amanb2000 / radioactivity_analysis

Measuring and analyzing radioactivity in air via beta decay curves.

Radioactivity_Analysis

Measuring and analyzing radioactivity in air via beta decay curves. Further scientific background can be found in the Jupyter Notebook Radioactivity_In_Air (see repository structure below).

Repository Structure

Primary data analysis is done using Jupyter Notebooks in the notebooks folder. Raw data can be found in the data folder and cached in the notebooks/cache folder.

To run the analysis yourself, instantiate your python environement of choice and run pip install -r requirements.txt to get all dependencies.

Dependencies

1. numpy
2. pandas
3. scikit-learn
4. jupyterlab
5. matplotlib
6. sklearn
7. scipy

Academic Background

Introduction

The goal of this experiment is to measure and identify radioactive substances in the air with high accuracy and specificity. Isotopes of interest include:

1. Uranium-238 => Lead-206
2. Thorium-232 => Lead-208
3. Uranian-235 => Lead-207

Equipment

1. Geiger-Mueller Tube: Radiation measurement.
   • Tube has ionized gas, any ionizing radiation causes a negative charge accumulation.
2. Picker-Scaler: Power supply.
3. DAQ/Computer Setup.

Data Collection

Session I: Cesium-137 Calibration Data for Voltage & Background Radiation Measurement

• Collecting data for Cesium-137 source.
• Plot data in python, taking plateau curve for data ⇒ determine operating voltage.
• Collecting data on background radiation.

Session 2:

• Air sampler used to retrieve air samples ⇒ analysis of the model.
• "Radioactivity in Air" program is used to collect data from Geiger-Mueller tubes (counts number of beta(?) particle emissions in a given time span).
• Note that the filter on the GM tube is thin enough to let in beta particles and thick enough to stop alpha particles.
  • Therefore, we are counting only beta particles.

The Model

Approach to Data Analysis

• Multiple levels of sophistication:
• EASY METHOD:
  • Constant background subtraction of your data
  • Use a semi-logarithmic plot, find rough half-life and total counting rate by the end of the sampling period.
• Full-Blown Analysis:
  • Observing gamma-rays from filter paper.

Parts of the Model

Model for counts registered in the detector:

Part I: Constant background from no sample:

1. Result of cosmic rays and gamma-rays in the room.
2. Average number of counts per time period is easily calculated from the background data collection.
3. The goal is <= 1 count per minute of error.
4. Determine the background under identical conditions to the experiment.
5. If you observe n counts, the standard error is \sqrt{n}

Part II: Decay of long-lived isotopes accumulated on filter paper:

1. Dominated by decay of "4n" series made from Thorium-232
   1. Results in Rn-220 (0.145s) → Po-216 (10.64h) → Bi-212 (1.00h) → Po-212 (0.3 us)
2. Half life of 10.64 hours for Po-216 will dominate activity.
3. Longer-lived components can be identified with semi-logarithmic plot of detector counts with background subtracted.
   1. Only for times after the half-hour activity from Rn-222 have decayed.
   2. Only the first half hour of data is relevant to the short-lived products of Radon-222.
4. In Python: Make a linear fit of data after the half-hour from the air sample.
   1. If good linear fit: subtract this component from the total data ⇒ corrected data ⇒ third part of the model.

Part III: Decay of products of Radon-222:

Secular Equilibrium:

• Radon-222 → Po-218. Po-218 is highly ionized ⇒ becomes attached to dust.
  • The next beta decay produce Pb-214 and B-214 (also highly ionized), with half lives of ~0.5 hours.
  • These products also attach themselves to dust particles.
• Number of isotopic breakdowns per second for each is proportional to the amount of that isotope that is present.
  • More Po-218 ⇒ more decay per second ⇒ build up an equilibrium depending on lifetime of Po-218 and the amount being formed from it's predecessor Radon.
  • At equilibrium: Number of Po-218 created = number of Po-218 destroyed per second.
    • This type of equilibrium continues down the chain...
  • This is called SECULAR EQUILIBRIUM ⇒ holds whenever parent has long half life compared to next generation(s).
• Therefore, MEASUREMENT OF DISINTEGRATION RATE OF ANY MEMBER OF THE SERIES ⇒ ACTIVITY OF RADON PRESENT.
• All beta activity in air can be accounted for given decay of Pb-214 and Bi-214 (after alpha particles were eliminated by that aluminum filter).
• If model fits data: If we determine the amount of Pb-214, we can get the CONCENTRATION OF RADON IN THE ATMOSPHERE.

