅= Scattering angle of the recoil electron
𝑃𝑣= Momentum of photon before collision
𝑃𝑣′= Momentum of photon after collision
𝑃𝑒= Momentum of the recoil electron.
Fig. 2) A schematic diagram representing Compton effect.

5. COMPTON WAVELENGTH SHIFT EQUATION
• The amount by which the light's
wavelength changes is called the Compton
shift, and is given by the equation:-
• 𝜆= Wavelength of incident photon
=
2𝜋ħ
hv
𝜆′
= Wavelength of scattered photon
=
2𝜋ħ
h𝑣′
∆𝜆= Difference between the two
wavelength
𝜆𝐶= Compton wavelength of the electron
𝜆𝐶 =
ℎ
𝑚𝑒𝑐
=
2𝜋ħc
𝑚𝑒𝑐2
= 0.0243 Å
• ∆𝜆 = 𝜆′
− 𝜆
= 𝜆𝐶 (1 − cos 𝜃)

6. COMPTON WAVELENGTH SHIFT EQUATION
Before Compton Interaction After Compton Interaction
Total Energy before interaction
hν + 𝑚𝑒𝑐2
Total Energy after interaction
hν + 𝐸𝐾𝑚𝑒𝑐2
Momentum before interaction (X-axis)
ℎ𝑣
𝑐
Momentum after Interaction
ℎ𝑣′
𝑐
cos 𝜃 + 𝑃𝑒 cos ɸ
Momentum before interaction (Y-axis)
0
Momentum after Interaction
ℎ𝑣′
𝑐
sin 𝜃 − 𝑃𝑒 sin ɸ

7. RELATIONSHIP BETWEEN SCATTERING ANGLE AND
RECOIL ANGLE
• cot
𝜃
2
= (1 + 𝜖) tan ɸ
• 𝜖 =
ℎ𝑣
𝑚𝑒𝑐2
• ε = incident photon energy hν normalized to
electron rest mass energy (𝑚𝑒𝑐2
= 0.511
keV)
• This figure the relationship between the
recoil electron ∅ and photon scattering
angle 𝜃 for various values of 𝜖
• Stating that for a given θ, the higher is the
incident photon energy hν or the higher is ε,
the smaller is the recoil electron angle ɸ.

8. SCATTERED PHOTON ENERGY AS FUNCTION OF
INCIDENT PHOTON ENERGY
AND PHOTON SCATTERING ANGLE
• ℎ𝑣′ = ℎ𝑣
1+𝜀 2−𝜀 𝜀+2 cos2 ɸ
( 1+𝜀 2−𝜀2 cos2 ɸ
• Scattered photon energy hν’ against the
incident photon energy hν for various
scattering angles 𝜃 in the range from 0° to
180°,
the following conclusion can be said:-

9. SCATTERED PHOTON ENERGY AS FUNCTION OF
INCIDENT PHOTON ENERGY
AND PHOTON SCATTERING ANGLE
1. For θ = 0; ɸ = ½ π
Energy of the scattered photon hν’ equals the
energy of the incident photon hν, Since in this
case no energy is transferred to the recoil
electron, (Which is just classical Thomson
scattering.)
2. For θ > 0;
The energy of the scattered photon saturates at
high values of hν.
The higher the scattering angle 𝜃, the lower is
the value of hν’.

10. SCATTERED PHOTON ENERGY AS FUNCTION OF
INCIDENT PHOTON ENERGY
AND PHOTON SCATTERING ANGLE
3. For 𝜃 =
1
2
𝜋; ɸ =
1
2
𝜋
Remember! -cot
𝜃
2
= 1 + 𝜖 tan ɸ
ℎ𝑣′
=
ℎ𝑣
1 + 𝜀
Which after a series of calculation, will give us 0.511
MeV.
4. For 𝜃 = 𝜋; ɸ = 0
ℎ𝑣′ =
ℎ𝑣
1 + 2𝜀
Which after a series of calculation, will give us 0.255
MeV.
5. The above two points shows that photons scattered
with 𝜃 >
1
2
𝜋, cannot exceed 511 Kev in Kinetic
Energy.
More over the maximum energy for back scatter
cannot exceed 0.255 MeV.*

11. COMPTON SCATTERING FUNCTION
• Compton scatter fraction 𝑓′𝐶 ℎ𝜈, 𝜃 ,
can be define as the ratio between the scattered
photon energy hν’ to the incident photon
energy hν. From the previous equation
we can write

