**What Is an Example of Nuclear Decay?**

Henri Becquerel’s discovery of nuclear decay in uranium in 1896 ushered in a new era of innovation in physics. In 1899, the emissions he measured (known as radiation) were categorized into two types (alpha and beta decay), and in 1903, a new class of radiation named gamma rays was identified. These radiations are produced by the decay of radioactive atomic nuclei, which transition from a high-energy, unstable state to a low-energy, stable ground state by emitting energy under the form of specific particles or waves.

This process, known as radioactive decay, varies from element to element, as does the nature of the emitted particles (alpha, beta or gamma) and their respective energy levels. In our lab, we set out to gain a better understanding of nuclear decay by obtaining precise measurements of decay rates for different radioactive samples using a scintillation counter and signal-processing electronic circuit.

Here, we show how radioactive decay counts follow the Poisson statistical distribution, how the radioactive decay plot of Ba137m can be used to determine its half-life, and how gamma-ray spectra calibration can be used to identify unknown features in decay spectra. Using a radioactive sample, scintillation detector, photomultiplier tube, preamp, spectroscopy amplifier, single and multi-channel analyzers, capacitor, counter and oscilloscope, we successfully measured radioactive decay counts for a variety of samples and used this data to resolve the aforementioned problems.** **Our results demonstrated that nuclear decay follows a Poisson distribution, that Ba137m does indeed have a half-life of approximately 153 seconds (we found it to be 154.7s±2.3s) , and that the most prominent feature of the Na22 spectrum was due to positron emission during + decay. We also calculated two close approximations of the electron mass me=9.25±0.31*10-31kg and me=9.09±0.18*10-31kg using distinctive features of Cs137 and Na22 gamma-ray spectra respectively.

**Introduction**

Measuring nuclear decay has been greatly simplified by the advent of more modern detectors and signal-processing circuits. Indeed, in this lab, we used a NaI scintillation detector, which absorbed gamma rays emitted by the radioactive sample and then relaxed by emitting photons. The photomultiplier tube converted these light pulses to current signals, the preamp converted them to voltage pulses, and these signals were then processed through various circuit components which filtered, measured and counted them. We began by measuring decay rates for a Cs137 source, then plotted the data and measured its mean and standard deviation to ensure it followed the expected Poisson statistical distribution that is characteristic of nuclear decay. Then, we measured the detector counts of a Ba137m sample as a function of time, and fit the resulting curve to obtain an experimental value for the radioactive half-life of Ba137m. We also measured background radiation levels in order to account for their effect on our barium decay data. Finally, we used the gamma-ray spectra of Co60 and Cs137 to calibrate the MAESTRO program’s gamma-ray spectrum channel scale into a photon-energy scale, which we used to identify features in a Na22 (sodium) source. We also determined the energy of Compton-scattered electrons in Cs137, which allowed us to calculate the mass of an electron. The high degree of accuracy of our results confirmed the successful tuning of our apparatus and validated our experimental approach.

**Theory**

A detailed understanding of the mechanics of radioactive decay was essential to the completion of this lab. It is important to note that, due to the principle of wave-particle duality, we interchangeably refer to specific particles and their corresponding radiation. For examples, gamma rays are simply high-energy photons. Radioactive decay is a natural process by which unstable atomic nuclei spontaneously emit particles to lower their overall energy and transition to more stable states. There exist specific ratios of neutrons to protons which allow for a stable nucleus, but when those ratios are not respected, nuclei are considered unstable and prone to radioactive decay.

If the number of neutrons in a nucleus is too great, a neutron spontaneously decays into a proton through emission of a W- boson through a process known as – decay. This boson in turn decays into an electron and antielectron-neutrino. This process, shown in figure 1, conserves the mass number A, which represents the total number of nucleons, but increases the atomic number Z (number of protons) by 1. However, if the number of protons is too great, a positively charged proton decays into a neutron through emission of a W+ boson, which itself decays into a positron and electron-neutrino, a process known as + decay.

The positron is the antiparticle of the electron, meaning they have identical mass, opposite charge, and annihilate when they collide, releasing gamma rays with energy equal to their combined energies. Both + decay and positron-electron annihilation are shown in figure 2. Lastly, many atoms decay into the unstable excited states of different elements, rather than directly decaying into a stable ground state. Hence, to restore stability, these excited states shed their excess energy by emitting gamma rays through a process known as gamma (γ) decay, also shown in figure 1. In cases where both surplus protons and neutrons exist in the nucleus, an atom emits an alpha particle made of two neutrons and two protons to restore its stability by changing into another element. Alpha decay is also the only decay type presented here that is not explored in this lab.

**Figure 1: During – decay (left), a neutron releases a W- boson, which in turn becomes an antielectron-neutrino and a – particle (electron). During decay (right), an excited unstable atom emits gamma rays to return to its lower energy ground state.**

**Figure 2: During + decay (left), a proton releases a W+ boson, which in turn becomes an electron neutrino and a + particle (positron). During positron-electron annihilation (right), the titular particles collide and annihilate, releasing two gamma rays with the same total energy (conservation of energy).**

Compton scattering also plays a role in this experiment and is responsible for the existence of recurring features on our gamma-ray spectra. Compton scattering describes the phenomenon by which a photon collides with an electron at rest, imparting it with some energy and setting it into motion. Given that this recoil electron has absorbed part of the photon’s energy, the deflected photon’s wavelength increases, since photon wavelength is inversely proportional to energy by the energy equation E=hf=hc . The change in photon energy increases with the angle of deflection, as shown by the equation E’=E1+(Emc2)(1-cos ) used in the prelab, where E’ is the energy of the scattered photon and E is the energy of the incident photon. From this equation, we can also deduce that the scattered photon energy is minimal at θ=180°, and that the scattered electron’s energy must therefore be maximal.

Knowledge of the Poisson statistical distribution is also key to this experiment. The Poisson distribution is a limiting case of the binomial distribution that is used when the probability of an event occurring approaches 0 but the number of trials approaches infinity. Hence, it applies well to modelling radioactive decay, because the probability of a single nucleus decaying at any given time is very small for elements with long half-lives, but the sheer number of nuclei being studied in a single sample allows for a high number of events to occur regardless. The Poisson distribution equation Pn=e-μ*nn! given in the prelab reading shows that the probability P(n) of n counts occurring during a given time period depends solely on the expected mean count number during that time. The same is true for the variance 2=μ and standard deviation = of the distribution, which are also purely dependent on the mean.

**Methods**

We began our work on nuclear decay by measuring count rates for a Cesium-137 source and using the experimental data we obtained to determine whether the number of photons detected during an arbitrary time duration followed a Poisson statistical distribution. In addition to the previous info on Poisson statistics, this experiment required a detailed understanding of the photon detection and signal processing apparatus, which is summarized below in Figure 3. Cesium-137 is an unstable isotope of cesium which spontaneously undergoes – decay, transforming a neutron into a proton through emission of an electron and antineutrino, which results in the stable isotope Barium-137. Energy is also emitted under the form of gamma rays, which are very high frequency electromagnetic waves, a portion of which are directed towards the NAI detector. The photons forming said gamma rays are absorbed by the detector’s NAI sodium iodide crystal, which enters an excited, high-energy state. It immediately returns to its ground state by emitting a number of visible spectrum photons proportional to the energy of the incident gamma ray, a process known as scintillation. These photons are reflected towards a photocathode plate, which marks the start of the photomultiplier tube.

**Figure 3: Block diagram of our gamma ray detection and signal-processing apparatus for the first two experiments.**

At the photocathode, the photons dislodge a proportional number of electrons via the photoelectric effect. These electrons then encounter a series of dynodes biased at successively higher voltages. As they approach the first dynode, the electrons move through a potential difference, and the corresponding electric field does work on each electron, increasing their kinetic energy. Upon encountering the dynode, each electron dislodges multiple other electrons, which are then attracted to the next dynode, where this process repeats, thus causing an exponential increase in electron numbers. These electrons are captured at the anode, where their combined charge is proportional to the initial gamma ray detected at the scintillator. Hence, increasing the photomultiplier tube voltage bias would increase the potential difference, in turn increasing electron kinetic energy, electron dislodging rates and final current. This explains the high suggested photomultiplier tube bias of V=500V, which we used for our first few experiments.

