Enrico Fermi

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Enrico Fermi
Enrico Fermi (1901–1954)
Born	29 September 1901 Rome, Italy
Died	28 November 1954 (aged 53) Chicago, Illinois, United States
Citizenship	Italy (1901–54) United States (1944–54)
Fields	Physics
Institutions	Scuola Normale Superiore University of Göttingen Leiden University University of Florence Sapienza University of Rome Columbia University University of Chicago
Alma mater	Scuola Normale Superiore
Doctoral advisor	Luigi Puccianti[1]
Doctoral students	Harold Agnew[2] Edoardo Amaldi Owen Chamberlain Geoffrey Chew Jerome Friedman Richard Garwin Marvin Goldberger Tsung-Dao Lee Ettore Majorana Arthur Rosenfeld Emilio Segrè Sam Treiman[1]
Other notable students	Jack Steinberger Chen Ning Yang
Known for	Nuclear chain reaction Fermi–Dirac statistics Theory of beta decay
Notable awards	Matteucci Medal (1926) Nobel Prize (1938) Hughes Medal (1942) Medal for Merit (1946) Franklin Medal (1947) ForMemRS (1950)[3] Barnard Medal for Meritorious Service to Science (1950) Rumford Prize (1953) Max Planck Medal (1954)
Spouse	Laura Fermi
Signature

Enrico Fermi (Italian: [enˈri.ko ˈfeɾ.mi]; 29 September 1901 – 28 November 1954) was an Italian physicist, best known for his work on Chicago Pile-1 (the first nuclear reactor), and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics. He is one of the men referred to as the "father of the atomic bomb".^[4] Fermi held several patents related to the use of nuclear power, and was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and the discovery of transuranic elements. He was widely regarded as one of the very few physicists to excel both theoretically and experimentally.

Fermi's first major contribution was to statistical mechanics. After Wolfgang Pauli announced his exclusion principle in 1925, Fermi followed with a paper in which he applied the principle to an ideal gas, employing a statistical formulation now known as Fermi–Dirac statistics. Today, particles that obey the exclusion principle are called "fermions". Later Pauli postulated the existence of an uncharged invisible particle emitted along with an electron during beta decay, to satisfy the law of conservation of energy. Fermi took up this idea, developing a model that incorporated the postulated particle, which he named the "neutrino". His theory, later referred to as Fermi's interaction and still later as weak interaction, described one of the four fundamental forces of nature. Through experiments inducing radioactivity with recently discovered neutrons, Fermi discovered that slow neutrons were more easily captured than fast ones, and developed the Fermi age equation to describe this. After bombarding thorium and uranium with slow neutrons, he concluded that he had created new elements; although he was awarded the Nobel Prize for this discovery, the new elements were subsequently revealed to be fission products.

Fermi left Italy in 1938 to escape new Italian Racial Laws that affected his Jewish wife Laura. He emigrated to the United States where he worked on the Manhattan Project during World War II. Fermi led the team that designed and built Chicago Pile-1, which went critical on 2 December 1942, demonstrating the first artificial self-sustaining nuclear chain reaction. He was on hand when the X-10 Graphite Reactor at Oak Ridge, Tennessee went critical in 1943, and when the B Reactor at the Hanford Site did so the next year. At Los Alamos he headed F Division, part of which worked on Edward Teller's thermonuclear "Super" bomb. He was present at the Trinity test on 16 July 1945, where he used his Fermi method to estimate the bomb's yield.

After the war, Fermi served under Oppenheimer on the influential General Advisory Committee, which advised the Atomic Energy Commission on nuclear matters and policy. Following the detonation of the first Soviet fission bomb in August 1949, he strongly opposed the development of a hydrogen bomb on both moral and technical grounds. He was among the scientists who testified on Oppenheimer's behalf at the 1954 hearing that resulted in the denial of the latter's security clearance. Fermi did important work in particle physics, especially related to pions and muons, and he speculated that cosmic rays arose through material being accelerated by magnetic fields in interstellar space. Many awards, concepts, and institutions are named after Fermi, including the Enrico Fermi Award, the Enrico Fermi Institute, the Fermi National Accelerator Laboratory, the Fermi Gamma-ray Space Telescope, the Enrico Fermi Nuclear Generating Station, and the synthetic element fermium.

Early life

Enrico Fermi was born in Rome on 29 September 1901. He was the third child of Alberto Fermi, a division head (Capo Divisione) in the Ministry of Railways, and Ida de Gattis, an elementary school teacher.^[5][6] His only sister, Maria, was two years older than he, and his brother, Giulio, was a year older. After the two boys were sent to a rural community to be wet nursed, Enrico rejoined his family in Rome when he was two and a half.^[7] While he came from a Roman Catholic family and was baptized in accord with his grandparents' wishes, the family was not religious, and Fermi was an agnostic throughout his adult life. As a young boy he shared his interests with his brother Giulio. They built electric motors and played with electrical and mechanical toys.^[8] Giulio died during the administration of anesthesia for an operation on a throat abscess in 1915.^[9]

One of Fermi's first sources for the study of physics was a book found at the local market of Campo de' Fiori in Rome. The 900-page book from 1840, Elementorum physicae mathematicae, was written in Latin by Jesuit Father Andrea Caraffa, a professor at the Collegio Romano. It covered mathematics, classical mechanics, astronomy, optics, and acoustics, to the extent that they were understood when it was written.^[10][11] Fermi befriended another scientifically inclined student, Enrico Persico,^[12] and the two worked together on scientific projects such as building gyroscopes and measuring the Earth's magnetic field. Fermi's interest in physics was further encouraged by his father's colleague Adolfo Amidei, who gave him several books on physics and mathematics that he read and assimilated quickly.^[13]

Scuola Normale Superiore in Pisa

Enrico Fermi as student in Pisa

Fermi graduated from high school in July 1918 and, at Amidei's urging, applied to the Scuola Normale Superiore in Pisa. Having lost one son, his parents were reluctant to let him move away from home for four years while attending the Sapienza University of Rome, but in the end they acquiesced. The school provided free lodging for students, but candidates had to take a difficult entrance exam that included an essay. The given theme was "Specific characteristics of Sounds". The 17-year-old Fermi chose to derive and solve the partial differential equation for a vibrating rod, applying Fourier analysis in the solution. The examiner, Professor Giuseppe Pittarelli from the Sapienza University of Rome, interviewed Fermi and concluded that his entry would have been commendable even for a doctoral degree. Fermi achieved first place in the classification of the entrance exam.^[14]

During his years at the Scuola Normale Superiore, Fermi teamed up with a fellow student named Franco Rasetti with whom he would indulge in light-hearted pranks and who would later become Fermi's close friend and collaborator. In Pisa, Fermi was advised by the director of the physics laboratory, Luigi Puccianti, who acknowledged that there was little that he could teach Fermi, and frequently asked Fermi to teach him something instead. Fermi's knowledge of quantum physics reached such a high level that Puccianti asked him to organize seminars on the topic.^[15] During this time Fermi learned tensor calculus, a mathematical technique invented by Gregorio Ricci and Tullio Levi-Civita that was needed to demonstrate the principles of general relativity.^[16] Fermi initially chose mathematics as his major, but soon switched to physics. He remained largely self-taught, studying general relativity, quantum mechanics, and atomic physics.^[17]

A light cone is a three-dimensional surface of all possible light rays arriving at and departing from a point in spacetime. Here, it is depicted with one spatial dimension suppressed. The time line is the vertical axis.

