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I'm not so experienced with nuclear physics, and I wanted to know how to deal with rates. For example if we suppose this reactions,

$${}^{12}C \rightarrow(p\gamma)_{\lambda}\rightarrow {}^{13}N \rightarrow (\beta^+)\rightarrow {}^{13}C$$

For the first reaction, you can search online the rates and construct the reaction equation,

$$\frac{n({}^{12}C)}{dt}=-n({}^{12}C)\lambda$$

First of all, this rate $\lambda$ is directly the S-factor you can found in tables? Or you need to do something more?

But for the second reaction, that is a beta decay, I'can not found rates online, so I supose beta decays should work differently and not like this,

$$\frac{n({}^{13}N)}{dt}=n({}^{12}C)\lambda-n({}^{13}N)\lambda_{\beta}$$

Because what will be the rate $\lambda_{\beta}$? Is related to the decay time $\tau_{\beta}$? Because I know that $1/\tau_{\beta}$ can be considered a rate, but I don't know, if you can work with that in the same way that you work with nuclear reactions.

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    $\begingroup$ ENDF and ENSDF are your sources online for nuclear data. $\endgroup$
    – Jon Custer
    Commented Dec 30, 2022 at 15:59
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    $\begingroup$ Regardless of the decay mechanism, there is a constant decay rate, which depends on which decay we're talking about, even when you compare two decays of the same type. If you want to calculate anything, you will indeed need to look up the rate of positron decay by nitrogen-$13$. Apparently its half-life is $9.97$ minutes, so $\lambda=\frac{\ln2}{t_{1/2}}=1.16\times10^{-3}\mbox{s}^{-1}$. $\endgroup$
    – J.G.
    Commented Dec 30, 2022 at 17:04

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In this part of your reaction, ${}^{12}\rm C \rightarrow(p\gamma)_{\lambda}\rightarrow {}^{13}N $, there is a production of $^{13}\rm N$ when a proton is absorbed into a $^{12}\rm C$ nucleus.
The rate of production if $^{13}\rm N$ depends on the flux and energy of the protons, the number (density) and absorption cross section of $^{12}\rm C$ nuclei.

If an equation of the form $\rm\dfrac{d\,n({}^{12}C)}{dt}=-n({}^{12}C)\lambda$ is used then $\lambda$ would depend on the flux and energy of the protons and the absorption cross section of $^{12}\rm C$ nuclei.

This is rather different from the reaction, $\rm^{13}N \rightarrow (\beta^+)\rightarrow {}^{13}C$ for which there is a constant$^{\bf\large \color{red}*}$ decay constant $\lambda_{\beta}=1.16\times10^{-3}\mbox{s}^{-1}$ corresponding to a tabulated half life of $9.97$ minutes.
$\bf \color{red}*$ It might be that the energy of the proton absorbed by the $\rm ^{12}C$ affects to a small degree $\lambda_{\beta}$?

Decays in which the decay constant do not vary are characterized by the Bateman equations as described in this text.

The reaction to which you refer is part of the CNO cycle in main sequence stars which fuses hydrogen into Helium.

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