vol III chap 12 sect 1
Previous: 11.3. Phases in a learning and research process.
12. Bosons or fermions in condensed matter.¶
Introduction.
When there are bosons and where fermions?
Next Physics Nobel Prizes have been awarded as critical explanations of previous question:
In 1972 to John Bardeen (1908-1991), Leon Neil Cooper (1930) and John Robert Schrieffer (1931-2019) “for their jointly developed theory of superconductivity, usually called the BCS-theory”.
In 1985 to Klaus von Klitzing (1943) “for the discovery of the quantized Hall effect”.
In 1996 to David M. Lee (1931), Douglas D. Osheroff (1945) and Robert C. Richardson (1937- 2013) “for their discovery of superfluidity in helium-3”.
In 1998 to Robert B. Laughlin (1950), Horst L. Störmer (1949) and Daniel C. Tsui (1939) “for their discovery of a new form of quantum fluid with fractionally charged excitations”.
In 2001 to Eric A. Cornell (1961), Wolfgang Ketterle (1957) and Carl E. Wieman (1951) “for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates”.
Learning objectives of Chapter 12.
After this Chapter you should be able to:
- Describe Physics Nobel Prize contributions for understanding boson condensation: superconductivity. superfluidity and Bose Einstein condensation.
- Describe Physics Nobel Prize contributions for understanding the Integral and Fractional Quantum Hall Effects.
- Analyze the differences among prescriptive learning and emergent learning and consider Albert Einstein productive learning trajectory.
Description of content of Chapter 12.
Section 12.1. Boson condensation.
We review three phenomena connected with the concept of boson condensation: superconductivity, superfluidity and Bose-Einstein condensation. We refer to Ketterle´s Nobel Lecture.
Section 12.2. Quantum Hall effects.
We deal with the behavior of electrons in quantum Hall effects when this quantization is described in terms of entire or fractional numbers. We refer to the Physics Nobel Prizes awarded in 1985 to Klaus von Klitzing “for the discovery of the quantized Hall effect” and in 1998 to Laughlin, Störmer and Tsui “for their discovery of a new form of quantum fluid with fractionally charged excitations”.
Section 12.3. Schools for education and learning trajectories.
We apply the Analysis of learning trajectories to the description of main characteristics of Albert Einstein life in connection with prescriptive and emergent learning.
12.1. Boson condensation.¶
Statistical mechanics is a branch of physics in which statistical calculations of microscopic variables are related to macroscopic observables. The probability that something happens in a physical system depends on the number of possible configurations describing the structure and dynamics of the system. How configurations are counted depends on what kind of assumptions and criteria are applied according to classical physics or to modern physics.
In classical mechanics the particles of a physical system can be distinguished from each other and are localizable by specifying simultaneously their vectors of position (\(r\)) and momentum (\(p = mv\)). To count configurations of particles with such properties we speak of classical Maxwell-Boltzmann statistics (MB).
In quantum mechanics the Heisenberg uncertainty principle (\(Δx Δp_x\) ~ \(h\)) must be satisfied and the positions of the N particles of the system are described in probabilistic terms by the wave function \(Ψ(r_1, r_2, r_3,... r_N)\). Two situations may arise with respect to the exchange of positions between two of the components of the system: the symmetric case characterizing bosons corresponds to Bose-Einstein statistics (BE) where \(Ψ(r_1, r_2, r_3,... r_N) = (+)Ψ(r_2, r_1, r_3,... r_N)\), while the antisymmetric case characterizing fermions corresponds to the so-called Fermi-Dirac (FD) statistic where \(Ψ(r_1, r_2, r_3,... r_N) = (-)Ψ(r_2, r_1, r_3,... r_N)\).
Fermions have half-integral spin, they are selective and obey the exclusion principle; a gas of fermions have at most one particle in each one-particle quantum states. Bosons have integral spin, they are gregarious and can condense in a quantum state described by one and the same wavefunction; below a certain critical temperature they form a Bose-Einstein condensate.
