2 + 2 = 5

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/2_%2B_2_%3D_5#See_also

Two Plus Two Make Five (1895), by Alphonse Allais, is a collection of absurdist short stories about anti-intellectualism as politics.

2 + 2 = 5 or two plus two equals five is a mathematical falsehood which is used as an example of a simple logical error that is obvious to anyone familiar with basic arithmetic.

The phrase has been used in various contexts since 1728, and is best known from the 1949 dystopian novel Nineteen Eighty-Four by George Orwell.

As a theme and as a subject in the arts, the anti-intellectual slogan 2 + 2 = 5 pre-dates Orwell and has produced literature, such as Deux et deux font cinq (Two and Two Make Five), written in 1895 by Alphonse Allais, which is a collection of absurdist short stories; and the 1920 imagist art manifesto 2 × 2 = 5 by the poet Vadim Shershenevich.

Self-evident truth and self-evident falsehood

Main article: Self-evidence

In establishing the mundane reality of the self-evident truth of 2 + 2 = 4, in De Neutralibus et Mediis Libellus (1652) Johann Wigand said: "That twice two are four; a man may not lawfully make a doubt of it, because that manner of knowledge is grauen [graven] into mannes [man's] nature."

In the comedy-of-manners play Dom Juan, or The Feast with the Statue (1665), by Molière, the libertine protagonist, Dom Juan, is asked in what values he believes, and answers that he believes "two plus two equals four".

In the 18th century, the self-evident falsehood of 2 + 2 = 5 was attested in the Cyclopædia, or an Universal Dictionary of Arts and Sciences (1728), by Ephraim Chambers: "Thus, a Proposition would be absurd, that should affirm, that two and two make five; or that should deny 'em to make four." In 1779, Samuel Johnson likewise said that "You may have a reason why two and two should make five, but they will still make but four."

In the 19th century, in a personal letter to his future wife, Anabella Milbanke, Lord Byron said: "I know that two and two make four—& should be glad to prove it, too, if I could—though I must say if, by any sort of process, I could convert 2 & 2 into five, it would give me much greater pleasure."

In Gilbert and Sullivan's Princess Ida (1884), the Princess comments that "The narrow-minded pedant still believes/That two and two make four! Why, we can prove,/We women—household drudges as we are –/That two and two make five—or three—or seven;/Or five-and-twenty, if the case demands!"

Politics, literature, propaganda

France

In the political pamphlet "What is the Third Estate?" (1789), Emmanuel-Joseph Sieyès provided the humanist bases for the French Revolution (1789–1799).

In the late 18th century, in the pamphlet What is the Third Estate? (1789), about the legalistic denial of political rights to the common-folk majority of France, Emmanuel-Joseph Sieyès, said: "Consequently, if it be claimed that, under the French constitution, 200,000 individuals, out of 26 million citizens, constitute two-thirds of the common will, only one comment is possible: It is a claim that two and two make five."

Using the illogic of "two and two make five", Sieyès mocked the demagoguery of the Estates-General for assigning disproportionate voting power to the political minorities of France—the Clergy (First Estate) and the French nobility (Second Estate)—in relation to the Third Estate, the numeric and political majority of the citizens of France.

In the 19th century, in the novel Séraphîta (1834), about the nature of androgyny, Honoré de Balzac said:

Thus, you will never find, in all Nature, two identical objects; in the natural order, therefore, two and two can never make four, for, to attain that result, we must combine units that are exactly alike, and you know that it is impossible to find two leaves alike on the same tree, or two identical individuals in the same species of tree. That axiom of your numeration, false in visible nature, is false likewise in the invisible universe of your abstractions, where the same variety is found in your ideas, which are the objects of the visible world extended by their interrelations; indeed, the differences are more striking there than elsewhere.

In the pamphlet "Napoléon le Petit" (1852), about the limitations of the Second French Empire (1852–1870), such as majority political support for the monarchist coup d'Ḗtat, which installed Napoleon III (r. 1852–1870), and the French peoples' discarding from national politics the liberal values that informed the anti-monarchist Revolution, Victor Hugo said: "Now, get seven million, five hundred thousand votes to declare that two-and-two-make-five, that the straight line is the longest road, that the whole is less than its part; get it declared by eight millions, by ten millions, by a hundred millions of votes, you will not have advanced a step."

In The Plague (1947), French philosopher Albert Camus declared that times came in history when those who dared to say that 2 + 2 = 4 rather than 2 + 2 = 5 were put to death.

Russia

Soviet propaganda: The "Arithmetic of an Alternative Plan: 2 + 2 plus the Enthusiasm of the Workers = 5" exhorts the workers of the Soviet Union to realise five years of production in four years' time (Iakov Guminer, 1931).

In the late 19th century, the Russian press used the phrase 2 + 2 = 5 to describe the moral confusion of social decline at the turn of a century, because political violence characterised much of the ideological conflict among proponents of humanist democracy and defenders of tsarist autocracy in Russia. In The Reaction in Germany (1842), Mikhail Bakunin said that the political compromises of the French Positivists, at the start of the July Revolution (1830), confirmed their middle-of-the-road mediocrity: "The Left says, 2 times 2 are 4; the Right, 2 times 2 are 6; and the Juste-milieu says, 2 times 2 are 5".