In September 1920, Fermi was admitted to the Physics department. Since there were only three students in the department—Fermi, Rasetti, and Nello Carrara—Puccianti let them freely use the laboratory for whatever purposes they chose. Fermi decided that they should research X-ray crystallography, and the three worked to produce a Laue photograph—an X-ray photograph of a crystal.^[18] During 1921, his third year at the university, Fermi published his first scientific works in the Italian journal Nuovo Cimento. The first was entitled "On the dynamics of a rigid system of electrical charges in translational motion" (Italian: Sulla dinamica di un sistema rigido di cariche elettriche in moto traslatorio). A sign of things to come was that the mass was expressed as a tensor—a mathematical construct commonly used to describe something moving and changing in three-dimensional space. In classical mechanics, mass is a scalar quantity, but in relativity it changes with velocity. The second paper was "On the electrostatics of a uniform gravitational field of electromagnetic charges and on the weight of electromagnetic charges" (Italian: Sull'elettrostatica di un campo gravitazionale uniforme e sul peso delle masse elettromagnetiche). Using general relativity, Fermi showed that a charge has a weight equal to U/c², where U was the electrostatic energy of the system, and c is the speed of light.^[17]

The first paper seemed to point out a contradiction between the electrodynamic theory and the relativistic one concerning the calculation of the electromagnetic masses, as the former predicted a value of 4/3 U/c². Fermi addressed this the next year in a paper "Concerning a contradiction between electrodynamic and the relativistic theory of electromagnetic mass" in which he showed that the apparent contradiction was a consequence of relativity. This paper was sufficiently well-regarded that it was translated into German and published in the German scientific journal Physikalische Zeitschrift in 1922.^[19] That year, Fermi submitted his article "On the phenomena occurring near a world line" (Italian: Sopra i fenomeni che avvengono in vicinanza di una linea oraria) to the Italian journal I Rendiconti dell'Accademia dei Lincei. In this article he examined the Principle of Equivalence, and introduced the so-called "Fermi coordinates". He proved that on a world line close to the time line, space behaves as if it were a Euclidean space.^[20][21]

Fermi submitted his thesis, "A theorem on probability and some of its applications" (Italian: Un teorema di calcolo delle probabilità ed alcune sue applicazioni), to the Scuola Normale Superiore in July 1922, and received his laurea at the unusually young age of 21. The thesis was on X-ray diffraction images. Theoretical physics was not yet considered a discipline in Italy, and the only thesis that would have been accepted was one on experimental physics. For this reason, Italian physicists were slow in embracing the new ideas like relativity coming from Germany. Since Fermi was quite at home in the lab doing experimental work, this did not pose insurmountable problems for him.^[21]

Fermi–Dirac statistics. Fermi function F() vs. energy , with μ = 0.55 eV and for various temperatures in the range 50 to 375 K (−223.2 to 101.9 °C).

While writing the appendix for the Italian edition of the book The Mathematical Theory of Relativity by August Kopff in 1923, Fermi was the first to point out that hidden inside the famous Einstein equation (E = mc²) was an enormous amount of nuclear potential energy to be exploited. "It does not seem possible, at least in the near future", he wrote, "to find a way to release these dreadful amounts of energy—which is all to the good because the first effect of an explosion of such a dreadful amount of energy would be to smash into smithereens the physicist who had the misfortune to find a way to do it."^[21]

Fermi decided to travel abroad, and spent a semester studying under Max Born at the University of Göttingen, where he met Werner Heisenberg and Pascual Jordan. Fermi then studied in Leiden with Paul Ehrenfest from September to December 1924 on a fellowship from the Rockefeller Foundation obtained through the intercession of the mathematician Vito Volterra. Here Fermi met Hendrik Lorentz and Albert Einstein, and became good friends with Samuel Goudsmit and Jan Tinbergen. From January 1925 to late 1926, Fermi taught mathematical physics and theoretical mechanics at the University of Florence, where he teamed up with Rasetti to conduct a series of experiments on the effects of magnetic fields on mercury vapour. He also participated in seminars at the Sapienza University of Rome, giving lectures on quantum mechanics and solid state physics.^[22] While giving lectures of new quantum mechanics based on remarkable accuracy of predictions of Schrödinger equation, the Italian physicist would often say, "It has no business to fit so well!"^[23]

After Wolfgang Pauli announced his exclusion principle in 1925, Fermi responded with a paper "On the quantisation of the perfect monoatomic gas" (Italian: Sulla quantizzazione del gas perfetto monoatomico), in which he applied the exclusion principle to an ideal gas. The paper was especially notable for Fermi's statistical formulation, which describes the distribution of particles in systems of many identical particles that obey the exclusion principle. This was independently developed soon after by the British physicist Paul Dirac, who also showed how it was related to the Bose–Einstein statistics. Accordingly, it is now known as Fermi–Dirac statistics.^[24] Following Dirac, particles that obey the exclusion principle are today called "fermions", while those that do not are called "bosons".^[25]

Professor in Rome

Professorships in Italy were granted by competition (Italian: concorso) for a vacant chair, the applicants being rated on their publications by a committee of professors. Fermi applied for a chair of mathematical physics at the University of Cagliari on Sardinia, but was narrowly passed over in favour of Giovanni Giorgi.^[26] In 1926, at the age of 24, he applied for a professorship at the Sapienza University of Rome. This was a new chair, one of the first three in theoretical physics in Italy, that had been created by the Minister of Education at the urging of Professor Orso Mario Corbino, who was the University's professor of experimental physics, the Director of the Institute of Physics, and a member of Benito Mussolini's cabinet. Corbino, who also chaired the selection committee, hoped that the new chair would raise the standard and reputation of physics in Italy.^[27] The committee chose Fermi ahead of Enrico Persico and Aldo Pontremoli,^[28] and Corbino helped Fermi recruit his team, which was soon joined by notable students such as Edoardo Amaldi, Bruno Pontecorvo, Ettore Majorana and Emilio Segrè, and by Franco Rasetti, whom Fermi had appointed as his assistant.^[29] They were soon nicknamed the "Via Panisperna boys" after the street where the Institute of Physics was located.^[30]

Fermi and his students (the Via Panisperna boys) in the courtyard of Rome University's Physics Institute in Via Panisperna, about 1934. From Left to right: Oscar D'Agostino, Emilio Segrè, Edoardo Amaldi, Franco Rasetti and Fermi

Fermi married Laura Capon, a science student at the University, on 19 July 1928.^[31] They had two children: Nella, born in January 1931, and Giulio, born in February 1936.^[32] On 18 March 1929, Fermi was appointed a member of the Royal Academy of Italy by Mussolini, and on 27 April he joined the Fascist Party. He later opposed Fascism when the 1938 racial laws were promulgated by Mussolini in order to bring Italian Fascism ideologically closer to German National Socialism. These laws threatened Laura, who was Jewish, and put many of Fermi's research assistants out of work.^{[33][34][35][36]}

