====== 6. Rate of radioactive decay ======
Chapter 6 from //BASICS OF NUCLEAR PHYSICS AND OF RADIATION DETECTION AND MEASUREMENT – An open-access textbook for nuclear and radiochemistry students// by Jukka Lehto

===== 6.1. Decay law – activity - decay rate =====

The rate of radioactive decay is a characteristic feature for each radionuclide. Decay rate, also called activity, is the number of nuclear transformations (decays) (dN) at a defined time difference (unit time) (dt) and it is referred to as A (activity):

𝐴=|−𝑑𝑁∕𝑑𝑡| [VI.I]

Radioactive decay is a stochastic phenomenon and we cannot know when a single nucleus will decay or what is the exact number of decays in unit time. If a fairly large number of radioactive nuclei are considered we can, however, know what fraction of nuclei will probably decay in unit time. This fraction, the probability of radioactive decay events, is characteristic for each radionuclide and it is called decay constant (//λ//). If, for example, the decay constant is 0.0001 s<sup>-1</sup> it means that among 100000 radioactive nuclei probably 10 nuclei will decay in one second and accordingly among 1000000 nuclei probably 100 nuclei. Thus radioactive decay rate is directly proportional to the number of radioactive nuclei.

𝐴=|−𝑑𝑁∕𝑑𝑡|=𝜆∙𝑁   [VI.II]

where N is the number of radioactive nuclei.

In the following we will see what is the number of radioactive nuclei (//N//) at defined time point (//t//) when their initial number //(N<sub>0</sub>)// is known at time point //t<sub>0</sub>//. From the equation VI.II we get 𝑑𝑁/𝑁=−𝜆∙𝑑𝑡 and its integration ∫𝑑𝑁/𝑁=∫−𝜆∙𝑑𝑡 yields //ln//⁡𝑁= −𝜆∙𝑡+𝐶. When considering time t = 0, when //N = N<sub>0</sub>// , the constant C gets a value //lnN<sub>0</sub>// and inserting this into the equation //ln//⁡𝑁= −𝜆∙𝑡+𝐶 yields 𝑙𝑛𝑁−𝑙𝑛𝑁<sub>0</sub>=−𝜆∙𝑡 and further 𝑙𝑛𝑁/𝑁<sub>0</sub>=−𝜆∙𝑡. Taking antilogarithm from both sides yields 𝑁/𝑁<sub>0</sub>=𝑒<sup>−𝜆𝑡</sup> and further

𝑁=𝑁<sub>0</sub>∙𝑒<sup>−𝜆𝑡</sup>  [VI.III],

which equation answers the question what is the number of radioactive nuclei (//N//) at certain time point (//t//) when we know the initial number on nuclei //(N<sub>0</sub>)// at time point //t<sub>0</sub>//.  Thus, to calculate this only the value of the decay constant (//λ//) is to be known.

The number of radioactive nuclei is not usually known and their number is also difficult to directly determine. Usually we are, however, more interested in development of activities with time. As seen from Equation VI.II the activity is directly proportional to the number of decaying nuclei, thus we can transform Equation VI.III to calculate activity (//A//) at a time point (//t//) just by replacing //N// with //A//:

𝐴=𝐴<sub>0</sub>∙𝑒<sup>−𝜆∙𝑡</sup>   [VI.IV]

===== 6.2. Half-life =====

Decay constants are known for all radionuclides and they are tabulated in various textbooks and databases. They are, however, not used in calculations of activities but instead half-lives (t½) are used for this purpose. Half-life is defined as the time in which half of the initial radioactive nuclei have decayed. Since the activity is directly proportional to the number of radioactive nuclei this means that also activity decreases to half within the time of half-life. In the following, the relation between the decay constant and the half-life will be shown. In addition, an equation by which activities can be calculated at desired time points using half-lives will be derived.

