Dosimetry

Abstract

Radioactivity is the manifestation of an unstable (radioactive) nucleus going through a single or a series of internal readjustments (decays), more or less complex, leading eventually to a more stable configuration. The probability λ of this process occurring within a unit interval of time is characteristic of the nuclear species and it is independent of the length of time during which the radioactive nucleus has been in existence. The value of λ (decay constant) has been found to be independent of the chemical and physical status of the atom; this observation has been attributed to the fact that the energies involved in the latter, though sufficient to affect the extranuclear electrons, are usually too small to influence the course of events within the nucleus¹. Hence, if a sufficiently large number N of radioactive atoms is in existence at any one time t, the number dN undergoing decay in a sufficiently short time dt can be expressed in differential form as :

$$dN = - \lambda N\,dt$$

(1)

the negative sign indicating that, through decay, the number of radioactive atoms is decreasing. Integration of this equation leads to the well known exponential law of decay

$$N\left( t \right) = N_0 e^{ - \lambda t} $$

(2)

which states that in a population of N ₀ radioactive atoms existing at time zero only a fraction e ^−λt will be present at a later time t. Although this number is adequate to describe the process, other quantities are sometimes used more frequently to indicate the rapidity of decay. Thus the time T within which N ₀ is reduced to one half is termed the half life; it can be easily calculated from (2) by putting e ^−λT = 0.5, namely

$$T = 0.693/\lambda .$$

(3)

Useful in many calculations is the mean (or average life) τ of the radioactive nucleus: since between time t and t+dt, λN(t) nuclei that have lived a time t have decayed, τ can be defined as

$$\tau = \frac{1} {{N_0 }}\int\limits_0^\infty {t\lambda \,N\left( t \right)dt = \lambda \int\limits_0^\infty {te^{ - \lambda t} dt = \frac{1} {\lambda }.} } $$

(4)

This expression indicates that during an average life the population has been reduced to e−1 = 0.368 of its initial value and that τ = 1.44 T. It is obvious from (1) that if the number dN(t)/dt of atoms decaying the unit time can be measured at any one time t, the population at that time can be simply calculated from

$$N\left( t \right) = \frac{{d\,N}} {{\lambda \,dt}} = \frac{{\tau \,dN}} {{dt}}.$$

(5)

Expression (5) indicates that the decay constant of a substance can be computed by measuring the variation in activity dN/dt without knowledge of the actual number of atoms actually present; namely, without the need of absolute standardization (vide infra) of the sample but by simple measurements in instruments of unknown but reproducible efficiency. Radioactive substances are usually measured in units of one curie (Ci), which is defined as “the quantity of any radioactive nuclide in which the number of disintegrations per second is 3.700 × 10⁻¹⁰”. It is worthwhile noting that the mass of a curie of a radioactive element varies tremendously throughout the atomic scale. In particular from (5) one obtains the number of atoms per curie :

$$N\left( t \right) = 3.7 \times 10^{10} \times \tau $$

and the mass M in grams

$$\begin{array}{*{20}c} M \hfill & { = \frac{{{\text{atomic}}\,{\text{number}}\, \times N\left( t \right)}} {{6.02 \times 10^{23} }}} \hfill \\ {} \hfill & { = 6.14 \times 10^{ - 14} \times \tau \times {\text{atomic}}\,{\text{number}}} \hfill \\\end{array} $$

where τ is given in seconds and 6.02 × 10²³ is Avogadro’s number¹.

Work performed under the auspices of the U.S. Atomic Energy Commission.