In Notes from Underground (1864), by Feodor Dostoevsky, the anonymous protagonist accepts the falsehood of "two plus two equals five", and considers the implications (ontological and epistemological) of rejecting the truth of "two times two makes four", and proposed that the intellectualism of free will—Man's inherent capability to choose or to reject logic and illogic—is the cognitive ability that makes humanity human: "I admit that twice two makes four is an excellent thing, but, if we are to give everything its due, twice two makes five is sometimes a very charming thing, too."

In the literary vignette "Prayer" (1881), Ivan Turgenev said that: "Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: 'Great God, grant that twice two be not four'." In God and the State (1882), Bakunin dismissed deism: "Imagine a philosophical vinegar sauce of the most opposed systems, a mixture of Fathers of the Church, scholastic philosophers, Descartes and Pascal, Kant and Scottish psychologists, all this a superstructure on the divine and innate ideas of Plato, and covered up with a layer of Hegelian immanence, accompanied, of course, by an ignorance, as contemptuous as it is complete, of natural science, and proving, just as two times two make five, the existence of a personal god." Moreover, the slogan "two plus two equals five", is the title of the collection of absurdist short stories Deux et deux font cinq (Two and Two Make Five, 1895), by Alphonse Allais; and the title of the imagist art manifesto 2 x 2 = 5 (1920), by the poet Vadim Shershenevich.

In 1931, the artist Yakov Guminer [ru] supported Stalin's shortened production schedule for the economy of the Soviet Union with a propaganda poster that announced the "Arithmetic of an Alternative Plan: 2 + 2 plus the Enthusiasm of the Workers = 5" after Stalin's announcement, in 1930, that the first five-year plan (1928–1933) instead would be completed in 1932, in four years' time.

George Orwell

The anti-Nazi propagandist George Orwell, at the BBC during the Second World War (1939–1945)

See also: Doublespeak

In Orwell's Nineteen Eighty-Four, it appears as a possible statement of Ingsoc (English Socialism). The Party (i.e. a political party) slogan "War Is Peace, Freedom Is Slavery, Ignorance Is Strength" is a dogma which the Party expects the citizens of Oceania to accept as true. Writing in his secret diary in the year 1984, the protagonist Winston Smith ponders if the Inner Party might declare "two plus two equals five" as fact, as well as whether or not belief in such a consensus reality substantiates the lie. About the falsity of "two plus two equals five", in the Ministry of Love, the interrogator O'Brien tells the thought criminal Smith that control over physical reality is unimportant to the Party, provided the citizens of Oceania subordinate their real-world perceptions to the political will of the Party; and that, by way of doublethink: "Sometimes, Winston. [Sometimes it is four fingers.] Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once".

George Orwell used the idea of 2 + 2 = 5 in an essay of January 1939 in The Adelphi; "Review of Power: A New Social Analysis by Bertrand Russell":

It is quite possible that we are descending into an age in which two plus two will make five when the Leader says so.

In propaganda work for the BBC (British Broadcasting Corporation) during the Second World War (1939–1945), Orwell applied the illogic of 2 + 2 = 5 to counter the reality-denying psychology of Nazi propaganda, which he addressed in the essay "Looking Back on the Spanish War" (1943), indicating that:

Nazi theory, indeed, specifically denies that such a thing as "the truth" exists. There is, for instance, no such thing as "Science". There is only "German Science", "Jewish Science", etc. The implied objective of this line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future, but the past. If the Leader says of such and such an event, "It never happened"—well, it never happened. If he says that "two and two are five"—well, two and two are five. This prospect frightens me much more than bombs—and, after our experiences of the last few years [the Blitz, 1940–41] that is not a frivolous statement.

In addressing Nazi anti-intellectualism, Orwell's reference might have been Hermann Göring's hyperbolic praise of Adolf Hitler: "If the Führer wants it, two and two makes five!" In the political novel Nineteen Eighty-Four (1949), concerning the Party's philosophy of government for Oceania, Orwell said:

In the end, the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?

The 1951 British edition of the text, published by Secker & Warburg, erroneously omitted the "5", thus rendering it simply as "2 + 2 =". This error, likely the result of a typesetting mistake, remained in all further editions of the text until the 1987 edition, whereafter a correction was made based on Orwell's original typescript. This misprint did not exist in the American editions of the text, with British students of the text in the meanwhile misinterpreting Orwell's original intentions.

Contemporary usage

Political graffiti in Havana, Cuba questioning government policy perceived to be "2+2=5"

In The Cult of the Amateur: How Today's Internet is Killing Our Culture (2007), the media critic Andrew Keen uses the slogan "two plus two equals five" to criticise the Wikipedia policy allowing any user to edit the encyclopedia — that the enthusiasm of the amateur for user generated content, peer production, and Web 2.0 technology leads to an encyclopedia of common knowledge, and not an encyclopedia of expert knowledge; that the "wisdom of the crowd" will distort what society considers to be the truth.

In 2020, a social media debate followed biostatistics PhD student Kareem Carr's statement, "If someone says 2+2=5, the correct response is, 'What are your definitions and axioms?' not a rant about the decline of Western civilization".

Room-temperature superconductor

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Room-temperature_superconductor

A room-temperature superconductor is a hypothetical material capable of displaying superconductivity above 0 °C (273 K; 32 °F), operating temperatures which are commonly encountered in everyday settings. As of 2023, the material with the highest accepted superconducting temperature was highly pressurized lanthanum decahydride, whose transition temperature is approximately 250 K (−23 °C) at 200 GPa.