//**Do not use upper-case T½ for half-life, use lower lower-case t½. T refers to temperature, t to time.**//

We consider a time difference equal to a half-life t = t<sub>½</sub>, during which time the number of radioactive nuclei decays to half, i.e. N = N<sub>0</sub>/2 . Inserting t = t<sub>½</sub> and N = N<sub>0</sub>/2 to Equation VI.III 𝑁=𝑁<sub>0</sub>∙𝑒<sup>−𝜆𝑡</sup> yields 𝑁<sub>0</sub>/2=𝑁<sub>0</sub>∙𝑒<sup>−𝜆∙𝑡<sub>½</sub></sup> and further 𝑒<sup>𝜆𝑡<sub>½</sub></sup>=2. Taking logarithms from both sides gives 𝜆𝑡<sub>½</sub>=𝑙𝑛2 and further
𝜆=𝑙𝑛2/𝑡<sub>½</sub>. Replacing //λ// in equations 𝑁=𝑁<sub>0</sub>∙𝑒<sup>−𝜆𝑡</sup>,
𝐴=𝐴<sub>0</sub>∙𝑒<sup>−𝜆𝑡</sup> with //ln2/t<sub>½</sub>//  yields


𝑁=𝑁<sub>0</sub>∙2<sup>-𝑡/𝑡<sub>½</sub></sup> [VI.V]

and


𝐴=𝐴<sub>0</sub>∙2<sup>-𝑡/𝑡<sub>½</sub></sup>  [VI.VI]

With the latter equation we can calculate activities using half-lives at any time points when we know the initial activity. When we want to calculate the initial activity at an earlier time point we use the inverse equation

𝐴<sub>0</sub>=𝐴∙2<sup>𝑡/𝑡<sub>½</sub></sup>   [VI.VII]

===== 6.3. Activity unit =====

The official SI unit of activity is Becquerel (Bq) and it means one decay in SI unit time, i.e. one second:

1 Bq = 1 decay <sup>s-1</sup> [VI.VIII]

Earlier Curie (Ci) was used as the activity unit. One Curie is  3.7×10<sup>10</sup> decays in second and thus 

1 Ci = 3.7×10<sup>10</sup> s<sup>-1</sup> = 3.7×10<sup>10</sup> Bq [VI.IX]

Curie unit was derived as the number of decays taking place in one gram of <sup>226</sup>Ra in one second using half-life of 1580 years (today it is known to be 1600 years).

Sometimes activities are expressed as a dps unit, meaning disintegrations per second which are equal to activities presented as Bequerels. In some instances, for example in liquid scintillation counting, activity is also presented as dpm units (disintegrations per minute). One dpm is 1/60 dps or 16.7 mBq.

===== 6.4. Specific activity - activity concentration =====

Specific activity is often used as a synonym to activity concentration, but strictly speaking they have different meanings. Specific activity refers to concentration of a radionuclide with respect to the total amount of the same element as the radionuclide. Thus, specific activity is its concentration in a unit mass or mole of the same element, for example, 5 kBq of <sup>137</sup>Cs per 1 g of Cs or 0.038 kBq of <sup>137</sup>Cs per 1 mole of Cs.

Activity concentration in turn is the concentration of a radionuclide in a unit mass or volume of any matter in question, for example, 5 kBq of <sup>137</sup>Cs per 1 kg of soil or 5 kBq of <sup>137</sup>Cs per 1 litre of water.

===== 6.5. The relation between the activity and the mass =====

Conversion of activities to masses or vice versa is based on the radioactive decay law

𝐴=𝜆×𝑁  [VI.X]

which shows the direct dependence of the activity on the number of decaying nuclei. Replacing 𝜆 by ln2/t<sub>1/2</sub> and N by (m/M)×N<sub>A</sub> (where m is the mass in grams, M the molar mass of the element and N<sub>A</sub> the Avogadro number) yields

𝐴=(𝑙𝑛2×𝑚×𝑁<sub>𝐴</sub>)/(𝑡<sub>½</sub>×𝑀) [VI.XI]

or the other way round

𝑚=(𝐴×𝑀×𝑡<sub>½</sub>)/(𝑙𝑛2×𝑁<sub>𝐴</sub>)  [VI.XII]

which can be used to convert activities to masses or vice versa.

