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As was explained in the question What is the relation between the half-time and the line-width of a radioactive nucleus?, the half-life $\tau$ of an unstable nucleus is related to the linewidth $\sigma_E$ of the resonance in its energy spectrum by the uncertainty relation $$\sigma_E\cdot\tau=\frac\hbar 2.$$ This is required by the Fourier relation between the two, and it has apparently been confirmed experimentally for relatively short decays.

How short are such short decays? What's the longest radioactive decay half-life that's been experimentally shown to coincide with an experimentally-observed linewidth?

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Emilio Pisanty
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2 Answers2

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About short-decays it is not easy to measure resonance widths. There are many reasons for this. Short-lived resonances are wide and their ranges of energies overlap at least partially with other resonances. I mean, neighbor resonances form a strong background. Therefore, comparisons with theoretical predictions are very difficult.

The ideal case for defining the resonance width is when the resonance is sharp, i.e. the width is much smaller than the central energy of the resonance, and also much smaller than the distance between the central energy in the resonance and the potential barrier. These conditions can be fulfilled by long-lived nuclides, however in this case the number of events detected in experiments is not big and the statistics is poor.

Thus, for your last question, the hope to give an answer is very small. The imprecision of the results of measurements is so big, that the half-life is usually given as log$(\text t_{1/2})$. And from one series of experiments to another series carried some years later (eventually by another team of experimenters), there may be differences of 10 times (e.g. see the report on the half-life measurement of $^{130} \rm Te$ ).

Moreover, in comparisons of the measured half-life with theoretical models, it is not the agreement with the resonance width that is tested, but the agreement with all sort of rules about the half-life, predicted by models, see for instance the following reference

G. Royer, "ALPHA DECAY POTENTIAL BARRIERS AND HALF-LIVES AND ANALYTICAL FORMULA PREDICTIONS FOR SUPERHEAVY NUCLEI", "Workshop on the State of the Art in Nuclear Cluster Physics (SOTANCP2008), Strasbourg : France (2008)".)

By the way, in cases that the law of decay obeys the exponential form $N(t) = N_0 \ e^{-\Gamma t}$ the relation between the half-life and the resonance width (usually denoted by $\Gamma$) is

$$ \Gamma \ \text t_{1/2} = \hbar \ ln\ 2$$

Urb
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Sofia
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  • This doesn't answer the question which is what the longest lifetime for which the width has be measured. – dmckee --- ex-moderator kitten Jan 30 '15 at 23:24
  • @dmckee : I don't know the last measurement done. I know the list in the Wikipedia and the works of Royer. – Sofia Jan 30 '15 at 23:29
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    Two comments. First, you've made a lot of edits without doing anything to address the question that was asked. Secondly, while width measurements are almost never made in atomic physics and rarely in nuclear physics they are a little industry in particle physics; the $J/\psi$ is one of the common examples that get's toted out for classroom work. As is the $Z^0$. The question isn't "Do people do this?" it is "What is the narrowest width measurement that can be convincingly made?" – dmckee --- ex-moderator kitten Jan 31 '15 at 02:27
  • @dmckee : I don't belong to particle physics domain. Second, yes, I use to improve my posts. But the main issue, I exactly address the question of the compatibility between width and half-life. This compatibility is *poor*. Even if you will find an element for which such a compatibility exists, it is as if you use an non-precise computer, and once it gives a correct result. – Sofia Jan 31 '15 at 05:26
  • @dmckee : I don't have access to the article that you mentioned about $^5He$, and if you can, maybe you can download it and send me too by email. But I saw that its half-life is ~$10^{-20}$sec. These are the *worst resonances* because they are the widest and one should use all sort of techniques to eliminate the overlapping. I didn't study $^5He$ in particular but I know the general problems. Also, I don't know where you live but not always I can answer you quickly, there are differences between the night hours between different regions. – Sofia Jan 31 '15 at 05:40
  • Sofia, the question asks for the longest lifetime which has been convincingly measured by line width. That means the narrowest line. If wide lines being a problem is a non-issue. Separating the intrinsic line width from detector effects and kinematic broadening could be. – dmckee --- ex-moderator kitten Jan 31 '15 at 20:19
  • @dmckee , no, I copy here the question: What's the longest radioactive decay half-life that's been experimentally shown to coincide with an experimentally-observed linewidth. I.e. half-life measurement that is compatible with width measurement. Well, what I said in my post was that the answer is disappointing, due to the not exact measurements on the half-lives. Well, trust me a bit, I work with such things. The exactity of the data is poor, the models don't succeed to reproduce the data very well. About line shapes I don't know experiments but I see the difficulties explained by experts. – Sofia Jan 31 '15 at 20:37
  • @dmckee I can show you material, but, please trust me a bit. I had long talks with G. Royer and other experts about such things. – Sofia Jan 31 '15 at 20:40
  • Let us continue this discussion in chat. – dmckee --- ex-moderator kitten Jan 31 '15 at 21:17
  • @dmckee : O.K., just wait a bit, I want to help someone. – Sofia Jan 31 '15 at 21:21
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Just to offer a random benchmark--because I haven't made a study of this question--$^5\mathrm{He}$ is described as having a linewidth of $0.60 \,\mathrm{MeV}$, which corresponds to a lifetime in the single digits of nanoseconds.

References:

Urb
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