COMPTON EFFECT PROBLEMS
Question 1
a) Compton scattering, discovered by Arthur Holly Compton, is the
inelastic scattering of a photon by a charged particle, usually an election.
Let us consider a photon of energy 𝐸0 = ℎ𝑓0 and momentum ⃗𝑝⃗⃗⃗⃗⃗⃗0
scattering a stationary electron. After they scatter the photon has energy
𝐸 = ℎ𝑓 and a momentum ⃗𝑝⃗⃗⃗⃗⃗ that makes an angle 𝜃 with ⃗𝑝⃗⃗⃗⃗⃗⃗0 . The
electron recoils with total energy 𝐸𝑒.
i.
Using conservation of energy and momentum give an expression for
1
− as a function of the electron mass and the angle 𝜃.
𝑝⃗⃗⃗
𝑝⃗⃗⃗
(3 Marks)
, the so-called
ii.
iii.
Deduce from i) an expression for
Compton formula.
(3 Marks)
𝜆0
𝑚𝑒𝑐 is called the Compton wavelength. Compute its value.
Can we observe the Compton effect with visible light, why? Hint:
Consider ∆𝜆 /𝜆0.
(5 Marks) iv. Explain why Compton needed Xrays (𝜆0~10−10 𝑚) to see its effects.
(2 Marks)
b) Shown in the figure below is a Compton scattering event in which a high
energy photon is deflected by its interaction with an electron. The
photon is deflected by the angle,∅, and the electron (initially at rest) is
rejected at the angle,𝜃.
i.
Let the initial energy of the photon by 5 keV. What is the initial
wavelength of the photon? What is the momentum of the initial
photon?
ii.
(4 Marks)
If the angle ∅ is 30 degrees, calculate the wavelength of the scattered
photon.
iii.
(3 Marks)
Find the momentum vector (yes, x and y components) of the photon
as it exits the collision zone.
iv.
Find the momentum vector of the electron as it exits the collision
zone.
v.
vi.
(2 Marks)
(2 Marks)
What is the direction,𝜃, of the electron?
What is the energy of the electron?
(1
What is the wavelength of the electron?
(1 Mark)
Mark) vii.
(1 Mark)
Question 2
a) The absorption coefficient of bone is 600 𝑚−2 for X-rays of energy 20
keV. A beam of such X-rays has an intensity of 20 W𝑚−2. Calculate the
intensity of the beam after passing through a 4.0 mm thickness of bone.
(5 Marks)
b) Moseley’s law
(i)
Using the data below, plot 𝑍
to verify the law.
(5Marks)
Z
(ii)
Element
𝛌 Å (𝑲𝜶)
12
Mg
9.888
15
P
6.155
17
Cl
4.729
20
Ca
3.360
22
Ti
2.750
24
Cr
2.291
26
Fe
1.937
Derive values for a and b and use Moseley’s equation to estimate
𝑲𝜶 wavelength of manganese (Z=25)
(8 Marks)
c) Moseley’s law is an empirical relationship between the frequency of
Xray and atomic number of target. For 𝐾𝛼 lines, the relationship is given
as
𝑣 = 𝑅(𝑍−1)2
Where v is frequency, R is Rydberg frequency =3.289 x 1015Hz, Z is the
atomic number of target. Now consider that you are operating an Xray tube with Cu target by applying an accelerating potential of 25 kV.
i. Draw a schematic of X-ray spectrum emitted by the tube; label
characteristic and continuous radiations.
(5 Marks) ii. Calculate
λ𝑆𝑊𝐿and using Moseley’s law determine λ𝐾𝛼
(6 Marks)
(iii)
Briefly, explain the origin of characteristic and continuous X-rays.
(4 Marks)
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