Radioactivity

Theory of Radioactive Transformations.

We have seen that the radioactive bodies spontaneously and continuously emit a great number of a- and 0-particles. In addition, new types of radioactive matter like the emanations and active deposits appear, and these are quite distinct in chemical and physical properties from the parent matter. The radiating power is an atomic property, for it is unaffected by combination of the active element with in active bodies, and is uninfluenced by the most powerful chemical and physical agencies at our command. In order to explain these results, Rutherford and Soddy in z go3 put forward a simple but comprehensive theory. Unlike the atoms of the ordinary elements the atoms of radioactive matter are unstable, and each second a definite fraction of the number of atoms present breaks up with explosive violence, in most cases expelling an a- or j3-particle with great velocity. Taking as a simple illustration that an « particle is expelled during the explosion, the resulting atom has decreased in mass and possesses chemical and physical properties entirely distinct from the parent atom. A new type of matter has thus appeared as a result of the transformation. The atoms of this new matter are again unstable and break up in turn, the process of successive disintegration of the atom continuing through a number of distinct stages. On this view, a substance like the radium emanation is derived from the transformation of ra dium. The atoms of the emanation are far more unstable than the atoms of radium, and break up at a much quicker rate. We shall now consider the law of radioactive transformation according to this theory. It is experimentally observed that in all simple radio active substances, the intensity of the radiation decreases in a geometrical progression with the time, i.e., I/I0 = where I is the intensity of the radiation at any time, t, the initial intensity, and X a constant. Now according to this theory, the intensity of the radiation is proportional to the number of atoms breaking up per second. From this it follows that the atoms of active matter present decrease in a geometrical progression with the time, i.e., where N is the number of atoms present at a time t, the initial number, and X the same constant as before. Dif ferentiating, we have dN/dt= —XN, i.e., X represents the frac tion of the total number of atoms present which break up per second. The radioactive constant X has a definite and charac teristic value for each type of matter. Since X is usually a very small fraction, it is convenient to distinguish the products by stat ing the time required for half the matter to be transformed. This will be called the period of the product, and is numerically equal to The average life of the atoms of a product before transformation is given by 1/X. As far as our observation has gone, the law of radioactive change is applicable to all radioactive mat ter without exception. It appears to be an expression of the law of probability, for the average number breaking up per second is proportional to the number present. Viewed from this point of view, the number of atoms breaking up per second should have a certain average value, but the number from second to second should vary within certain limits according to the theory of prob ability. The theory of this effect was first put forward by Schweidler, and has since been verified by a number of experi menters, including Kohlrausch, Meyer, and Regener and H. Geiger. This variation in the number of atoms breaking up from moment to moment becomes marked with weak radioactive mat ter, where only a few atoms break up per second. The variations observed are in good agreement with those to be expected from the theory of probability. This effect does not in any way in validate the law of radioactive change. On an average the number of atoms of any simple kind of matter breaking up per second is proportional to the number present. We shall now consider how the amount of radioactive matter which is supplied at a constant rate from a source varies with the time. For clearness, we shall take the case of the production of emanation by radium. The rate of transformation of radium is so slow compared with that of the emanation that we may assume without sensible error that the number of atoms of radium breaking up per second, i.e., the supply of fresh emanation, is on the average constant over the interval required. Suppose that initially radium is completely freed from emanation. In consequence of the steady supply, the amount of emanation present increases, but not at a constant rate, for the emanation is in turn breaking up. Let q be the number of atoms of emanation produced by the radium per second and N the number present after an interval t, then dN/dt= q— X N where X is the radioactive constant of the emanation. It is obvious that a steady state will ultimately be reached when the number of atoms of emanation supplied per second are on the average equal to the atoms which break up per second. If be the maxi

mum number, Inte grating the above equation, it follows that = I If a curve be plotted (fig. I) with N as ordinates and time as abscissae, it is seen that the re covery curve is complementary to the decay curve. The ab scissae represent multiples of the ( time T for half the product to be transformed. The activity, which is proportional to the number of atoms present, falls to in the time T, in time 2T, and so on.

This process of production and disappearance of active matter holds for all the radioactive bodies. We shall now consider some special cases of the variation of the amount of active matter with time which have proved of great importance in the analysis of radioactive changes.

(a) Suppose that initially the matter A is present, and this changes into B and B into C, it is required to find the number of atoms P, Q and R of A, B and C present at any subsequent time t.

Let Xi, X3 be the constants of transformation of A, B and C respectively. Suppose n be the number of atoms of A initially present. From the law of radioactive change it follows: it will be seen from (3) that the value of Q, initially zero, increases to a maximum and then decays ; finally, according to an exponential law, with the period of the more slowly transformed product, whether A or B.

(b) A primary source supplies the matter A at a constant rate, and the process has continued so long that the amounts of the products A, B, C have reached a steady limiting value. The pri mary source is then suddenly removed. It is required to find the amounts of A, B and C remaining at any subsequent time t.

In this case of equilibrium, the number of particles of A supplied per second from the source is equal to the number of particles which change into B per second, and also of B into C. This requires the relation where 'Po, Qo, Ro are the initial number of particles of A, B, C present, and Xi, X2, X3 are their constants of transformation.

Using the same notation as in case (I), but remembering the new initial conditions, it can easily be shown that the number of particles P, Q and R of the matter A, B and C existing at the time t after removal are given by The curves expressing the rate of variation of P, Q, R with time are in these cases very different from case (i).

(c) The matter A is supplied at a constant rate from a primary source. Required to find the number of particles of A, B and C present at any time t later, when initially A, B, and C were absent.

This is a converse case from case (2) and the solutions can be obtained from general considerations. Initially suppose A, B and C are in equilibrium with the primary source which supplied A at a constant rate. The source is then removed and the amounts of A, B and C vary according to the equation given in case (2). The source after removal continues to supply A at the same rate as before. Since initially the product A was in equilibrium with the source, and the radioactive processes are in no way changed by the removal of the source, it is clear that the amount of A present in the two parts in which the matter is distributed is un changed. If be the amount of A produced by the source in the time t, and P the amount remaining in the part removed, then where Po is the equilibrium value. Thus The ratio can be written down from the solution given in case (2). Similarly the corresponding values of may be at once derived. It is obvious in these cases that the curve plotted with P/Po as ordinates and time as abscissae is comple mentary to the corresponding curve with as ordinates. This simple relation holds for all recovery and decay curves of radioactive products in general.

We have so far considered the variation in the number of atoms of successive products with time when the periods of the products are known. In practice, the variation of the number of atoms is deduced from measurements of activity, usually made by the electric method. Using the same notation as before, the activity of any product is proportional to its rate of breaking up, i.e., to where P is the number of atoms present. If two products are present, the activity is the sum of two corresponding terms and X2Q. In practice, however, no two products emit a- or i3 particles with the same velocity. The difference in ionizing power of a single a-particle from the two products has thus to be taken into account. If under the experimental conditions, the ioniza tion produced by an a-particle from the second product is K times that from the first product, the activity observed is pro portional to In this way, it is possible to compare the theoretical activity curves of a mixture of products with those deduced experimentally.