Tuesday, July 10, 2012

Using energy and momentum conservation principles in physics : Compton Scattering




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 ϕ 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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