This current pulse is transmitted to the preamp, which uses a capacitor to convert the current into a proportionally sized voltage pulse. However, the voltage pulses output by the preamp were irregular in size and timing since the Cesium-137 atoms decayed at random and produced gamma rays with differing energy levels, which would in turn dislodge more or less electrons at the photocathode. We greatly amplified this output signal by passing it through a spectroscopy amplifier, then connected it to a single-channel analyzer. Using this analyzer, we could define a desired voltage range for amplifier output voltage pulses, and consequently discard pulses outside the chosen range. Since pulse height was dependent on voltage, this essentially allowed us to filter which gamma ray signals were studied. We set the minimum delay time between subsequent pulses to be 1.0μs, then set our channel lower end E to be 0.5V and defined a channel width ∆E=5.0V, allowing us to detect signals with amplitudes of up to 5.5V. This reduced the number of observed pulses on the oscilloscope, and produced pulses with a standardized width and size, allowing for easier counting. The final circuit component was an National Instruments data acquisition board which we used to count pulse numbers for various time durations.

After finishing our circuit installation, we began taking data using the Count3.vi LabVIEW program, which counted the number of pulses received by the counter over a chosen time period known as an “event”, repeated for a chosen number of events. We tested this program by setting event duration and number of events to t=10ms and 1024 respectively, then repeated this procedure for an event length of t=0.1s to obtain a dataset of radioactive decay rates over 102.4 seconds. We calculated our mean decay count and standard deviation to be μ=106.2 counts/event and σ=10.21 counts/event respectively, then calculated the theoretical Poisson standard deviation of σ==10.3 counts/event, which was within 1% of our experimental value. This strongly supports the Poisson distribution of our decay data.

Then, we reduced our event time to t=10ms and took 36 separate measurements of mean decay rates over 1024 events. From these measurements, we created a collection of means, of which we then calculated both the mean μ=10.62 counts/event and standard deviation σ=0.089 counts/event. We then calculated the standard error of the mean of the first of our 36 datasets and obtained SE==0.31710.7=0.0990 counts/event, which was approximately 10.5% larger than the standard deviation of the means. This still provides evidence that nuclear decay follows an approximately Poisson distribution, although it is less convincing than our previous result. For our final counting experiment, we took the data from our original dataset with t=0.1s, sorted it into a histogram and overlaid it with the plot of a theoretical Poisson distribution with the same mean, creating the plot shown in figure 4. The correspondence between the histogram and ideal distribution was very visible, and when accounting for sources of error in experimental measurement, this data provided further evidence for the Poisson distribution of nuclear decay rates.

**Figure 4: This plot shows a histogram of voltage pulse counts per event for our dataset of event length t=0.1s. The overlaid theoretical fit shows that nuclear decay counts approximately follow a Poisson distribution.**

Once these measurements were completed, we exchanged our Cs137 sample, which had a half-life of 30.08 years, for a sample of excited Barium-137m (Ba137m) with a half-life of 153 seconds (per the prelab). In most cases (94.7%), Cesium-137 decays into this metastable excited barium atom, which then undergoes gamma decay, changing into its ground state of Ba137 by emitting a photon with energy E=661.77 keV (per the prelab). By chemically separating the Barium-137m from the Cesium sample using an elution solution, we obtained a more unstable sample with shorter half-life, which we set about experimentally measuring through the use of our decay count apparatus.

We recorded decay counts for the excited barium sample over 1024 events of duration t=1s, the first few dozen of which were dedicated to measuring the background radiation present before the introduction of the sample. This later allowed us to account for background radiation levels when fitting our data. We graphed the barium decay data in figure 5, then fit the resulting curve in Prism9 using a one-phase exponential decay function, which calculated a half-life of t1/2=τ*ln 2 =168.7s±0.7s, representing an increase of approximately 10% from our theoretical value. This value was particularly concerning as it did not fall within the given 95% confidence interval, also shown in figure 5. However, when accounting for pile-up effects, we obtained a half-life value of 154.7±2.3s, which was only 1.1% larger than our theoretical value. The reasoning and methods that produced this correction are further explored in the Analysis section.

**Figure 5: Graph of decay counts/s vs time for the Ba137m sample after subtracting background decay data. The exponential fit produced a large half-life value, which is likely a result of pile-up effects, which are further explored in our analysis section.**

We previously mentioned our measurement of background decay data, which we conducted to ensure that such effects were negligeable during the barium decay experiment. In order to better approximate the magnitude of background decay, we took experimental data for 360 events with event duration t=1s with no sample present. Then, we took the mean of the resulting counts, which we found to be =36 counts/s, and subtracted it from each data point in our original barium decay dataset. In doing so, we treated the background as a constant which influenced our experiment equally at all stages, a decision which is justified by the graph shown in figure 6, which demonstrates that the radiation background is approximately constant at the timescales we studied.