Mathematical Model of Daughters of Radon-222

• After filtering is stopped, the few alpha-emitting Po-218 atoms don't affect much, especially after the first 10-15 minutes where it increases the amount of Pb-214.
• Subsequent counting rate = only because of Pb-214 and Bi-214.
• Rest of the model will neglect any build-up of Pb-214 from Po-218 on the filter paper.

...

• Variation of counting rate over time = f(decay constant_Pb-214, decay constant_Bi-214).
  • A_B(t) = actual activity of Pb-214 (RaB) at time t.
  • A_C(t) = actual activity of Bi-214 (RaC) at time t.
• The ratio of isotope B (Pb-214) and isotope C (Bi-214) at time zero is:

$$R = \frac{A_c(0)}{A_B(0)}$$

• Observed counting rate = real activity x efficiency (efficiency = \varepsilon)
  • efficiency is not 1 because low-energy beta particles tend to be stopped by the aluminum foil.
• Beta particle energy spectra ranges from 0 -> E_max
  • E_max is higher for Bi-214 than for Pb-214
  • \varepsilon is therefore larger for Bi-214
  • For 27 um thick foil and window of density 1.5mg/cm^2, we have:

$$\varepsilon_C = 0.95, \varepsilon_B = 0.80 ⇒ \varepsilon_C/\varepsilon_B = 0.1$$

• If we say

$$A^{obs}_B(t) = \varepsilon_B A_B(t); ,,, A^{obs}_C(t) = \varepsilon_CA_C(t)$$

• Where obs denotes that it is what we actually count in the experiment and non-obs is what is really happening/being emitted, then

$$\frac{A^{obs}_C(0)}{A^{obs}_B(0)} = \frac{\varepsilon_C}{\varepsilon_B} \frac{A_C(0)}{A_B(0)} = \frac{\varepsilon_C}{\varepsilon_B}R$$

• Recall that Po-218 is neglected because of its short lifetime...
• The following relatonships are for

$$\lambda = \frac{\ln 2}{\text{half-life}}$$

$$A_B^{obs}(t) = A_B^{obs}(0)e^{-\lambda_B t}$$

$$A_C^{obs}(t) = \frac{\varepsilon_C}{\varepsilon_B} A_B^{obs}(0)\lbrack \frac{\lambda_C}{\lambda_C - \lambda_B} e^{-\lambda_Bt} + ( R - \frac{\lambda_C}{\lambda_C - \lambda_B} ) e^{-\lambda_C t} \rbrack$$

$$A_B^{obs}(t)+A_C^{obs}(t) = A_B^{obs}(0) \times \lbrack(1 + \frac{\varepsilon_C}{\varepsilon_B}\frac{\lambda_C}{\lambda_C-\lambda_B})e^{-\lambda_B t} + \frac{\varepsilon_C}{\varepsilon_B} ( R - \frac{\lambda_C}{\lambda_C - \lambda_B} ) e^{-\lambda_C t} \rbrack$$

• the total observed counting rate is (A_b, obs(t) + A_c,obs(t)

Comparison of Model to Data

• Part III ⇒ predicted decay curves given A_B,obs and R
• Goal of the advanced model is to get a good fit (after applying steps 1-2)
• First approximation of A_B,obs is from extrapolating curve back to t=0, noting that

$$A_B^{obs}(0) + A_C^{obs}(0) = A_B^{obs}(0)[1+\frac{\varepsilon_C}{\varepsilon_B}]$$

• First approximation of R based on graph from Whyte and Taylor that states that R is a function of sampling time: