The counter-intuitive time scales in atomic physics and nuclear physics

Question

Compare atomic physics and nuclear physics. The interaction in the latter is much stronger than that in the former.

However, the typical spontaneous emission time scale in atomic physics is on the order of $10^{-6}$ s. In contrast, in nuclear physics, many radioactive nucleus have a half-time of $10^6$ years or even more.

This is counter-intuitive in view of the Fermi golden rule, right? By the rule, the decay rate is proportional to the interaction strength squared and the density of states. Both are larger in nuclear physics, right?

Of course, the golden rule does not always apply, but at least it can give the right order of magnitude, right?

Answer 1

The slow nuclear transitions have a potential barrier so they proceed by tunnelling and the rate is supressed by a factor of $e^{-E/E_0}$, where $E$ is the barrier height and $E_0$ is some characteristic energy. Your golden rule calculation is giving you the rate in the absence of a barrier.

Potential barriers are rare in atomic physics, but frequent in chemistry. For example if you mix hydrogen and oxygen at low temperature the mixture will be stable indefinitely. Add enough energy locally to jump over the barrier (e.g. a spark) and bang!

Answer 2

the typical spontaneous emission time scale in atomic physics is on the order of 10^−6 s. In contrast, in nuclear physics, many radioactive nucleus have a half-time of 10^6 years or even more.

I beg to humbly disagree with your picture of generalization of atomic and nuclear time scale of events and putting up a contrast/relation with the atomic and nuclear interaction strengths.

One should look up the whole area of Nuclear terrain and various events happening (reactions/decay/fission/fusion/capture/stripping/pick-up...) at variety of time scales and thus come to certain generalizations:

e,g, The W , Z boson particles are very short-lived, with a half-life of about 3×10^−25 s.

In Compound nuclear reactions either a low energy particle/projectile is absorbed or a higher energy particle transfers energy to the nucleus, leaving it with too much energy to be fully bound together.

On a time scale of about 10−19 seconds, particles, usually neutrons,do come out or one can say are "boiled" off. That is, it remains together till a neutron gets enough energy to escape to escape the mutual attraction.

In Atomic situations charged particles rarely come out because of the coulomb barrier

Your present approach to describing reaction rates is based on the time required for the concentration of a reactant to decrease to one-half its initial value. This period of time is called **the half-life of the reaction, written as t(1/2). **Should this alone determine the time scale of nuclear interactions/events?

Let us ponder over!

Another example:

We present a method for detecting short-lived particles in nuclear emulsions. Production and decay vortices as close as 0.5 μm can be resolved. This corresponds to a lifetime of the order of 10^−15 s. The principle is to fit straight lines to the tracks and make a chi-square test of the hypothesis of a single vertex. This technique may be useful for detecting decays of heavy quarks.

Detection of short-lived particles in nuclear emulsions http://www.sciencedirect.com/science/article/pii/0029554X79902982