KAHIBARO
Table of Contents
Meaning of Activity
In radioactivity, activity tells us how fast a radioactive sample is decaying. It is the number of nuclear decays that occur per unit time. If a sample has a high activity, many nuclei decay each second. If it has a low activity, only a few decay each second.
Activity does not tell us how dangerous a source is by itself. It only tells us the decay rate. The type of radiation, its energy, and how it interacts with matter are separate topics.
Definition of Activity
Suppose a sample contains $N$ radioactive nuclei at a given time $t$. As time passes, $N$ decreases because some nuclei decay. The activity $A$ is defined as the rate at which the number of undecayed nuclei decreases:
$$
A = -\frac{dN}{dt}
$$
The minus sign appears because $N$ gets smaller with time, while activity is taken as a positive quantity.
Using the radioactive decay law,
$$
N(t) = N_0 e^{-\lambda t}
$$
we get
$$
A(t) = \lambda N(t)
$$
and therefore
$$
A(t) = \lambda N_0 e^{-\lambda t} = A_0 e^{-\lambda t}
$$
where $A_0 = \lambda N_0$ is the initial activity.
Important formulas for activity:
$$
A = -\frac{dN}{dt}
$$
$$
A = \lambda N
$$
$$
A(t) = A_0 e^{-\lambda t}
$$
Activity decreases exponentially with time, just like the number of undecayed nuclei.
Physical Interpretation
The formula $A = \lambda N$ is very useful. It says that activity depends on two things, the decay constant $\lambda$, and the number of radioactive nuclei still present.
If a sample has more undecayed nuclei, it has more chances for decay, so its activity is larger. As the sample decays and $N$ becomes smaller, the activity also becomes smaller.
A large decay constant means each nucleus has a greater probability of decaying per unit time, so the activity is higher for the same number of nuclei.
Units of Activity
The SI unit of activity is the becquerel, abbreviated Bq.
$$
1 \,\text{Bq} = 1 \,\text{decay per second}
$$
An older unit is the curie, abbreviated Ci. It is much larger than the becquerel.
$$
1 \,\text{Ci} = 3.7 \times 10^{10} \,\text{Bq}
$$
This means one curie corresponds to $3.7 \times 10^{10}$ decays every second.
| Unit | Meaning |
|---|---|
| $1\,\text{Bq}$ | 1 decay per second |
| $1\,\text{Ci}$ | $3.7 \times 10^{10}$ decays per second |
Always remember:
$$
1\,\text{Bq} = 1\,\text{s}^{-1}
$$
Activity is a rate, not an amount of material.
Activity and Number of Nuclei
Because
$$
A = \lambda N
$$
we can move between activity and number of radioactive nuclei if $\lambda$ is known:
$$
N = \frac{A}{\lambda}
$$
This is helpful in practice. Sometimes we measure activity directly with a detector, then infer how many radioactive nuclei are present.
If the sample contains very many atoms but only some are radioactive, activity depends only on the radioactive ones.
How Activity Changes with Time
Since activity follows exponential decay, it falls by the same factor over equal time intervals that correspond to the half-life.
If the half-life is $T_{1/2}$, then after one half-life:
$$
A = \frac{A_0}{2}
$$
After two half-lives:
$$
A = \frac{A_0}{4}
$$
After three half-lives:
$$
A = \frac{A_0}{8}
$$
So activity becomes smaller and smaller, but in the ideal mathematical model it never becomes exactly zero.
| Time | Activity |
|---|---|
| $0$ | $A_0$ |
| $T_{1/2}$ | $A_0/2$ |
| $2T_{1/2}$ | $A_0/4$ |
| $3T_{1/2}$ | $A_0/8$ |
Relation to Mass
A larger mass of a radioactive isotope usually means more radioactive nuclei, so it usually means greater activity. But activity is not determined by mass alone. Two samples with the same mass can have very different activities if they are different isotopes, because their decay constants are different.
For a pure radioactive isotope of molar mass $M$, the number of nuclei is
$$
N = \frac{m}{M} N_A
$$
where $m$ is the sample mass and $N_A$ is Avogadro's number. Then
$$
A = \lambda \frac{m}{M} N_A
$$
This shows that activity is proportional to mass for a given isotope.
Example
Suppose a sample contains
$$
N = 2.0 \times 10^{12}
$$
radioactive nuclei, and its decay constant is
$$
\lambda = 4.0 \times 10^{-6}\,\text{s}^{-1}
$$
Then the activity is
$$
A = \lambda N
$$
$$
A = (4.0 \times 10^{-6})(2.0 \times 10^{12})
$$
$$
A = 8.0 \times 10^6\,\text{s}^{-1}
$$
Since $1\,\text{Bq} = 1\,\text{s}^{-1}$,
$$
A = 8.0 \times 10^6\,\text{Bq}
$$
So the sample has an activity of $8.0$ MBq.
Activity Curve
The activity of a sample decreases smoothly with time.
The graph has the same shape as the graph of $N(t)$ because activity is directly proportional to $N$.
Measuring Activity
In experiments, activity is often estimated by counting radiation events detected in a certain time. If a detector records many counts per second, the source likely has higher activity. In real measurements, detector efficiency matters, so the measured count rate is not always equal to the true activity.
This means activity is a property of the source, while count rate is what the detector records.
Do not confuse activity with count rate.
Activity is the true decay rate of the source.
Count rate is the number of events detected per unit time, and it can be smaller than the activity.
Summary
Activity is the decay rate of a radioactive sample. It is defined by
$$
A = -\frac{dN}{dt}
$$
and for exponential decay it satisfies
$$
A = \lambda N
$$
so it also decreases exponentially:
$$
A(t) = A_0 e^{-\lambda t}
$$
Its SI unit is the becquerel, with
$$
1\,\text{Bq} = 1\,\text{decay/s}
$$
Activity is one of the most important quantities in radioactivity because it tells us how rapidly a sample is undergoing nuclear decay.