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We know the law of radioactivity:

$$N=N_0e^{-\lambda t}$$

where $\lambda$ is the exponential decay constant. My question is: This constant depends of what?

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The constant is a function of the stability of the nucleus, and is experimentally determined for every isotope. In other words - every kind of nucleus has its own value of $\lambda$ and there is no way (that I know) to get an accurate value for it, other than measurement.

But there are some nuclear physicists roaming who will put me out of my misery, I'm sure...

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  • $\begingroup$ It can be theoretically predicted with some accuracy. The transition probability is a function of the density of states near the final nuclear energy and the squared norm of the matrix element of the quantum transition operator. "Fermi's Golden Rule." $\endgroup$ – user22620 Nov 20 '14 at 0:11
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The transition probability per unit time of a nucleus from an initial state i to a final state f, representing the decayed system, is modeled by Fermi's Golden Rule: $$\lambda=T_{i\rightarrow f} = \frac{2\pi}{\hbar}\left|\left\langle i\left|H'\right|f\right\rangle\right|^2\rho$$ Where $T_{i\rightarrow f}$ is the transition probability from state $i$ to state $f$ per unit time, $H'$ is the matrix element of the the transition operator, and $\rho$ is the state density about the final nuclear energy.

The experimental measurement of a decay constant provides a benchmark for validation of theoretical models of the physics of nucleon-nucleon interactions and nuclear energy structure. In some rare cases, the decay probabilities are so minute that the Golden Rule provides a useful a priori estimate of the decay likelihood, which can guide the design of experimental measurements of such rare decays.

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  • $\begingroup$ How does one come up with $H'$ for a particular nucleus? Is it always of the same form? I just don't know how to turn this equation into a number - say for the probability of Co-57 decaying. Could you point me to an example of the actual calculation? How accurate are the results? $\endgroup$ – Floris Nov 20 '14 at 1:35
  • $\begingroup$ More importantly, how does $T_{i\to f}$ relate to $\lambda$ in the question? $\endgroup$ – Kyle Kanos Nov 20 '14 at 1:48
  • $\begingroup$ It's the same, per Krane. Edited. $\endgroup$ – user22620 Nov 20 '14 at 1:54
  • $\begingroup$ I can't point you to a calculation, I will look into it. I suspect that the structure of the transition probability has been proven from QM, but the matrix for large nuclei cannot currently be calculated. $\endgroup$ – user22620 Nov 20 '14 at 1:55
  • $\begingroup$ Here's an investigation into a nuclear matrix element: sciencedirect.com/science/article/pii/S0370269312013160 . I think that these are theoretical constructs that we either currently lack the theoretical tools to provide exacting estimates of, or they cannot be expressed in closed form. Hopefully a QM person will chime in, I am just an engineer. $\endgroup$ – user22620 Nov 20 '14 at 2:09
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Here is a table of isotopes versus lifetimes the color code of the lifetimes on the right hand column:

isotope table

Isotope half-lives. Note that the darker more stable isotope region departs from the line of protons (Z) = neutrons (N), as the element number Z becomes larger

Modeling a nucleus is a many body problem and also a many forces problem. There exists the nuclear force ( strong), the weak and the electromagnetic, leading to sequential decays. As most many body problems the models have to follow the data rather than be predictive.

The nuclear force will give short lifetimes, the electromagnetic ( electron capture for example) a bit longer and the weak the longest of all, as basic inputs. BUT the particular shells of the nucleus filled, the binding energies per nucleon and the ratio of protons to neutrons will have a strong role too, modifying the intrinsic lifetimes of the underlying interactions.

The nuclear shell model allows for the possibility to use fermi's golden rule as given in the answer by user22620, but the specifics of the nuclide under study have to be taken into account, no general solution.

Here is a power point presentation for the essentials of nuclear physics for those interested further.

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There are many types of nuclear decay, and many techniques for estimating half-lives.

  • For beta decay of states in spherical nuclei, calculation of decay rates is a classic application of the (spherical) nuclear shell model.

  • For gamma decay, there are generic estimates that are based on the energy and multipolarity of the transition. (The term to google on is "Weisskopf units.") These are usually good to within one or two orders of magnitude. For better precision, you can use more specialized techniques. E.g., the spherical shell model works for a spherical nucleus. For a collectively rotating deformed nucleus, a rough rule of thumb is that the strength of an in-band E2 transition is $\sim Z$ in Weisskopf units.

  • Alpha-decay half-lives approximately follow the rule that the log of the half-life varies linearly with $E^{-1/2}$, where $E$ is the decay energy. Odd nuclei tend to have huge hindrance factors in alpha decay rates compared to their even-even neighbors. Decay of an odd nucleus often requires that the alpha carry away angular momentum, but that adds a centrifugal barrier. There is also a selection rule that says parity shouldn't change.

  • For spontaneous fission, one uses the deformed nuclear shell model to calculate the potential energy as a function of some parameter $\beta$ that describes the deformation. You then have a quantum-mechanical tunneling problem, and you can use the WKB approximation to estimate the tunneling probability.

There are many other cases, e.g., a superdeformed nucleus (shaped like an ellipsoid with a 2:1:1 axis ratio) can decay to a normally-deformed state, and one technique for estimating the decay rate would be the one used for fission, but with the tunneling going from superdeformation to normal deformation (decreasing $\beta$) rather than from normal deformation to scission (increasing $\beta$). Nuclear structure physics is not a unified, well understood field with simple methods that work in all cases. It's a hodge-podge of approximations.

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