## Radon Decay Chain

A System Dynamics Based Model

Francesco Pagni

francesco.pagni-eq3g@poste.it

Dear reader, glad to be here!. This was my first autonomous exercise, using Vensim. It is similar to the famous tank exercise. We do have n tanks (n levels), corresponding to A, B, C,. box variables. The output of a tank is the input of the next one.
In this case, the box variable A represents the number of type "a" unstable (radioactive) atomic nucleus, B of type "b" and so on.

Radioactive decay is the set of various processes by which unstable atomic nuclei (nuclides) emit subatomic particles (radiation).

Decay is said to occur in the parent nucleus and produces a daughter nucleus.

parent nucleus = daughter nucleus + radiation

In this case a = b+radiation, b= c+radiation and so on.

If one decay event occurs the number of parent nucleus decreases by 1 and the number of daughter nucleus increases by 1. In the case a=b+radiation, A= A-1 and B= B+1.
The variation of the level B versus time dB/dt [number/second] is equal to the disintegration per second of radionuclide "a" [number/second] minus the disintegration of radionuclide "b" itself [number/second].

The generic causal loop diagram describing decay is:

The flow diagram is (N is the number of doughter nuclei, N0 is its initial value, "Parent dps" is the number of parent decay events per second and "Daughter dps" is the number of daughter decay events per second).
Please, programme the flow diagram using Vensim and run the model in the case the radionuclide has not a parent (Parent dps=0 second-1). You'll discover the famous exponential rule

N(t)=N0 exp(-ln(2)*t/DoughterHalfLife)

You should obtain following results.

Let us play with a simplified version of Radon 222 decay chain.
Radon-222 is a chemically inert gas emanating from the soil and stone; it diffuses through the air and decays with the emission of an alpha particle.The decay chain of Radon-222 contains seven radioactive nuclides and one stable nuclide. Let us take only 3 of them:

Rn-222, (T0=3.8 days) =..= Pb-214 (T2=26.8 min) = .. = Pb-210 (22 yr)

The flow diagram can be built cloning the "Decay cell" described before and linking clones together.