Editing Positronium

{{Short description|Bound state of an electron and positron}}
{{distinguish|Protonium}}
{{About|the exotic atom|the hydrogen isotope|Isotopes of hydrogen#Hydrogen-1 (protium)}}
[[File:Positronium.svg|thumb|200px|right|An [[electron]] and [[positron]] orbiting around their common [[centre of mass]]. An s state has zero angular momentum, so orbiting around each other would mean going straight at each other until the pair of particles is either scattered or annihilated, whichever occurs first. This is a [[Bound state|bound quantum state]] known as '''positronium'''.]]
{{Antimatter}}

'''Positronium''' ('''Ps''') is a system consisting of an [[electron]] and its [[antimatter|anti-particle]], a [[positron]], bound together into an [[exotic atom]], specifically an [[onium]].  Unlike hydrogen, the system has no [[proton]]s.  The system is unstable: the two particles annihilate each other to predominantly produce two or three [[gamma-ray]]s, depending on the relative spin states.  The [[energy level]]s of the two particles are similar to that of the [[hydrogen atom]] (which is a bound state of a [[proton]] and an electron). However, because of the reduced mass, the [[frequency|frequencies]] of the [[spectral line]]s are less than half of those for the corresponding hydrogen lines.

==States==

The mass of positronium is 1.022&nbsp;MeV, which is twice the electron mass minus the binding energy of a few eV. The lowest energy orbital state of positronium is 1S, and like with hydrogen, it has a [[hyperfine structure]] arising from the relative orientations of the spins of the electron and the positron.

The [[Singlet state|''singlet'' state]], {{SubatomicParticle|para-positronium}},  with [[Antiparallel vectors|antiparallel]] [[Spin (physics)|spin]]s ([[spin quantum number|''S'']]&nbsp;=&nbsp;0, ''M<sub>s</sub>''&nbsp;=&nbsp;0) is known as ''para''-positronium (''p''-Ps). It has a mean lifetime of {{val|0.12|ul=ns}} and decays preferentially into two gamma rays with energy of {{val|511|ul=keV}} each (in the [[center-of-mass frame]]). ''Para''-positronium can decay into any even number of photons (2, 4, 6, ...), but the probability quickly decreases with the number: the [[branching ratio]] for decay into 4 photons is {{val|1.439|(2)|e=-6}}.<ref name="hep-ph0310099">
{{cite journal
 |last1=Karshenboim | first1=Savely G.
 |date=2003
 |title=Precision Study of Positronium: Testing Bound State QED Theory
 |doi=10.1142/S0217751X04020142
 |journal=International Journal of Modern Physics A
 |volume=19
 |issue=23
 |pages=3879–3896
 |arxiv=hep-ph/0310099
 |bibcode = 2004IJMPA..19.3879K | s2cid=14848837
 }}</ref>

''Para-''positronium lifetime in vacuum is approximately<ref name="hep-ph0310099"/>
<math display="block">t_0 = \frac{2 \hbar}{m_\mathrm{e} c^2 \alpha^5} = 0.1244 ~\mathrm{ns}.</math>

The [[Triplet state|''triplet'' state]]s, <sup>3</sup>S<sub>1</sub>, with [[Parallel (geometry)|parallel]] spins (''S''&nbsp;=&nbsp;1, ''M<sub>s</sub>''&nbsp;=&nbsp;−1, 0, 1) are known as ''ortho''-positronium (''o''-Ps), and have an energy that is approximately 0.001 eV higher than the singlet.<ref name="hep-ph0310099"/> These states have a mean lifetime of {{val|142.05|0.02|u=ns}},<ref name="Badertscher">
{{cite journal |first1=A.| last1=Badertscher| first2=P.| last2=Crivelli| first3=W.|last3=Fetscher| first4=U.|last4=Gendotti|first5=S. N.| last5=Gninenko
 |first6=V.|last6=Postoev| first7=A.|last7=Rubbia| first8=V.|last8=Samoylenko| first9=D.|last9=Sillou
 |year=2007
 |title=An Improved Limit on Invisible Decays of Positronium
 |journal=[[Physical Review D]]
 |volume=75 |pages=032004
 |doi=10.1103/PhysRevD.75.032004
 |arxiv=hep-ex/0609059
 |bibcode=2007PhRvD..75c2004B
 |issue=3 |s2cid=9001914}}</ref> and the leading decay is three gammas. Other modes of decay are negligible; for instance, the five-photons mode has branching ratio of ≈{{val||e=-6}}.<ref name="hep-ph9911410">
{{Cite book
 |last1=Czarnecki |first1=Andrzej |last2=Karshenboim |first2=Savely G.
 |date=2000
 |chapter=Decays of Positronium
 |editor1-last=Levchenko | editor1-first=B. B.
 |editor2-last=Savrin | editor2-first=V. I.
 |title=Proceedings of the International Workshop on High Energy Physics and Quantum Field Theory (QFTHEP)
 |volume=14
 |pages=538–544
 |arxiv=hep-ph/9911410
 |bibcode = 1999hep.ph...11410C
}}</ref>

