Theory of Nuclear Decay Simulation
Radioactive decay is typically modeled as an exponential process. The probability of a nucleus decaying in a short time interval \( \Delta t \) is proportional to the length of the interval. Mathematically, the number of undecayed nuclei \( N(t) \) at time \( t \) follows the equation:
$$ N(t) = N_0 e^{-\lambda t} $$
where:
- \( N_0 \) is the initial number of nuclei,
- \( N(t) \) is the number of undecayed nuclei at time \( t \),
- \( \lambda \) is the decay constant, which defines the likelihood of decay per unit time.
Decay Constant and Half-Life
The decay constant \( \lambda \) is related to the half-life \( T_{1/2} \) of a radioactive substance by the equation:
$$ T_{1/2} = \frac{\ln(2)}{\lambda} $$
where \( T_{1/2} \) is the time required for half of the nuclei in a sample to decay.
Random Numbers in Simulation
Each nucleus decays independently, which means that the decay of one nucleus does not influence the decay of another. To simulate this random process, we can use random numbers to determine whether each nucleus decays within a given time interval.
For a small time interval \( \Delta t \), the probability \( P \) that a nucleus will decay during this interval is given by:
$$ P = \lambda \Delta t $$
A random number \( r \) is generated for each nucleus in the range \([0, 1]\). If \( r < P \), the nucleus is considered to have decayed in that time interval; otherwise, it remains undecayed.
Simulation Process
The simulation proceeds by repeating the random-number check over successive time intervals:
- Start with an initial number of nuclei \( N_0 \).
- For each time step, generate a random number for each undecayed nucleus.
- If the random number is less than \( P \), mark the nucleus as decayed.
- Track the number of remaining nuclei and plot \( N(t) \) against time \( t \).