During their time in Rome, Fermi and his group made important contributions to many practical and theoretical aspects of physics. In 1928, he published his Introduction to Atomic Physics (Italian: Introduzione alla fisica atomica), which provided Italian university students with an up-to-date and accessible text. Fermi also conducted public lectures and wrote popular articles for scientists and teachers in order to spread knowledge of the new physics as widely as possible.^[37] Part of his teaching method was to gather his colleagues and graduate students together at the end of the day and go over a problem, often from his own research.^[37][38] A sign of success was that foreign students now began to come to Italy. The most notable of these was the German physicist Hans Bethe,^[39] who came to Rome as a Rockefeller Foundation fellow, and collaborated with Fermi on a 1932 paper "On the Interaction between Two Electrons" (German: Über die Wechselwirkung von Zwei Elektronen).^[37]

At this time, physicists were puzzled by beta decay, in which an electron was emitted from the atomic nucleus. To satisfy the law of conservation of energy, Pauli postulated the existence of an invisible particle with no charge and little or no mass that was also emitted at the same time. Fermi took up this idea, which he developed in a tentative paper in 1933, and then a longer paper the next year that incorporated the postulated particle, which Fermi called a "neutrino".^[40][41][42] His theory, later referred to as Fermi's interaction, and still later as the theory of the weak interaction, described one of the four fundamental forces of nature. The neutrino was detected after his death, and his interaction theory showed why it was so difficult to detect. When he submitted his paper to the British journal Nature, that journal's editor turned it down because it contained speculations which were "too remote from physical reality to be of interest to readers".^[41] Thus Fermi saw the theory published in Italian and German before it was published in English.^[29]

In the introduction to the 1968 English translation, physicist Fred L. Wilson noted that:

Fermi's theory, aside from bolstering Pauli's proposal of the neutrino, has a special significance in the history of modern physics. One must remember that only the naturally occurring β emitters were known at the time the theory was proposed. Later when positron decay was discovered, the process was easily incorporated within Fermi's original framework. On the basis of his theory, the capture of an orbital electron by a nucleus was predicted and eventually observed. With time much experimental data has accumulated. Although peculiarities have been observed many times in β decay, Fermi's theory always has been equal to the challenge.

The consequences of the Fermi theory are vast. For example, β spectroscopy was established as a powerful tool for the study of nuclear structure. But perhaps the most influential aspect of this work of Fermi is that his particular form of the β interaction established a pattern which has been appropriate for the study of other types of interactions. It was the first successful theory of the creation and annihilation of material particles. Previously, only photons had been known to be created and destroyed.^[42]

In January 1934, Irène Joliot-Curie and Frédéric Joliot announced that they had bombarded elements with alpha particles and induced radioactivity in them.^[43][44] By March, Fermi's assistant Gian-Carlo Wick had provided a theoretical explanation using Fermi's theory of beta decay. Fermi decided to switch to experimental physics, using the neutron, which James Chadwick had discovered in 1932.^[45] In March 1934, Fermi wanted to see if he could induce radioactivity with Rasetti's polonium-beryllium neutron source. Neutrons had no electric charge, and so would not be deflected by the positively charged nucleus. This meant that they needed much less energy to penetrate the nucleus than charged particles, and so would not require a particle accelerator, which the Via Panisperna boys did not have.^[46][47]

Enrico Fermi between Franco Rasetti (left) and Emilio Segrè in academic dress

Fermi had the idea to resort to replacing the polonium-beryllium neutron source with a radon-beryllium one, which he created by filling a glass bulb with beryllium powder, evacuating the air, and then adding 50 mCi of radon gas, supplied by Giulio Cesare Trabacchi.^[48][49] This created a much stronger neutron source, the effectiveness of which declined with the 3.8-day half-life of radon. He knew that this source would also emit gamma rays, but, on the basis of his theory, he believed that this would not affect the results of the experiment. He started by bombarding platinum, an element with a high atomic number that was readily available, without success. He turned to aluminium, which emitted an alpha particle and produced sodium, which then decayed into magnesium by beta particle emission. He tried lead, without success, and then fluorine in the form of calcium fluoride, which emitted an alpha particle and produced nitrogen, decaying into oxygen by beta particle emission. In all, he induced radioactivity in 22 different elements.^[50] Fermi rapidly reported the discovery of neutron-induced radioactivity in the Italian journal La Ricerca Scientifica on 25 March 1934.^[49][51][52]

Beta decay. A neutron decays into a proton, and an electron is emitted. In order for the total energy in the system to remain the same, Pauli and Fermi postulated that a neutrino () was also emitted

The natural radioactivity of thorium and uranium made it hard to determine what was happening when these elements were bombarded with neutrons but, after correctly eliminating the presence of elements lighter than uranium but heavier than lead, Fermi concluded that they had created new elements, which he called hesperium and ausonium.^[53][47] The chemist Ida Noddack criticised this work, suggesting that some of the experiments could have produced lighter elements than lead rather than new, heavier elements. Her suggestion was not taken seriously at the time because her team had not carried out any experiments with uranium, and its claim to have discovered masurium (technetium) was disputed. At that time, fission was thought to be improbable if not impossible on theoretical grounds. While physicists expected elements with higher atomic numbers to form from neutron bombardment of lighter elements, nobody expected neutrons to have enough energy to split a heavier atom into two light element fragments in the manner that Noddack suggested.^[54][53]

The Via Panisperna boys also noticed some unexplained effects. The experiment seemed to work better on a wooden table than a marble table top. Fermi remembered that Joliot-Curie and Chadwick had noted that paraffin wax was effective at slowing neutrons, so he decided to try that. When neutrons were passed through paraffin wax, they induced a hundred times as much radioactivity in silver compared with when it was bombarded without the paraffin. Fermi guessed that this was due to the hydrogen atoms in the paraffin. Those in wood similarly explained the difference between the wooden and the marble table tops. This was confirmed by repeating the effect with water. He concluded that collisions with hydrogen atoms slowed the neutrons.^[55][47] The lower the atomic number of the nucleus it collides with, the more energy a neutron loses per collision, and therefore the less collisions that are required to slow a neutron down by a given amount.^[56] Fermi realised that this induced more radioactivity because slow neutrons were more easily captured than fast ones. He developed a diffusion equation to describe this, which became known as the Fermi age equation.^[55][47]

In 1938 Fermi received the Nobel Prize in Physics at the age of 37 for his "demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons".^[57] After Fermi received the prize in Stockholm, he did not return home to Italy, but rather continued on to New York City along with his family, where they applied for permanent residency. The decision to move to America and become US citizens was primarily a result of the racial laws in Italy.^[33]

Manhattan Project

Soon after Fermi's arrival in New York City on 2 January 1939,^[58] he was offered five different chairs, and chose to work at Columbia University,^[59] where he had already given summer lectures in 1936.^[60] He received the news that in December 1938, the German chemists Otto Hahn and Fritz Strassmann had detected the element barium after bombarding uranium with neutrons,^[61] which Lise Meitner and her nephew Otto Frisch correctly interpreted as the result of nuclear fission. Frisch confirmed this experimentally on 13 January 1939.^[62][63] The news of Meitner and Frisch's interpretation of Hahn and Strassmann's discovery crossed the Atlantic with Niels Bohr, who was to lecture at Princeton University. Isidor Isaac Rabi and Willis Lamb, two Columbia University physicists working at Princeton, found out about it and carried it back to Columbia. Rabi said he told Enrico Fermi, but Fermi later gave the credit to Lamb:^[64]