At standard atmospheric pressure, cuprates currently hold the temperature record, manifesting superconductivity at temperatures as high as 138 K (−135 °C). Over time, researchers have consistently encountered superconductivity at temperatures previously considered unexpected or impossible, challenging the notion that achieving superconductivity at room temperature was infeasible. The concept of "near-room temperature" transient effects has been a subject of discussion since the early 1950s.

Reports

Since the discovery of high-temperature superconductors ("high" being temperatures above 77 K (−196.2 °C; −321.1 °F), the boiling point of liquid nitrogen), several materials have been claimed, although not confirmed, to be room-temperature superconductors.

Corroborated studies

In 2014, an article published in Nature suggested that some materials, notably YBCO (yttrium barium copper oxide), could be made to briefly superconduct at room temperature using infrared laser pulses.

In 2015, an article published in Nature by researchers of the Otto Hahn Institute suggested that under certain conditions such as extreme pressure H
₂S transitioned to a superconductive form H
₃S at 150 GPa (around 1.5 million times atmospheric pressure) in a diamond anvil cell. The critical temperature is 203 K (−70 °C) which would be the highest T_c ever recorded and their research suggests that other hydrogen compounds could superconduct at up to 260 K (−13 °C).

Also in 2018, researchers noted a possible superconducting phase at 260 K (−13 °C) in lanthanum decahydride (LaH
₁₀) at elevated (200 GPa) pressure. In 2019, the material with the highest accepted superconducting temperature was highly pressurized lanthanum decahydride, whose transition temperature is approximately 250 K (−23 °C).

Though not room temperature, a rare earth 'infinite layer' nickelate was recently discovered that superconducted at the unheard of (for nickelates) temperature of 44K at ambient pressure. This material is stable in air unlike cuprates, and other nickelates may have even higher critical temperatures. The current theory is that these materials leverage very unusual physics including pair density waves (PDW) that may not be as sensitive to the normal pitfalls of high temperature superconductors like low critical current.

Uncorroborated studies

In 1993 and 1997, Michel Laguës and his team published evidence of room temperature superconductivity observed on Molecular Beam Epitaxy (MBE) deposited ultrathin nanostructures of bismuth strontium calcium copper oxide (BSCCO, pronounced bisko, Bi₂Sr₂Ca_n−1Cu_nO_2n+4+x). These compounds exhibit extremely low resistivities orders of magnitude below that of copper, strongly non-linear I(V) characteristics and hysteretic I(V) behavior.

In 2000, while extracting electrons from diamond during ion implantation work, South African physicist Johan Prins claimed to have observed a phenomenon that he explained as room-temperature superconductivity within a phase formed on the surface of oxygen-doped type IIa diamonds in a 10⁻⁶ mbar vacuum.

In 2003, a group of researchers published results on high-temperature superconductivity in palladium hydride (PdH_x: x > 1) and an explanation in 2004. In 2007, the same group published results suggesting a superconducting transition temperature of 260 K, with transition temperature increasing as the density of hydrogen inside the palladium lattice increases. This has not been corroborated by other groups.

In March 2021, an announcement reported superconductivity in a layered yttrium-palladium-hydron material at 262 K and a pressure of 187 GPa. Palladium may act as a hydrogen migration catalyst in the material.

On 31 December 2023, "Global Room-Temperature Superconductivity in Graphite" was published in the journal Advanced Quantum Technologies, claiming to demonstrate superconductivity at room temperature and ambient pressure in highly oriented pyrolytic graphite with dense arrays of nearly parallel line defects.

Retracted or unreliable studies

A magnet levitating above a superconductor (at −200 °C) that is exhibiting the Meissner effect.

In 2012, an Advanced Materials article claimed superconducting behavior of graphite powder after treatment with pure water at temperatures as high as 300 K and above. So far, the authors have not been able to demonstrate the occurrence of a clear Meissner phase and the vanishing of the material's resistance.

In 2018, Dev Kumar Thapa and Anshu Pandey from the Solid State and Structural Chemistry Unit of the Indian Institute of Science, Bangalore claimed the observation of superconductivity at ambient pressure and room temperature in films and pellets of a nanostructured material that is composed of silver particles embedded in a gold matrix. Due to similar noise patterns of supposedly independent plots and the publication's lack of peer review, the results have been called into question. Although the researchers repeated their findings in a later paper in 2019, this claim is yet to be verified and confirmed.

Since 2016, a team led by Ranga P. Dias has produced a number of retracted or challenged papers in this field. In 2016 they claimed observation of solid metallic hydrogen in 2016. In October 2020, they reported room-temperature superconductivity at 288 K (at 15 °C) in a carbonaceous sulfur hydride at 267 GPa, triggered into crystallisation via green laser. This was retracted in 2022 after flaws in their statistical methods were identified and led to questioning of other data. In 2023 he reported superconductivity at 294 K and 1 GPa in nitrogen-doped lutetium hydride, in a paper widely met with skepticism about its methods and data. Later in 2023 he was found to have plagiarized parts of his dissertation from someone else's thesis, and to have fabricated data in a paper on manganese disulfide, which was retracted. The lutetium hydride paper was also retracted. The first attempts to replicate those results failed.

