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Ch. 21.4 Nuclear Decay Simulation

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Program 1

Nuclear Decay Simulation

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 Mathematical Problem Formulation

                                        
        

    
Radioactive Decay Simulation

    

        A radioactive substance starts with an initial quantity of \( N_0 = 10,000 \) atoms, and each atom has a probability of decaying based on a decay constant \( \lambda = \frac{0.693}{5.27} \), where 5.27 represents the half-life of the substance in years. 
    

    

        The decay process is simulated over a period of 50 time units, with a time step of 1 unit.
    

                                    

                                    
                                                                        

                                        
 Theoretical Background & Computational Mechanics

                                        
    

  
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,
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  • \( N(t) \) is the number of undecayed nuclei at time \( t \),
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  • \( \lambda \) is the decay constant, which defines the likelihood of decay per unit time.
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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 \( N_0 \).
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  • For each time step, generate a random number for each undecayed nucleus.
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  • If the random number is less than \( P \), mark the nucleus as decayed.
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  • Track the number of remaining nuclei and plot \( N(t) \) against time \( t \).
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                            Program 2
                            
Nuclear Decay Simulation

                        

                        

                            
                            
                            
                        

                    


                    
                    

                        
                        

                            

                                 Python 3.11
                                Ready
                            

                            

                                
                                
                            

                            
                            

                                

                                    
                                    
                                    
                                    
                                

                                

                                    
                                

                            

                        


                        
                        

                            

                                
                                
                                                                
                                                            


                            
                            

                                

                                

                            


                            
                            

                                

                                    
Click "Run Code" to execute script and render figures.

                                

                            


                            
                                                        

                                

                                                                        

                                        
 Mathematical Problem Formulation

                                        
Nuclear Decay Simulation: Compare with different decay constants.

                                    

                                    
                                                                    

                            

                                                    

                    

                

                            
            
            
            
            
            

                

                    

                         Theoretical Principles & Computational Physics Guide
                    

                    

                        Nuclear Decay Simulation: Statistical Physics & Stochastic Modeling                    

                    

                         Dr. Alorika Chatterjee & Dr. Anirban Shaw
                         Science & Physics Learners
                         Python 3 / NumPy / SciPy
                    

                


                
                

                    

                         1. Core Principles & Theoretical Framework
                    

                    

                        
                    

                        
 Stochastic Processes: Simulates random physical phenomena including Brownian diffusion, radioactive decay, and thermal fluctuations.

                        
 Ensemble Averages: Computes macroscopic thermodynamic observables as statistical ensemble averages over microscopic configurations.

                        
 Monte Carlo Convergence: Statistical error scales inversely with the square root of trial count: $\sigma \propto 1/\sqrt{N}$.

                    
                    

                


                
                

                    

                         2. Mathematical Formulation & Algorithmic Pipeline
                    

                    

                        
                    
Random walk diffusive spreading and Monte Carlo error scaling:

                    
$$ \langle x(t) \rangle = 0, \quad \langle x^2(t) \rangle = 2 D t, \quad \text{Statistical Uncertainty: } \sigma_N = \frac{\sigma}{\sqrt{N}} $$
                    

                


                
                

                    

                         3. Numerical Mechanics & Implementation Logic
                    

                    

                        
                    
    
                        
  • Random Number Seeding: Initialize pseudo-random generators with fixed seeds to ensure reproducible simulation results.
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  • Ensemble Sampling: Execute $M$ independent trials of $N$ steps each to build accurate probability distributions.
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  • Histogramming: Bin particle final positions to verify convergence toward Gaussian/Poisson distributions.
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                         4. Relevant Computational Practice Exercises
                    

                    

                                                    

                                Exercise 1
                                

                                    Simulate $M = 1000$ independent random walk trajectories and confirm that root-mean-square displacement scales as $\sqrt{t}$.                                

                                
                            

                                                    

                                Exercise 2
                                

                                    Apply Monte Carlo integration to estimate $\pi$ using hit-or-miss circle sampling and plot error vs trial count $N$.                                

                                
                            

                                                    

                                Exercise 3
                                

                                    Model radioactive nuclear decay for $N_0 = 10^4$ nuclei and compare against the continuous decay law $N(t) = N_0 e^{-\lambda t}$.                                

                                
                            

                                            

                


                
                

                     Standard Literature & References:
                    
    
                                                    
  • F. Reif, "Fundamentals of Statistical and Thermal Physics", Waveland Press (2009).
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  • Mark Newman, "Computational Physics", University of Michigan (2013).
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