I remember very vividly the first month, January, 1939, that I started working at the Pupin Laboratories because things began happening very fast. In that period, Niels Bohr was on a lecture engagement at the Princeton University and I remember one afternoon Willis Lamb came back very excited and said that Bohr had leaked out great news. The great news that had leaked out was the discovery of fission and at least the outline of its interpretation. Then, somewhat later that same month, there was a meeting in Washington where the possible importance of the newly discovered phenomenon of fission was first discussed in semi-jocular earnest as a possible source of nuclear power.^[65]

Noddack was proven right after all. Fermi had dismissed the possibility of fission on the basis of his calculations, but he had not taken into account the binding energy that would appear when a nuclide with an odd number of neutrons absorbed an extra neutron.^[54] For Fermi, the news came as a profound embarrassment, as the transuranic elements that he had partly been awarded the Nobel Prize for discovering had not been transuranic elements at all, but fission products. He added a footnote to this effect to his Nobel Prize acceptance speech.^[66][64]

Illustration of Chicago Pile-1, the first nuclear reactor to achieve a self-sustaining chain reaction. Designed by Fermi. it consisted of uranium and uranium oxide in a cubic lattice embedded in graphite.

The scientists at Columbia decided that they should try to detect the energy released in the nuclear fission of uranium when bombarded by neutrons. On 25 January 1939, Fermi was a member of the experimental team at Columbia University which conducted the first nuclear fission experiment in the United States, in the basement of Pupin Hall; the other members of the team were Herbert L. Anderson, Eugene T. Booth, John R. Dunning, G. Norris Glasoe, and Francis G. Slack.^[67] The next day, the Fifth Washington Conference on Theoretical Physics began in Washington, D.C. under the joint auspices of The George Washington University and the Carnegie Institution of Washington. There, the news on nuclear fission was spread even further, which fostered many more experimental demonstrations.^[68]

Few weeks after French scientists Hans von Halban, Lew Kowarski and Frédéric Joliot-Curie,^[69] Fermi and Anderson demonstrated that uranium bombarded by neutrons emitted more neutrons than it absorbed.^[70][71] Leó Szilárd obtained 200 kilograms (440 lb) of uranium oxide from Canadian radium producer Eldorado Gold Mines Limited, allowing Fermi and Anderson to conduct experiments with fission on a much larger scale.^[72] Fermi and Szilárd collaborated on a design of a device to achieve a self-sustaining nuclear reaction—a nuclear reactor. Due to the rate of absorption of neutrons by the hydrogen in water, it was unlikely that a self-sustaining reaction could be achieved with natural uranium and water as a neutron moderator. Fermi suggested, based on his work with neutrons, that uranium oxide could be used in the form of blocks, with graphite as a moderator instead of water. This would reduce the rate of capture of the neutrons, and make it theoretically possible to achieve a self-sustaining chain reaction. Szilárd then came up with what proved to be a workable design, a pile of uranium oxide blocks surrounded by graphite bricks.^[73] Szilárd, Anderson and Fermi jointly published a paper on "Neutron Production in Uranium" ^[72] but their work habits and personalities were different, and Fermi had trouble working with Szilárd.^[74]

Fermi was the first to warn military leaders about the potential impact of nuclear energy, giving a lecture on the subject at the Navy Department on 18 March 1939. The response fell short of what he had hoped for, although the Navy agreed to provide $1,500 towards further research at Columbia.^[75] In August 1939, three Hungarian physicists—Szilárd, Eugene Wigner and Edward Teller—prepared the Einstein–Szilárd letter, which they persuaded Einstein to sign, warning President Franklin D. Roosevelt of the probability that Germany was planning to build an atomic bomb. Because of the German invasion of Poland on 1 September, it was October before they could arrange for the letter to be personally delivered. Roosevelt was sufficiently concerned that he assembled the S-1 Uranium Committee to investigate the matter.^[76]

Fermi's ID badge photo from Los Alamos

The S-1 Uranium Committee provided money for Fermi to buy graphite,^[77] and he built a pile of graphite bricks on the seventh floor of the Pupin laboratory.^[78] By August 1941, he had six tons of uranium oxide and thirty tons of graphite, which he used to build a still larger pile in the Schermerhorn Hall at Columbia.^[79] When the S-1 Uranium Committee next met on 18 December 1941, there was a heightened sense of urgency in the wake of the attack on Pearl Harbor and the subsequent United States entry into World War II. While most of the effort thus far had been directed at three different processes for producing enriched uranium, S-1 Uranium Committee member Arthur Compton determined that plutonium was a feasible alternative which could be mass-produced in nuclear reactors by the end of 1944.^[80] To achieve this, he decided to concentrate the plutonium work at the University of Chicago. Fermi reluctantly moved, and his team became part of the new Metallurgical Laboratory there.^[81]

Given the number of unknown factors involved in creating a self-sustaining nuclear reaction, it seemed inadvisable to do so in a densely populated area. Compton arranged with Colonel Kenneth Nichols, the head of the Army's Manhattan District, for land to be acquired in the Argonne Forest about 20 miles (32 km) from Chicago, and Stone & Webster was contracted to develop the site. This work was halted by an industrial dispute. Fermi then persuaded Compton that he could build a reactor in the squash court under Stagg Field at the University of Chicago. Construction of the pile began on 6 November 1942, and Chicago Pile-1 went critical on 2 December.^[82] The shape of the pile was intended to be roughly spherical, but as work proceeded Fermi calculated that criticality could be achieved without finishing the entire pile as planned.^[83]

This experiment was a landmark in the quest for energy, and it was typical of Fermi's approach. Every step was carefully planned, every calculation meticulously done.^[82] When the first self-sustained nuclear chain reaction was achieved, Compton made a coded phone call to James B. Conant, the chairman of the National Defense Research Committee.

I picked up the phone and called Conant. He was reached at the President's office at Harvard University. "Jim," I said, "you'll be interested to know that the Italian navigator has just landed in the new world." Then, half apologetically, because I had led the S-l Committee to believe that it would be another week or more before the pile could be completed, I added, "the earth was not as large as he had estimated, and he arrived at the new world sooner than he had expected."