12. COMPTON SCATTERING FUNCTION
• Using this equation, we can plot a graph against
scattering angle 𝜃 for various incident photon
of energies ℎ𝜈 in the range form 10 MeV to 100
MeV

13. COMPTON SCATTERING FUNCTION
• Using this equation, The following features are
notable.:-
1. hν = hν’ for all hν at 𝜃 = 0
𝑓’𝑐 (hν,𝜃) 𝜃 = 0 = 1
for all hν from 0 to infinity.
2. For a given hν, as 𝜃 increases, the Compton
scattering factor decreases gradually to level of
at
3. For a given scattering angle 𝜃, the larger is the
incident photon energy, the smaller is the
Compton Scattering Factor 𝑓′𝐶 (ℎ𝜈, 𝜃)

14. ENERGY TRANSFER TO COMPTON RECOIL ELECTRON
• Kinetic energy of the Compton (recoil)
electron 𝐸𝑘
𝑐
(ℎ𝜈, 𝜃) depends on photon
energy hν and photon scattering angle 𝜃.
The relationship is determined using
conservation of energy.
From which we get;

15. ENERGY TRANSFER TO COMPTON RECOIL ELECTRON
• We insert the equation to get;
For a Photon energy hν, the recoil electron kinetic
energy ranges
from a minimum value of
(𝐸𝑘
𝑐
)min = 0 for scattering angle 𝜃 = 0
(forward scattering)
corresponding to electron recoil angle ɸ = ½ 𝜋,
To a maximum value of Scattering angle = 𝜃 = 0
(Backscattering)
corresponding to electron recoil angle ɸ = 𝜃
The Figure, with solid curves, shows a plot of 𝑓𝑐(ℎ𝜈, 𝜃) against
scattering angle 𝜃 for various incident photon energies in the
range from 10 keV to 100 MeV.

16. ENERGY TRANSFER TO COMPTON RECOIL ELECTRON
• We insert the equation to get;
For a Photon energy hν, the recoil electron kinetic
energy ranges
from a minimum value of
(𝐸𝑘
𝑐
)min = 0 for scattering angle 𝜃 = 0
(forward scattering)
corresponding to electron recoil angle ɸ = ½ 𝜋,
To a maximum value of Scattering angle = 𝜃 = 𝜋
(Backscattering)
corresponding to electron recoil angle ɸ =0
The ratio of the kinetic energy of the
Compton (recoil) electron 𝐸𝑘
𝑐
(ℎ𝜈, 𝜃) to the
energy of the incident photon ℎ𝜈,
represents the fraction of the incident
photon energy that is transferred to the
Compton electron in a Compton effect.
This is called the Compton energy transfer
fraction 𝑓𝑐(ℎ𝜈, 𝜃).

17. DIFFERENTIAL ELECTRONIC CROSS SECTION FOR
COMPTON SCATTERING.
• Oskar Klein and Yoshio Nishina in 1928, derived an expression for the cross-section
of Compton interaction between a proton and a free electron.
And is given as
• The Differential Klein–Nishina electronic cross section
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑Ω
, which is

18. AND

19. THE KLEIN-NISHINA FACTOR IS PLOTTED AGAINST THE SCATTERING ANGLE 𝜃 FOR VARIOUS
VALUES OF THE ENERGY PARAMETER
𝜖 =
ℎ𝑣
𝑚𝑒𝑐2
ε = incident photon energy
hν normalized to electron
rest mass energy (𝑚𝑒𝑐2 =
0.511 keV)

20. • KLEIN–NISHINA ATOMIC FORM FACTOR FOR
COMPTON EFFECT 𝐹𝐾𝑁AGAINST SCATTERING ANGLE
𝜃
• So basically,
1. At low 𝜀, the probability of forward
scattering and backward scattering is
equal.
2. As 𝜀 increases, scattering becomes
increasingly more forward peek and
backscattering rapidly diminishes.

