Visualizing Lifetime Distributions for Nuclear Decay Cascades
Visualizing Lifetime Distributions for Nuclear Decay Cascades
by Srinath Rangan
Mentor: Lauren C. Cooper
Mentor: Lauren C. Cooper
The purpose of the project was to find a unique way to observe the decay patterns of any given isotope. An algorithm was written to utilize different calculation methods in order to generate graphics that showcase the mathematics behind radioactive decay. This includes calculations for one-decay, chain-decay, and alternative-decay processes that build off the work of physicists Ernest Rutherford and Harry Bateman and the Bateman mathematical model.
Producing the Half-Life Distributions Compared to Decay Products
This section covers how the program is able to produce automated graphs given any radioactive isotope by identifying its children nucleotides.
Proton & Neutron Representation
Proton & Neutron Representation
In order to show how the the molecules undergo radioactive decay, a visual representation was created using large polyhedrons in Graphics3D. The program used a manipulate function that the user controls in order to produce a series of different decay products.
Defining Shapes to Show Protons and Neutrons
Defining Shapes to Show Protons and Neutrons
Generating Random Points in a Closed 3D Area
Generating Random Points in a Closed 3D Area
Calculations for Deriving the Decay Compare Model
Calculations for Deriving the Decay Compare Model
Getting the HalfLives & Decay Constants
Getting the HalfLives & Decay Constants
Creating the Decay Charts
Creating the Decay Charts
One Parent to Two Children
One Parent to Two Children
One Parent to Three Children
One Parent to Three Children
One Parent to Series of Chain Reactions
One Parent to Series of Chain Reactions
Decay chain series follow a unique pattern that could be written for a chain of n daughter nuclides. The function below plots a chain of length one but rather than approaching one as some of the functions above, the amount of the daughter nuclide decreases to zero since there are further decay products being created. I would like to explore the recursive differential equations for this further to obtain more accurate results.
decayChart4[isotope_]:=Manipulate[Style[Row[{Module[{y=parentDecay3[isotope,NumberOfHalfLifes],z=chainSolver2[isotope,NumberOfHalfLifes]},Plot[{parentDecay3[isotope,t],chainSolver2[isotope,t]},{t,0,10},PlotRange{0,1},PlotLegendsPlaced[{IsotopeData[isotope,"Name"],IsotopeData[IsotopeData[isotope,"DaughterNuclides"][[1]],"Name"]},Below],Epilog{PointSize[Large],Point[{NumberOfHalfLifes,y}],Point[{NumberOfHalfLifes,z}]},PlotLabelIsotopeData[isotope,"Name"]<>": "<>ToString@(y*100)<>"%, "<>IsotopeData[IsotopeData[isotope,"DaughterNuclides"][[1]],"Name"]<>": "<>ToString@(z*100)<>"%",ImageSize500,AxesLabel{Style["HalfLife",FontSize14,Bold],Style["Percentage",FontSize14,Bold]}]],Show[{blueImages[[Round[100*parentDecay3[isotope,NumberOfHalfLifes]]+1]],yellowImages[[Round[100*chainSolver2[isotope,NumberOfHalfLifes]]+1]]}]}],ImageSizeMultipliers{0.34,0.34}],{NumberOfHalfLifes,0,10}]
If the granddaughter of Actinium-235 were shown below, you would see the percentage of the granddaughter increase as the daughter nuclide increases
More Examples
More Examples
Developing the Decay Chain Graph
Developing the Decay Chain Graph
Getting Nodes and Predecessors
Getting Nodes and Predecessors
Defining Utility Functions
Defining Utility Functions
Sample Tree with Information Chart
Sample Tree with Information Chart
Generating the Buttons and Functions
Generating the Buttons and Functions
Generating Vertices with Buttons
Generating Vertices with Buttons
Generating Decay Graph with Buttons
Generating Decay Graph with Buttons
Writing Function for Button Graph Generation
Writing Function for Button Graph Generation
Example
Example
Final Product with Graph & Chart Integration
Putting it All Together
Putting it All Together
Example
Example
Future Endeavors
Future Endeavors
Some possible extensions that this program could have is making functions for n-chain reactions. For any potential given chain process, I would like to calculate all of the successive decays. However, this program currently doesn’t fully exploit the differential recursive functions outlined in the Bateman model. Additionally, the graphical representations of the decay shown with the molecules’ behavior over a certain number of half-lives could be updated to showcase more animations that further explain the behavior of the decay pattern. It was difficult to find examples where the decay constants of all the successive decay products have the same order of magnitude, so it was challenging to find good examples of n-chain decay that doesn’t drop off after two or three isotopes.