Cherenkov radiation

Manjunath.R

#16/1, 8th Main Road, Shivanagar, Rajajinagar, Bangalore560010, Karnataka, India

*Corresponding Author Email: manjunath5496@gmail.com

*Website: http://www.myw3schools.com/

Abstract

Cherenkov radiation is the electromagnetic radiation emitted when a charged particle (such as an
electron) passes through an insulator at a constant speed greater than the speed of light in that
medium. In this article, we provide a simple, concise discussion about "Cherenkov radiation"
which demonstrates the characteristic blue glow of an underwater nuclear reactor.

1
 Pavel Alekseyevich Cherenkov was a Soviet physicist who shared the Nobel Prize in physics in
1958 with Ilya Frank and Igor Tamm for the discovery of Cherenkov radiation, made in 1934.

In some situations, photon behaves like a wave, while in others, it behaves like a particle. The
photons can be thought of as both waves and particles. In 1924 a French physicist Louis de
Broglie developed a formula to relate this dual wave and particle behavior:

hc
E = hυ, c = λυ, E= = mc2,
λ

where E and m are the energy and mass of the photon, υ and λ are the frequency and wavelength
of the photon, h is the Planck constant, c is the speed of light. From this we obtain the definition
of the photon wavelength through the Planck constant and the momentum of the photon:
h
λ=
mc

This equation is used to describe the wave properties of matter, specifically, the wave nature of
the electron:

2
 h
λe =
m𝑒 v

m0
where λe is wavelength, h is Planck's constant, me = is the relativistic mass of the
2
√1−v2
c

electron, moving at a velocity v.

h
pe =
λ𝑒

From this it follows that,

dp𝑒 dλ𝑒 h
=− ×
dt dt λ2𝑒

dp𝑒 p2𝑒 dλ𝑒

= ×−
dt h dt
Sir Isaac Newton first presented his three laws of motion in the "Principia Mathematica
Philosophiae Naturalis" in 1686. His second law defines a force exerted on the electron to be
equal to the rate of change in momentum of the electron:

dp𝑒
F=
dt
p2𝑒 dλ𝑒
F= ×−
h dt

According to the law that nothing may travel faster than the speed of light – i.e., according to the
Albert Einstein's law of variation of mass with velocity (the most famous formula in the
world. In the minds of hundreds of millions of people it is firmly associated with the menace of
atomic weapons. Millions perceive it as a symbol of relativity theory):

m0
me =
2
√1−v2
c

3
That the electron's mass me in motion at speed v is the mass m0 at rest divided by the factor

v2
√1 − implies: the mass of the electron is not constant; it varies with changes in its velocity.
c2

Differentiating the above equation, we get:

mev dv + v2 dme = c2 dme

dme (c2 – v2) = me v dv

In relativistic mechanics (the arguably most famous cult of modern physics, which has a highly
interesting history which dates back mainly to Albert Einstein and may be a little earlier to H.
Poincaré), we define the energy mec2 which a moving electron possess to be = m0 c2 + Ek .

mec2 = m0 c2 + Ek

dm𝑒 c2 dEk
= = Fv
dt dt
dp𝑒 dm𝑒 dv
F= = v+ me
dt dt dt
v2
F=F + me a
c2
m𝑒 a
F= v2
1− 2
c

So as v approaches c, the bottom term approaches zero and therefore the force approaches
infinity. It requires an infinite amount of force to accelerate the electron to the speed of light.

Because:

m0
me =
2
√1−v2
c

Therefore:

m3𝑒 a
F=
m20

For non-relativistic case (v << c), the above equation reduces to F = m0a.

4
 p2𝑒 dλ𝑒 m3𝑒 a
×− =
h dt m20

From this it follows that,

m20 v2 dλ𝑒
a= ×−
hm𝑒 dt

Thus, we have the formula for the calculation of the acceleration of the electron.

Cherenkov radiation is the electromagnetic radiation emitted when a charged particle (such as
an electron) travels in a medium with speed v such that:

c
<v<c
n

where c is speed of light in vacuum, and n is the refractive index of the medium. We define the
ratio between the speed of the particle and the speed of light as:

v
β=
c

Using simple trigonometric relation one can

determine the Cherenkov angle:

1
cosθ =
nβ

c
cosθ =
nv

v2 1
=
c2 n2 cos2 θ

Since the charged particle is relativistic, we can use the relation:

5
 m20 v2 dλ𝑒
a= ×−
hm𝑒 dt

The heavier the charged particle,

the higher kinetic energy it must
v2 √c2 −v2 dλ𝑒 possess to be able to emit
a= ×−
c2 λC dt Cherenkov radiation.

h
where λC = is the Compton wavelength of the electron.
m0 c

υC √1−β2 dλ𝑒
a= ×−
n2 cos2 θ dt

where υC is the Compton frequency of the electron.

The emission of Cherenkov radiation depends on the refractive index n of

c
cosθ =
nv the medium or the velocity v of the particle in that medium.

Ee2 – E02 = (mec2 + m0c2) (mec2 − m0c2)

Since:

(mec2 − m0c2) = Ek

Ee = √E02 + p𝑒2 c 2

Therefore:

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 p2𝑒 m2𝑒 v2
Ek = =
(m𝑒 + m0 ) (m𝑒 + m0 )

m2𝑒 c2
Ek = 2
n cos2θ (m𝑒+ m0)

m0
The mass me of an electron moving with a velocity v is given by me = where: m0 = rest
2
√1−v2
c
mass of electron and c = speed of light.

m0 c
√c 2 − v 2 = m𝑒

√c2 −v2 m0 c
=
v m𝑒 v

c2 λ𝑒
√ −1=
v2 λC

λC λC
λe = =
2 √n cos2 θ−1
2
√ c2 −1
v

References:

 Light – The Physics of the Photon, by Ole Keller.

 University Physics with Modern Physics by Hugh D. Young.

 Isaac Newton and the Laws of Motion by Andrea Gianopoulos.

 An Introduction to Cherenkov Radiation by H Alaeian.

 Relativity by Albert Einstein.