Radioactive Decay and Half-Life Calculation

Example in Nuclear Physics

Problem Statement:

A radioactive substance undergoes exponential decay, which can be described by the equation:
$$
N(t) = N_0 e^{-\lambda t}
$$

  • $N(t)$ is the number of undecayed nuclei at time $t$.
  • $N_0$ is the initial number of nuclei.
  • $\lambda$ is the decay constant.
  • $t$ is time.

The half-life ($T_{1/2}$) of the substance is related to $\lambda$ by:
$$
T_{1/2} = \frac{\ln(2)}{\lambda}
$$

Task:

  1. Simulate radioactive decay for a sample of $1000$ nuclei over $10$ half-lives.
  2. Plot the number of remaining nuclei as a function of time.
  3. Verify that at $T_{1/2}$, half of the original nuclei remain.

Python Implementation

Below is the $Python$ code to solve and visualize the decay process.

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import numpy as np
import matplotlib.pyplot as plt

# Constants
N0 = 1000  # Initial number of nuclei
half_life = 5  # Half-life in arbitrary time units
lambda_ = np.log(2) / half_life  # Decay constant

time = np.linspace(0, 10 * half_life, 1000)  # Time from 0 to 10 half-lives
N_t = N0 * np.exp(-lambda_ * time)  # Decay equation

# Plot results
plt.figure(figsize=(8, 5))
plt.plot(time, N_t, label=f"Radioactive Decay (Half-life = {half_life} units)", color='blue')
plt.axvline(half_life, linestyle='--', color='red', label=f"T_1/2 = {half_life} units")
plt.axhline(N0/2, linestyle='--', color='green', label=f"N = {N0//2} at T_1/2")

plt.xlabel("Time")
plt.ylabel("Remaining Nuclei")
plt.title("Exponential Radioactive Decay")
plt.legend()
plt.grid()
plt.show()

Explanation of the Code

  1. Initialization:

    • The initial number of nuclei is set to $1000$.
    • The half-life is defined as $5$ arbitrary time units.
    • The decay constant $\lambda$ is calculated using $\lambda = \ln(2)/T_{1/2}$.

  2. Simulation:

    • Time is generated as an array from $0$ to $10$ half-lives.
    • The number of remaining nuclei is computed using the exponential decay formula.

  3. Visualization:

    • A plot is generated showing the decay curve.
    • A vertical dashed line marks $T_{1/2}$.
    • A horizontal dashed line marks $N_0/2$, verifying that half of the nuclei remain at $T_{1/2}$.

This provides a clear illustration of radioactive decay over time.