On July 23, 2023, a Korean team claimed that Cu-doped lead apatite, which they named LK-99, was superconducting up to 370 K, though they had not observed this fully. They posted two preprints to arXiv, published a paper in a journal, and submitted a patent application. The reported observations were received with skepticism by experts due to the lack of clear signatures of superconductivity. The story was widely discussed on social media, leading to a large number of attempted replications, none of which had more than qualified success. By mid-August, a series of papers from major labs provided significant evidence that LK-99 was not a superconductor, finding resistivity much higher than copper, and explaining observed effects such as magnetic response and resistance drops in terms of impurities and ferromagnetism in the material.

Theories

Metallic hydrogen and phonon-mediated pairing

Theoretical work by British physicist Neil Ashcroft predicted that solid metallic hydrogen at extremely high pressure (~500 GPa) should become superconducting at approximately room temperature, due to its extremely high speed of sound and expected strong coupling between the conduction electrons and the lattice-vibration phonons.

A team at Harvard University has claimed to make metallic hydrogen and reports a pressure of 495 GPa. Though the exact critical temperature has not yet been determined, weak signs of a possible Meissner effect and changes in magnetic susceptibility at 250 K may have appeared in early magnetometer tests on an original now-lost sample. A French team is working with doughnut shapes rather than planar at the diamond culette tips.

Organic polymers and exciton-mediated pairing

In 1964, William A. Little proposed the possibility of high-temperature superconductivity in organic polymers.

Other hydrides

In 2004, Ashcroft returned to his idea and suggested that hydrogen-rich compounds can become metallic and superconducting at lower pressures than hydrogen. More specifically, he proposed a novel way to pre-compress hydrogen chemically by examining IVa hydrides.

In 2014–2015, conventional superconductivity was observed in a sulfur hydride system (H
₂S or H
₃S) at 190 K to 203 K at pressures of up to 200 GPa.

In 2016, research suggested a link between palladium hydride containing small impurities of sulfur nanoparticles as a plausible explanation for the anomalous transient resistance drops seen during some experiments, and hydrogen absorption by cuprates was suggested in light of the 2015 results in H
₂S as a plausible explanation for transient resistance drops or "USO" noticed in the 1990s by Chu et al. during research after the discovery of YBCO.

It has been predicted that ScH
₁₂ (scandium dodecahydride) would exhibit superconductivity at room temperature – T_c between 333 K (60 °C) and 398 K (125 °C) – under a pressure expected not to exceed 100 GPa.

Some research efforts are currently moving towards ternary superhydrides, where it has been predicted that Li
₂MgH
₁₆ (dilithium magnesium hexadecahydride) would have a T_c of 473 K (200 °C) at 250 GPa.

Spin coupling

It is also possible that if the bipolaron explanation is correct, a normally semiconducting material can transition under some conditions into a superconductor if a critical level of alternating spin coupling in a single plane within the lattice is exceeded; this may have been documented in very early experiments from 1986. The best analogy here would be anisotropic magnetoresistance, but in this case the outcome is a drop to zero rather than a decrease within a very narrow temperature range for the compounds tested similar to "re-entrant superconductivity".

In 2018, support was found for electrons having anomalous 3/2 spin states in YPtBi. Though YPtBi is a relatively low temperature superconductor, this does suggest another approach to creating superconductors.

"Quantum bipolarons" could describe how a material might superconduct at up to nearly room temperature.

Metallic hydrogen

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Metallic_hydrogen

Metallic hydrogen is a phase of hydrogen in which it behaves like an electrical conductor. This phase was predicted in 1935 on theoretical grounds by Eugene Wigner and Hillard Bell Huntington.

At high pressure and temperatures, metallic hydrogen can exist as a partial liquid rather than a solid, and researchers think it might be present in large quantities in the hot and gravitationally compressed interiors of Jupiter and Saturn, as well as in some exoplanets.

Theoretical predictions

A diagram of Jupiter showing a model of the planet's interior, with a rocky core overlaid by a deep layer of liquid metallic hydrogen (shown as magenta) and an outer layer predominantly of molecular hydrogen. Jupiter's true interior composition is uncertain. For instance, the core may have shrunk as convection currents of hot liquid metallic hydrogen mixed with the molten core and carried its contents to higher levels in the planetary interior. Furthermore, there is no clear physical boundary between the hydrogen layers—with increasing depth the gas increases smoothly in temperature and density, ultimately becoming liquid. Features are shown to scale except for the aurorae and the orbits of the Galilean moons.

Hydrogen under pressure

Though generally placed atop the alkali metal column in the periodic table, hydrogen does not, under ordinary conditions, exhibit the properties of an alkali metal. Instead, it forms diatomic H₂ molecules, similar to halogens and some nonmetals in the second period of the periodic table, such as nitrogen and oxygen. Diatomic hydrogen is a gas that, at atmospheric pressure, liquefies and solidifies only at very low temperature (20 K and 14 K respectively).

In 1935, physicists Eugene Wigner and Hillard Bell Huntington predicted that under an immense pressure of around 25 GPa (250,000 atm; 3,600,000 psi), hydrogen would display metallic properties: instead of discrete H₂ molecules (which consist of two electrons bound between two protons), a bulk phase would form with a solid lattice of protons and the electrons delocalized throughout. Since then, producing metallic hydrogen in the laboratory has been described as "the holy grail of high-pressure physics".

The initial prediction about the amount of pressure needed was eventually shown to be too low. Since the first work by Wigner and Huntington, the more modern theoretical calculations point toward higher but potentially achievable metallization pressures of around 400 GPa (3,900,000 atm; 58,000,000 psi).