21. DIFFERENTIAL ELECTRONIC CROSS SECTION PER UNIT
SCATTERING ANGLE
DIFFERENTIAL
ELECTRONIC
CROSS SECTION
PER UNIT
SCATTERING
ANGLE
• It is important to consider the directional
distribution of scattered photons and
recoil electrons in the form of the cross
section per unit scattering angle 𝜃 and
per unit recoil angle ɸ.
• The differential electronic cross section
per unit scattering angle is obtained from
differential electronic cross section per
unit solid angle
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑Ω
𝑑Ω = 2𝜋 sin𝜃 𝑑𝜃

22. DIFFERENTIAL ELECTRONIC CROSS SECTION PER UNIT
SCATTERING ANGLE
• The differential cross section
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝜃
is plotted
against scattering angle 𝜃
• Differential electronic cross section per unit
scattering angle 𝜃 for Compton effect (
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝜃
)
against scattering angle 𝜃 (solid curves) and
differential electronic cross sections per unit
recoil angle ɸ for Compton effect
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝜑
against recoil angle 𝜑 (dashed curves)
for four values of ε (0, 0.1, 1, and 10), the
incident photon energy hν normalized to the
rest mass energy of the electron 𝑚𝑒𝑐2
. The
cross sections are drawn on a cartesian plot

23. DIFFERENTIAL CROSS SECTION PER UNIT RECOIL
ANGLE
• The differential electronic cross section
per unit recoil angle
𝑑𝑒𝜎𝐶
𝑑𝜑
is determined from
the differential electronic cross section
per unit scattering angle
𝑑𝑒𝜎𝐶
𝑑𝜃
which is
written as follows.
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑ɸ
=
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝜃
𝑑𝜃
𝑑ɸ
After Differentiating
Both Sides

24. • After incorporating the previous
equations, it is expressed with the
following two equations
DIFFERENTIAL CROSS SECTION PER UNIT RECOIL
ANGLE

25. DIFFERENTIAL ELECTRONIC CROSS SECTION PER UNIT
SCATTERING ANGLE
• The differential cross section
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝜃
is plotted
against scattering angle 𝜃
• Differential electronic cross section per unit
scattering angle 𝜃 for Compton effect (
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝜃
)
against scattering angle 𝜃 (solid curves) and
differential electronic cross sections per unit
recoil angle ɸ for Compton effect
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝜑
against recoil angle 𝜑 (dashed curves)
for four values of ε (0, 0.1, 1, and 10), the
incident photon energy hν normalized to the
rest mass energy of the electron 𝑚𝑒𝑐2
. The
cross sections are drawn on a cartesian plot

26. The Following in noticed:-
1. The area under the 𝑑𝑒
𝜎𝐶
𝐾𝑁
𝑑ɸ
decreases
with increasing 𝜀.
2. With increasing 𝜀, the electron curve
becomes more and more
asymmetrical and for large ε exhibits
only one peak which moves to
increasingly smaller angles ɸ.
DIFFERENTIAL CROSS SECTION PER UNIT RECOIL
ANGLE

27. DIFFERENTIAL CROSS SECTION PER UNIT RECOIL
ANGLE
The angular distribution of recoil electrons is
present only in the forward hemisphere; however, it
is zero in forward direction ɸ = 0 and exhibits
maxima at values of ɸ which depend on photon
energy hν.
In other words, larger the ε, the
smaller is the angle at which maximum occurs.

28. DIFFERENTIAL KLEIN–NISHINA ENERGY TRANSFER
CROSS SECTION
• The differential electronic energy transfer coefficient
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑡𝑟
𝑑Ω
for the Compton
effect is calculated by multiplying the differential electronic cross section of
&
We GET
𝜖 =
ℎ𝑣
𝑚𝑒𝑐2
ε = incident photon energy
hν normalized to electron
rest mass energy (𝑚𝑒𝑐2 =
0.511 keV)

29. ENERGY DISTRIBUTION FOR RECOIL ELECTRON
• The Energy distribution for recoil electron is given by the equation:-
• Where;
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑Ω
=
𝑟𝑒
2
2
1 + cos2 𝜃 𝐹𝐾𝑁
𝑑Ω
𝑑𝜃
= 2𝜋 sin 𝜃
𝑑𝜃
𝑑𝐸𝑘
= ℎ𝜈
2𝜀sin2 𝜃
2
1 + 2𝜀sin2 θ
2

30. ENERGY DISTRIBUTION FOR RECOIL ELECTRON
The differential electronic cross section
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝐸𝐾
is plotted against the kinetic energy
𝐸𝑘
𝐶
of the recoil electron for various values
of incident photon electron.
The following features can be recognized :
-
1. The distribution of kinetic energies
given to the Compton recoil electrons
is essentially flat from zero almost to
the maximum electron kinetic energy
𝐸𝐾
𝐶
𝑀𝐴𝑋
where a sharp increase in
concentration occurs.
2. 𝐸𝐾
𝐶
𝑀𝐴𝑋
= ℎ𝑣
2𝜀
1+2𝜀

31. TOTAL ELECTRONIC KLEIN–NISHINA CROSS SECTION
FOR COMPTON SCATTERING
• This is calculated by integrating the differential electronic cross section per unit
solid angle over the whole solid angle to get :-
The numerical value of 𝑒𝜎𝐶
𝐾𝑁
can be obtained by determining the area under the
𝑑𝑒𝜎𝐶
𝐾𝑁
𝑑𝜃
curve for a given 𝜀.