''Ortho''-positronium lifetime in vacuum can be calculated approximately as:<ref name="hep-ph0310099"/>
<math display="block">t_1 = \frac{\frac{1}{2} 9 h}{2 m_\mathrm{e} c^2 \alpha^6 (\pi^2 - 9)} = 138.6 ~\mathrm{ns}.</math>

However more accurate calculations with corrections to [[Big O notation|O]](α<sup>2</sup>) yield a value of {{val|7.040|ul=us}}<sup>&minus;1</sup> for the decay rate, corresponding to a lifetime of {{val|142|u=ns}}.<ref name=Kat/><ref name=adk>{{cite journal|last1=Adkins|first1=G. S.|last2=Fell|first2=R. N.|last3=Sapirstein|first3=J.|title=Order α<sup>2</sup> Corrections to the Decay Rate of Orthopositronium|journal=Physical Review Letters|date=29 May 2000|volume=84|issue=22|pages=5086–5089|doi=10.1103/PhysRevLett.84.5086|pmid=10990873|arxiv = hep-ph/0003028 |bibcode = 2000PhRvL..84.5086A |s2cid=1165868}}</ref>

Positronium in the 2S state is [[metastable]] having a lifetime of {{val|1100|u=ns}} against [[annihilation]].<ref name="atom-ph150305755">
{{cite journal
 |last1=Cooke| first1=D. A. |last2=Crivelli| first2=P. | first3=J. |last3=Alnis| first4=A. |last4=Antognini| first5=B. |last5=Brown| first6=S.
 |last6=Friedreich| first7=A. |last7=Gabard| first8=T. W. |last8=Haensch| first9=K. |last9=Kirch| first10=A. |last10=Rubbia| first11=V. |last11=Vrankovic
 |year=2015
 |title=Observation of positronium annihilation in the 2S state: towards a new measurement of the 1S-2S transition frequency
 |doi=10.1007/s10751-015-1158-4
 |journal=Hyperfine Interact.
 |volume=233
 |issue=1–3
 |pages=67–73
 |arxiv=1503.05755 |bibcode=2015HyInt.233...67C| s2cid=89605682 }}</ref> The positronium created in such an excited state will quickly cascade down to the ground state, where annihilation will occur more quickly.

=== Measurements ===
Measurements of these lifetimes and energy levels have been used in [[precision tests of quantum electrodynamics]], confirming [[quantum electrodynamics]] (QED) predictions to high precision.<ref name="hep-ph0310099" /><ref>
{{cite journal
 |last1=Rubbia | first1=A.
 |date=2004
 |title=Positronium as a probe for new physics beyond the standard model
 |doi=10.1142/S0217751X0402021X
 |journal=International Journal of Modern Physics A
 |volume=19
 |issue=23
 |pages=3961–3985
 |arxiv=hep-ph/0402151
 |bibcode = 2004IJMPA..19.3961R | citeseerx=10.1.1.346.5173
 | s2cid=119442567
}}</ref><ref>
{{cite journal
 |last1=Vetter |first1=P.A. |last2=Freedman |first2=S.J.
 |date=2002
 |title=Branching-ratio measurements of multiphoton decays of positronium
 |journal=[[Physical Review A]]
 |volume=66 |pages=052505
 |doi=10.1103/PhysRevA.66.052505
 |bibcode = 2002PhRvA..66e2505V
 |issue=5 |osti=821022 |url=https://digital.library.unt.edu/ark:/67531/metadc736097/
}}</ref>

Annihilation can proceed via a number of channels, each producing [[gamma rays]] with total energy of {{val|1022|ul = keV}} (sum of the electron and positron mass-energy), usually 2 or 3, with up to 5 gamma ray photons recorded from a single annihilation.

