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\section{Introduction}  \subsection{Muon Decay}  \subsection{Gamma Ray Spectroscopy}   \par The radioactive decay of a nucleus for elements () and () was studied by detecting gamma rays which were emitted in response to the decay. Gamma rays are detected through the use of a NaI:TI scintillator crystal. The crystal produces a fast moving free electron which will lose its energy through the excitation of ions in its path as it travels through the crystal. The excitation results in the emission of visible light, which is directed to the photosensitive surface of a photomultiplier tube. The photons then eject electrons via the photoelectric effect. These electrons are collected within the photomultiplier and then amplified in order to yield a current pulse. This current pulse is converted to a voltage pulse, the height of which is directly proportional to the number of photoelectrons. Since the number of photonelectrons is proportional to the number of photons reaching the photomultiplier, and the number of photons is proportional to the initial energy of the freed electron, the height of the voltage pulse is therefore proportional to the initial energy of the freed electron.  \par Thus when a source such as () is placed near the scintillator the photomultiplier will produce a series of voltage pulse, each of which correspond to the decay of a nucleus. These voltage pulses are analyzed using a multi-channel analyzer. The multi-channel analzer sorts the pulses according the their height, and then counts them to give a spectrial energy distribution of the freed electron. The spectral distribution for () is show in Figure ().  \par  For this lab, the only peak we are interested in from Figure \ref{fig:Callibration}, is the photopeak, which is produced by the photoeletric effect. Consider a gamma ray striking an ion in a scintillator crystal. The gamma ray should be absorbed by the ion, and all of it's energy is transferred to a bound electron. The bound electron is thus freed, and begins to move rapidly through the crystal. The energy of the gamma ray () is far greater than the energy of the electron bound to the ion, therefore the energy of the freed electron assumes the value of the energy of the gamma ray. The photoelectric effect will then cause a voltage peak in the spectrum seen by the photomultiplier, known as the photopeak. The height of the voltage peak should correspond to the energy of the initial gamma ray. \par Though not used in this lab, it is also important at least mention the two other processes by which (), in order to explain the presence of other structure seen in the spectral distributions for various isotopes. The Compton Edge, seen in Figure (), occurs via a process known as Compton scattering. In this scenario, energy is not absorbed by an ion in the scintillator crystal, and is instead scattered through an angle () by an electron. The gamma ray then moves off with a reduced energy and a change in momentum. The electron will carry away some of the gamma rays energy. The energy of the electron, which is the energy lost by the gamma ray will vary depending on the angle (). The photomultiplier will produce a spectrum (Compton spectrum) which corresponds to the angle dependent energies. However, the energy produced by the Compton scattered electrons is essentially a constant, thus, the Compton spectrum will appear as a flat plateau up to a feature known as the Compton Edge, which corresponds to the largest possible scattering angle (). When (), the energy of the electron will be at a maximum. After the Compton Edge, the spectrum will drop off quite sharply.   \par The compton scattering previously explained only applies to gamma rays that were scattered by electrons in the scintillator. Gamma rays can also be scattered into the scintillator from outside interactions, which will result in a peak in the photomultiplier spectrum known as the backscatter peak. In this scenario, the signal detected is from the scattered gamma ray and not from the electron. The resulting energy peak is known as the backscatter peak.   \par The last interaction is known as pair production. An incoming gamma ray with an energy above, (), which is the rest mass of an electron positiron pair, the gamma ray can spontanesouly create an electron-positron pair and be totally absorbed. If the electron and positron lose all of their kinetic energy while inside the scintillator they would produce aa voltage pulse correspoding to (), where () is the E is the gamma ray energy. Interaction between the electron and positron in the crystal can further complicate the spectrum. Since it is not explicity important to our experiment, further reading on the spectrum seen for pair production can be found, ().