# Why some nuclei with “magic” numbers of neutrons have a half-life less than their neighbor isotopes?

It's easy to find the "magic" numbers of neutrons on the diagrams of alpha-decay energy: 82, 126, 152, 162. Such "magic" nuclei should be more stable than their neighbors.  But why some nuclei with "magic" numbers of neutrons have a half-life less than their neighbor isotopes with odd numbers of neutrons?

Examples for "magic" number 126:

A half-life of "magic" Po-210 is 138 days, whereas a half-life of neighbor isotope Po-209 is 102 years.

A half-life of "magic" Ra-214 is 2.46 sec, whereas a half-life of neighbor isotope Ra-213 is 2.74 min.

Examples for "magic" number 152:

A half-life of "magic" Cm-248 is 348 thousand years, whereas a half-life of neighbor isotope Cm-247 is 16 million years!

A half-life of "magic" Cf-250 is 13 years, whereas a half-life of neighbor isotope Cf-249 is 351 years.

P.S. Source of the diagrams data

## 3 Answers

You're assuming that nuclei with exactly a magic number of neutrons are more stable than all their non-magic neighbors in the chart of nuclides, but there's no reason to think that.

If a nucleus has a magic number of neutrons, that means one shell is completely full, and the next shell is empty. Therefore the next neutron you add (going to magic+1) will have significantly greater single-particle energy, so that nucleus should be less stable than the magic one. This is born out by both of your examples for the well-known magic number 126:

• Po-211 (N = 126+1): 0.5 seconds
• Ra-215 (N = 126+1): 1.5 milliseconds

However, if you remove one neutron, then nothing changes about the other (magic−1) neutrons; they are still in the same shells as in the magic case. No extra stability is caused by having a completely full shell as opposed to an almost full one. (In this respect, the word "magic" is misleading.)

Here's another way to think of it: In alpha decay (which is the main decay branch for all of your examples), two neutrons are removed from the nucleus. It will always be the two most energetic neutrons that are removed (those in the highest shells). In a magic nucleus, the lower shell is completely full, so two neutrons are removed from that lower shell. In a magic+1 nucleus, there is a lone neutron in the upper shell, so when two neutrons are removed, one comes from the upper shell and one from the lower shell. Since one comes from the upper shell, more energy is released in the alpha decay and in general the half-life will be shorter.

However, in a magic−1 nucleus, there are no neutrons in the upper shell, so both neutrons come from the lower shell. This is the same situation as the magic case, so there's no reason to expect the decay energies or half-lives to be drastically different. (Of course they won't be exactly the same, but the differences come from other, more subtle effects.)

BTW, 152 is not a "canonical" or "universal" magic number. It shows up on the plot you have there of some specific elements, but if you look at other elements, the gap between the shells occurs at a different place. 126 is a universal magic number, but at 152 the situation is more complicated because changing the neutron/proton ratio also shifts the shells relative to each other. This is why the thing I said about magic+1 always being less stable than magic doesn't hold for one of those. Nuclear structure is really complicated.

• @Keenan, you said "However, in a magic−1 nucleus, there are no neutrons in the upper shell, so both neutrons come from the lower shell. This is the same situation as the magic case, so there's no reason to expect the decay energies or half-lives to be drastically different". Difference between 138 days and 102 years is not significant? – voix Dec 5 '10 at 10:31
• @Keenan, one might assume that it is harder to remove two neutrons from a completely full shell than from the "magic-1" shell as in the case of beryllium: energy of Be-8 alpha-decay is plus 0.1 Mev, whereas one of Be-7 is minus 1.6 MeV. But why so easy to remove two neutrons from the "magic+1" shell? – voix Dec 5 '10 at 10:33
• The difference between 138 days and 102 years is only 2 orders of magnitude. If you want to get more precise than that, you'll have to use a better model than the "idiot model" presented here. The two biggest errors with what you're assuming are (1) nuclear structure is in reality way more complicated than uninteracting protons and neutrons filling perfectly fixed shells, and (2) there is really no simple relationship between decay energy and decay half-life. In general they're negatively correlated, but half-life is not determined by energy; other factors go into it. – Keenan Pepper Dec 5 '10 at 18:38
• It is in general NO HARDER to remove two neutrons from a completely full shell than from an almost-full shell. Why would it be? The 7Be and 8Be example is confusing. 8Be is not magic, because it has 4 protons and 4 neutrons, and 4 is NOT a magic number. What were you trying to show with that? – Keenan Pepper Dec 5 '10 at 18:48
• @Keenan, it's HARDER to remove two NEUTRONS from a completely full NEUTRON shell than from an almost-full NEUTRON shell. An almost-full neutron shell is more neutron-deficit than a completely full one. And maybe that is the answer. – voix Dec 7 '10 at 14:25

Have you considered the problem in light of the semi-empirical mass formula?

$$m = Zm_p + Nm_n - \frac{E_B}{c^2}$$

where the binging energy is

$$E_B = a_VA - a_SA^{2/3} - a_C\frac{Z(Z-1)}{A^{1/3}} - A_A\frac{(A - 2Z)^2}{A} +\delta(A,Z)$$

and all the coefficients are experimentally tuned values.

What you are looking for is a pair where the mass of the post decay nucleus is "much" lighter than the pre-decay nucleus. And the important aspects are the difference in the nucleon masses, and the changes in the Coulomb ($a_C$) and Asymmetry ($a_A$) terms.

The wikipedia article I liked has numeric values. You'll note that this kind of consideration suggests the island and rock of stability.

• according to this formula even nucleus is more stable than odd one. And it doesn't concern "magic" nuclei. As for beta-stability, "magic" Cm-248 (half-life: 348 thousand years) is beta-stable, Cm-247 (half-life: 16 million years) is not. – voix Nov 30 '10 at 18:53
• @voix: Sorry are the beta thing...I've had beta decay on my mind of late. – dmckee Nov 30 '10 at 19:12

As @Keenan rightly pointed out, the main decay branch for all of my examples is alpha-decay. Alpha particles consist of two protons and two neutrons. And it's easy to remove two neutrons from a completely full ("magic") neutron shell than from a "magic-1" neutron shell. Because a "magic-1" neutron shell is more neutron-deficit than a "magic" one. Maybe that is the answer.

BTW, "magic" nuclei No(264,102) and Rf(266,104) are also beta-stable, so they should have a long enough half-life. But "magic-1" nuclei No(263,102) and Rf(265,104) should be even more stable.