Introduction: Energy Scales There are several energies that characterize gamma ray uorescence. The first is the excitation energy, E0, of the nuclear transition being studied. This is usually the energy difference between a nucleus in an excited state and in its ground state. When a nucleus decays from an excited state, the excitation energy is often radiated away as a gamma ray. In practice, the energies of such gamma rays are typically less than 1 MeV. [1] Due to quantum mechanics, there is a small spread in the possible energy values of the emitted gamma ray. The uncertainty principle states that Mom 2 h (1) where h is Planck's constant and At is the time taken to measure the energy level. It is important to note that At is NOT the same as the lifetime of the state, 1:. Assuming one can measure the the excited state for no more than the lifetime 1: we get an lower bound on the energy spread AEO . This energy spread is called the linewidth and is denoted F. Experimentally it is determined by measuring the width of an energy peak at its half maximum. The energy scale for F is usually many orders of magnitude smaller than EL. [1] The next energy scale of interest is determined by the experiment being done. An example is the case of hyperfine splitting where the normal energy levels of an excited nucleus are modified in the presence of a magnetic field from its orbiting electrons. The magnetic field will cause each energy level of the nucleus to change slightly and thus create slightly different transitions. This slight change is the called the experimental energy. Typically the experimental energy is extremely small compared to the energy level. In order to observe the experimental energy (and thus have an interesting experiment) the linewidth must be narrow enough to observe the shifting or splitting. The final energy scale in gamma ray uorescence experiments is the energy lost to a recoiling nucleus. When a nucleus emits a gamma ray, conservation of momentum requires that it recoil in the opposite direction. Likewise, when a nucleus absorbs a gamma ray, it absorbs momentum and begins to move (recoil) in the same direction. The energy associated with the recoiling nucleus is called the recoil energy. Since the nuclei we are concerned with will be moving nonrelatvistically, we may calculated the recoil energy, R, as 2 R = (pnucleus )2 = (ppkoton) =5 (E0)2 (2) 2 M 2M 2M6 where M is the mass of the nucleus and c is the speed of light. Exercise 1a: Exercise 1b: Exercise 1c: Derive an expression for Fusing Eq. 1. Assuming an excited nucleus of Fe57 has a lifetime of 0.1us, determine the minimum linewidth. What happens to the linewidth as successively shorter times are used to measure the the energy of the excited state Assume you have a nucleus of F c57 with a transition with energy 1E}J = 14.4 keV. Determine R and the ratio of R to E0. Explain why it was reasonable to approximate ppm\" S Eo/c in the last step of Equation 2. Was it reasonable to assume the nucleus was nonrelativistic? Explain. Assume the hyperne splitting in a Fe57 nucleus is roughly 10'7 eV. Compare the experimental energy of the hyperne splitting to linewidth and recoil energy from previous parts. Would you be able to experimentally observe the hyperfine splitting? Explain why or why not