Lets say you have some unstable isotope which decays via beta decay. For example, lets say you want to calculate the amount of beta radiation that is emitted from the decay of $^{87}\textrm{Kr}\to ^{87}\textrm{Rb}\to^{87}\textrm{Sr}$ where both decays are via beta- decay. I'm eventually trying to calculate the amount of beta radiation that a person would be exposed to after some shielding, but first I need to figure out the amount of radiation that is actually emitted. I know the half-lives of each isotopes and the $Q$-value for each decay, but I can't find how to do the calculation.

  • $\begingroup$ What you are looking for is a description of exponential decay. This article en.wikipedia.org/wiki/Exponential_decay has a pretty basic description. $\endgroup$ – user16622 Jun 25 '16 at 3:31
  • $\begingroup$ Nuclear decay is possible in multiple channels, just as a chemical reaction sometimes has multiple pathways. So, you need to identify the decay pathway of interest, and when you do, the products of the decay are the 'radiation' as well as daughter nuclei. How much is EMITTED depends on whether you can collect the radiant products outside the lump of isotope (they may be reabsorbed rather than exiting). $\endgroup$ – Whit3rd Jun 25 '16 at 6:56
  • $\begingroup$ You can not calculate radiation dose to human, after or before shielding using the Q value. Absorbed dosed to a given medium depends on so may other factors and there is no simple way or equation of calculating it ! $\endgroup$ – Manoj Jun 27 '16 at 12:59

Wikipedia's article on radioactive decay gives the equation that describes a two-step decay chain, together with the corresponding solutions. In your case, you'll have $$\begin{align} N_{\text{Kr}}(t) &= N_{\text{Kr}}(0)e^{-\lambda_{\text{Kr}}t} & N_{\text{Rb}}(t) &= N_{\text{Kr}}(0)\frac{\lambda_{\text{Kr}}}{\lambda_{\text{Rb}} - \lambda_{\text{Kr}}}\bigl(e^{-\lambda_{\text{Kr}}t} - e^{-\lambda_{\text{Rb}}t}\bigr) \end{align}$$ What you want is the total activity rate, the number of decays per unit time: $$\begin{align}A(t) &= A_{\text{Kr}}(t) + A_{\text{Rb}}(t) \\ &= -\frac{\mathrm{d}N_{\text{Kr}}(t)}{\mathrm{d}t} - \frac{\mathrm{d}N_{\text{Rb}}(t)}{\mathrm{d}t} \\ &= N_{\text{Kr}}(0)\frac{\lambda_{\text{Kr}}\lambda_{\text{Rb}}}{\lambda_{\text{Rb}} - \lambda_{\text{Kr}}}\bigl(e^{-\lambda_{\text{Kr}}t} - e^{-\lambda_{\text{Rb}}t}\bigr)\end{align}$$ You can then convert this into whatever measure of radioactivity you need.

  • $\begingroup$ I realized I asked the wrong question. I know how to calculate the radioactivity. What I want to calculate is the radiation dosage. Is there a straightforward way to do this, or should I make a new question? $\endgroup$ – TheStrangeQuark Jun 25 '16 at 14:37
  • $\begingroup$ What do you mean by radiation dosage? There are several different ways of measuring the amount of radiation conveyed to a body. If you want to know about that, yes, you should ask a new question, but make sure to be quite clear about what quantity you're looking for. $\endgroup$ – David Z Jun 25 '16 at 15:28

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