1. Transformation Kinetic
Radioactive Decays
Transformation Kinetic

2. Half-Life
Different radioactive atoms are transformed at different rates and each radionuclide has its own characteristic transformation rate. For example, when the activity of is measured daily over a period of about 3 months and the percentage of the initial activity is plotted as a function of time, the curve shown in Figure 4-10 is obtained. The data show that one-half of the is gone in 14.3 days, half of the remainder in another 14.3 days, half of what is left during the following 14.3 days, and so on.

4. The time required for any given radionuclide to decrease to one-half of its original quantity is a measure of the speed with which it undergoes radioactive transformation. This period of time is called the half-life and is characteristic of the particular radionuclide.
Each radionuclide has its own unique rate of transformation, and no operation, either chemical or physical, is known that will change the transformation rate; the decay rate of a radionuclide is an unalterable property of that nuclide.
Half-lives of radionuclides range from microseconds to billions of years.

5. remaining after n half-lives is given by the relationship:
From the definition of the half-life, it follows that the fraction of a radionuclide
remaining after n half-lives is given by the relationship:
(1)
where A0 is the original quantity of activity and A is the activity left after n half-lives.

6. If the decay data for any radionucide are plotted on semilog paper, with the activity measurements recorded on the logarithmic axis and time on the linear axis, a straight line is obtained. If time is measured in units of half-lives, the straight line shown in the following Figure is obtained.

8. The fact that the graph of activity versus time, when drawn on semilog paper, is a straight line tells us that the quantity of activity left after any time interval is given by the following equation:
where A0 is the initial quantity of activity, A is the amount left after time t, λ is the transformation rate constant (also called the decay rate constant, or simply the decay constant), and e is the base of the system of natural logarithms. The transformation rate constant is the fractional decrease in activity per unit time and is defined as:
(2)

9. (3)
where N is a number of radioactive atoms and ∆N is the number of these atoms that are transformed during a time interval t. The fraction ∆N /N is the fractional decrease in the number of radioactive atoms during the time interval ∆t. A negative sign is given to λ to indicate that the quantity N is decreasing.

10. The quantitative relationship between half-life T and decay rate constant λ may be found by setting A/A0 in Eq. (2) equal to 1/2 and solving the equation for t. In this case, of course, the time is the half-life T:
(4)

11. Average Life
Although the half-life of a particular radionuclide is a unique, reproducible characteristic of that nuclide, it is nevertheless a statistical property and is valid only because of the very large number of atoms involved. (One microgram of radium contains Any particular atom of a radionuclide may be transformed at any time, from zero to infinity. For some applications, as in the case of dosimetry of internally deposited radioactive material (to be discussed later), it is convenient to use the average life of the radioisotope. The average life is defined simply as the: sum of the lifetimes of the individual atoms divided by the total number of atoms originally present.

12. (5)
where N0 is the number of radioactive atoms in existence at time t = 0. Since
we have
This expression, when integrated by parts by letting t=u and λ e(-λt)= dv, shows the value for the mean life of a radioisotope to be:
(7)

13. If the expression for the transformation (decay) constant in terms of the half-life, , of the radioisotope:
is substituted into Eq. (5), the relationship between the half-life and the mean life is found to be

14. Activity
The Becquerel

15. The unit for quantity of radioactivity must be based on the number of radioactive decays occurring within a prescribed time in the radioactive material. Activity is defined as “the number of decays within a given”. Two units for measuring the activity are used. The SI unit is called the Becquerel, symbolized by Bq, and is defined as “that quantity of radioactive material in which one atom is transformed per second (tps)”. Very often, the term disintegration is used instead of transformation and the Becquerel is defined in term of disintegrations per second, dps.
1 Bq = 1 tps = 1 dps

16. In the case of a pure beta emitter, for example, 1 Bq, or 1 tps, does, in fact, result in one beta particle per second. In the case of a more complex radionuclide, however, such as , (see the Figure), each transformation releases one beta particle and two gamma rays; the total number of radiations, therefore, is

17. For many purposes, the Becquerel is a very small quantity of activity and, therefore, the following multiples of the Becquerel are commonly used:

18. The Curie
The curie, symbolized by Ci, is the unit for quantity of radioactivity that was used before the adoption of the SI units and the Becquerel. The curie, which originally was defined as the activity of 1 g of , is now more explicitly defined
as: “the activity of that quantity of radioactive material in which 3.7× atoms are transformed in one second. The curie is related to the Becquerel by:
1 Ci = 3.7 × Bq.

19. For health physics as well as for many other purposes, the curie is a very large amount of activity. Submultiples of the curie, as listed below, are therefore used: