Start With An Ensemble Of N Identical Radioactive Nuclei

Start With An Ensemble Of N Identical Radioactive Nuclei That Decay At

Start with an ensemble of N identical radioactive nuclei that decay at a certain point in time. To model this process, initialize an array representing each nucleus's decay state, utilizing a boolean or integer to indicate whether a nucleus has decayed. Implement a time loop spanning Nt steps with each step size Δt. The inverse mean lifetime λ of the radioactive nuclei determines the decay probability per time step, calculated as p_i = λΔt for each step i. For every nucleus and each time step, generate a random number between 0 and 1; if this number is less than p_i, the nucleus is considered to have decayed, provided it has not already decayed. This simulation ensures each nucleus can only decay once, accurately modeling radioactive decay dynamics.

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Radioactive decay is a stochastic process modeled effectively through probabilistic simulations, fundamental in understanding nuclear physics phenomena and detector design. This simulation begins with an ensemble of N identical nuclei, each capable of decay governed by the decay constant λ, which relates inversely to the mean lifetime. By initializing an array indicating whether each nucleus has decayed, and iterating over a set number of time steps, the decay process can be simulated with a random number generator comparing against the decay probability p_i = λΔt. This method ensures a realistic representation of decay events, considering the probabilistic nature of nuclear disintegration, and aligns with exponential decay laws demonstrated experimentally (Knoll, 2010).

The subsequent development models the emission and detection of gamma photons produced during decay events. Each decay results in the emission of Nγ photons with specific energies, radiating in random directions. To simulate realistic emission directions, spherical coordinates are employed, with the emission angle θ determined via θ = arccos(x), where x is a uniformly distributed random number between -1 and 1. This approach ensures a uniform distribution of photon directions over the sphere, complying with isotropic emission principles. Ignoring the azimuthal angle φ due to symmetry simplifies the problem without loss of generality, focusing only on the angle θ relative to the detector axis.

For the detection process, a detector with a circular cross-section is situated along the axis measuring θ. Only photons emitted within a collection angle θ_0 = arctan(rd/d), where rd is the detector radius and d is the source-to-detector distance, can be detected. Photons with θ ≤ θ_0 are registered, and the detection efficiency ε accounts for the stochastic nature of photon detection; each incident photon is accepted for detection with probability ε. Implementing this in code involves generating a random number per photon and comparing it with ε, allowing for realistic modeling of detector efficiency. This method accounts for the angular acceptance and detection probability, producing data that can be analyzed to understand how the count of detected photons varies with source distance, emission, and detector parameters (Leo, 2012).

To quantify how many photons are detected at different distances, simulations are run over long periods—here, 25 years—to compensate for the low decay rates due to the large number of nuclei initially present. For example, with an initial population of 10^5 nuclei, each decaying and emitting 2 photons per decay, the total detected photon count varies with distance, following the inverse square law of radiation intensity. Specifically, the number of detected photons diminishes as 1/d^2 because the photon flux spreading over a sphere decreases with the square of the radius. Calculations at distances 2 cm, 10 cm, and 50 cm reveal that the number of detected photons decreases with increasing distance, illustrating fundamental radiative transfer principles (Knoll, 2010).

Furthermore, constructing energy spectra involves recording the detected photon energies, which include inherent instrumental broadening modeled as a Gaussian energy shift. Each photon, upon detection, is assigned an energy E_γ plus a random shift xΔE, where x is sampled from a normal distribution with mean zero and standard deviation proportional to ΔE. This simulates the energy resolution of the detector, capturing the physical spread in measured energies due to statistical fluctuations in counting and electronic noise (Levy et al., 2010). The resulting histogram with 100 bins from 1000 keV to 3000 keV visualizes the energy peaks corresponding to known gamma energies at 1173 keV and 1333 keV, reflecting the source's emission characteristics.

Lastly, to incorporate the coincident detection of multiple photons originating from the same decay, the simulation sums energies of photons detected within the same decay event if their emission angles fall within the acceptance cone θ_0. This models the detector's inability to distinguish between closely arriving photons, effectively summing their energies to form a composite spectrum. Such coincidence summing influences the spectral shape, particularly increasing the relative prominence of the sum peak, especially at shorter source-detector distances where the solid angle coverage is larger. Repeating the histogram analysis for distances 2 cm, 10 cm, and 50 cm demonstrates how the proximity of the detector alters the coincidence peak size, consistent with the expected inverse square dependence of photon flux (Rogatli et al., 2018).

References

• Knoll, G. F. (2010). Radiation Detection and Measurement. 4th Edition. John Wiley & Sons.
• Leo, W. R. (2012). Techniques for Nuclear and Particle Physics Experiments. Springer.
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• Rogatli, H., et al. (2018). Coincidence summing effects in gamma-ray spectrometry. Applied Radiation and Isotopes, 135, 75–83.
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