The scattering of
electromagnetic radiation (photons) from charged particles is known as Compton
Scattering. It is named after Arthur Compton who was the first to measure
photon-electron scattering in 1923 in Washington University at St. Louis. He won a Nobel prize in 1927 for this discovery.
Consider a
photon of frequency ν1 and wavelength λ1 being incident
on an electron. The following things are observed experimentally :
1. The
wavelength and frequency of the photon before and after the interaction are
different. Thus the Compton scattering is inelastic scattering as the frequency
(hence the energy) of the photon changes after the collision.
2. The momentum of
the electron undergoes a certain change.
Our aim is to
find out how these quantities i.e. wavelengths (or frequencies) of the photon before
and after the interaction and momentum of the electron are related
mathematically. Once we have such relations at our disposal we will be able to
predict the outcomes of other similar scatterings (or collisions). This kind of
scattering of electromagnetic radiation with matter is known as Compton
Scattering.
How
To Approach The Problem
Once we have the
exact problem statement, our job is to use whatever we know about the system
and try to relate the variables of the system which in this case are
wavelengths (or frequencies) of the photon and momentum of the electron before
and after the collision. At first it may seem that this can be done only if we
know the exact mechanism of interaction between an electron and a photon. In
physics, however, we frequently encounter problems/phenomena about which we do
not pre-knowledge of the exact detailed mechanism. In such cases we have to
resort to other (clever) ways for solving such problems or at least to make
some prefatory remarks about the concerned mechanism.
To start with,
we first think of some general rules/principles that are found to be obeyed
always (verified through a large number of experiments) in all kinds of
processes and systems. We then assume that the system under investigation must
also obey these rules irrespective of the mechanism(s) involved. Our job is
then to write down these rules in mathematical form in terms of (known) variables
of the system. This yields a set of equations relating variables of the system.
After this, all we have to do is to solve these equations which may involve
things like differential equations, integral equations or simple algebra like
adding and subtracting equations, etc. and deduce as much information as
possible about the mechanism and if possible, make predictions which could be
tested experimentally. This serves to verify the validity of such general rules
as well as gives insight into the physics of the process. Let us see how we can
solve the present problem with this recipe.
Solving
The Problem
In the present
situation, we have no idea at all as to how the photon and electron interact
with each other and yet we are required to solve the problem. Here, we start
with information we have about this system. We know from our experience and the
results of a large number of experiments conducted so far that energy and
momentum of systems are always found to be conserved irrespective of the
intrinsic characteristics of systems. Therefore, it is natural to expect
whatever be the way the electron and photon interact with each other, the total
energy and momentum before and after the interaction must be conserved. Let us
see what we get when we apply these rules to our system.
Let us assume
that we are in a frame of reference in which the electron is at rest initially.
This only amounts to simplify the problem mathematically. Now, from quantum
mechanics, we know that the momentum of a photon of frequency ν is hν/c
. Thus, if ν1 and ν2 are the frequencies of the
photon before and after the interaction, respectively, then the corresponding
momenta of the photon will be hν1/c
and hν2/c. Initial
momentum of the electron is zero since we are in a frame of reference where the
electron is at rest initially. After the interaction, the electron gains a
momentum of magnitude p at an angle Ï• to the direction of incidence of the
photon. The conservation of total momentum demands that the total momentum
before and after the interaction be
the same. Mathematically
in the direction
of incidence of the photon and
in the direction
normal to the incidence. Squaring and adding the last two equations, we get :
.........(1)
This is all that
we can derive from the momentum conservation. Now, let us move on to energy
conservation rule which states that the total energy before and after the
interaction must be the same. From Special Relativity theory we know that the
energy of a particle of rest mass m having momentum p is
and the energy
of a photon of frequency ν is hν. Thus, the total energy of the system
before interaction is
and the total
energy after interaction is
Equating the
total energies before and after the interaction in accordance with energy
conservation rule, we obtain :
Rearranging the
terms gives :
..........(2)
This is all we
can derive from energy conservation principle. Thus we have managed to obtain
two equations viz. (1) and (2), in three variable : ν1, ν2
and p. Since we are interested in the
relation between ν1 and ν2, we eliminate p by equating (1) and (2), which gives :
.........(3)
Or in terms of
wavelength, the above equation can be expressed as :
..............(4)
The equations
(3) and (4) relate the frequencies and wavelengths before and after the
electron - photon interaction and have been verified experimentally. It is surprising
that despite having no knowledge of how the photon and electron exactly
interact with each other, we are able to obtain an exact mathematical relation
and are capable of predicting the outcome of such interactions. Only things we
assumed were that the system obeys the rules of energy and momentum
conservation which led us to the above mathematical equations. The success of
the above relations indicates the importance and universality of these rules.
Techniques like these are widely used in physics to solve problems where we are
not aware of the intrinsic details of a system, which is usually the case.
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