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Freeman J. Dyson in his "Time without end: Physics and biology in an open universe", Lecture 2: Physics, part G: All matter decays to iron, claimed that on a long enough time scale "Elements heavier than iron will decay to iron by various processes such as fission and alpha emission". I'm not able to find why or how will that happen. All question related to this part of paper seem to accept this as a obvious fact. Am I missing something?

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    $\begingroup$ This is just another way of saying that Iron has the highest binding energy per nucleon, and is thus the most stable element. $\endgroup$
    – Physiker
    Feb 25 at 9:01
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    $\begingroup$ But just because it has highest binding energy per nucleon, doesn't mean that all elements heavier than it will fission into it right? because if there's a stable gold isotope it will not readily decay into iron for no reason even after 10^1500 years right? $\endgroup$ Feb 25 at 9:05
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    $\begingroup$ On long enough time scales is the key here. First of all, $10^{1500}$ is a ridiculously large number. To give you a flavour, the Universe is only $10^{18}$ seconds old. So, one suspects that Gold can indeed turn into Iron over such long time periods. $\endgroup$
    – Physiker
    Feb 25 at 9:08
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    $\begingroup$ @Physiker gold turns to iron and what - that's the problem with this proposal (see my answer). If we're considering baryon number violating processes, then iron isn't stable either. Otherwise gold does not decay $\endgroup$
    – AXensen
    Feb 26 at 5:54
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    $\begingroup$ Possible duplicate, worth checking: physics.stackexchange.com/q/540246/226902 (I am not flagging this as a duplicate because this question is better received than the older one). $\endgroup$
    – Quillo
    Feb 26 at 8:16

2 Answers 2

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Nuclear decay is a quantum process and as long as their exists a final state with lower energy than the initial state, and electric charge, baryon and lepton number are conserved, a nucleus will always eventually decay either directly or via quantum tunnelling. "Eventually", however, may be a very, very, very long time.

Because protons are electrically charged, there is a Coulomb repulsive force in a nucleus proportional to the number of protons squared ($Z^2$) fighting against the attractive short-range strong nuclear force that is proportional to the number of nucleons ($A$). The net binding energy due these forces has a minimum around iron.

"Stable" nuclei above iron don't immediately decay because there is a potential barrier between the initial and final states. For example, for a nucleus to undergo spontaneous fission, it must first start to elongate before splitting, but this elongated state has higher energy than the initial state. This makes the decay impossible classically, but quantum tunnelling makes it possible with a rate that is exponentially sensitive to the height of the barrier which depends on the ratio of Coulomb repulsion to nuclear attraction, i.e. $Z^2/A$.

From Wikipedia we can see how the spontaneous fission lifetime depends on $Z^2/A$ for heavy radioactive elements such as uranium.

Half-life vs Z^2/A for heavy nuclei such as uranium

If we very, very naively extrapolate the green line down to elements close to iron, we find lifetimes $\sim 10^{125}$ seconds, more than a hundred orders of magnitude longer than the current age of the universe. This extrapolation is, of course, at best an order-of-magnitude estimate of the order-of-magnitude and might better be written as $\sim 10^{10^{\sim2}}$ seconds.

Update in response to comments and other answer

As @AXensen notes in their very nice answer, the above order-of-magnitude for the lifetimes may be a wild underestimate for lower mass isotopes.

Since Nickel-62 is actually the most tightly bound nucleus, not iron, it would have been better for Dyson to have referred to the "iron region of the table of isotops" instead of "iron". The atomic mass binding energy per nucleon is quite flat around iron, which makes it impossible for individual atoms in this region to ever decay via spontaneous fission or alpha decay.

As @AXensen also notes, however, tunnelling can still cause groups of atoms to transform. For example, an individual deuterium atom is absolutely stable, but a deuterium molecule is not. If my reading is correct, J.D. Jackson's 1957 paper on muon-catalyzed fusion implies that the lifetime of a normal (2-electron) deuterium molecule is around $10^{50}$ seconds for $\mathrm{D_2}\rightarrow \mathrm{^4He}$. Atoms of heavier elements tend to be inside stars or eventually clump together into dust particles, so collective tunnelling transformations are possible even when individual atomic decays are not. The rate for such tunnelling, however, will be much, much, much longer than my above estimate for spontaneous fission. Even then, some atoms - such as the nobel gas krypton - that don't tend to clump, would never decay.

