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[Phys-L] Bateman (was survey meters)

Batemen equations, to which Bernard refers (see below), can be
simplified when one is not interested in radioactivity associated with
predecessors whose half-lives are relatively short. Consider this
sequence:

0 --> 1 --> 2 --> 3 --> 4 --> 5 --> 6 --> 7 --> 8 -->

where numbers refer to generations of isotopes. Suppose we are
interested in the time dependence of radioactivity from the product #5
when the initial activity, and the half-life To of the starter, are
given. Initial amount of products #1, 2, 3 and 4 must be zeros in this
formalism, as indicated by Bernard.

Suppose that To=100 years while T1, T2, T3 and T4 are much shorter. In
such case, to a very good approximation, one can say that the result is
essentially the same as for a much easier case:

0 --> 5 --> 6

The starter, as always, decays exponentially. The #5 grows accordingly
and then decays exponentially at the same rate as the original. This
particular problem does not ask about radioactivity from #6, 7, 8, etc.
The half-life of #5 is presumably given, for example, 200 minutes or
200 years.
Ludwik


On Oct 21, 2005, at 2:33 AM, Bernard Cleyet wrote:

Bateman solved the diff. eq. for a chain of N radioactive products.
It's given in Friedlander and Kennedy (1955, P. 135). Unfortunately it
is for at T = 0 only the parent isotope alone is present. F & K give
instructions for constructing the general case. The previous six pp.
deal w/ growth, decay, transient, and secular equilibrium.

table 8.1 (P. 491 Evans) is the short lived decay products of Rn 222.

It includes a row giving the excess activity of the daughters when
transient equilib. exists. The previous pages discuss the growth (and
decay) of daughter and granddaughter products under various conditions.


It's late so I'll not continue, but I think one may plug and plot the
equations instead of doing it numerically.

BTW, this is a question much discussed, e.g.:


http://hps.org/publicinformation/ate/q1466.html

bc

John Mallinckrodt wrote:

[Oops. Sorry. Let's try that again.]

Brian wrote:


If instead of using a spreadsheet, I synthesize the effect
of a double decay cascade like this,
where the first species decays at one third the rate of
the daughter species,......

'exponential decay of double series
count = 1000
for a = 1 to 1000
count = count*0.99
count2 = count* 0.97
print a; " " ;count; " "; count2
next a
end

.... a single exponential of this form:
count2 = 970* exp(-0.001005 *a)

leads to a fit with exceptionally good statistics
anovar: F = 9.6E20, SE estimate = 1.4E-7
using just a single exponential equation.

Therefore, I expect I am misunderstanding some prior comments
about the possibility of pulling details of a double decay out of an
exponential time series. I cannot do this.
Or was my synthetic dataset faulty?


I'm not sure I follow the above, but I have put my spreadsheet on the
web at

<http://www.csupomona.edu/~ajm/special/radon.xls>

It starts with 1000 Po-218 nuclei (direct daughters of Rn-222), 8080
Pb-214 nuclei, and 6000 Bi-214 nuclei at t = -20 min. This insures
that the sequence is very close to secular equilibrium. Every minute
the spreadsheet calculates the number of decays of the Po-218, the
Pb-214, and the Bi-214, and accumulates the product of the final
decay as Pb-210 which has a long half life. The Po-218 is
replenished (artificially in the model, but by the decay of Rn-222 in
the real world), maintaining the secular equilibrium, until t = 0.
After that the Po-218 is allowed to decay, mimicking the removal of
its Rn-222 source.

The spreadsheet calculates and plots the number of decays per minute
for each short-lived nuclei and also plots the composite decay rate
for the Pb-214 AND Bi-214 decays. You will see that the composite
decay begins relatively slowly (perhaps like a 50 minute half life
and looks more like 30 minutes after a couple of hours.
--
John "Slo" Mallinckrodt

Professor of Physics, Cal Poly Pomona
<http://www.csupomona.edu/~ajm>

and

Lead Guitarist, Out-Laws of Physics
<http://www.csupomona.edu/~hsleff/OoPs.html>


Ludwik Kowalski
Let the perfect not be the enemy of the good.