The annihilation into a [[neutrino]]–antineutrino pair is also possible, but the probability is predicted to be negligible. The branching ratio for ''o''-Ps decay for this channel is {{val|6.2|e=-18}} ([[electron neutrino]]–antineutrino pair) and {{val|9.5|e=-21}} (for other flavour)<ref name="hep-ph9911410" /> in predictions based on the Standard Model, but it can be increased by non-standard neutrino properties, like relatively high [[magnetic moment]]. The experimental upper limits on branching ratio for this decay (as well as for a decay into any "invisible" particles) are <{{val|4.3|e=-7}} for ''p''-Ps and <{{val|4.2|e=-7}} for ''o''-Ps.<ref name="Badertscher" />

== Energy levels ==
{{main|Bohr model#Electron energy levels}}

While precise calculation of positronium energy levels uses the [[Bethe–Salpeter equation]] or the [[Breit equation]], the similarity between positronium and hydrogen allows a rough estimate.  In this approximation, the energy levels are different because of a different effective mass, ''μ'', in the energy equation (see [[Bohr model#Electron energy levels|electron energy levels]] for a derivation):
<math display="block">E_n = -\frac{\mu q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2},</math>
where:

* {{math|''q''<sub>e</sub>}} is the [[Elementary charge|charge magnitude]] of the electron (same as the positron),
* {{mvar|h}} is the [[Planck constant]],
* {{math|''ε''<sub>0</sub>}} is the [[electric constant]] (otherwise known as the permittivity of free space),
* {{mvar|μ}} is the [[reduced mass]]: <math display="block">\mu = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = \frac{m_\mathrm{e}^2}{2m_\mathrm{e}} = \frac{m_\mathrm{e}}{2},</math> where {{math|''m''<sub>e</sub>}} and {{math|''m''<sub>p</sub>}} are, respectively, the mass of the electron and the positron (which are ''the same'' by definition as antiparticles).

Thus, for positronium, its reduced mass only differs from the electron by a factor of 2. This causes the energy levels to also roughly be half of what they are for the hydrogen atom.

So finally, the energy levels of positronium are given by
<math display="block"> E_n = -\frac{1}{2} \frac{m_\mathrm{e} q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2} = \frac{-6.8~\mathrm{eV}}{n^2}.</math>

The lowest energy level of positronium ({{math|1=''n'' = 1}}) is {{val|-6.8|u=eV}}. The next level is {{val|-1.7|u=eV}}. The negative sign is a convention that implies a [[bound state]].  Positronium can also be considered by a particular form of the [[Two-body Dirac equations|two-body Dirac equation]]; Two particles with a [[Coulomb's law|Coulomb interaction]] can be exactly separated in the (relativistic) [[center-of-momentum frame]] and the resulting ground-state energy has been obtained very accurately using [[finite element method]]s of [[Janine Shertzer]].<ref name="Shertzer"/>  Their results lead to the discovery of anomalous states.<ref>
{{cite journal
 |last=Patterson |first=Chris W.
 |date=2019
 |title=Anomalous states of Positronium
 |journal=[[Physical Review A]]
 |volume=100 |issue=6|pages=062128
 |doi=10.1103/PhysRevA.100.062128
 |arxiv=2004.06108
 |bibcode=2019PhRvA.100f2128P
 |s2cid=214017953
}}</ref><ref>
{{cite journal
 |last=Patterson |first=Chris W.
 |date=2023
 |title=Properties of the anomalous states of Positronium
 |journal=[[Physical Review A]]
 |volume=107 |issue=4|pages=042816
 |doi=10.1103/PhysRevA.107.042816
 |arxiv=2207.05725
|bibcode=2023PhRvA.107d2816P
 }}</ref>
The Dirac equation whose Hamiltonian comprises two Dirac particles and a static Coulomb potential is not relativistically invariant.  But if one adds the {{math|{{sfrac|''c''<sup>2''n''</sup>}}}} (or {{math|''α''<sup>2''n''</sup>}}, where {{mvar|α}} is the [[fine-structure constant]]) terms, where {{math|''n'' {{=}} 1,2...}}, then the result is relativistically invariant.  Only the leading term is included.  The {{math|''α''<sup>2</sup>}} contribution is the Breit term; workers rarely go to {{math|''α''<sup>4</sup>}} because at {{math|''α''<sup>3</sup>}} one has the Lamb shift, which requires quantum electrodynamics.<ref name="Shertzer">
{{cite journal
 |last1=Scott |first1=T.C. |last2=Shertzer |first2=J. |author2-link= Janine Shertzer |last3=Moore |first3=R.A.
 |date=1992
 |title=Accurate finite element solutions of the two-body Dirac equation
 |journal=[[Physical Review A]]
 |volume=45 |pages=4393–4398
 |doi=10.1103/PhysRevA.45.4393
 |bibcode=1992PhRvA..45.4393S |pmid=9907514 |issue=7
}}</ref>

