Start With An Ensemble Of N Identical Radioactive Nuc 943104

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 give you an idea of a possible way to go about this, you could start with an array of integers (or bool) for determining whether a particle has decayed or not. Create a time loop of Nt time steps with step size Δt. The inverse-mean-lifetime λ of a radioactive sample is the probability of decay per unit time; for each time step i, the probability of decay is pi = λΔt. For each nucleus, generate a random number between 0 and 1; if the number is less than pi and the nucleus hasn't decayed before, mark it as decayed. Repeat this process through the time steps until all nuclei have either decayed or the simulation ends.

Building upon this, develop a virtual detector that models gamma photon emission, including angular acceptance. When each nucleus decays, it emits Nγ photons with different energies traveling in random directions, characterized by angles θ and φ in spherical coordinates. To simplify, align the detector along the θ measurement axis, ignoring φ due to symmetry. The detector has a circular cross-section with radius rd at distance d from the source; photons emitted within a collection angle θ0, where tan θ0 = rd/d, will be detected. Generate the emission angle θ for each photon by producing a random number x between -1 and 1 and calculating θ = arccos x, which ensures uniform distribution over the sphere's surface.

Implement a detector function that registers only photons with θ

Conduct simulations with a detector of radius 10 cm, efficiency 0.3, and a source emitting 2 photons per decay, starting with 100,000 nuclei with decay constant λΔt over 25 years at distances of 2 cm, 10 cm, and 50 cm. Analyze how the number of detected photons varies with distance, expecting fewer detected photons at larger distances due to geometric and efficiency factors. The number of detected photons should decline with increasing distance following the inverse square law, modulated by the detector’s solid angle coverage and efficiency.

In addition, generate energy spectra for the detected photons, incorporating an inherent energy detection width (ΔE = 20 keV). For each detected photon, record its energy as Eγ plus a correction term ΔE × x, where x is drawn from a normal distribution generated by a Gaussian random function. Use the randn function (in MATLAB) or a custom Gaussian random number generator (shown in the prompt) for this purpose. This models the energy resolution of the detector.

Simulate the emission of two gamma-ray energies, E1 = 1173 keV and E2 = 1333 keV, from the initial nuclei. For each photon detected, store the energy with the added Gaussian shift and plot histograms with 100 bins ranging from 1000 keV to 3000 keV. This spectral analysis reveals the characteristic peaks of the emitted gamma photons, broadened by the detector’s resolution.

Finally, account for coincidence detection where photons originating from the same nucleus and falling within the acceptance angle are summed instead of recorded separately, mimicking the detector’s inability to distinguish closely arriving photons. For simulations at the same distances (2 cm, 10 cm, 50 cm), create histograms displaying the summed energies, observing how the proximity of the detector affects the height and shape of the coincidence peaks. Closer distances produce larger coincidence events due to higher photon flux within the acceptance angle, showing the impact of geometrical and timing considerations on spectral measurements.

These comprehensive simulations demonstrate key aspects of radioactive decay, gamma-ray detection, energy resolution, and coincidence effects in nuclear physics experiments, highlighting the importance of geometry, efficiency, and energy resolution in spectral analysis and detector performance.

Paper For Above instruction

Radioactive decay phenomena and gamma-ray detection are fundamental topics in nuclear physics, providing crucial insights into nuclear structure, decay mechanisms, and detector technologies. This paper explores the simulation of radioactive decay, photon emission, and detection processes, emphasizing angular acceptance, energy resolution, and coincidence detection effects. Such simulations help in understanding detector efficiency, the influence of geometry, and the spectral characteristics of emitted gamma radiation, essential for designing and interpreting nuclear experiments.

The modeling begins with the stochastic nature of radioactive decay, utilizing a Monte Carlo approach. An initial ensemble of N nuclei is represented by an array indicating their decay status. The decay probability per time step, pi = λΔt, is derived from the inverse-mean-lifetime λ of the sample. By generating a uniform random number for each nucleus at each time step and comparing it with pi, one determines whether the nucleus has decayed without decay repetition. This process continues over Nt time steps, leading to a decay distribution consistent with exponential decay laws.

Building on this, the simulation incorporates gamma photon emission characteristics. Each decay emits Nγ photons with distinct energies, directed randomly in space described by spherical coordinates (θ, φ). Due to symmetry, φ can be ignored by aligning the detector along the θ-axis, focusing on the angular acceptance. The detector's geometrical acceptance depends on the collection angle θ0, derived from its physical dimensions and distance d via tan θ0 = rd/d. Photons emitted within this angle are candidates for detection. To generate emission angles θ uniformly over the sphere, a random number x in [-1, 1] is used to compute θ = arccos x, ensuring uniform volumetric distribution rather than simply choosing θ uniformly between 0 and π.

The detector function evaluates whether each photon falls within the acceptance cone and applies an efficiency factor via a probabilistic acceptance. The detection probability accounts for detector imperfections, modeled as a random acceptance step with a given efficiency. Simulating photon detection at various distances (2 cm, 10 cm, 50 cm) reveals how geometric factors influence the count of detected photons, aligning with the inverse square law. As distance increases, the solid angle subtended by the detector decreases, reducing detection rates proportionally.

Further, the energy spectrum of the detected photons is constructed by adding a Gaussian-distributed random shift to each photon’s intrinsic energy, reflecting the finite energy resolution. The Gaussian distribution is generated using the randn function in MATLAB or a custom function in C/C++, implementing the probability density p(x) = e^(-x²/2). Such energy broadening results in characteristic peaks corresponding to known gamma energies, providing insights into the spectral resolution and detector calibration. Histograms of these energies, with specified bin ranges and widths, display the photon populations and the expected line shapes.

The model extends to coincidence detection, where multiple photons emitted from the same decay event and arriving within a short time window are summed rather than individually recorded. This simulates pile-up effects common in nuclear spectroscopy, affecting the spectral shape and peak intensities. Running simulations at different distances shows how event rates and coincidence events vary with geometrical considerations, impacting the measured spectra. Closer detectors yield higher coincidence counts, leading to prominent summed-energy peaks, while larger distances diminish these effects.

Overall, this comprehensive simulation approach synthesizes concepts from stochastic decay processes, photon emission and detection physics, energy resolution, and coincidence phenomena. These models are invaluable for designing efficient detectors, interpreting spectral data accurately, and understanding the limitations posed by geometry, efficiency, and resolution. Such insights support continued advancement in nuclear instrumentation and experimental techniques.

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