As expected, a variation of 36 counts/s was negligeable considering the high-count rates obtained at the beginning of the decay process, and the background subtract had no effect on our fit results. That being said, we incorporated this corrected data into figure 5 in the name of increased accuracy. We also investigated the option of including our relative confidence in the data through a weighting process, which consisted of dividing each data point by the root of the corresponding count value. This procedure, which increased the weight of lower count data points in the subsequent fit, produced a significantly less accurate fit, and was therefore abandoned.

**Figure 6: This graph of background radiation counts vs time demonstrates that background radiation is negligeable when compared to the count levels described in the previous graph, and that it remains approximately constant over time.**

For our third experiment, we modified our initial signal-processing circuit by replacing the single-channel analyzer and counter with a PHA box and multi-channel analyzer, as shown in figure 7. The PHA box contained a 0.01μF capacitor, which we AC coupled to the analyzer to improve signal resolution and minimize DC noise. This multi-channel analyzer, unlike its predecessor, could simultaneously count voltage pules of different amplitudes, which allowed us to create an entire spectrum of gamma-ray counts for photons of different energies. The MAESTRO program presented an x-axis composed of discrete “channels”, each corresponding (as in our first experiment) to a defined voltage range. We minimized low voltage noise by removing the first 10 channels from our spectrum, which we also capped at 512 channels to avoid overly extending our x-axis.

Then, we placed a new Cobalt-60 (Co60) source sample near the detector and measured decay counts for the chosen range of channels over a period of 120s, thus observing the characteristic double peak described in the prelab documentation. However, most of our data was concentrated in the first half of the x-axis, so we increased the spectroscopy amplifier FINE gain to 5.5 for a total gain of 15.5 and increased the PMT bias to 530V from 500V. Increasing PMT bias increased electron dislodging in the photomultiplier tube, thus increasing output current, and the corresponding voltage output by the preamplifier was further multiplied by the increase in gain at the spectroscopy amplifier. This had the overall effect of better spacing data along the x-axis (due to the detection of a wider range of photons, notably at higher energies). Using this data, we plotted the graph shown in figure 8.

**Figure 7: Block diagram of our gamma ray detection and signal-processing apparatus for the gamma-ray spectroscopy experiment.**

**Figure 8: This gamma-ray spectroscopy plot for Co60 indicates that the highest count rates were obtained at channels 395 and 447. Cross-referencing these channel numbers with known peak energies helped us to calibrate the original channel scale into a photon energy scale.**

After obtaining data for Co60, we repeated our previous measurement process for a sample of Cs137, making sure to keep our photomultiplier tube bias and amplifier gain constant. We once again recorded decay data over 120 seconds, then plotted the results to create the graph shown in figure 9. The large central feature of this gamma-ray spectrum had its peak located at channel 227. According to the National Nuclear Data Center website, the peak of the decay count curve for Cs137 corresponds to photons with an energy of 661.7 keV. In addition, we found that the curve peaks for the Co60 spectrum corresponded to photons with energies of 1163.2keV and 1322.5keV respectively. Then, we used these values to calibrate the original channel scale into a photon energy scale by plotting channel values against their corresponding photon energies and using the equation of the resulting line, which is shown in figure 10. Hence, we established that the photon energy Y at any point was given as a function of its channel number X by the equation Y=3.05X-30.3. This equation was essential in calculating the mass of an electron from our cesium spectrum.

**Figure 9: This gamma-ray spectroscopy plot for Cs137 indicates that the highest count rate was obtained at channel 227, which corresponds to a photon energy of 661.7 keV. **

**Figure 10: We used this energy vs channel number plot to calibrate the MAESTRO software’s channel scale into an electron energy scale. **

Indeed, the small feature labelled with a 1 in figure 9 is an example of a Compton edge produced by backscattering (reflection at a 180-degree angle) of photons off electrons at various stages of the signal detection process. Said photons’ new energies are described by the Compton scattering equation described in our theory section. These photons are reabsorbed by the NAI crystal then reemitted, and this effect happens sufficiently frequently that a peak is created on the spectrum. The peak shown at 1 is at channel 71, which corresponds to a gamma ray energy of 71*3.05-30.3=186 keV. Using the Compton scattering equation formulation given in our theory section, we calculated a corresponding electron mass of me=9.25±0.31*10-31kg, which is within 1.6% of the accepted value, thus validating our experimental approach and calibration procedure.