== Formation and decay in materials ==
After a radioactive atom in a material undergoes a [[beta decay|β<sup>+</sup> decay]] (positron emission), the resulting high-energy positron slows down by colliding with atoms, and eventually annihilates with one of the many electrons in the material. It may however first form positronium before the annihilation event. The understanding of this process is of some importance in [[positron emission tomography]]. Approximately:<ref name="Harpen2003">{{cite journal|last1=Harpen|first1=Michael D.|title=Positronium: Review of symmetry, conserved quantities and decay for the radiological physicist|journal=Medical Physics|volume=31|issue=1|year=2003|pages=57–61|issn=0094-2405|doi=10.1118/1.1630494|pmid=14761021}}</ref><ref name="pmid30641509">{{cite journal|vauthors=Moskal P, Kisielewska D, Curceanu C, Czerwiński E, Dulski K, Gajos A | display-authors=etal| title=Feasibility study of the positronium imaging with the J-PET tomograph. | journal=Phys Med Biol | year= 2019 | volume= 64 | issue= 5 | pages= 055017 | pmid=30641509 | doi=10.1088/1361-6560/aafe20 | arxiv=1805.11696| bibcode=2019PMB....64e5017M| doi-access=free }}</ref>
* ~60% of positrons will directly annihilate with an electron without forming positronium. The annihilation usually results in two gamma rays. In most cases this direct annihilation occurs only after the positron has lost its excess kinetic energy and has thermalized with the material.
* ~10% of positrons form ''para''-positronium, which then promptly (in ~0.12 ns) decays, usually into two gamma rays.
* ~30% of positrons form ''ortho''-positronium but then annihilate within a few nanoseconds by 'picking off' another nearby electron with opposing spin. This usually produces two gamma rays. During this time, the very lightweight positronium atom exhibits a strong zero-point motion, that exerts a pressure and is able to push out a tiny nanometer-sized bubble in the medium.
* Only ~0.5% of positrons form ''ortho''-positronium that self-decays (usually into ''three'' gamma rays). This natural decay rate of ''ortho''-positronium is relatively slow (~140 ns decay lifetime), compared to the aforementioned pick-off process, which is why the three-gamma decay rarely occurs.

== History ==

[[File:Positronium Beam.jpg|thumb|upright=1.4|The Positronium Beam at [[University College London]], a lab used to study the properties of positronium.<ref>{{Cite journal|last1=N.|first1=Zafar|last2=G.|first2=Laricchia|last3=M.|first3=Charlton|last4=T.C.|first4=Griffith|date=1991|title=Diagnostics of a positronium beam|url=https://inis.iaea.org/search/search.aspx?orig_q=RN:23020661|journal=Journal of Physics B|language=en|volume=24|issue=21|page=4661|doi=10.1088/0953-4075/24/21/016|bibcode=1991JPhB...24.4661Z|s2cid=250896764 |issn=0953-4075}}</ref>]]

