7.1 The experimental determination of excited states

So far we have for the most part been considering atomic nuclei in their quantum ground states. Most nuclei on Earth have been in their ground states since the time of its creation. However, almost all nuclei possess excited states of higher energy (and therefore less binding energy) than their ground state. There are many ways of exhibiting these excited states, and determining their energies and quantum numbers. One method is to scatter energetic protons of known momentum Pi from the nucleus of interest and to observe their angle of scattering #θ and final momentum pf. This process is illustrated in Fig. 7.1.

To conserve momentum, the recoiling nucleus has momentum (Pi — pf cos #θ) in the direction of the incoming proton and pf sin θ in the perpendicular direction. Taking all momenta and energies to be nonrelativistic, the difference E between the initial and final kinetic energies of the system is

where m*A is the mass of the recoiling target nucleus. By conservation of energy E must be the excitation energy given to the nucleus. In terms of the initial and final proton kinetic energies Ei and Ef, equation (7.1) becomes

In equations (7.1) and (7.2), mA = mA + E/c2 may be replaced by the mass mA of the nucleus in its ground state, with little error.

In practice, a mono-energetic beam of protons is directed at a target containing the nucleus in question. If the target is a solid it is generally made so thin that the probability of a proton scattering more than once off a nucleus is small.