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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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