The Croatian physicist [[Stjepan Mohorovičić]] predicted the existence of positronium in a 1934 article published in ''[[Astronomische Nachrichten]]'', in which he called it the "electrum".<ref>
{{cite journal
 |last=Mohorovičić |first=S.
 |date=1934
 |journal=[[Astronomische Nachrichten]]
 |volume=253 |pages=93–108
 |doi=10.1002/asna.19342530402
 |title=Möglichkeit neuer Elemente und ihre Bedeutung für die Astrophysik
 |issue=4
|bibcode = 1934AN....253...93M }}</ref> Other sources incorrectly credit [[Carl David Anderson|Carl Anderson]] as having predicted its existence in 1932 while at [[Caltech]].<ref name="DeutschObit">
{{cite press release
 |publisher=MIT
 |year=2002
 |title=Martin Deutsch, MIT physicist who discovered positronium, dies at 85
 |url=http://web.mit.edu/newsoffice/2002/deutsch.html
}}</ref> It was experimentally discovered by [[Martin Deutsch]] at [[MIT]] in 1951 and became known as positronium.<ref name="DeutschObit"/>  Many subsequent experiments have precisely measured its properties and verified predictions of quantum electrodynamics. 

A discrepancy known as the ortho-positronium lifetime puzzle persisted for some time, but was resolved with further calculations and measurements.<ref>{{Cite news|url = http://physicsworld.com/cws/article/news/2003/may/28/positronium-puzzle-is-solved|title = Positronium puzzle is solved|last = Dumé |first = Belle|date = May 23, 2003|work = [[Physics World]]}}</ref> Measurements were in error because of the lifetime measurement of unthermalised positronium, which was produced at only a small rate. This had yielded lifetimes that were too long. Also calculations using relativistic quantum electrodynamics are difficult, so they had been done to only the first order.  Corrections that involved higher orders were then calculated in a non-relativistic quantum electrodynamics.<ref name="Kat">{{cite journal|last1=Kataoka|first1=Y.|last2=Asai|first2=S.|last3=Kobayashi|first3=t.|title=First Test of O(α<sup>2</sup>) Correction of the Orthopositronium Decay Rate|journal=Physics Letters B|volume=671|issue=2|pages=219–223|url=https://www.icepp.s.u-tokyo.ac.jp/papers/ps/icepp-report/ut-icepp-08-09.pdf|year=2009|bibcode=2009PhLB..671..219K|arxiv=0809.1594|doi=10.1016/j.physletb.2008.12.008}}</ref>

In 2024, the [[AEgIS experiment|AEgIS]] collaboration at [[CERN]] was the first to cool positronium by laser light, leaving it available for experimental use. The substance was brought to {{Convert|-100|C}} using [[laser cooling]].<ref>
{{cite journal
 |last=Glöggler |first=L. T.
 |date=2024
 |journal=[[Physical Review Letters]]
 |volume=132 |pages=083402 
 |doi=10.1103/PhysRevLett.132.083402
 |title=Positronium Laser Cooling via the 13S−23P Transition with a Broadband Laser Pulse
 |issue=8
 |pmid=38457696
 |bibcode = |doi-access=free
 |hdl=11311/1261341
 |hdl-access=free
 }}</ref><ref>{{Cite news |last=Ghosh |first=Pallab |date=2024-02-22 |title=Antimatter: Scientists freeze positronium atoms with lasers |url=https://www.bbc.com/news/science-environment-68349448 |access-date=2024-02-23 |work=[[BBC]] |language=en-GB}}</ref>

== Exotic compounds ==

Molecular bonding was predicted for positronium.<ref>
{{cite journal
 |title=Signature of the existence of the positronium molecule
 |arxiv=physics/9804023
 |last1=Usukura | first1=J.
 |last2=Varga | first2=K.
 |last3=Suzuki | first3=Y.
 |date=1998
 | doi=10.1103/PhysRevA.58.1918
 | volume=58
 | issue=3
 | journal=Physical Review A
 | pages=1918–1931
 |bibcode=1998PhRvA..58.1918U|s2cid=11941483
 }}</ref>  Molecules of [[positronium hydride]] (PsH) can be made.<ref>{{cite web|url=http://www.sc.doe.gov/bes/accomplishments/files/BES_Accomp_FY1992.pdf|page=9|title="Out of This World" Chemical Compound Observed|url-status=dead|archive-url=https://web.archive.org/web/20091012090920/http://www.sc.doe.gov/bes/accomplishments/files/BES_Accomp_FY1992.pdf|archive-date=2009-10-12}}</ref>  Positronium can also form a cyanide and can form bonds with halogens or lithium.<ref name="Saito">{{cite journal|last=Saito|first=Shiro L.|date=2000|title=Is Positronium Hydride Atom or Molecule?|journal=Nuclear Instruments and Methods in Physics Research B|volume=171|pages=60–66|doi=10.1016/s0168-583x(00)00005-7|bibcode = 2000NIMPB.171...60S|issue=1–2}}</ref>

