Editing Stable nuclide (section)

=== Physical magic numbers and odd and even proton and neutron count<span class="anchor" id="Proton and neutron count parity"></span><span class="anchor" id="Odd and even proton and neutron count"></span> ===
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{{See also|Even and odd atomic nuclei}}

Stability of isotopes is affected by the ratio of protons to neutrons, and also by presence of certain [[Magic number (physics)|magic numbers]] of neutrons or protons which represent closed and filled quantum shells. These quantum shells correspond to a set of energy levels within the [[Nuclear shell model|shell model]] of the nucleus; filled shells, such as the filled shell of 50 protons for tin, confers unusual stability on the nuclide. As in the case of tin, a magic number for ''Z'', the atomic number, tends to increase the number of stable isotopes for the element.

Just as in the case of electrons, which have the lowest energy state when they occur in pairs in a given orbital, nucleons (both protons and neutrons) exhibit a lower energy state when their number is even, rather than odd. This stability tends to prevent beta decay (in two steps) of many even–even nuclides into another even–even nuclide of the same mass number but lower energy (and of course with two more protons and two fewer neutrons), because decay proceeding one step at a time would have to pass through an odd–odd nuclide of higher energy. Such nuclei thus instead undergo [[double beta decay]] (or are theorized to do so) with half-lives several orders of magnitude larger than the [[age of the universe]]. This makes for a larger number of stable even–even nuclides, which account for 150 of the 251 total. Stable even–even nuclides number as many as three [[isobar (nuclide)|isobars]] for some mass numbers, and up to seven isotopes for some atomic numbers.

Conversely, of the 251 known stable nuclides, only five have both an odd number of protons ''and'' odd number of neutrons: hydrogen-2 ([[deuterium]]), [[lithium-6]], [[boron-10]], [[nitrogen-14]], and [[tantalum-180m]]. Also, only four naturally occurring, radioactive odd–odd nuclides have a half-life >10{{sup|9}} years: [[potassium-40]], [[vanadium-50]], [[lanthanum-138]], and [[lutetium-176]]. Odd–odd [[primordial nuclide]]s are rare because most odd–odd nuclei [[beta-decay]], because the decay products are even–even, and are therefore more strongly bound, due to [[Semi-empirical mass formula#Pairing term|nuclear pairing effects]].<ref>{{cite book| last=Various| editor=Lide, David R.| year=2002| title=Handbook of Chemistry & Physics| edition=88th| publisher=CRC| url=http://www.hbcpnetbase.com/| access-date=2008-05-23| isbn=978-0-8493-0486-6| oclc=179976746| archive-date=2017-07-24| archive-url=https://web.archive.org/web/20170724011402/http://www.hbcpnetbase.com/| url-status=dead}}</ref>

Yet another effect of the instability of an odd number of either type of nucleon is that odd-numbered elements tend to have fewer stable isotopes. Of the 26 [[monoisotopic element]]s (those with only one stable isotope), all but one have an odd atomic number, and all but one has an even number of neutrons: the single exception to both rules is [[beryllium]].

The end of the stable elements occurs after [[lead]], largely because nuclei with 128 neutrons—two neutrons above the [[magic number (physics)|magic number]] 126—are extraordinarily unstable and almost immediately alpha-decay.<ref name=n126sig>{{cite journal |last1=Kelkar |first1=N. G. |last2=Nowakowski |first2=M. |date=2016 |title=Signature of the ''N''&nbsp;{{=}}&nbsp;126 shell closure in dwell times of alpha-particle tunneling |journal=Journal of Physics G: Nuclear and Particle Physics |volume=43 |number=105102 |doi=10.1088/0954-3899/43/10/105102 |arxiv=1610.02069|bibcode=2016JPhG...43j5102K }}</ref> This contributes to the very short half-lives of [[astatine]], [[radon]], and [[francium]]. A similar phenomenon occurs to a much lesser extent with 84 neutrons—two neutrons above the magic number 82—where various isotopes of [[lanthanide]] elements alpha-decay.