# 2.2.1 Radioactive Decay

References: Sorenson and Phelps; Hendee
Created October 6, 1995

Unit of the "Curie" was originally defined as the activity of 1 gm of the naturally occuring element radium. This isotope turns out to have been Radium-226 with a half-life of about 1600 years.
1 Curie = 1 Ci = 3.7 x 10**10 disintegrations/sec = 3.7 x 10**10 Bequerels
(Bq)

1 rutherford = 1 Rf = 1 MBq

The element Bi-209 (Z=83) is stable. All other nuclides with Z > 82 (lead) are unstable. (Element number 110 was recently discovered). At least 27 known elements are totally radioactive, i.e. they have no stable isotopes. Unstable nuclei undergo trqnsitions defined as radioactive decay in order to attain an energetically favorable quantum mechanical state.

## 2.2.1.1 Mathematics of Radioactive Decay

### There are four known forces at work in our world:

- Strong Nuclear Force: relative strength =1
- Electrostatic Force: relative strength = 10**-2
- Weak Nuclear Force: relative strength = 10**-13
- Gravitational Force: relative strength = 10**-39

The weak force is responsible for radioactive decay. Radioactive decay is a game of chance. Once cannot pick out a single nucleus and predict how long it will be until it undergoes radioactive decay. But *in sufficient numbers*, the **probability** of decay becomes well defined.
If we have enough radioactive nuclei, then we can talk about the likelihood of decay in terms of probability distributions and well-defined statistics such as mean lifetime. If we have sufficient numbers of radioactive nuclei, the rate of decay, i.e. the number of decays per unit time, is directly proportional to the number of nuclei:

which means if you double the number of nuclei in your sample you will double the rate at which you observe decays occuring within the sample. Here lambda is a constant of proportionality known as the **decay constant**and the negative sign means the the number of nuclei is decreasing as radioactive decay takes place.

The decay constant is different for each radioactive nucleus. This is because each radioactive nucleus has a unique internal makeup of nucleons.

### 2.2.1.2 Equation for Radioactive Decay

Separating variables in the above equation, we get the following:

### 2.2.1.3 Radioactivity

The symbol A is often substituted:

where A is defined as the radioactivity or simply activity of the sample.
### 2.2.1.4 Half-Life

The halflife t-sub-1/2 is defined as the time interval over which the activity of the sample has decayed to one half its original value. The following derivation defines the half-life in terms of the decay constant:

### 2.2.1.5 Mean Life Tau

We integrate the time period over which each nucleus lives over all nuclei which ultimately decay. This gives us the mean lifetime tau associated with this nuclide:

In order to solve this integral, we must integrate by parts:

### 2.2.1.6 Decay Factor

The fraction of activity remaining is known as the decay factor:

The decay factor is often used in tables to readily provide a factor with which to compute activity after a given decay time.
### 2.2.1.7 Half-Lives and Absorbed Dose

We often hear that short lived nuclides are better for minimizing absorbed dose to the patient. Here we demonstrate the reason for this:

If we need the same amount of activity in order to acquire an acceptable image in a reasonable amount of time, then this determines the number of radioactive nuclei within the body. If we assume that all of these nuclei remain in the body until they decay, then the shorter the halflife, the fewer decays will occur within the body.

Return to Main Table of Contents

Return to Section 2 Table of Contents

Proceed to 2.2.2 - Specific Activity

Douglas J. Wagenaar, Ph.D., wagenaar@nucmed.bih.harvard.edu