"Is that so," was Conant's excited response. "Were the natives friendly?" "Everyone landed safe and happy."^[84]

Fermi (centre), with Ernest O. Lawrence (left) and Isidor Isaac Rabi (right)

To continue the research where it would not pose a public health hazard, the reactor was disassembled and moved to the Argonne site, where Fermi directed research on reactors and other fundamental sciences, revelling in the myriad of research opportunities that the reactor provided by an abundance of neutrons.^[85] The laboratory soon branched out from physics and engineering into using the reactor for biological and medical research. Initially, Argonne was run by Fermi as part of the University of Chicago, but it became a separate entity with Fermi as its director in May 1944.^[86]

Just in case something went wrong, Fermi was on hand at Oak Ridge to witness the air-cooled X-10 Graphite Reactor go critical on 4 November 1943. The technicians woke him early so that he could see it happen.^[87] Getting X-10 operational was another milestone in the plutonium project. It provided data on reactor design, training for DuPont staff in reactor operation, and produced the first small quantities of reactor-bred plutonium.^[88] Fermi became an American citizen in July 1944, the earliest date the law allowed.^[89]

In September 1944, Fermi inserted the first uranium fuel slug into the B Reactor at the Hanford Site, the production reactor designed to breed plutonium in large quantities. Like X-10, it had been designed by Fermi's team at the Metallurgical Laboratory, and built by DuPont, but it was much larger, and was water-cooled. Over the next few days, 838 tubes were loaded, and the reactor went critical. Shortly after midnight on 27 September, the operators began to withdraw the control rods to initiate production. At first all appeared to be well, but around 03:00, the power level started to drop and by 06:30 the reactor had shut down completely. The Army and DuPont turned to Fermi's team for answers. The cooling water was investigated to see if there was a leak or contamination. The next day the reactor suddenly started up again, only to shut down once more a few hours later. The problem was traced to neutron poisoning from xenon-135, a fission product with a half-life of 9.2 hours. Fortunately, DuPont had deviated from the Metallurgical Laboratory's original design in which the reactor had 1,500 tubes arranged in a circle, and had added an additional 504 tubes to fill in the corners. The scientists had originally considered this over-engineering a waste of time and money, but Fermi realized that by loading all 2,004 tubes, the reactor could reach the required power level and efficiently produce plutonium.^[90][91]

The FERMIAC, an analog device invented by Enrico Fermi to implement studies of neutron transport

In mid-1944, Robert Oppenheimer persuaded Fermi to join his Project Y in Los Alamos, New Mexico.^[92] Arriving in September, Fermi was appointed an associate director of the laboratory, with broad responsibility for nuclear and theoretical physics, and was placed in charge of F Division, which was named after him. F Division consisted of four branches: the F-1 Super and General Theory under Teller, which was responsible for the development of a thermonuclear "Super" bomb; the F-2 Water Boiler under L. D. P. King, which looked after the "water boiler" research reactor; F-3 Super Experimentation under Egon Bretscher; and the F-4 Fission Studies under Anderson.^[93] Fermi observed the Trinity test on 16 July 1945, and conducted an experiment dropping strips of paper to estimate the bomb's yield. He simply measured how far they were blown by the explosion using his paces, and came up with a figure of ten kilotons of TNT; the actual yield was about 18.6 kilotons.^[94]

Along with Oppenheimer, Compton and Ernest Lawrence, Fermi was part of the scientific panel that advised the Interim Committee on target selection. The panel agreed with the committee that atomic bombs be used without warning against an industrial target.^[95] Like others at the Los Alamos Laboratory, Fermi found out about the atomic bombings of Hiroshima and Nagasaki from the public address system in the technical area. Fermi did not believe that atomic bombs would scare people into not starting wars, nor did he think that the time was ripe for world government. He therefore did not join the Association of Los Alamos Scientists.^[96]

Post-war work

Fermi became a professor at the University of Chicago on 1 July 1945,^[97] although he did not depart the Los Alamos Laboratory with his family until 31 December 1945.^[98] The Metallurgical Laboratory became the Argonne National Laboratory on 1 July 1946, the first of the national laboratories established by the Manhattan Project.^[99] The short distance between Chicago and Argonne allowed Fermi to work at both places. At Argonne he continued experimental physics, investigating neutron scattering with Leona Marshall.^[100] He also discussed theoretical physics with Maria Mayer, helping her develop insights into spin–orbit coupling that would lead to her receiving the Nobel Prize.^[101]

The Manhattan Project was replaced by Atomic Energy Commission (AEC) on 1 January 1947.^[102] Fermi served on the AEC General Advisory Committee, an influential scientific committee chaired by Robert Oppenheimer.^[103] He also liked to spend a few weeks of each year at the Los Alamos National Laboratory,^[104] where he collaborated with Nicholas Metropolis,^[105] and with John von Neumann on Rayleigh–Taylor instability, the science of what occurs at the border between two fluids of different densities.^[106]

Following the detonation of the first Soviet fission bomb in August 1949, Fermi, along with Isidor Rabi, wrote a strongly worded report for the committee, opposing the development of a hydrogen bomb on moral and technical grounds.^[107] Nonetheless, Fermi continued to participate in work on the hydrogen bomb at Los Alamos as a consultant. Along with Stanislaw Ulam, he calculated that not only would the amount of tritium needed for Teller's model of a thermonuclear weapon be prohibitive, but a fusion reaction could still not be assured to propagate even with this large quantity of tritium.^[108] Fermi was among the scientists who testified on Oppenheimer's behalf at the Oppenheimer security hearing in 1954 that resulted in denial of Oppenheimer's security clearance.^[109]

In his later years, Fermi continued teaching at the University of Chicago. His PhD students in the post-war period included Owen Chamberlain, Geoffrey Chew, Jerome Friedman, Marvin Goldberger, Tsung-Dao Lee, Arthur Rosenfeld and Sam Treiman.^[1][110] Jack Steinberger was a graduate student.^[111] Fermi conducted important research in particle physics, especially related to pions and muons. He made the first predictions of pion-nucleon resonance,^[105] relying on statistical methods, since he reasoned that exact answers were not required when the theory was wrong anyway.^[112] In a paper co-authored with Chen Ning Yang, he speculated that pions might actually be composite particles.^[113] The idea was elaborated by Shoichi Sakata. It has since been supplanted by the quark model, in which the pion is made up of quarks, which completed Fermi's model, and vindicated his approach.^[114]

Fermi wrote a paper "On the Origin of Cosmic Radiation" in which he proposed that cosmic rays arose through material being accelerated by magnetic fields in interstellar space, which led to a difference of opinion with Teller.^[112] Fermi examined the issues surrounding magnetic fields in the arms of a spiral galaxy.^[115] He mused about what is now referred to as the "Fermi paradox": the contradiction between the presumed probability of the existence of extraterrestrial life and the fact that contact has not been made.^[116]

Toward the end of his life, Fermi questioned his faith in society at large to make wise choices about nuclear technology. He said:

Some of you may ask, what is the good of working so hard merely to collect a few facts which will bring no pleasure except to a few long-haired professors who love to collect such things and will be of no use to anybody because only few specialists at best will be able to understand them? In answer to such question[s] I may venture a fairly safe prediction. History of science and technology has consistently taught us that scientific advances in basic understanding have sooner or later led to technical and industrial applications that have revolutionized our way of life. It seems to me improbable that this effort to get at the structure of matter should be an exception to this rule. What is less certain, and what we all fervently hope, is that man will soon grow sufficiently adult to make good use of the powers that he acquires over nature.^[117]

Fermi died at age 53 of stomach cancer in his home in Chicago,^[4] and was interred at Oak Woods Cemetery.^[118]

A commemorative plaque remembers him in the Basilica of Santa Croce, Florence, a church also known as "Temple of Italian Glories", for the many burials of artists, scientists and prominent figures in Italian history.^[119]

