Editing Extended periodic table (section)

=====Relativistic Dirac equation=====
[[File:1s negative continuum.svg|thumb|right|540px|Energy eigenvalues for the 1s, 2s, 2p<sub>1/2</sub> and 2p<sub>3/2</sub> shells from solutions of the [[Dirac equation]] (taking into account the finite size of the nucleus) for ''Z''&nbsp;=&nbsp;135–175 (–·–), for the Thomas-Fermi potential (—) and for ''Z''&nbsp;=&nbsp;160–170 with the self-consistent potential (---)<ref name=Fricke/>]]
The [[Theory of relativity|relativistic]] [[Dirac equation]] gives the ground state energy as

:<math>E=\frac{m c^2}{\sqrt{1+\dfrac{Z^2 \alpha^2}{\bigg({n-\left(j+\frac12\right)+\sqrt{\left(j+\frac12\right)^2-Z^ 2\alpha^2}\bigg)}^2}}},</math>

where ''m'' is the rest mass of the electron.<ref>{{cite web |title=Solution of the Dirac Equation for Hydrogen |url=https://quantummechanics.ucsd.edu/ph130a/130_notes/node501.html}}</ref> For ''Z''&nbsp;>&nbsp;137, the wave function of the Dirac ground state is oscillatory, rather than bound, and there is no gap between the positive and negative energy spectra, as in the [[Klein paradox]].<ref>{{cite book|first1=J. D.|last1= Bjorken|first2=S. D.|last2= Drell|year=1964|title=Relativistic Quantum Mechanics|url=https://archive.org/details/relativisticquan0000bjor|url-access=registration|publisher=[[McGraw-Hill]]}}</ref> More accurate calculations taking into account the effects of the finite size of the nucleus indicate that the binding energy first exceeds 2''mc''<sup>2</sup> for ''Z''&nbsp;>&nbsp;''Z''<sub>cr</sub> probably between 168 and 172.<ref name=gamowstates/> For ''Z''&nbsp;>&nbsp;''Z''<sub>cr</sub>, if the innermost orbital (1s) is not filled, the electric field of the nucleus will [[pair production|pull an electron out of the vacuum]], resulting in the spontaneous emission of a [[positron]].<ref>{{cite journal|first1=W. |last1=Greiner|first2= S. |last2=Schramm |year=2008|title=Resource Letter QEDV-1: The QED vacuum |journal=[[American Journal of Physics]] |volume=76 |issue=6|pages=509 |doi=10.1119/1.2820395|bibcode=2008AmJPh..76..509G}}, and references therein</ref><ref>{{cite journal|last1=Wang|first1=Yang|last2=Wong|first2=Dillon|last3=Shytov|first3=Andrey V.|last4=Brar|first4=Victor W.|last5=Choi|first5=Sangkook|last6=Wu|first6=Qiong|last7=Tsai|first7=Hsin-Zon|last8=Regan|first8=William|last9=Zettl|first9=Alex|author9-link=Alex Zettl|last10=Kawakami|first10=Roland K.|last11=Louie|first11=Steven G.|last12=Levitov|first12=Leonid S.|last13=Crommie|first13=Michael F.|title=Observing Atomic Collapse Resonances in Artificial Nuclei on Graphene|journal=Science|date=May 10, 2013|volume=340|issue=6133|pages=734–737|doi=10.1126/science.1234320|arxiv = 1510.02890 |bibcode = 2013Sci...340..734W|pmid=23470728|s2cid=29384402}}</ref> This diving of the 1s subshell into the negative continuum has often been taken to constitute an "end" to the periodic table,<ref name=PT172/><ref name="rsc"/><ref>{{Cite journal|last1=Indelicato|first1=Paul|last2=Bieroń|first2=Jacek|last3=Jönsson|first3=Per|date=2011-06-01|title=Are MCDF calculations 101% correct in the super-heavy elements range?|url=https://dspace.mah.se/handle/2043/12984|journal=Theoretical Chemistry Accounts|language=en|volume=129|issue=3–5|pages=495–505|doi=10.1007/s00214-010-0887-3|issn=1432-881X|hdl=2043/12984|s2cid=54680128|hdl-access=free}}</ref> but in fact it does not impose such a limit, as such resonances can be interpreted as [[Gamow state]]s. Nonetheless, the accurate description of such states in a multi-electron system, needed to extend calculations and the periodic table past ''Z''<sub>cr</sub>&nbsp;≈&nbsp;172, are still open problems.<ref name=gamowstates>{{cite journal |last1=Smits |first1=O. R. |last2=Indelicato |first2=P. |first3=W. |last3=Nazarewicz |first4=M. |last4=Piibeleht |first5=P. |last5=Schwerdtfeger |date=2023 |title=Pushing the limits of the periodic table—A review on atomic relativistic electronic structure theory and calculations for the superheavy elements |url= |journal=Physics Reports |volume=1035 |issue= |pages=1–57 |doi=10.1016/j.physrep.2023.09.004 |access-date=|arxiv=2301.02553 |bibcode=2023PhR..1035....1S }}</ref>