Liquid metallic hydrogen

Helium-4 is a liquid at normal pressure near absolute zero, a consequence of its high zero-point energy (ZPE). The ZPE of protons in a dense state is also high, and a decline in the ordering energy (relative to the ZPE) is expected at high pressures. Arguments have been advanced by Neil Ashcroft and others that there is a melting point maximum in compressed hydrogen, but also that there might be a range of densities, at pressures around 400 GPa, where hydrogen would be a liquid metal, even at low temperatures.

Geng predicted that the ZPE of protons indeed lowers the melting temperature of hydrogen to a minimum of 200 to 250 K (−73 to −23 °C) at pressures of 500–1,500 GPa (4,900,000–14,800,000 atm; 73,000,000–218,000,000 psi).

Within this flat region there might be an elemental mesophase intermediate between the liquid and solid state, which could be metastably stabilized down to low temperature and enter a supersolid state.

Superconductivity

Main article: Superconductivity

Further information: Room-temperature superconductor

In 1968, Neil Ashcroft suggested that metallic hydrogen might be a superconductor, up to room temperature (290 K or 17 °C). This hypothesis is based on an expected strong coupling between conduction electrons and lattice vibrations.

As a rocket propellant

Metastable metallic hydrogen may have potential as a highly efficient rocket propellant; the metallic form would be stored, and the energy of its decompression and conversion to the diatomic gaseous form when released through a nozzle used to generate thrust, with a theoretical specific impulse of up to 1700 seconds (for reference, the current most efficient chemical rocket propellants have an I_sp less than 500 s), although a metastable form suitable for mass-production and conventional high-volume storage may not exist. Another significant issue is the heat of the reaction, which at over 6000 K is too high for any known engine materials to be used. This would necessitate diluting the metallic hydrogen with water or liquid hydrogen, a mixture that would still provide a significant performance boost over current propellants.

Possibility of novel types of quantum fluid

Presently known "super" states of matter are superconductors, superfluid liquids and gases, and supersolids. Egor Babaev predicted that if hydrogen and deuterium have liquid metallic states, they might have quantum ordered states that cannot be classified as superconducting or superfluid in the usual sense. Instead, they might represent two possible novel types of quantum fluids: superconducting superfluids and metallic superfluids. Such fluids were predicted to have highly unusual reactions to external magnetic fields and rotations, which might provide a means for experimental verification of Babaev's predictions. It has also been suggested that, under the influence of a magnetic field, hydrogen might exhibit phase transitions from superconductivity to superfluidity and vice versa.

Lithium alloying reduces requisite pressure

In 2009, Zurek et al. predicted that the alloy LiH₆ would be a stable metal at only one quarter of the pressure required to metallize hydrogen, and that similar effects should hold for alloys of type LiH_n and possibly "other alkali high-hydride systems", i.e. alloys of type XH_n, where X is an alkali metal. This was later verified in AcH₈ and LaH₁₀ with T_c approaching 270 K leading to speculation that other compounds may even be stable at mere MPa pressures with room-temperature superconductivity.

Experimental pursuit

Shock-wave compression, 1996

In March 1996, a group of scientists at Lawrence Livermore National Laboratory reported that they had serendipitously produced the first identifiably metallic hydrogen for about a microsecond at temperatures of thousands of kelvins, pressures of over 100 GPa (1,000,000 atm; 15,000,000 psi), and densities of approximately 0.6 g/cm³. The team did not expect to produce metallic hydrogen, as it was not using solid hydrogen, thought to be necessary, and was working at temperatures above those specified by metallization theory. Previous studies in which solid hydrogen was compressed inside diamond anvils to pressures of up to 250 GPa (2,500,000 atm; 37,000,000 psi), did not confirm detectable metallization. The team had sought simply to measure the less extreme electrical conductivity changes they expected. The researchers used a 1960s-era light-gas gun, originally employed in guided missile studies, to shoot an impactor plate into a sealed container containing a half-millimeter thick sample of liquid hydrogen. The liquid hydrogen was in contact with wires leading to a device measuring electrical resistance. The scientists found that, as pressure rose to 140 GPa (1,400,000 atm; 21,000,000 psi), the electronic energy band gap, a measure of electrical resistance, fell to almost zero. The band gap of hydrogen in its uncompressed state is about 15 eV, making it an insulator but, as the pressure increases significantly, the band gap gradually fell to 0.3 eV. Because the thermal energy of the fluid (the temperature became about 3,000 K or 2,730 °C due to compression of the sample) was above 0.3 eV, the hydrogen might be considered metallic.

Other experimental research, 1996–2004

Many experiments are continuing in the production of metallic hydrogen in laboratory conditions at static compression and low temperature. Arthur Ruoff and Chandrabhas Narayana from Cornell University in 1998, and later Paul Loubeyre and René LeToullec from Commissariat à l'Énergie Atomique, France in 2002, have shown that at pressures close to those at the center of the Earth (320–340 GPa or 3,200,000–3,400,000 atm) and temperatures of 100–300 K (−173–27 °C), hydrogen is still not a true alkali metal, because of the non-zero band gap. The quest to see metallic hydrogen in laboratory at low temperature and static compression continues. Studies are also ongoing on deuterium. Shahriar Badiei and Leif Holmlid from the University of Gothenburg have shown in 2004 that condensed metallic states made of excited hydrogen atoms (Rydberg matter) are effective promoters to metallic hydrogen, however these results are disputed.