We ended our experiment by measuring decay counts for a Na22 sodium sample and plotting its corresponding gamma-ray spectrum, which is shown in figure 11. This spectrum possesses a single large central feature with a peak at channel 177, which we calculated to be approximately 3.05*177-30.3=510 keV using our calibration scale. However, when converting this value to kilograms, we obtained a mass of m=510*1.78266*10-33kg=9.09±0.18*10-31kg, which is only 0.2% smaller than the electron mass me=9.109*10-31kg. Indeed, we know from our prelab reading that Na22 undergoes + decay to transition into the excited state of Ne22 (neon), a process which involves the release of positrons. These positrons annihilate against the electrons in the materials surrounding the detector, releasing two equal-energy gamma rays which divide the previous electron-positron total energy between them.

One of these gamma rays enters the detector, eventually producing the large peak described previously, while the other is emitted in the opposite direction and never enters the detector. This also explains the peak’s position, as its energy is equivalent to that of half the electron-positron pair. Since these have equal energy, the peak represents gamma rays with the energy of a single electron, explaining the mass-equivalence described previously. The smaller feature has maximal amplitude at channel 427 (equivalent to 1272 keV), which corresponds within 0.5% to the energy difference between the excited and ground states of neon resulting from nuclear decay of Na22. Hence, we attribute this feature’s existence to the gamma decay of the excited neon state.

**Figure 11: This gamma-ray spectroscopy plot for Na22 showcases the effects of both positron annihilation outside the NAI detector and gamma decay of the resulting excited Ne22 sample. **

**Analysis**

The principal motivation for our data analysis will be the determination and justification of the previously stated experimental uncertainties. As stated in our methods section, the exponential regression of our barium decay dataset produced a half-life of 168.7s±0.7s, which was approximately 10% larger than our expected value of 153s. Neither the subtraction of background decay counts nor the weighting of the datapoints succeeded in producing a better fit, so we investigated other possible sources of error, particularly pile-up effects. Indeed, we had set the single-channel analyzer delay time to 1μs, which meant that any voltage pulses detected during the 1μs delay time after a previous reading would be ignored.

We hypothesized that this effect would disproportionately effect data representing high decay count rates, since a larger number of voltage pulses would collide. Inversely, the probability of two pulses being generated within 1μs would be negligeable for low decay counts, given that each event had a duration of 1s. To test this hypothesis, we disregarded decay rates of over 1700 counts/s and conducted another identical analysis, the graph of which is shown in figure 12. This time, we obtained a half-life of 154.7s±2.3s, where 154.7s is only 1.1% larger than the theoretical half-life of 153s. In addition, our experimental uncertainty of only 1.5% contained the theoretical value. This validated both our data-collection methods and hypothesis, as this pile-up was inherent to the measurement process and not due to human error.

**Figure 12: Graph of decay counts/s vs time for the Ba137m sample after removing data points with counts of 1700/s and higher. The exponential fit produced a significantly more accurate value, indicating that pile-up effects were decreasing the accuracy of our previous fit.**

Th primary uncertainties of interest in the gamma-ray spectroscopy portion of our lab related to the slope of our calibration plot and our calculations of the mass of the electron. The simple linear regression shown in figure 10 calculated a 95% confidence interval of 3.00 keV/channel to 3.10 keV/channel for our slope value. In addition, there is an uncertainty inherent to these channels, as they represent a range of possible energies, and the exact peak count energy cannot be accurately determined. Suppose a total uncertainty of 1 channel exists in the measurement of any channel. Then we can express the channel number used in the calculation of the Compton edge electron mass as 71±0.5. Therefore, in a worst-case scenario where slope=3.10 keV/channel and the Compton edge was at channel 71.5, we could expect a peak energy of 71.5*3.10-30.3= 191 keV, which corresponds to a new mass value of 9.56*10-31kg. Hence, the electron mass is written me=9.25*10-31kg±0.31*10-31kg, which represents an uncertainty of 3.4% containing the theoretical value.