Atoms with atomic numbers above ''Z''<sub>cr</sub>&nbsp;≈&nbsp;172 have been termed ''supercritical'' atoms. Supercritical atoms cannot be totally ionised because their 1s subshell would be filled by spontaneous pair creation in which an electron-positron pair is created from the negative continuum, with the electron being bound and the positron escaping. However, the strong field around the atomic nucleus is restricted to a very small region of space, so that the [[Pauli exclusion principle]] forbids further spontaneous pair creation once the subshells that have dived into the negative continuum are filled. Elements 173–184 have been termed ''weakly supercritical'' atoms as for them only the 1s shell has dived into the negative continuum; the 2p<sub>1/2</sub> shell is expected to join around element 185 and the 2s shell around element 245. Experiments have so far not succeeded in detecting spontaneous pair creation from assembling supercritical charges through the collision of heavy nuclei (e.g. colliding lead with uranium to momentarily give an effective ''Z'' of 174; uranium with uranium gives effective ''Z''&nbsp;=&nbsp;184 and uranium with californium gives effective ''Z''&nbsp;=&nbsp;190).<ref>{{cite book|last1=Reinhardt|first1=Joachim|title = Nuclear Physics: Present and Future|pages=195–210|last2=Greiner|first2=Walter|doi=10.1007/978-3-319-10199-6_19|date=2015|chapter=Probing Supercritical Fields with Real and with Artificial Nuclei|isbn=978-3-319-10198-9}}</ref>

Even though passing ''Z''<sub>cr</sub> does not mean elements can no longer exist, the increasing concentration of the 1s density close to the nucleus would likely make these electrons more vulnerable to [[electron capture|''K'' electron capture]] as ''Z''<sub>cr</sub> is approached. For such heavy elements, these 1s electrons would likely spend a significant fraction of time so close to the nucleus that they are actually inside it. This may pose another limit to the periodic table.<ref name=colloq>{{cite journal |title=Colloquium: Superheavy elements: Oganesson and beyond |first1=S. A. |last1=Giuliani |first2=Z. |last2=Matheson |first3=W. |last3=Nazarewicz |first4=E. |last4=Olsen |first5=P.-G. |last5=Reinhard |first6=J. |last6=Sadhukhan |first7=B. |last7=Schtruempf |first8=N. |last8=Schunck |first9=P. |last9=Schwerdtfeger |date=2019 |journal=Reviews of Modern Physics |volume=91 |issue=1 |pages=011001-1–011001-25 |doi=10.1103/RevModPhys.91.011001|bibcode=2019RvMP...91a1001G |s2cid=126906074 |doi-access=free }}</ref>

Because of the factor of ''m'', [[muonic atom]]s become supercritical at a much larger atomic number of around 2200, as [[muon]]s are about 207 times as heavy as electrons.<ref name=gamowstates/>