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    $\begingroup$ "as long as their exists a final state with lower energy" is a key. There are many isotopes heavier than iron for which there is no final state with lower energy. That's why there are stable isotopes heavier than iron. There may be isotopes that we expect to be unstable but we haven't observed it yet, like bismuth 209 used to be. Most of the isotopes we call stable are definitely stable forever, because we have measured their mass to sufficient precision that we know there is no possible decay, it's not just that the decays are too long to observe as your extrapolation argument suggests. $\endgroup$
    – AXensen
    Feb 26 at 8:29
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    $\begingroup$ If the stability is proportional to A, why istopes with a higher amount of neutrons are more likely to be radioactive? Or is there something I'm missing? $\endgroup$ Feb 26 at 14:40
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    $\begingroup$ @MindwinRememberMonica Stability comes from having the "right" number of neutrons, either too few or too many make an isotope radioactive. Adding neutrons reduces the chances of strong fission/alpha decays caused by coulomb repulsion withing the nucleus, but since bare neutrons are heavier than bare protons, adding them also increases the likelihood of beta decay. $\endgroup$ Feb 26 at 16:48
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Just because iron has the lowest mass per nucleon does not imply that everything heavier than iron can decay to it. It is possible that an isotope has less binding energy per nucleon than iron, but the other particle (take alpha for example) it would have to emit to emit to reach iron has more mass than the difference between iron's mass and our isotope's mass. Notice that unless you're splitting into two irons, the other thing you're emitting also has less binding energy per nucleon than iron, so when you include both decay products it's not completely obvious that that end product will always be lower total mass.

For example, Nickel 60 has a mass of 59.93079AMU, iron 56 (lowest mass per nucleon) has a mass of 55.93494AMU, and helium 4 has a mass of 4.00260AMU. Helium plus iron is more than nickel by 0.00675AMU. It will never decay in this way. And any other manner of decaying you might imagine would be much worse.

This is why there are stable isotopes above iron.

Based on the phrasing "by various processes such as fission and alpha emission," I honestly think Dyson just forgot that alpha particles also have mass that shouldn't be ignored. But he also mentions that things lighter than iron will fuse via quantum tunneling. And I think this mechanism for tunneling-fusion could also give a mechanism for nickel 60 to turn into iron (for example).

Consider a nickel 60 atom sitting next to chromium 52 atom. An alpha particle could quantum tunnel from the nickel 60 to the chromium 52 to make two irons. This takes a little longer than Dyson's back of the envelope estimate though - with an energy gap of 0.00675AMU$\times c^2$ and a distance of 2 angstroms, using the WKB approximation I get a quantum tunneling timescale of $10^{65,000}\,\mathrm{s}$, where he gets a timescale of $10^{1,500}\,\mathrm{s}$ for his fusion process. It may be true then that some gravitationally bound systems like brown dwarfs would turn into mostly iron through what could be described as "the exchange of a virtual proton/alpha particle/neutron" - if you ignore the mechanism I'll describe in the last paragraph.

Note however, that neutron stars have a stronger bonding per nucleon than iron. And black holes obviously do too, which decay to mostly light through hawking radiation.

In our current understanding of the universe, the standard model, even iron decays eventually because of something called a chiral anomaly. This can violate baryon number conservation - turning protons into positrons. It's not clear to me that this iron formation thing will occur before the decay of the larger elements because of this process.

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    $\begingroup$ Thanks for making these good points, which I shouldn't have glossed over since they were vaguely lurking in the back of my mind. I will update my answer shortly. $\endgroup$ Feb 26 at 16:54
  • $\begingroup$ A good counterexample is worth a thousand arguments :-) $\endgroup$ Feb 26 at 17:22

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