Decay of Radioactivity

1

The Decay Constant

If one has a sample containing N radioactive atoms of a certain radionuclide, the average decay rate, Δ N /Δ t , for that sample is given by:

where λ is the decay constant for the radionuclide. The decay constant has a characteristic value for each radionuclide. It is the fraction of the atoms in a sample of that radionuclide undergoing radioactive decay per unit of time during a period that is so short that only a small fraction decay during that interval. Alternatively, it is the probability that any individual atom will undergo decay during the same period. The units of λ are (time) ⁻¹ . Thus 0.01 sec ⁻¹ means that, on the average, 1% of the atoms undergo radioactive decay each second. In Equation 4-1 the minus sign indicates that Δ N /Δ t is negative; that is, N is decreasing with time.

Equation 4-1 is valid only as an estimate of the average rate of decay for a radioactive sample. From one moment to the next, the actual decay rate may differ from that predicted by Equation 4-1 . These statistical fluctuations in decay rate are described in Chapter 9 .

Some radionuclides can undergo more than one type of radioactive decay (e.g., ¹⁸ F: 97% β ⁺ , 3% electron capture). For such types of “branching” decay, one can define a value of λ for each of the possible decay modes, for example, λ ₁ , λ ₂ , λ ₃ , and so on, where λ ₁ is the fraction decaying per unit time by decay mode 1, λ ₂ by decay mode 2, and so on. The total decay constant for the radionuclide is the sum of the branching decay constants:

The fraction of nuclei decaying by a specific decay mode is called the branching ratio (B.R.). For the i ^th decay mode, it is given by:

2

Definition and Units of Activity

The quantity Δ N /λ t , the average decay rate, is the activity of the sample. It has dimensions of disintegrations per second (dps) or disintegrations per minute (dpm) and is essentially a measure of “how radioactive” the sample is. The Systeme International (SI) unit of activity is the becquerel (Bq). A sample has an activity of 1 Bq if it is decaying at an average rate of 1 sec ⁻¹ (1 dps). Thus:

where λ is in units of sec ⁻¹ . The absolute value is used to indicate that activity is a “positive” quantity, as compared with the change in number of radioactive atoms in Equation 4-1 , which is a negative quantity. Commonly used multiples of the becquerel are the kilobecquerel (1 kBq = 10 ³ sec ⁻¹ ), the megabecquerel (1 MBq = 10 ⁶ sec ⁻¹ ), and the gigabecquerel (1 GBq = 10 ⁹ sec ⁻¹ ).

The traditional unit for activity is the curie (Ci), which is defined as 3.7 × 10 ¹⁰ dps (2.22 × 10 ¹² dpm). Subunits and multiples of the curie are the millicurie (1 mCi = 10 ⁻³ Ci), the microcurie (1 µCi = 10 ⁻³ mCi = 10 ⁻⁶ Ci), the nanocurie (1 nCi = 10 ⁻⁹ Ci), and the kilocurie (1 kCi = 1000 Ci). Equation 4-1 may be modified for these units of activity:

The curie was defined originally as the activity of 1 g of ²²⁶ Ra; however, this value “changed” from time to time as more accurate measurements of the ²²⁶ Ra decay rate were obtained. For this reason, the ²²⁶ Ra standard was abandoned in favor of a fixed value of 3.7 × 10 ¹⁰ dps. This is not too different from the currently accepted value for ²²⁶ Ra (3.656 × 10 ¹⁰ dps/g).

SI units are the “official language” for nuclear medicine and are used in this text; however, because traditional units of activity still are used in day-to-day practice in many laboratories, we sometimes also indicate activities in these units as well. Conversion factors between traditional and SI units are provided in Appendix A .

The amounts of activity used for nuclear medicine studies typically are in the MBq-GBq range (10s of µCi to 10s of mCi). Occasionally, 10s of gigabecquerels (curie quantities) may be acquired for long-term supplies. External-beam radiation sources (e.g., ⁶⁰ Co therapy units) use source strengths of 1000s of GBq [1000 GBq = 1 terraBq (TBq) = 10 ¹² Bq]. At the other extreme, the most sensitive measuring systems used in nuclear medicine can detect activities at the level of a few becquerels (nanocuries).