Impact and legacy

As a person, Fermi seemed simplicity itself. He was extraordinarily vigorous and loved games and sport. On such occasions his ambitious nature became apparent. He played tennis with considerable ferocity and when climbing mountains acted rather as a guide. One might have called him a benevolent dictator. I remember once at the top of a mountain Fermi got up and said: "Well, it is two minutes to two, let's all leave at two o'clock"; and of course, everybody got up faithfully and obediently. This leadership and self-assurance gave Fermi the name of "The Pope" whose pronouncements were infallible in physics. He once said: "I can calculate anything in physics within a factor 2 on a few sheets: to get the numerical factor in front of the formula right may well take a physicist a year to calculate, but I am not interested in that." His leadership could go so far that it was a danger to the independence of the person working with him. I recollect once, at a party at his house when my wife cut the bread, Fermi came along and said he had a different philosophy on bread-cutting and took the knife out of my wife's hand and proceeded with the job because he was convinced that his own method was superior. But all this did not offend at all, but rather charmed everybody into liking Fermi. He had very few interests outside physics and when he once heard me play on Teller's piano he confessed that his interest in music was restricted to simple tunes.

Egon Bretscher^[3]

Legacy

Fermi received numerous awards in recognition of his achievements, including the Matteucci Medal in 1926, the Nobel Prize for Physics in 1938, the Hughes Medal in 1942, the Franklin Medal in 1947, and the Rumford Prize in 1953.
He was awarded the Medal for Merit in 1946 for his contribution to the Manhattan Project.^[120] In 1999, Time named Fermi on its list of the top 100 persons of the twentieth century.^[121] Fermi was widely regarded as an unusual case of a 20th-century physicist who excelled both theoretically and experimentally. The historian of physics, C. P. Snow, wrote that "if Fermi had been born a few years earlier, one could well imagine him discovering Rutherford's atomic nucleus, and then developing Bohr's theory of the hydrogen atom. If this sounds like hyperbole, anything about Fermi is likely to sound like hyperbole".^[122]

Fermi was known as an inspiring teacher, and was noted for his attention to detail, simplicity, and careful preparation of his lectures.^[123] Later, his lecture notes were transcribed into books.^[124] His papers and notebooks are today in the University of Chicago.^[125] Victor Weisskopf noted how Fermi "always managed to find the simplest and most direct approach, with the minimum of complication and sophistication."^[126] Fermi's ability and success stemmed as much from his appraisal of the art of the possible, as from his innate skill and intelligence. He disliked complicated theories, and while he had great mathematical ability, he would never use it when the job could be done much more simply. He was famous for getting quick and accurate answers to problems that would stump other people. Later on, his method of getting approximate and quick answers through back-of-the-envelope calculations became informally known as the "Fermi method", and is widely taught.^[127]

Fermi was fond of pointing out that Alessandro Volta, working in his laboratory, could have had no idea where the study of electricity would lead.^[128] Fermi is generally remembered for his work on nuclear power and nuclear weapons, especially the creation of the first nuclear reactor, and the development of the first atomic and hydrogen bombs. His scientific work has stood the test of time. This includes his theory of beta decay, his work with non-linear systems, his discovery of the effects of slow neutrons, his study of pion-nucleon collisions, and his Fermi–Dirac statistics. His speculation that a pion was not a fundamental particle pointed the way towards the study of quarks and leptons.^[129]

Things named after Fermi

The sign at Enrico Fermi Street in Rome

Many things have been named in Fermi's honour. These include the Fermilab particle accelerator and physics lab in Batavia, Illinois, which was renamed in his honour in 1974,^[130] and the Fermi Gamma-ray Space Telescope, which was named after him in 2008, in recognition of his work on cosmic rays.^[131] Three nuclear reactor installations have been named after him: the Fermi 1 and Fermi 2 nuclear power plants in Newport, Michigan, the Enrico Fermi Nuclear Power Plant at Trino Vercellese in Italy,^[132] and the RA-1 Enrico Fermi research reactor in Argentina.^[133] A synthetic element isolated from the debris of the 1952 Ivy Mike nuclear test was named fermium, in honour of Fermi's contributions to the scientific community. It follows the element einsteinium, which was discovered with it.^[134][135] Since 1956, the United States Atomic Energy Commission has named its highest honour, the Fermi Award, after him. Recipients of the award include well-known scientists like Otto Hahn, Robert Oppenheimer, Edward Teller and Hans Bethe.^[136]

Bibliography

• Introduzione alla Fisica Atomica (in Italian). Bologna: N. Zanichelli. 1928. OCLC 9653646.
• Fisica per i Licei (in Italian). Bologna: N. Zanichelli. 1929. OCLC 9653646.
• Molecole e cristalli (in Italian). Bologna: N. Zanichelli. 1934. OCLC 19918218.
• Thermodynamics. New York: Prentice Hall. 1937. OCLC 2379038.
• Fisica per Istituti Tecnici (in Italian). Bologna: N. Zanichelli. 1938.
• Fisica per Licei Scientifici (in Italian). Bologna: N. Zanichelli. 1938.
• Elementary particles. New Haven: Yale University Press. 1951. OCLC 362513.

For a full list of his papers, see pages 75–78 in ^[3]

Patents

• US Patent 2206634, "Process for the Production of Radioactive Substances", issued July 1940
• US Patent 2836554, "Air Cooled Neutronic Reactor", issued April 1950
• US Patent 2524379, "Neutron Velocity Selector", issued October 1950
• US Patent 2852461, "Neutronic Reactor", issued September 1953
• US Patent 2708656, "Neutronic Reactor", issued May 1955
• US Patent 2768134, "Testing Material in a Neutronic Reactor", issued October 1956
• US Patent 2780595, "Test Exponential Pile", issued February 1957
• US Patent 2798847, "Method of Operating a Neutronic Reactor", issued July 1957
• US Patent 2807581, "Neutronic Reactor", issued September 1957
• US Patent 2807727, "Neutronic Reactor Shield", issued September 1957
• US Patent 2813070, "Method of Sustaining a Neutronic Chain Reacting System", issued November 1957
• US Patent 2837477, "Chain Reacting System", issued June 1958
• US Patent 2931762, "Neutronic Reactor", issued April 1960
• US Patent 2969307, "Method of Testing Thermal Neutron Fissionable Material for Purity", issued January 1961

Pauli exclusion principle

From Wikipedia, the free encyclopedia

The Pauli exclusion principle is the quantum mechanical principle that says that two identical fermions (particles with half-integer spin) cannot occupy the same quantum state simultaneously. In the case of electrons, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers (n, ℓ, m_ℓ and m_s). For two electrons residing in the same orbital, n, ℓ, and m_ℓ are the same, so m_s must be different and the electrons have opposite spins. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925.

A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles. This means that the wave function changes its sign if the space and spin co-ordinates of any two particles are interchanged.

Integer spin particles, bosons, are not subject to the Pauli exclusion principle: any number of identical bosons can occupy the same quantum state, as with, for instance, photons produced by a laser and Bose–Einstein condensate.