===== 6.6. Determination of half-lives =====

The determination of half-lives can be accomplished in two ways:

  * For radionuclides decaying with such a fast rate that we can observe the decrease in a reasonable time the half-lives can be determined from their activities as a function of time as shown below in Fig. VI.1.

When representing graphically the equation 𝐴=𝐴<sub>0</sub>∙2<sup>-𝑡/𝑡<sub>½</sub></sup> we get an exponential curve (Figure VI.1,left side). Taking logarithms from both sides yields the equation 𝑙𝑛𝐴=−𝑙𝑛2/𝑡<sub>½</sub>∙𝑡+𝑙𝑛𝐴<sub>0</sub>, the graphical representation of which is line with a slope of −𝑙𝑛2/𝑡<sub>½</sub> and the y-axis intersection is 𝑙𝑛𝐴<sub>0</sub>,
i.e. activity at time t<sub>0</sub> (Figure VI.1, right side). The half-life is obtained by fitting a line to the logarithms of observed activity values and calculating the half-life from the slope. If, for example, in Figure VI.1 time were in years, the half-life of the nuclide would result by solving the equation −0.693=−𝑙𝑛2/(𝑡½) into 1 year.

{{:textbook:nrctextbook:determination_of_activity_fig_6_1.png?400|}}

Figure VI.1. Activity (A) as a function of time (t). Left side presents activity in linear scale and right side in logarithmic scale.

The method described above in Fig. VI.1 can also be used to determine half-lives of two coexisting radionuclides supposing that they differ enough from each other. Figure VI.2 shows the total activity curve of two radionuclides as a function of time both in a linear and a logarithmic activity scale. In the first phase, when there are still both radionuclides present, the logarithmic curve shape resembles an exponential one. As the shorter-lived radionuclide has decayed the curve turns into a line. This line represents the decay of the longer-lived radionuclide and its half-life can be 
calculated from the slope of this line. To calculate the half-life of the shorter-lived radionuclide the line is extrapolated to time point zero and the extrapolated activity values of the line are subtracted from total activity curve. This yields another line representing the decay of the shorter-lived radionuclide for which the half-life is calculated from its slope.

{{:textbook:nrctextbook:individual_and_total_activities_of_two_coexisting_radionuclides_as_a_function_of_time_fig_6_2.png?400|}}

Figure VI.2. Individual and total activities of two coexisting radionuclides as a function of time. Top: activity in linear scale. Bottom: activity in logarithmic scale. Diamond (blue): Radionuclide with a half-life of 3 hours. Square (red): radionuclide with a half-life of 12 hours. Triangle (green): total activity.

  * For long-lived radionuclides for which the activity decreases so slowly that we cannot observe its decrease the half-life is determined by measuring both the activity and the mass of the radionuclide and calculating half-life from the equation 𝑡<sub>½</sub>=(𝑙𝑛2×𝑚×𝑁<sub>𝐴</sub>)/(𝐴×𝑀).

For example, when the half-life is 106 years the activity decreases by 0.00007% in one year. Small differences in activity like this cannot be measured. To determine the half-life of long-lived radionuclides we need to measure the mass (m) of the radionuclide and count rate (R) obtained from the activity measurement. In addition, we also need to accurately know the counting efficiency (E) of the measurement system. If, for example, we have 1.27 mg of <sup>232</sup>Th and the count rate obtained from its measurement is 2.65 cps and counting efficiency of the measurement system 
is 51.5% (0.515) the activity of the sample is A= R/E = 2.65 s<sup>-1</sup>/0.515 = 5.15 Bq. The number of thorium atoms in the sample is 1.27·10<sup>-3</sup> g × 6.023·10<sup>23</sup> atoms/mole /232.0 g/mole = 3.295·10<sup>18</sup>.
Now the half-life can be calculated from the equation t½ = (ln2/A) x N = 0.693 × 3.295·10<sup>18</sup> /5.15 s<sup>-1</sup> = 4.44·10<sup>17</sup> s = 1.41·10<sup>10</sup> a. This method can also in principle be used to measure half-lives of shorter lived radionuclides but in their case the accurate measurement of the mass may either completely exclude the use of this method or at least results in inaccurate value.