Similarly, we can express the channel number used in the calculation of the sodium peak electron mass as 177±0.5. Therefore, in a worst-case scenario where slope=3.10 keV/channel and the Compton edge was at channel 71.5, we could expect a peak energy of 177.5*3.10-30.3= 520 keV, which corresponds to a new mass value of 9.27*10-31kg. Hence, the electron mass is written me=9.09*10-31kg±0.18*10-31kg, which represents an uncertainty of 2% easily containing the theoretical value. The small values of our worst-case uncertainties, as well as the fact that they easily contain the theoretical values we sought, confirm the validity of our experimental approach.

**Results & Discussion**

Over the course of this lab, we obtained evidence that nuclear decay counts follow a Poisson distribution by conducting three separate counting experiments. Then, we measured the decay counts of an excited Ba137m sample returning to its ground state. Though our initial half-life value was erroneous, accounting for pile-up effects yielded a value of t1/2 with a percentage error of only 1.1%. Finally, we combined experimental measurements and publicly available information on gamma-ray energy spectra to calibrate a photon-energy scale.

We then used this scale to calculate two approximations of the mass of an electron to within 1.6% and 0.2% respectively (using a Compton edge feature and Na22 spectrum peak) and find evidence of positron annihilation. In addition, our respective uncertainties of 1.5%, 3.4% and 2% were relatively low considering they applied to worst-case scenarios. Our success in obtaining experimental data to a high degree of accuracy can be primarily attributed to the long sampling times used for all experiments. Indeed, we took 1024 data points in the barium decay experiment, which proved essential when we were required to remove several hundred to account for pile-up effects, as we still possessed enough low-decay count data to produce an accurate fit. Similarly, running the MAESTRO software for 120s ensured that count numbers were sufficiently different to easily distinguish peak data points, which in turn reduced uncertainty values during later calculations.

Though our results were both accurate and relatively precise, we can consider some ways in which they could have been improved. In our Poisson statistics experiment, we could have taken an average of the standard errors of multiple runs (rather than a single one), which might have produced a lower value more comparable to the standard deviation of the set of means. In the barium decay experiment, a 1μs delay was necessary to ensure proper functionality of the single-channel analyzer, we could consider the effects of lowering the duration of said delay, which would conceivably reduce the scale of pile-up effects. Should this negatively affect analyzer functionality, we could imagine using another analyzer geared towards higher-volume pulse detection, which would improve the initial half-life value and reduce the amount of data removed during corrections. Also, we could easily improve the accuracy of our calibration in the gamma-ray spectroscopy experiment by studying a greater number of different radioactive samples, which would provide us with a greater variety of distinctive features to use in our calibration plot. Indeed, our current reliance on only three data points increases experimental uncertainties and amplifies the effects of any measurement errors.

**Conclusion**

This series of experiments allowed us to apply our knowledge of nuclear decay theory and Poisson statistics to the measurement of decay rates across a variety of experiments. It also acquainted us with a variety of detection and signal-processing apparatus, namely a NAI scintillation detector, photomultiplier tube, preamp, signal amplifier, single and multi-channel analyzers, and a counter. By repeatedly taking decay rate data, we confirmed that nuclear decay follows a Poisson distribution. Using a rapidly decaying barium sample, we calculated then corrected a half-life measurement by accounting for pile-up effects. We also observed evidence of positron annihilation and calculated multiple electron mass approximations by calibrating a photon-energy scale. Then, we completed an in-depth analysis of our uncertainties and explored how we might further refine our measurements during a future experiment. A potential future avenue of interest would be measuring the radioactive decay of a material with slightly higher half-life and lower decay counts than Barium-137m. If our originally calculated half-life was closer to the corresponding theoretical value than in the barium experiment’s case, this would provide further evidence of the importance of pile-up effects (which would be less noticeable for consistently lower counts). Taking a greater number of measurements and studying a larger variety of radioactive samples would also likely improve the accuracy of the first and third experiments respectively. This lab was all the more important given that understanding how to measure and model nuclear decay patterns has important implications in many fields. In astronomy, light curves produced by the decay of certain isotopes are used to date stellar bodies, and in medicine, the detection of radioisotopes intentionally ingested by patients can simplify the imaging of certain key organs. However, the implementation of such technologies requires specific knowledge of certain decay patterns that would be difficult to obtain from our lab alone.