Overview

The Pauli exclusion principle governs the behavior of all fermions (particles with "half-integer spin"), while bosons (particles with "integer spin") are not subject to it. Fermions include elementary particles such as quarks (the constituent particles of protons and neutrons), electrons and neutrinos. In addition, protons and neutrons (subatomic particles composed from three quarks) and some atoms are fermions, and are therefore subject to the Pauli exclusion principle as well. Atoms can have different overall "spin", which determines whether they are fermions or bosons — for example helium-3 has spin 1/2 and is therefore a fermion, in contrast to helium-4 which has spin 0 and is a boson. As such, the Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability, to the chemical behavior of atoms.

"Half-integer spin" means that the intrinsic angular momentum value of fermions is (reduced Planck's constant) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics fermions are described by antisymmetric states. In contrast, particles with integer spin (called bosons) have symmetric wave functions; unlike fermions they may share the same quantum states. Bosons include the photon, the Cooper pairs which are responsible for superconductivity, and the W and Z bosons. (Fermions take their name from the Fermi–Dirac statistical distribution that they obey, and bosons from their Bose–Einstein distribution).

History

In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more chemically stable than those with odd numbers of electrons. In the famous 1916 article "The Atom and the Molecule" by Gilbert N. Lewis, for example, the third of his six postulates of chemical behavior states that the atom tends to hold an even number of electrons in the shell and especially to hold eight electrons which are normally arranged symmetrically at the eight corners of a cube (see: cubical atom). In 1919 chemist Irving Langmuir suggested that the periodic table could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set of electron shells around the nucleus.^[1] In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable "closed shells".

Pauli looked for an explanation for these numbers, which were at first only empirical. At the same time he was trying to explain experimental results of the Zeeman effect in atomic spectroscopy and in ferromagnetism. He found an essential clue in a 1924 paper by Edmund C. Stoner which pointed out that for a given value of the principal quantum number (n), the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the closed shell of the noble gases for the same value of n. This led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state, if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as electron spin.

Connection to quantum state symmetry

The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as a sum of states in which one particle is in state and the other in state :

and antisymmetry under exchange means that A(x,y) = −A(y,x). This implies A(x,y) = 0 when x=y, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantity A(x,y) is not a matrix but an antisymmetric rank-two tensor.

Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component:

is necessarily antisymmetric. To prove it, consider the matrix element:

This is zero, because the two particles have zero probability to both be in the superposition state . But this is equal to

The first and last terms on the right hand side are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:

.

or

Pauli principle in advanced quantum theory

According to the spin-statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics. In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. Since, nonrelativistically, particles can have any statistics and any spin, there is no way to prove a spin-statistics theorem in nonrelativistic quantum mechanics.

In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantum nonlinear Schrödinger equation. In momentum space the exclusion principle is valid also for finite repulsion in a Bose gas with delta function interactions,^[2] as well as for interacting spins and Hubbard model in one dimension, and for other models solvable by Bethe ansatz. The ground state in models solvable by Bethe ansatz is a Fermi sphere.

Consequences

Atoms and the Pauli principle

The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborate electron shell structure of atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. An electrically neutral atom contains bound electrons equal in number to the protons in the nucleus.
Electrons, being fermions, cannot occupy the same quantum state as other electrons, so electrons have to "stack" within an atom, i.e. have different spins while at the same electron orbital as described below.

An example is the neutral helium atom, which has two bound electrons, both of which can occupy the lowest-energy (1s) states by acquiring opposite spin; as spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli principle. However, the spin can take only two different values (eigenvalues). In a lithium atom, with three bound electrons, the third electron cannot reside in a 1s state, and must occupy one of the higher-energy 2s states instead. Similarly, successively larger elements must have shells of successively higher energy. The chemical properties of an element largely depend on the number of electrons in the outermost shell; atoms with different numbers of shells but the same number of electrons in the outermost shell have similar properties, which gives rise to the periodic table of the elements.

Solid state properties and the Pauli principle

In conductors and semiconductors, there are very large numbers of molecular orbitals which effectively form a continuous band structure of energy levels. In strong conductors (metals) electrons are so degenerate that they cannot even contribute much to the thermal capacity of a metal. Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.

Stability of matter

The stability of the electrons in an atom itself is unrelated to the exclusion principle, but is described by the quantum theory of the atom. The underlying idea is that close approach of an electron to the nucleus of the atom necessarily increases its kinetic energy, an application of the uncertainty principle of Heisenberg.^[3] However, stability of large systems with many electrons and many nucleons is a different matter, and requires the Pauli exclusion principle.^[4]

It has been shown that the Pauli exclusion principle is responsible for the fact that ordinary bulk matter is stable and occupies volume. This suggestion was first made in 1931 by Paul Ehrenfest, who pointed out that the electrons of each atom cannot all fall into the lowest-energy orbital and must occupy successively larger shells. Atoms therefore occupy a volume and cannot be squeezed too closely together.^[5]

A more rigorous proof was provided in 1967 by Freeman Dyson and Andrew Lenard, who considered the balance of attractive (electron–nuclear) and repulsive (electron–electron and nuclear–nuclear) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle.^[6] The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsive exchange interaction, which is a short-range effect, acting simultaneously with the long-range electrostatic or coulombic force. This effect is partly responsible for the everyday observation in the macroscopic world that two solid objects cannot be in the same place at the same time.

Astrophysics and the Pauli principle

Dyson and Lenard did not consider the extreme magnetic or gravitational forces which occur in some astronomical objects. In 1995 Elliott Lieb and coworkers showed that the Pauli principle still leads to stability in intense magnetic fields such as in neutron stars, although at a much higher density than in ordinary matter.^[7] It is a consequence of general relativity that, in sufficiently intense gravitational fields, matter collapses to form a black hole.

Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of white dwarf and neutron stars. In both types of body, atomic structure is disrupted by large gravitational forces, leaving the constituents supported by "degeneracy pressure" alone. This exotic form of matter is known as degenerate matter. In white dwarfs atoms are held apart by electron degeneracy pressure. In neutron stars, subject to even stronger gravitational forces, electrons have merged with protons to form neutrons. Neutrons are capable of producing an even higher degeneracy pressure, albeit over a shorter range. This can stabilize neutron stars from further collapse, but at a smaller size and higher density than a white dwarf. Neutrons are the most "rigid" objects known; their Young modulus (or more accurately, bulk modulus) is 20 orders of magnitude larger than that of diamond. However, even this enormous rigidity can be overcome by the gravitational field of a massive star or by the pressure of a supernova, leading to the formation of a black hole.

Degenerate matter

From Wikipedia, the free encyclopedia

 

Degenerate matter^[1][2] in physics is a collection of free, non-interacting particles with a pressure and other physical characteristics determined by quantum mechanical effects. It is the analogue of an ideal gas in classical mechanics. The degenerate state of matter, in the sense of deviant from an ideal gas, arises at extraordinarily high density (in compact stars) or at extremely low temperatures in laboratories.^[3][4] It occurs for matter particles such as electrons, neutrons, protons, and fermions in general and is referred to as electron-degenerate matter, neutron-degenerate matter, etc. In a mixture of particles, such as ions and electrons in white dwarfs or metals, the electrons may be degenerate, while the ions are not.

