RMPH 2013 Problem 2
Physics · momentum & collisions · modern-physics · quantum-mechanics
Problem
Romanian Master of Physics 2013
Theoretical Problem No. 2 (10 points)
Compton scattering
An electron storage ring contains circulating electrons with high energy produced by an
accelerator and rotating in an appropriate magnetic field. X - ray photons are directed so as
to collide with electrons stored in storage ring. Phenomenon that occurs is known as Inverse
Compton Scattering.
A photon of wavelength λ is scattered by a moving, free electron. As a result the electron
i
stops and the resulting photon of wavelength λ is scattered at an angle θ=60° with
0
respect to the direction of the incident photon; this photon is again scattered by a second
free electron at rest. In this second scattering process a photon with wavelength of
λ =1,25×10−10m emerges at an angle θ=60° with respect to the direction of the photon of
f
wavelength λ .
0
To characterize the photons and the electrons during the processes use the following
notation:
initial photon – final first first second Second
photon after the photon electron electron electron electron
first before after before after
scattering collision collision collision collision
r r r r r
moment p p p moment p 0 0 p
i 0 f 1e 2e
energy E E E energy E E E E
i 0 f 1e 0e 0e 2e
r r
wavelength λ λ λ speed v 0 0 v
i 0 f 1e 2e
The following constants are known:
h = 6,6×10−34J⋅s - Planck’s constant
m =9,1×10−31kg - rest mass of the electron
0
c =3,0×108m/ s - speed of light in vacuum
Task No. 1 - First collision
1.a. Draw simple sketches marking the moments of the electron and photon before and after
the first collision. Clearly specify the coordinate system used.
1.b. Express the energy and moment of the electron implied in the first collision as a function
r
of the initial speed of the electron v and its rest mass m .
1e 0
1.c. Express the energy and wavelength of the photon after the first collision as a function of
( )
the wavelength of the initial photon λ, the scattering angle θ and Λ =h m ⋅c .
i 0
Task No. 2 - Second collision
2.a. Draw simple sketches marking moment of the electron and photon before and after the
second collision. Clearly specify the coordinate system used.
2.b. Express the energy and wavelength of the photon after the second collision as a
function of the wavelength of the photon before the collision λ , the scattering angle θ and
0
( )
Λ = h m ⋅c .
0
Theoretical Problem No. 2 Page 1 of 6
Romanian Master of Physics 2013
( )
2.c. Express the kinetic energy T = E −E and momentum p of the electron after the
2 2e 0e 2e
second collision as a function of the photon wavelengths after the collision λ,m , c and h.
f 0
Task No. 3 - Quantitative description of processes
Using the values of the given physical constants and numerical values of λ and θ
f
determine the expressions and numerical values for:
3.a. the De Broglie wavelength of the initial electron;
3.b. the energy and frequency of the initial photon;
3.c. the speed of the second electron after collision;
3.d. the variation of the photon wavelength after each collision process.
© The Problem is proposed by:
Delia DAVIDESCU, PhD
Adrian DAFINEI, PhD
Theoretical Problem No. 2 Page 2 of 6
Romanian Master of Physics 2013
ANSWER SHEET
Theoretical Problem No. 2 (10 points)
Compton scattering
Task No. 1 - First collision
1.a. Draw simple sketches marking the moments of the electron and photon before and after
the first collision. Clearly specify the coordinate system used.
0,50p
1.b. Express the energy and moment of
the electron implied in the first collision as
0,80p
a function of the initial speed of the
r
electron v and its rest mass m .
1e 0
1.c. Express the energy and wavelength
of the photon after the first collision as a
function of the wavelength of the initial 1,20p
photon λ, the scattering angle θ and
i
( )
Λ =h m ⋅c .
0
Theoretical Problem No. 2 Page 3 of 6
Romanian Master of Physics 2013
Task No. 2 - Second collision
2.a. Draw simple sketches marking moment of the electron and photon before and after the
second collision. Clearly specify the coordinate system used.
0,50p
2.b. Express the energy and wavelength
of the photon after the second collision as
a function of the wavelength of the photon 1,20p
before the collision λ , the scattering
0
( )
angle θ and Λ = h m ⋅c .
0
2.c. Express the kinetic energy
( )
T = E −E and momentum p of
2 2e 0e 2e
0,80p
the electron after the second collision as
a function of the photon wavelengths after
the collision λ,m , c and h.
f 0
Theoretical Problem No. 2 Page 4 of 6
Romanian Master of Physics 2013
Task No. 3 - Quantitative description of processes
3.a. Determine the expression for the de
1,00p
Broglie wavelength of the initial electron
Determine the numerical value for
1,00p
the de Broglie wavelength of the initial
electron
3.b. Determine the expressions for the
0,50p
energy and for the frequency of the initial
photon
Determine the numerical values for
0,50p
the energy and for the frequency of the
initial photon
3.c. Determine the expression for the
0,50p
speed of the second electron after
collision
Determine the numerical value for the
0,50p
speed of the second electron after
collision
Theoretical Problem No. 2 Page 5 of 6
Romanian Master of Physics 2013
3.d. Determine the expression for the
0,50p
variation of the photon wavelength after
each collision process
Determine the numerical value for the
0,50p
variation of the photon wavelength after
each collision process
Theoretical Problem No. 2 Page 6 of 6
Browse the problem library