Pulsed laser heating experiment, 2008

The theoretically predicted maximum of the melting curve (the prerequisite for the liquid metallic hydrogen) was discovered by Shanti Deemyad and Isaac F. Silvera by using pulsed laser heating. Hydrogen-rich molecular silane (SiH₄) was claimed to be metallized and become superconducting by M.I. Eremets et al.. This claim is disputed, and their results have not been repeated.

Observation of liquid metallic hydrogen, 2011

In 2011 Eremets and Troyan reported observing the liquid metallic state of hydrogen and deuterium at static pressures of 260–300 GPa (2,600,000–3,000,000 atm). This claim was questioned by other researchers in 2012.

Z machine, 2015

In 2015, scientists at the Z Pulsed Power Facility announced the creation of metallic deuterium using dense liquid deuterium, an electrical insulator-to-conductor transition associated with an increase in optical reflectivity.

Claimed observation of solid metallic hydrogen, 2016

On 5 October 2016, Ranga Dias and Isaac F. Silvera of Harvard University released claims in a pre-print manuscript of experimental evidence that solid metallic hydrogen had been synthesized in the laboratory at a pressure of around 495 gigapascals (4,890,000 atm; 71,800,000 psi) using a diamond anvil cell. A revised version was published in Science in 2017.

In the preprint version of the paper, Dias and Silvera write:

With increasing pressure we observe changes in the sample, going from transparent, to black, to a reflective metal, the latter studied at a pressure of 495 GPa... the reflectance using a Drude free electron model to determine the plasma frequency of 30.1 eV at T = 5.5 K, with a corresponding electron carrier density of 6.7×10²³ particles/cm³, consistent with theoretical estimates. The properties are those of a metal. Solid metallic hydrogen has been produced in the laboratory.

— Dias & Silvera (2016)

In June 2019 a team at the Commissariat à l'énergie atomique et aux énergies alternatives (French Alternative Energies & Atomic Energy Commission) claimed to have created metallic hydrogen at around 425GPa.

W. Ferreira et al. (including Dias and Silvera) repeated their experiments multiple times after the Science article was published, finally publishing in 2023 and finding metallisation of hydrogen between 477 and 491 gigapascals (4,710,000 and 4,850,000 atm). This time, the pressure was released to assess the question of metastability. Metallic hydrogen was not found to be metastable to zero pressure.

Experiments on fluid deuterium at the National Ignition Facility, 2018

In August 2018, scientists announced new observations regarding the rapid transformation of fluid deuterium from an insulating to a metallic form below 2000 K. Remarkable agreement is found between the experimental data and the predictions based on quantum Monte Carlo simulations, which is expected to be the most accurate method to date. This may help researchers better understand giant gas planets, such as Jupiter, Saturn and related exoplanets, since such planets are thought to contain a lot of liquid metallic hydrogen, which may be responsible for their observed powerful magnetic fields.

Degenerate matter

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Degenerate_matter

Degenerate matter occurs when the Pauli exclusion principle significantly alters a state of matter at low temperature. The term is used in astrophysics to refer to dense stellar objects such as white dwarfs and neutron stars, where thermal pressure alone is not enough to prevent gravitational collapse. The term also applies to metals in the Fermi gas approximation.

Degenerate matter is usually modelled as an ideal Fermi gas, an ensemble of non-interacting fermions. In a quantum mechanical description, particles limited to a finite volume may take only a discrete set of energies, called quantum states. The Pauli exclusion principle prevents identical fermions from occupying the same quantum state. At lowest total energy (when the thermal energy of the particles is negligible), all the lowest energy quantum states are filled. This state is referred to as full degeneracy. This degeneracy pressure remains non-zero even at absolute zero temperature. Adding particles or reducing the volume forces the particles into higher-energy quantum states. In this situation, a compression force is required, and is made manifest as a resisting pressure. The key feature is that this degeneracy pressure does not depend on the temperature but only on the density of the fermions. Degeneracy pressure keeps dense stars in equilibrium, independent of the thermal structure of the star.

A degenerate mass whose fermions have velocities close to the speed of light (particle kinetic energy larger than its rest mass energy) is called relativistic degenerate matter. The concept of degenerate stars, stellar objects composed of degenerate matter, was originally developed in a joint effort between Arthur Eddington, Ralph Fowler and Arthur Milne.

Concept

Main article: Fermi-Dirac statistics

Quantum mechanics uses the word 'degenerate' in two ways: degenerate energy levels and as the low temperature ground state limit for states of matter. The electron degeneracy pressure occurs in the ground state systems which are non-degenerate in energy levels. The term "degeneracy" derives from work on the specific heat of gases that pre-dates the use of the term in quantum mechanics.

Degenerate matter exhibits quantum mechanical properties when a fermion system temperature approaches absolute zero. These properties result from a combination of the Pauli exclusion principle and quantum confinement. The Pauli principle allows only one fermion in each quantum state and the confinement ensures that energy of these states increases as they are filled. The lowest states fill up and fermions are forced to occupy high energy states even at low temperature.

While the Pauli principle and Fermi-Dirac distribution applies to all matter, the interesting cases for degenerate matter involve systems of many fermions. These cases can be understood with the help of the Fermi gas model. Examples include electrons in metals and in white dwarf stars and neutrons in neutron stars. The electrons are confined by Coulomb attraction to positive ion cores; the neutrons are confined by gravitation attraction. The fermions, forced in to higher levels by the Pauli principle, exert pressure preventing further compression.