In a quantum mechanical description, free 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. The pressure (called degeneracy pressure or Fermi pressure) remains nonzero even near absolute zero temperature.^[3][4] Adding particles or reducing the volume forces the particles into higher-energy quantum states. This requires a compression force, and is made manifest as a resisting pressure. The key feature is that this degeneracy pressure does not depend on the temperature and only on the density of the fermions. It keeps dense stars in equilibrium independent of the thermal structure of the star.

Degenerate matter is also called a Fermi gas or a degenerate gas. A degenerate state with velocities of the fermions close to the speed of light (particle energy larger than its rest mass energy) is called relativistic degenerate matter.

Degenerate matter was first described for a mixture of ions and electrons in 1926 by Ralph H. Fowler,^[5] showing that at densities observed in white dwarfs the electrons (obeying Fermi–Dirac statistics, the term degenerate was not yet in use) have a pressure much higher than the partial pressure of the ions.

Concept

Imagine that a plasma is cooled and compressed repeatedly. Eventually, it will not be possible to compress the plasma any further, because the Pauli exclusion principle states that two fermions cannot share the same quantum state. When in this state, since there is no extra space for any particles, we can also say that a particle's location is extremely defined. Therefore, since (according to the Heisenberg uncertainty principle) ΔpΔx ≥ ħ/2 where Δp is the uncertainty in the particle's momentum and Δx is the uncertainty in position, then we must say that their momentum is extremely uncertain since the particles are located in a very confined space. Therefore, even though the plasma is cold, the particles must be moving very fast on average. This leads to the conclusion that in order to compress an object into a very small space, tremendous force is required to control its particles' momentum.

Unlike a classical ideal gas, whose pressure is proportional to its temperature (P = nkT/V, where P is pressure, V is the volume, n is the number of particles—typically atoms or molecules—k is Boltzmann's constant, and T is temperature), 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 is given by P = K(n/V)5/3, where K depends on the properties of the 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 P = K′(n/V)4/3, where K′ again depends on the properties of the particles making up the gas.^[6]

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, but at extremely high densities the degeneracy pressure usually dominates.

Exotic examples of degenerate matter include neutronium, strange matter, metallic hydrogen and white dwarf matter. 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. In metals it is useful to treat the conduction electrons alone as a degenerate, free electron gas while the majority of the electrons are regarded as occupying bound quantum states. This contrasts with degenerate matter that forms the body of a white dwarf, where all the electrons would be treated as occupying free particle momentum states.

Degenerate gases

Degenerate gases are gases composed of fermions that have a particular configuration that usually forms at high densities. Fermions are particles with half-integer spin. Their behavior is regulated by a set of quantum mechanical rules called the Fermi–Dirac statistics. One particular rule is the Pauli exclusion principle, which states that there can be only one fermion occupying each quantum state, which also applies to electrons that are not bound to a nucleus but merely confined to a fixed volume, such as in the deep interior of a star. Such particles as electrons, protons, neutrons, and neutrinos are all fermions and obey Fermi–Dirac statistics.

A fermion gas in which all energy states below some 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. The electron gas in ordinary metals and in the interior of white dwarf stars constitute two examples of a degenerate electron gas. Most stars are supported against their own gravitation by normal thermal gas pressure. White dwarf stars are supported by the degeneracy pressure of the electron gas in their interior, while for neutron stars the degenerate particles are neutrons.

Electron degeneracy

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 the matter becomes a degenerate gas when the electrons are all stripped from their parent atoms. In the core of a star, once hydrogen burning in nuclear fusion reactions stops, it 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 the ordinary gas laws. White dwarfs are luminous not because they are generating any 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. Pressure is only increased 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 pressure is increased, and the particles become spaced closer together, 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 compositions similar to the sun. The mass cutoff changes with the chemical composition of the object, as this affects the ratio of mass to number of electrons present. Celestial objects below this limit are white dwarf stars, formed by the collapse of the cores of stars that run out of fuel. During collapse, 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 (supported by neutron degeneracy pressure) or a black hole may be formed instead.

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. 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 modeled as a correction to the equations of state of electron-degenerate matter.

Neutron degeneracy

Neutron degeneracy is analogous to electron degeneracy and is demonstrated in neutron stars, which are primarily supported by the pressure from a degenerate neutron gas.^[7] This happens when a stellar core above 1.44 solar masses, the Chandrasekhar limit, collapses and is not halted by the degenerate electrons. As the star collapses, the Fermi energy of the electrons increases to the point where it is energetically favorable for them to combine with protons to produce neutrons (via inverse beta decay, also termed electron capture and "neutralization"). The result of this collapse is an extremely compact star composed of nuclear matter, which is predominantly a degenerate neutron gas, sometimes called neutronium, 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. In the case of neutron stars and white dwarf stars, this 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. This results in a star with a diameter on the order of a thousandth that of a white dwarf.

There is an upper limit to the mass of a neutron-degenerate object, the Tolman–Oppenheimer–Volkoff limit, which is analogous to the Chandrasekhar limit for electron-degenerate objects. The precise limit is unknown, as it depends on the equations of state of nuclear matter, for which a highly accurate model is not yet available. Above this limit, a neutron star may collapse into a black hole, or into other, denser forms of degenerate matter (such as quark matter) if these forms exist and have suitable properties (mainly related to degree of compressibility, or "stiffness", described by the equations of state).

Quark degeneracy

At densities greater than those supported by neutron degeneracy, quark matter is expected to occur. Several variations of this have been proposed that represent quark-degenerate states. Strange matter is a degenerate gas of quarks that is often assumed to contain strange quarks in addition to the usual up and down quarks. Color superconductor materials are degenerate gases of quarks in which quarks pair up in a manner similar to Cooper pairing in electrical superconductors. The equations of state for the various proposed forms of quark-degenerate matter vary widely, and are usually also poorly defined, due to the difficulty of modeling strong force interactions.

Quark-degenerate matter may occur in the cores of neutron stars, depending on the equations of state of neutron-degenerate matter. It may also occur in hypothetical quark stars, formed by the collapse of objects above the Tolman–Oppenheimer–Volkoff mass limit for neutron-degenerate objects. Whether quark-degenerate matter forms at all in these situations depends on the equations of state of both neutron-degenerate matter and quark-degenerate matter, both of which are poorly known.

Preon degeneracy hypothesis

Preons are subatomic particles proposed to be the constituents of quarks, which become composite particles in preon-based models. If preons exist, preon-degenerate matter might occur at densities greater than that which can be supported by quark-degenerate matter. The expected properties of preon-degenerate matter depend very strongly on the model chosen to describe preons, and the existence of preons is not assumed by the majority of the scientific community, due to conflicts between the preon models originally proposed and experimental data from particle accelerators.

Singularity

At densities greater than those supported by any degeneracy, gravity overwhelms all other forces. To the best of our current understanding, the body collapses to form a black hole. In the frame of reference that is co-moving with the collapsing matter, all the matter ends up in an infinitely dense singularity at the center of the event horizon. In the frame of reference of an observer at infinity, the collapse asymptotically approaches the event horizon.

As a consequence of relativity, the extreme gravitational field and orbital velocity experienced by infalling matter around a black hole would "slow" time for that matter relative to a distant observer.