Theory

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Theory: 

Pair production and annihilation phenomena arise from the mass energy relationship from special relativistic dynamics and the probabilistic/wave nature of particles in quantum mechanics.  Although a complete characterization of electron-positron pair production would require elements from quantum electrodynamics (QED), an adequate background of the process can be explained using special relativity and a few aspects of nuclear physics.

Relativistic Dynamics and Conservation Laws

            Initially a radioactive source emits a gamma ray with energy above the threshold of 1.022 MeV for production to take place.  This incident gamma ray interacts with a heavy nucleus and spontaneously splits into an electron and a positron.

  

 This process can be made more precise using four vector algebra.  Initially the gamma ray is incident on the nucleus at rest with a four momentum vector given by (in natural units):

                                                                                                                                 (1.1)

Here MN is the nuclear mass while Eg and Pg are the incident gamma ray energy and momentum respectively.  The final state contains an electron, positron at rest and the nucleus with some momentum and energy leading to a final momentum vector:

,                                                                                                                              (1.2)

where me represents the electron and positron mass while EN and PN are the final energy and momentum of the nucleus.  When equations 1.1 and 1.2 are squared (using the relativistic metric) and set equal to each other the result is:

                                                                                                  (1.3)

or by noting that Pg is equal to Eg in natural units, and that we have:

                                                                                                                                 (1.4)

If MN is set equal to zero then EN also goes to zero showing an inconsistency in equation 1.4.  In other words the equation expresses the necessity of an absorber nucleus for pair production to take place.  The deeper underlying principle behind this is the vertex theorem of QED stating that two photons are required to produce a particle-antiparticle pair.

 

 

This analysis of energy-momentum conservation explains why pair production will only take place in the electric field of the nucleus.  The incident gamma ray must interact with a photon from the electric field to produce the particle-antiparticle pair.

Absorber Attenuation and Thickness Dependence

The materials used in this experiment to provide absorption of incident gamma rays were rectangular slabs of solid materials.  The attenuation of an incident beam of gamma rays is due to several effects including photoelectric absorption, Compton scattering, and pair production.  Each process contributes to the overall attenuation of the incident beam.  A quantitative analysis of the physics involved provides a relationship between the incident beam intensity and the detectable pair production events.

                          

            Consider the diagram shown above.  A radioactive source emits gamma rays that are absorbed in the target slab.  The gamma ray intensity I of the source is related to the distance y in the target slab by:

,                                                                                                                                              (1.5)

where I0 is the initial intensity of the source, and m is the overall attenuation coefficient.[5]  Thus the rate of photon interactions in the source will be given by:

.                                                                                                                                       (1.6)

Since the total attenuation coefficient m is given by the sum of the attenuation coefficients mi for each process the rate of events for a particular interaction (i.e. photo-electric, Compton, or pair production) will be:

.                                                                                                                                    (1.7)

For each pair production event the created positron will interact with an electron in the material to produce two back-to-back 511 keV photons.  The one photon will travel a distance x, while the other will travel a distance L-x.  Thus the event yield will be further attenuated to give:

.                                                                                                  (1.8)

Including the possibility of two gamma ray energies (as is the case for 60Co) and the efficiencies of the detectors e1 and e2 we have finally the full event yield equation:

.                                                                                            (1.9)

 

Optimizing this equation provides a means to determine the best thickness for each material to provide the maximum pair production yield in the detectors.

Last Updated: 05/04/2009