1

The Decay Factor

With the passage of time, the number N of radioactive atoms in a sample decreases. Therefore the activity A of the sample also decreases (see Equation 4-4 ). Figure 4-1 is used to illustrate radioactive decay with the passage of time.

Suppose one starts with a sample containing N (0) = 1000 atoms * of a radionuclide having a decay constant λ = 0.1 sec ⁻¹ . During the first 1-sec time interval, the approximate number of atoms decaying is 0.1 × 1000 = 100 atoms (see Equation 4-1 ). The activity is therefore 100 Bq, and after 1 sec there are 900 radioactive atoms remaining. During the next second, the activity is 0.1 × 900 = 90 Bq, and after 2 sec, 810 radioactive atoms remain. During the next second the activity is 81 Bq, and after 3 sec 729 radioactive atoms remain. Thus both the activity and the number of radioactive atoms remaining in the sample are decreasing continuously with time. A graph of either of these quantities is a curve that gradually approaches zero.

* N ( t ) is symbolic notation for the number of atoms present as a function of time t. N (0) is the number N at a specific time t = 0, that is, at the starting point.

An exact mathematical expression for N ( t ) can be derived using methods of calculus. * The result is:

* The derivation is as follows:

from which follows Equation 4-6 .

Thus N ( t ), the number of atoms remaining after a time t , is equal to N (0), the number of atoms at time t = 0, multiplied by the factor e ^{−λ t} . This factor e ^{−λ t} , the fraction of radioactive atoms remaining after a time t , is called the decay factor (DF). It is a number equal to e —the base of natural logarithms (2.718 …)—raised to the power −λ t . For given values of λ and t , the decay factor can be determined by various methods as described in Section C later in this chapter. Note that because activity A is proportional to the number of atoms N (see Equation 4-4 ), the decay factor also applies to activity versus time:

The decay factor e ^{−λ t} is an exponential function of time t . Exponential decay is characterized by the disappearance of a constant fraction of activity or number of atoms present per unit time interval. For example if λ = 0.1 sec ⁻¹ , the fraction is 10% per second. Graphs of e ^{−λ t} versus time t for λ = 0.1 sec ⁻¹ are shown in Figure 4-2 . On a linear plot, it is a curve gradually approaching zero; on a semilogarithmic plot, it is a straight line. It should be noted that there are other processes besides radioactive decay that can be described by exponential functions. Examples are the absorption of x- and λ-ray beams (see Chapter 6 , Section D) and the clearance of certain tracers from organs by physiologic processes (see Chapter 22 , Section B.2).

When the exponent in the decay factor is “small,” that is, λ t ≲ 0.1, the decay factor may be approximated by e ^{−λ t} ≈ 1 − λ t . This form may be used as an approximation in Equations 4-6 and 4-7 .

2

Half-Life

As indicated in the preceding section, radioactive decay is characterized by the disappearance of a constant fraction of the activity present in the sample during a given time interval. The half-life (T _1/2 ) of a radionuclide is the time required for it to decay to 50% of its initial activity level. The half-life and decay constant of a radionuclide are related as ^*

where ln 2 ≈ 0.693. Usually, tables or charts of radionuclides list the half-life of the radionuclide rather than its decay constant. Thus it often is more convenient to write the decay factor in terms of half-life rather than decay constant:

* The relationships are derived as follows:

from which follow Equations 4-8 and 4-9 .

3

Average Lifetime

The actual lifetimes of individual radioactive atoms in a sample range anywhere from “very short” to “very long.” Some atoms decay almost immediately, whereas a few do not decay for a relatively long time (see Fig. 4-2 ). The average lifetime τ of the atoms in a sample has a value that is characteristic of the nuclide and is related to the decay constant λ by *

* The equation from which Equation 4-11 is derived is:

Combining Equations 4-9 and 4-11 , one obtains

The average lifetime for the atoms of a radionuclide is therefore longer than its half-life, by a factor 1/ln 2 (≈1.44). The concept of average lifetime is of importance in radiation dosimetry calculations (see Chapter 22 ).