In June 4, 1924 Einstein received a letter from Satyendra Nath Bose (1894-1977) explaining that a recent paper written by him was rejected for publication. He had worked on a derivation of Planck´s law of black-body radiation by introducing a new procedure for the coarse-grained counting in the phase space. Accompanying his letter Bose sent a translation in English of his rejected paper and asked Einstein, that in case he might think the work had sufficient merit, to arrange it for a publication in a German journal.
Einstein did the translation and submitted the paper including the following translator´s note: “In my opinion, Bose´s derivation of the Planck formula constitutes an important advance. The method used here also yields the quantum theory of the ideal gas, as I shall discuss elsewhere in more detail”. Probably Einstein was thinking on his paper Quantum Theory of the Monatomic Ideal Gas to be published in the journal Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 1924, 261–267.
Bose described certain rules for determining if two photons could be considered as identical or as different particles. Einstein applied these rules to the atoms of a gas and considered that at very low temperatures most of these atoms were in the same quantum state, the fundamental state where the energy attains the minimum possible value. In those circumstances no one could differentiate the components of such group of atoms. This process is similar to what happens when a gas forms a drop by changing the phase from vapor to liquid. There is a condensation in both systems.
Bose-Einstein condensation is a new state of matter, a state of microscopic quantum coherence. When the atoms are cooled down near the absolute zero, their quantum wave functions begin to overlap and the atoms lose their individual identities, behaving more like a single super-atom than an agglomeration of distinct atoms glued together. All these atoms occupy the same position with an almost null momentum. They satisfy Heisenberg´s indeterminacy principle.
At the time of the publication of Bose and Einstein papers the possibility of observing boson condensation was considered as having “a purely imaginary character”, also with no possible connections with unknown effects like superconductivity or superfluidity. These effects were experimentally discovered before their satisfactory theoretical explanations were available. Let us follow the sequence in time of this conceptual trajectory whose main steps are the condensates formed by bosons in these conditions: pairs of electrons in superconductivity, pair of atoms of He-3 in superfluidity and atoms of alkali gases like rubidium and sodium in Bose-Einstein condensation.
In 1908 Heike Kamerlingh Onnes (1853-1926) obtained liquid helium for the first time; afterwards it will be used as refrigerant. In 1911 he discovered superconductivity when he observed the sudden nullification of the electrical resistance of mercury at 4.2° K. (see Figure 12.1).
1913 Physics Nobel Prize awarded to Onnes.
WORK: “When different substances are cooled to very low temperatures, their properties change. In 1908 Heike Kamerlingh Onnes used an ingenious apparatus to cool helium to liquid form. Fluid helium was carefully studied and also became an important aid for the cooling of different substances and charting their properties at low temperatures. In 1911 Kamerlingh Onnes discovered that the electrical resistance of mercury completely disappeared at temperatures a few degrees above absolute zero. The phenomenon became known as superconductivity.”
MLA style: Heike Kamerlingh Onnes – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/1913/onnes/facts/
NOBEL LECTURE: Investigations into the properties of substances at low temperatures, which have led, amongst other things, to the preparation of liquid helium by Onnes.
MLA style: Heike Kamerlingh Onnes – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sun. 18 Jun 2023. https://www.nobelprize.org/prizes/physics/1913/onnes/lecture/
To make clear the experimental conditions of the measurements of the electrical resistance, Kammerling Onnes mentioned in his 1913 Nobel Lecture: “The disappearance did not take place gradually but (compare Fig. 17) abruptly. From 1/500 the resistance at 4.2°K drops to a millionth part. At the lowest temperature, 1.5°K, it could be established that the resistance had become less than a thousand-millionth part of that at normal temperature.”
(Source: taken from Kammerling Onnes´s Nobel Lecture Investigations into the properties of substances at low temperatures, which have led, amongst other things, to the preparation of liquid helium.)
Figure 12.1. Experimental demonstration of superconductivity in mercury.
In 1926 Willem Keesom (1876-1956), a student of Kamerlingh Onnes, invented a method to freeze liquid helium and years later he discovered a phase transition in Helium at 2.19° K.
