Help with population of various isotopes in decay chains (Bateman equations)

In summary, the teacher is trying to create a decay chain simulator for his pupils to use in class, but is having difficulty with the equations required to calculate the population of each isotope at each stage in time. He has taken information from [Cetnar, Annals of Nuclear Energy 33, 640 (2006)] to help him. The expressions for the population of Po-211, Ti-207, and Pb-207 are all valid up to [Po], [Ti],... equal to 0, but the one for Tl-207 is not.
  • #1
alby
4
0
(Apologies for cross-posting this in the Nuclear and Differential Equations forums. I'm new to posting on Physics Forums and not entirely sure where it belongs. Mods, please feel free to delete/move as appropriate.)

I am trying to create a decay chain simulator in Excel that my pupils can use to create graphs similar to those created by the Nucleonica Decay Engine: the idea being that they can find a decay chain and enter the isotopes and their half-lives and Excel will create the graph.

The problem I'm having is with the equations required to calculate the population of each isotope at each stage in time. My initial approach was far too simplistic and whilst it created correct *looking* graphs, the numbers weren't correct.

As an example, I'm using the last four stages of the actinium series uranium decay chain:
[tex] {}^{211}\mathrm{Bi}\rightarrow {}^{211}\mathrm{Po}\rightarrow {}^{207}\mathrm{Tl}\rightarrow {}^{207}\mathrm{Pb} [/tex]

I can calculate the population of the Bi-211 at time t with:
[tex] N_{Bi}=N_0 e^{-\lambda t} [/tex]

and the population of Po-211 using:
[tex] N_{Po} = \frac{\lambda_{Po}}{\lambda_{Bi} - \lambda_{Po}} N_0(e^{-\lambda_{Bi}t}-e^{\lambda_{Po}t}) [/tex]

The problem is with the equations for the population of Tl-207 and Pb-207. I know I should be looking at Bateman's work on the subject, but as a lowly teacher I can't access the literature.
 
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  • #2
Seems like the rate equations are simply
[tex] \dot{Po}=-\lambda_{Po}Po+\lambda_{Bi}Bi[/tex]
and so on. For example, the one for Ti would be
[tex] \dot{Ti}=-\lambda_{Ti}Ti+\lambda_{Po}Po[/tex]
By the way, the solution I get for Po is
[tex]Po=N_{0}\frac{\lambda_{Bi}}{\lambda_{Po}-\lambda_{Bi}}[e^{-\lambda_{Bi}t}-e^{-\lambda_{Po}t}][/tex]
All of the equations are valid up to [Po], [Bi],... equal to 0. There is no negative number of samples
 
  • #3
Sorry, but I couldn't get the tex in my reply to display properly without including these three expressions. Please ignore them:

[tex] 1 [/tex]
[tex] 1 [/tex]
[tex] 1 [/tex]

I have taken the following info from [Cetnar, Annals of Nuclear Energy 33, 640 (2006)].

An expression for the population of the nth nuclide in a serial decay chain (derived by Bateman):

[tex]
N_n(t) = \frac{N_1(0)}{\lambda_n} \sum^{n}_{i=1} \lambda_i \alpha_i \exp{[-\lambda_i t]},
[/tex]

where

[tex]
\alpha_i = \prod^{n}_{j=1, j \neq i} \frac{\lambda_j}{(\lambda_j - \lambda_i)}.
[/tex]

Hopefully the notation is obvious -- I'm struggling to get my tex expressions to display properly in this post. The N_1 is the parent population, N_2 the daughter population, and N_3 the grand-daughter population (in your case Bi211, Po211, and Pb207, respectively).

Note that this gives a different expression for the Po population you were attempting to calculate:

[tex]
N_{Po} = \frac{\lambda_{Po}}{\lambda_{Bi} - \lambda_{Po}}N_0(e^{-\lambda_{Bi}t}-e^{\lambda_{Po}t})
[/tex]

which agrees with that stated by Gato.

I get the impression that you are treating Tl207 as a daughter nuclide of Po211, whereas it is actually a daughter nuclide of Bi211, and so its population can be calculated in more or less the same way as the population of Po211 (I guess that to get the correct populations you have to scale the lifetime of Bi211 by the fraction of mutations that end in Po211 or Tl207; i.e., that in your calculation of N_Po you don't use [tex] \lambda_{Bi} [/tex], but instead use [tex] 0.00276 \lambda_{Bi} [/tex] as only 0.276% of Bi211 mutations end in Po211 (according to a figure in http://en.wikipedia.org/wiki/Decay_chain)).

To calculate the population of Pb207 I would use the expression separately for each path down the decay chain from Bi211 to Pb207 (i.e., Bi211 -> Po211 -> Pb207, and Bi211 -> Tl207 -> Pb207), and then take the sum of the two expected populations to get the actual population of Pb207. This is just a suggestion of how it can be done; I haven't tried it, and can't say with certainty that it is correct. Do let us know if this works for you.
 
  • #4
You mention the Nucleonica decay engine - but why don't just use this?. The people at Nucleonica (www.nucleonica.com) have solved all the problems you mention.
 
  • #5
"Why bother learning to play the piano? Other people can play the piano for you."
 

1. How do you calculate the population of isotopes in a decay chain?

The population of isotopes in a decay chain can be calculated using the Bateman equations, which take into account the decay rates of each isotope in the chain as well as the initial population of each isotope. By solving these equations, you can determine the population of each isotope at any given time.

2. What is the significance of understanding the population of isotopes in decay chains?

Understanding the population of isotopes in decay chains is important for a variety of reasons. It can help us better understand the behavior of radioactive materials and their potential effects on the environment and human health. It is also crucial for various applications in fields such as nuclear medicine and environmental science.

3. How does the half-life of an isotope affect its population in a decay chain?

The half-life of an isotope is a key factor in determining its population in a decay chain. Isotopes with shorter half-lives will decay at a faster rate, resulting in a smaller population over time. On the other hand, isotopes with longer half-lives will decay at a slower rate, leading to a larger population that persists for a longer period of time.

4. Can you give an example of a decay chain and how to calculate the population of isotopes using the Bateman equations?

One example of a decay chain is the uranium-238 decay chain, which includes multiple isotopes such as thorium-234 and radon-222. To calculate the population of isotopes in this chain, you would need to know the initial population of each isotope and their respective decay rates. By plugging these values into the Bateman equations, you can determine the population of each isotope at any given time.

5. Are there any limitations to using the Bateman equations for calculating the population of isotopes in decay chains?

While the Bateman equations are a useful tool for calculating the population of isotopes in decay chains, they do have some limitations. For instance, they assume that the decay rates of each isotope remain constant over time, which may not always be the case. Additionally, they do not take into account any external factors that may influence the decay process, such as temperature or chemical reactions.

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