The first observation of [[di-positronium]] ({{chem2|Ps2}}) [[molecule]]s—molecules consisting of two positronium atoms—was reported on 12 September 2007 by David Cassidy and Allen Mills from [[University of California, Riverside]].<ref>
{{cite journal
 |last1=Cassidy |first1=D.B. |last2=Mills |first2=A.P. (Jr.)
 |date=2007
 |title=The production of molecular positronium
 |journal=[[Nature (journal)|Nature]]
 |volume=449 |pages=195–197
 |doi=10.1038/nature06094
 |pmid=17851519
 |issue=7159
 |bibcode = 2007Natur.449..195C |s2cid=11269624 }}</ref><ref>
{{cite journal
 |author=Surko, C.
 |date=2007
 |title=A whiff of antimatter soup
 |journal=Nature
 |volume=449
 |issue=7159
 |pages=153–155
 |doi=10.1038/449153a|pmid=17851505
 |s2cid=8153916 
|doi-access=free
 }}</ref><ref>
{{cite web
 |url=http://www.physorg.com/news108822085.html
 |publisher=[[Physorg.com]]
 |title=Molecules of positronium observed in the lab for the first time
 |access-date=2007-09-07
}}</ref>

Unlike [[muonium]], positronium does not have a nucleus analogue, because the electron and the positron have equal masses.<ref name=barnabas/> Consequently, while muonium tends to behave like a light isotope of hydrogen,<ref>{{cite journal |last1=Rhodes |first1=Christopher J. |date=2012 |title=Muonium–the second radioisotope of hydrogen: a remarkable and unique radiotracer in the chemical, materials, biological and environmental sciences |journal=Science Progress |volume=95 |issue=2 |pages=101–174 |doi=10.3184/003685012X13336424471773 |pmid=22893978 |pmc=10365539 }}</ref> positronium shows large differences in size, polarisability, and binding energy from hydrogen.<ref name=barnabas>{{cite journal |last1=Barnabas |first1=Mary V. |last2=Venkateswaran |first2=Krishnan |last3=Walker |first3=David C. |date=January 1989 |title=Comparison of muonium and positronium with hydrogen atoms in their reactions towards solutes containing amide and peptide linkages in water and micelle solutions |journal=Canadian Journal of Chemistry |volume=67 |issue=1 |pages=120–126 |doi=10.1139/v89-020 |doi-access=free |bibcode=1989CaJCh..67..120B }}</ref>

== Natural occurrence ==

The [[timeline of the early universe|events]] in the early universe leading to [[baryon asymmetry]] predate [[recombination (cosmology)|the formation of atoms]] (including exotic varieties such as positronium) by around a third of a million years, so no positronium atoms occurred then.

Likewise, the naturally occurring positrons in the present day result from high-energy interactions such as in [[cosmic ray]]–atmosphere interactions, and so are too hot (thermally energetic) to form electrical bonds before [[annihilation]].

== See also ==
* [[Breit equation]]
* [[Antiprotonic helium]]
* [[Di-positronium]]
* [[Exciton]] — solid-state analog
* [[Protonium]]
* [[Quantum electrodynamics]]
* [[Two-body Dirac equations]]
* [[Quarkonium]]

== References ==
{{Reflist}}

== External links ==
* [https://www.feynmanlectures.caltech.edu/III_18.html#Ch18-S3 The annihilation of positronium - The Feynman Lectures on Physics]
* [https://www.universetoday.com/10584/the-search-for-positronium/ The Search for Positronium]
* [https://news.mit.edu/2002/deutsch Obituary of Martin Deutsch, discoverer of Positronium]

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