The allocation or distribution of fermions into quantum states ranked by energy is called the Fermi-Dirac distribution. Degenerate matter exhibits the results of Fermi-Dirac distribution.

Degeneracy pressure

Unlike a classical ideal gas, whose pressure is proportional to its temperature $${\displaystyle P=k_{\mathrm{B}}\frac{NT}{V},}$$ where P is pressure, k_B is the Boltzmann constant, N is the number of particles (typically atoms or molecules), T is temperature, and V is the volume, the pressure exerted by degenerate matter depends only weakly on its temperature. In particular, the pressure remains nonzero even at absolute zero temperature. At relatively low densities, the pressure of a fully degenerate gas can be derived by treating the system as an ideal Fermi gas, in this way $${\displaystyle P=\frac{(3{\pi }^2{)}^{2/3}{\hslash }^2}{5m}{\left(\frac{N}{V}\right)}^{5/3},}$$ where m is the mass of the individual particles making up the gas. At very high densities, where most of the particles are forced into quantum states with relativistic energies, the pressure is given by $${\displaystyle P=K{\left(\frac{N}{V}\right)}^{4/3},}$$ where K is another proportionality constant depending on the properties of the particles making up the gas.

Pressure vs temperature curves of a classical ideal gas and quantum ideal gases (Fermi gas, Bose gas), for a given particle density.

All matter experiences both normal thermal pressure and degeneracy pressure, but in commonly encountered gases, thermal pressure dominates so much that degeneracy pressure can be ignored. Likewise, degenerate matter still has normal thermal pressure; the degeneracy pressure dominates to the point that temperature has a negligible effect on the total pressure. The adjacent figure shows the thermal pressure (red line) and total pressure (blue line) in a Fermi gas, with the difference between the two being the degeneracy pressure. As the temperature falls, the density and the degeneracy pressure increase, until the degeneracy pressure contributes most of the total pressure.

While degeneracy pressure usually dominates at extremely high densities, it is the ratio between degenerate pressure and thermal pressure which determines degeneracy. Given a sufficiently drastic increase in temperature (such as during a red giant star's helium flash), matter can become non-degenerate without reducing its density.

Degeneracy pressure contributes to the pressure of conventional solids, but these are not usually considered to be degenerate matter because a significant contribution to their pressure is provided by electrical repulsion of atomic nuclei and the screening of nuclei from each other by electrons. The free electron model of metals derives their physical properties by considering the conduction electrons alone as a degenerate gas, while the majority of the electrons are regarded as occupying bound quantum states. This solid state contrasts with degenerate matter that forms the body of a white dwarf, where most of the electrons would be treated as occupying free particle momentum states.

Exotic examples of degenerate matter include neutron degenerate matter, strange matter, metallic hydrogen and white dwarf matter.

Degenerate gases

Degenerate gases are gases composed of fermions such as electrons, protons, and neutrons rather than molecules of ordinary matter. The electron gas in ordinary metals and in the interior of white dwarfs are two examples. Following the Pauli exclusion principle, there can be only one fermion occupying each quantum state. In a degenerate gas, all quantum states are filled up to the Fermi energy. Most stars are supported against their own gravitation by normal thermal gas pressure, while in white dwarf stars the supporting force comes from the degeneracy pressure of the electron gas in their interior. In neutron stars, the degenerate particles are neutrons.

A fermion gas in which all quantum states below a given energy level are filled is called a fully degenerate fermion gas. The difference between this energy level and the lowest energy level is known as the Fermi energy.

Electron degeneracy

Main articles: Electron degeneracy pressure and White dwarf

In an ordinary fermion gas in which thermal effects dominate, most of the available electron energy levels are unfilled and the electrons are free to move to these states. As particle density is increased, electrons progressively fill the lower energy states and additional electrons are forced to occupy states of higher energy even at low temperatures. Degenerate gases strongly resist further compression because the electrons cannot move to already filled lower energy levels due to the Pauli exclusion principle. Since electrons cannot give up energy by moving to lower energy states, no thermal energy can be extracted. The momentum of the fermions in the fermion gas nevertheless generates pressure, termed "degeneracy pressure".

Under high densities, matter becomes a degenerate gas when all electrons are stripped from their parent atoms. The core of a star, once hydrogen burning nuclear fusion reactions stops, becomes a collection of positively charged ions, largely helium and carbon nuclei, floating in a sea of electrons, which have been stripped from the nuclei. Degenerate gas is an almost perfect conductor of heat and does not obey ordinary gas laws. White dwarfs are luminous not because they are generating energy but rather because they have trapped a large amount of heat which is gradually radiated away. Normal gas exerts higher pressure when it is heated and expands, but the pressure in a degenerate gas does not depend on the temperature. When gas becomes super-compressed, particles position right up against each other to produce degenerate gas that behaves more like a solid. In degenerate gases the kinetic energies of electrons are quite high and the rate of collision between electrons and other particles is quite low, therefore degenerate electrons can travel great distances at velocities that approach the speed of light. Instead of temperature, the pressure in a degenerate gas depends only on the speed of the degenerate particles; however, adding heat does not increase the speed of most of the electrons, because they are stuck in fully occupied quantum states. Pressure is increased only by the mass of the particles, which increases the gravitational force pulling the particles closer together. Therefore, the phenomenon is the opposite of that normally found in matter where if the mass of the matter is increased, the object becomes bigger. In degenerate gas, when the mass is increased, the particles become spaced closer together due to gravity (and the pressure is increased), so the object becomes smaller. Degenerate gas can be compressed to very high densities, typical values being in the range of 10,000 kilograms per cubic centimeter.

