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Re: [Phys-l] models of radioactivity

I made a computer model recently of a carnival game (one where you throw darts at a board and try to pop balloons) which is exactly analogous to the non-infinitely divisible nuclear decay model. Everything maps over exactly, except that the balloon model has discrete time, rather than a continuous time. It was written for an entirely different context (taphonomic destruction of archaeological sites) but is general enough that students will be able to figure it out.

The "decay" of the balloon numbers (along with the half-life and so on) was eminently obvious with more than about 20 balloons left on the board, and still noticeable (despite the discreteness of the events) with only 10 balloons left on the board.

I'm quite confident that for 10 or more nuclei, the Poisson process model is a good approximation to the continuous version which is usually presented to students. For n <10, things get hairier, especially when there are competing processes (such as regeneration of the decaying nuclei from some parent atoms). This is because of a rescaling effect on the decay rates, compared to the populations---I call this the "cheating operator" in the carnival game.

I'm happy to send a quick write-up of the simulation to anyone who'd like to read it.

/************************************
Down with categorical imperative!
flutzpah@yahoo.com
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________________________________
From: Brian Whatcott <betwys1@sbcglobal.net>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Monday, April 20, 2009 11:12:24 AM
Subject: Re: [Phys-l] models of radioactivity

Though Jack's treatment follows the same path as Hugh's and the MIT
introductory treatment at the cited URL, I should probably underline a
caution given in the MIT URL I mentioned, as follows:

"Please notice that these models only make sense when the initial
population is quite large, since the only possible changes in population
here are integers. In other words, a given nucleus either decays or
doesn't, so at any time the number in each population is an integer.
Thus, if we make dp smaller than 1, we leave the realm in which the
model makes sense, and enter an artificial mathematical domain. Thus,
keeping dp finite, as we do in numerical calculations, rather than
letting it go to 0, as we do in formal differentiation is more
compatible with the model."

Brian W



Jack Uretsky wrote:
The time dependence of a population of decaying atoms is assumed to follow
the Poisson law
N(t) =N(0)e^{-pt).
the half-lkife T' is defined to follow N(T') =N(0)/2, or
e^{-pT'} =1/2
Taking the ln of each side of the last equation gives:
pT' =ln2, which leads immediately to the quoted equation.

On Sat, 18 Apr 2009, Brian Whatcott wrote:


John Denker wrote:

On 04/17/2009 10:25 PM, Hugh Haskell wrote:



we know that half-life (T) can be
expressed as

T = (ln 2)/p (1)


What do you mean by "we", Kemosabe?


We?
The folks at MIT giving an introduction to modeling radio-active
half-life, for example.
See this version
<http://www-math.mit.edu/~djk/calculus_beginners/chapter12/section02.html>

Hugh is in fact illustrating a comparable example to the difference
between compounding capital at time intervals, say weekly, monthly,
quarterly etc., and
compounding capital continuously. This is a standard introductory
element of
teaching exponential versus discrete time models, I thought?

Brian W
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