6 Radioactivity 6.1 Introduction The stable isotopes are located in a narrow band of the nuclear chart called the -stability line, alongside of which nuclei unstable by + or - emission are located. For A > 150 the emission of an -particle is energetically favorable, and in this region one finds several - emitters. Heavy nuclei also release energy if divided in two nearly equal parts and can, for this reason, fission spontaneously. A radioactive substance, which contains some unstable isotope, is in permanent transformation by the action of one or more of these processes. The physics of each of them will be studied later. In this chapter we are only interested in the statistical aspect of the problem: we want to find the evolution in time of the number of atoms, and of the activity of radioactive substances, in a way independent of the process in action. Thus let us call N the number of atoms of an unstable isotope at the instant t. If we admit that there is a fixed probability dt of occurrence of a certain process in a nucleus in the time interval dt, the change dN in the number of atoms during the interval dt can be written as dN(t) = -N dt. (6.1) is called the disintegration or decay constant of that isotope for the process in question. The rate at which an amount of a substance sample disintegrates is measured by the activity A(t) of that amount. From (6.1) we obtain A(t) = - dN = N. dt (6.2) The number N of atoms at a given instant t can be easily obtained by the integration of (6.1): N(t) = N 0 e -t , (6.3) where N 0 is the number of atoms at the initial instant. Using (6.3) we can evaluate the half-life of a radioactive sample, defined as the time necessary for the decay of half the