[corpofdiscovery]

Americium-241 has a half-life of about 432 years, and then decays into neptunium, which decays and etc... What would be the exact amount of atoms of elements in the decay chain of Am-241 existing if you had .29 micrograms after 5 years.

The decay chain is here:
http://metadata.berkeley.edu/nuclear-forensics/Am241.svg

The math is just too complex for me on this one.

[Enthalpy]

An exact solution would involve math not completely trivial. Though, you can get a satisfactory answer by simplifying the case, because the decay times differ a lot.

432 years, 2 microseconds, 27 days, 159,000 years.

What happens to neptunium formed from americium decay?
From 2µs and 432yr, what is the proportion of neptunium versus americium, since you wait 5 years, much more than 2µs? From 27d ans 432yr, the protactinium versus americium?

You can go on to uranium, which accumulates, and evaluate its concentration over time - this one isn't a fixed proportion over americium.

The thorium concentration is quadratic over time over the year time scale, and radium cubic, as well as the followers up to ²⁰⁹Bi which accumulates. I'm not convinced you get one atom of ²⁰⁵Tl within 5yr with this chain.

- Is it Berkeley.edu that puts "Americurium"?
- Some computed concentrations will be so low that initial impurity will overwhelm them.

General methods:
- If you have an electronics background, then all methods for linear systems apply: linear networks, fedback systems... Using fluence graphs, Laplace transformation and the like.
- If you have a background for linear algebra, you can write the set of linear equations with the formal s variable (Laplace transformation), solve, go the inverse Laplace.
- With 15 nodes, the general methods are not really accessible to hand solving. Either have a software that does it, and better, already knows all existing nuclides and decay modes - or go simplified methods by hand, as suggested previously.

[corpofdiscovery]

Thanks for the ideas. Just a heads up that Neptunium takes about 2 million years to decay, not 2 microseconds.

Here's a better decay chain, and it doesn't contain the mysterious element Americurium.
http://en.wikipedia.org/wiki/Decay_chain#Neptunium_series

Also, does anyone know any examples of a good, free software that would model this?

[Enthalpy]

2Myr: I should have reacted.

If you're asked to do it by hand, my bet is that a simplified answer is expected, with amounts increasing like t, t², t³...

I know no software for it, but there is necessarily some. Maybe Matlab has some plug-in for it.
I got 176,000 hits with the Internet search words
"decay chain" software
Or you could misuse any simulator like the free Spice (meant for electronic circuits), where you represent nuclide amounts by capacitor voltages for instance, and add some resistors and controlled current sources to represent the decays.

[Enthalpy]

Using the free electric simulator Spice to model a decay chain, explained on the joined example image:

• The amounts (can be moles) of each nuclide is represented by a capacitor charge, which equals the voltage if the capacitance is 1F.
• Resistors introduce one decay mode each. The ohmic value can be an exponential time in seconds.
• Controlled current sources inject in each daughter nuclide the current that flows through the parent's decay resistor.

Once the decay chain is modelled by an electric circuit, the user can give a set of voltages for the initial amounts of nuclides, and observe the evolution over time.

Expected behaviour of Spice on that use:

• Its algorithm is very robust, and some hundred nodes are nothing for it. It also accepts huge and tiny values for components, currents, voltages.
• When I used it during the Paleolithic, it wasn't so good on very different time scales for one analysis.
• It draws nice curves over a huge dynamic range.
• The user must manage himself the electric equivalents.

The equivalent circuit can be created for each decay chain when needed, or a few people could once enter one big circuit with all nuclides (several 100), and the user would enter only his initial amounts of the present nuclides.

Marc Schaefer, aka Enthalpy