42-3 Radioactive Decay

42-3 Radioactive Decay#

Prompts

  • Why does radioactive decay follow an exponential law? What does it mean that each nucleus has the same probability to decay per unit time?

  • Derive the relation between decay constant \(\lambda\) and half-life \(T_{1/2}\). If a sample has half-life 10 years, what fraction remains after 30 years?

  • Define activity \(R\). How does \(R\) change with time? Why is \(R = \lambda N\)?

  • Given \(N_0\) and \(\lambda\), how do you find the number of decays in a time interval? Can you work through an example?

  • The becquerel (Bq) is 1 decay/s. A sample has activity 1000 Bq. What does that tell you about \(N\) and \(\lambda\)?

Lecture Notes#

Overview#

  • Radioactive decay is a random process: each unstable nucleus has the same probability per unit time to decay, independent of its age. This leads to exponential decay of the number of undecayed nuclei.

  • The decay constant \(\lambda\) characterizes the decay rate; the half-life \(T_{1/2} = (\ln 2)/\lambda\) is the time for half the nuclei to decay.

  • Activity \(R\) is the decay rate (decays per second); \(R = \lambda N\) and \(R\) also decreases exponentially. The SI unit is the becquerel (Bq).

The Decay Law#

Radioactive decay is probabilistic: we cannot predict when a specific nucleus will decay, but each nucleus of a given nuclide has the same probability \(\lambda\,dt\) to decay in a short time \(dt\). The constant \(\lambda\) is the decay constant (units: s\(^{-1}\)).

The rate of change of the number \(N\) of undecayed nuclei is

(443)#\[ \frac{dN}{dt} = -\lambda N \]

The minus sign indicates that \(N\) decreases. This differential equation has the solution

(444)#\[ N = N_0 e^{-\lambda t} \]

where \(N_0\) is the number at \(t = 0\).

Why exponential?

Because each nucleus decays independently with the same probability per unit time, the fraction that decays in \(dt\) is constant: \(dN/N = -\lambda\,dt\). Integrating gives \(N \propto e^{-\lambda t}\). No memory, no aging — a nucleus “born” 1 second ago has the same decay probability as one that has survived for years.

Half-Life and Mean Lifetime#

The half-life \(T_{1/2}\) is the time for half the nuclei to decay. Setting \(N(T_{1/2}) = N_0/2\) in Eq. (444):

(445)#\[ T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda} \]

After \(n\) half-lives, the fraction remaining is \((1/2)^n\).

The mean lifetime (or mean life) is \(\tau = 1/\lambda\). It is the average time a nucleus survives before decaying.

Activity#

The activity \(R\) is the number of decays per unit time:

(446)#\[ R = -\frac{dN}{dt} = \lambda N \]

Since \(N = N_0 e^{-\lambda t}\), the activity also decays exponentially:

(447)#\[ R = R_0 e^{-\lambda t},\quad R_0 = \lambda N_0 \]

Quantity

Symbol

Relation

Decay constant

\(\lambda\)

Probability per unit time

Half-life

\(T_{1/2}\)

\((\ln 2)/\lambda\)

Mean lifetime

\(\tau\)

\(1/\lambda\)

Activity

\(R\)

\(\lambda N\)

Units: The SI unit of activity is the becquerel (Bq): \(1\ \text{Bq} = 1\) decay/s. The curie (Ci) is \(3.7\times 10^{10}\) Bq.

Measuring activity

We measure \(R\) (counts per second, corrected for detector efficiency), not \(N\) directly. From \(R = \lambda N\), we can infer \(N\) if \(\lambda\) is known, or vice versa.

Summary#

  • Decay law: \(dN/dt = -\lambda N\); solution \(N = N_0 e^{-\lambda t}\).

  • Half-life: \(T_{1/2} = (\ln 2)/\lambda\); mean lifetime \(\tau = 1/\lambda\).

  • Activity: \(R = \lambda N = -\frac{dN}{dt}\); \(R\) decays exponentially.

  • Unit: becquerel (Bq) = 1 decay/s.