There is an upper limit to the mass of an electron-degenerate object, the Chandrasekhar limit, beyond which electron degeneracy pressure cannot support the object against collapse. The limit is approximately 1.44 solar masses for objects with typical compositions expected for white dwarf stars (carbon and oxygen with two baryons per electron). This mass cut-off is appropriate only for a star supported by ideal electron degeneracy pressure under Newtonian gravity; in general relativity and with realistic Coulomb corrections, the corresponding mass limit is around 1.38 solar masses. The limit may also change with the chemical composition of the object, as it affects the ratio of mass to number of electrons present. The object's rotation, which counteracts the gravitational force, also changes the limit for any particular object. Celestial objects below this limit are white dwarf stars, formed by the gradual shrinking of the cores of stars that run out of fuel. During this shrinking, an electron-degenerate gas forms in the core, providing sufficient degeneracy pressure as it is compressed to resist further collapse. Above this mass limit, a neutron star (primarily supported by neutron degeneracy pressure) or a black hole may be formed instead.

Neutron degeneracy

Main article: Neutron star

Neutron degeneracy is analogous to electron degeneracy and exists in neutron stars, which are partially supported by the pressure from a degenerate neutron gas. Neutron stars are formed either directly from the supernova of stars with masses between 10 and 25 M_☉ (solar masses), or by white dwarfs acquiring a mass in excess of the Chandrasekhar limit of 1.44 M_☉, usually either as a result of a merger or by feeding off of a close binary partner. Above the Chandrasekhar limit, the gravitational pressure at the core exceeds the electron degeneracy pressure, and electrons begin to combine with protons to produce neutrons (via inverse beta decay, also termed electron capture). The result is an extremely compact star composed of "nuclear matter", which is predominantly a degenerate neutron gas with a small admixture of degenerate proton and electron gases.

Neutrons in a degenerate neutron gas are spaced much more closely than electrons in an electron-degenerate gas because the more massive neutron has a much shorter wavelength at a given energy. This phenomenon is compounded by the fact that the pressures within neutron stars are much higher than those in white dwarfs. The pressure increase is caused by the fact that the compactness of a neutron star causes gravitational forces to be much higher than in a less compact body with similar mass. The result is a star with a diameter on the order of a thousandth that of a white dwarf.

The properties of neutron matter set an upper limit to the mass of a neutron star, the Tolman–Oppenheimer–Volkoff limit, which is analogous to the Chandrasekhar limit for white dwarf stars.

Proton degeneracy

Sufficiently dense matter containing protons experiences proton degeneracy pressure, in a manner similar to the electron degeneracy pressure in electron-degenerate matter: protons confined to a sufficiently small volume have a large uncertainty in their momentum due to the Heisenberg uncertainty principle. However, because protons are much more massive than electrons, the same momentum represents a much smaller velocity for protons than for electrons. As a result, in matter with approximately equal numbers of protons and electrons, proton degeneracy pressure is much smaller than electron degeneracy pressure, and proton degeneracy is usually modelled as a correction to the equations of state of electron-degenerate matter.

Quark degeneracy

Main articles: Quark star and Strange star

At densities greater than those supported by neutron degeneracy, quark-degenerate matter may occur in the cores of neutron stars, depending on the equations of state of neutron-degenerate matter. There is no observational evidence to support this conjecture and theoretical models that predict de-confined quark matter are only valid at masses higher than any observed neutron star.

History

In 1914 Walther Nernst described the reduction of the specific heat of gases at very low temperature as "degeneration"; he attributed this to quantum effects. In subsequent work in various papers on quantum thermodynamics by Albert Einstein, by Max Planck, and by Erwin Schrödinger, the effect at low temperatures came to be called "gas degeneracy". A fully degenerate gas has no volume dependence on pressure when temperature approaches absolute zero.

Early in 1927 Enrico Fermi and separately Llewellyn Thomas developed a semi-classical model for electrons in a metal. The model treated the electrons as a gas. Later in 1927, Arnold Sommerfeld applied the Pauli principle via Fermi-Dirac statistics to this electron gas model, computing the specific heat of metals; the result became Fermi gas model for metals. Sommerfeld called the low temperature region with quantum effects a "wholly degenerate gas".

The concept of degenerate stars, stellar objects composed of degenerate matter, was originally developed in a joint effort between Arthur Eddington, Ralph Fowler and Arthur Milne. Eddington had suggested that the atoms in Sirius B were almost completely ionised and closely packed. Fowler described white dwarfs as composed of a gas of particles that became degenerate at low temperature; he also pointed out that ordinary atoms are broadly similar in regards to the filling of energy levels by fermions. In 1926, Milne proposed that degenerate matter is found in the cores of ordinary stars, not only in compact stars. In 1927 Ralph H. Fowler applied Fermi's model to the puzzle of the stability of white dwarf stars. This approach was extended to relativistic models by later studies and with the work of Subrahmanyan Chandrasekhar became the accepted model for star stability.