32. TOTAL ELECTRONIC KLEIN–NISHINA CROSS SECTION
FOR COMPTON SCATTERING
• 2 Cases;
1. For small incident photon energies :-
Which is almost equal to Thomson’s results =
2. For very large incident photon energies :-

33. TOTAL ELECTRONIC KLEIN–NISHINA CROSS SECTION
FOR COMPTON SCATTERING
• At low photon energies (𝐸𝜎𝐶
𝐾𝑁
) is
approximately equal to Thompson cross
section, which is independent of Photon
energy, with its value of 0.665b.
• Intermediate Photon energy
(𝐸𝜎𝐶
𝐾𝑁
)decreases gradually with photon
energy.
• The Compton electronic cross section
(𝑒𝜎𝐶
𝐾𝑁
) is independent of atomic number Z
of the absorber, since in the Compton
theory the electron is assumed to be free
and stationary, i.e., the electron’s binding
energy to the atom is assumed to be
negligible in comparison with the photon
energy hν.

34. ELECTRONIC ENERGY TRANSFER CROSS SECTION
FOR COMPTON EFFECT
• This is obtained by integrating the differential electronic energy transfer cress
section
d(𝑒𝜎𝑐
𝐾𝑁)𝑡𝑟
𝑑𝛺
of overall photon scattering angle from 0 to 180, and also over
recoil angle , from 0 to 90.

35. BINDING ENERGY EFFECTS CORRECTIONS
• Compton atomic cross section aσC plotted
against incident photon energy hν for
various absorbers ranging from hydrogen
to lead. The dashed curves represent
(𝑎𝜎𝐶
𝐾𝑁
)
data calculated with Klein–Nishina free-
electron relationship
• The solid curves represent the (𝑎𝜎𝐶) data
that incorporate the binding effects of
orbital electrons

36. BINDING ENERGY EFFECTS CORRECTIONS
• For a given Z of the absorber, the lower is
the incident photon energy hν, the larger
is the discrepancy between the measured
(𝑎𝜎𝐶) and the calculated (𝑎𝜎𝐶
𝐾𝑁
) .
• For a given incident photon energy hν, the
higher is the atomic number Z of the
absorber, the more pronounced is the
discrepancy

37. BINDING ENERGY EFFECTS CORRECTIONS
• Incoherent scattering function S(x, Z)
plotted against the momentum transfer
variable x where x =
sin
𝜃
2
𝜆
for various
absorbers in the range from hydrogen to
lead
• For large values of x the incoherent
scattering function S(x, Z) saturates at Z

38. COMPTON ATOMIC CROSS SECTION AND MASS
ATTENUATION COEFFICIENT

39. COMPTON MASS ENERGY TRANSFER COEFFICIENT
• The Compton mass energy transfer
coefficient
𝜎𝐶 𝑡𝑟
𝜌
is calculated from the
mass attenuation coefficient
𝜎𝐶
𝜌
using the
standard relationship
𝑓𝑐- Mean energy transfer fraction for the
Compton effect
Figure shows the (𝑎𝜎𝐶) and (𝑎𝜎𝐶
𝐾𝑁
)data for lead in addition, it
also shows the binding energy effect on the Compton atomic
energy transfer coefficients of lead by displaying (a𝜎𝐶)𝑡𝑟and
(𝑎𝜎𝐶
𝐾𝑁
)𝑡𝑟

40. IMPORTANT FORMULAS
Compton Wavelength-Shift
Relationship
between
Scattering angle
and recoil angle
Scattered Photon Energy
as Function of Incident
Photon Energy
and Photon Scattering
Angle
Compton Scattering
Function
Energy transfer to
Compton recoil electron

41. IMPORTANT FORMULAS
Differential electronic Cross
Section for Compton Scattering.
Differential Electronic
Cross Section per unit
Scattering angle
Differential Klein–Nishina
Energy Transfer Cross
Section

42. THANK YOU!