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Description adapted from http://www.compadre.org/osp/items/detail.cfm?ID=10577 by Wolfgang Christian

Three State Decay Theory (Parent Daughter G-daughter Radioactive Decay)

If the number of nuclides is large, the upper state (parent) N1, intermediate state (daughter) N2, and final state (Granddaughter) N3 populations obey the following coupled ordinary differential equations (ODEs)

where ?1 is the upper state decay rate and ?2 is the intermediate state decay rate. The nature of the decay is governed by these decay constants. The Three State Decay model displays both the continuous ODE solution for these nuclide populations as well as a stochastic (probabilistic) solution that assumes integer nuclide populations and uses probability to determine if a radionuclide survives for a time ?t without decaying.

Discrete decay model

The differential equation model produces continuous decay curves but hides the random and discrete nature of the underlying processes. A stochastic model that "rolls the dice" to determine if a radionuclide decays shows that the continuous model is only an approximation. The simulation computes the probably of decay between states 1->2, 2->3, and 1->3 during the finite time interval ?t. When the simulation is run, these probabilities are applied to each radioactive nucleus to determine if it decays. Note that there is a finite probability that a nuclide decays directly from state 1 to the stable state 3. Nature does not, however, skip state 2. We merely waited too long and the nuclide finished its decay before it was observed in state 2.

Three State Nuclear Decay

The Three State Nuclear Decay model extends the Two State Nuclear Decay model to simulate the radioactive decay of atomic nuclei in which the parent nucleus first decays into an intermediate state before decaying into a stable state. Although the decay of both the parent and intermediate nucleus (radionuclides) is spontaneous and unpredictable, the probability of decay of each radionuclide is constant and is usually known. The model displays a color-coded sample with N1 parent nuclides, N2 intermediate state nuclides, and N3 stable state nuclices. Users can set the initial numbers N1 and N2, the decay the decay constants k1 and k2. the time interval between measurements ?t before the simulation is run. The simulation counts the number of decay events ?N1 and ?N2 within ?t and stops when all nuclides are in the stable state.

Check boxes display a plot and a table showing the time evolution of each state as well as the number of decay events in each ?t time interval. The data plot allows users to compare the data generated by the random decay model with a differential equation-based model as described on the Theory page.

The Three State Decay model is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_nuclear_ThreeStateNuclearDecay.jar file will run the program if Java is installed. Other decay models are available. They can be found by searching the OSP Collection for radioactivity.

Credits:

The Original Three State Nuclear Decay model was developed by Wolfgang Christian using version 4.3.2 of the Easy Java Simulations (EJS) authoring and modeling tool. You can examine and modify a compiled EJS model if you run the program by double clicking on the model's jar file. Right-click within the running program and select "Open EJS Model" from the pop-up menu to copy the model's XML description into EJS. You must, of course, have EJS installed on your computer.

Information about EJS is available at: and in the OSP ComPADRE collection .