In 1933 Walther Meissner (1882 – 1974) and Robert Ochsenfeld (1901 –1993) discovered what is now called the Meissner effect: a superconductor placed in a weak external magnetic field expels the magnetic field when it is cooled at the transition temperature. Due to this effect a magnet can levitate above a cooled superconductor.
In 1934 Piotr Kapitsa (1894-1984) developed a method for producing liquid helium in large quantities, and in 1937 he discovered that at very low temperatures liquid helium can flow through narrow channels with complete absence of friction, without resistance. Since then, this property was called superfluidity.
The inert gas helium exists in two forms or isotopes: He-3 whose nucleus has two protons and one neutron, and He-4 with a nucleus with two protons and two neutrons. To balance the atomic charge of the complete atom, both isotopes have two electrons in the surrounding electronic shell. The total number of particles serves to classify helium atoms: those called fermions have an unpair number (3) as in He-3 and those called bosons have a pair number (4) as in He-4.
Nearly the absolute zero (-273.15° C) there is no more randomness in the motion of atoms, they now move together in a coordinated fashion. Thereafter atoms do not acquire energy by friction and then do not dissipate energy. The liquid is no more a classic liquid, it is a quantum superfluid, meaning it can flow without energy dissipation.
In 1935, the brothers Fritz London (1900-1954) and Heinz London (1907-1970) explained the Meissner effect as a consequence of the minimization of the electromagnetic free energy carried by a superconducting current.
In 1939 Fritz London proposed an interpretation of the He3 – He-4 phase transition as a Bose-Einstein condensation.
In 1941 Lev Landau (1908-1968) studied the movement of superfluid liquid helium and considered the quantized states of motion of the whole liquid by introducing the concept of quasiparticles. The superfluid liquid is a kind of Bose-Einstein condensate of helium atoms.
In 1950 Landau in collaboration with Vitaly Ginzburg (1916-2009) proposed the phenomenological Ginzburg- Landau theory to explain superconductivity. This theory is a combination of Landau's theory of second-order phase transition with a Schrödinger-like wave equation.
Landau was recognized with the Physics Nobel Prize in 1962 and Ginzburg shared with Alexei A. Abrikosov (1928-2017) and Anthony L. Legett (1938) the 2003 Physics Nobel Prize “for pioneering contributions to the theory of superconductors and superfluids”. For more details see their Nobel Lectures: Type II superconductors and the vortex lattice by Abrikisov, On Superconductivity and Superfluidity by Ginzburg and Superfluid 3-He: the early days as seen by a theorist by Leggett.
MLA style: Alexei Abrikosov – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Tue. 6 Feb 2024. https://www.nobelprize.org/prizes/physics/2003/abrikosov/lecture/
MLA style: Vitaly L. Ginzburg – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Tue. 10 Oct 2023. https://www.nobelprize.org/prizes/physics/2003/ginzburg/lecture/
MLA style: Anthony J. Leggett – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Tue. 6 Feb 2024. https://www.nobelprize.org/prizes/physics/2003/leggett/lecture/
1962 Physics Nobel Prize awarded to Landau.
WORK (Landau): “When certain substances are cooled to very low temperatures, their properties undergo radical changes. At temperatures a couple of degrees above absolute zero, helium becomes superfluid and the liquid flows without friction. One of Lev Landau’s many contributions within theoretical physics came in 1941, when he applied quantum theory to the movement of superfluid liquid helium. Among other things, he introduced the concept of quasiparticles as the equivalent of sound vibrations and vortexes. This allowed him to develop his theoretical explanation for superfluidity.”
MLA style: Lev Landau – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/1962/landau/facts/
2003 Physics Nobel Prize awarded to Ginzburg.
WORK (Ginzburg): “When certain substances are cooled to extremely low temperatures, they become superconductors, conducting electrical current entirely without resistance. With one type of superconductivity, the magnetic field is forced away from the conductor, but with another type of superconductivity, the magnetic field is admitted into the conductor. In 1950 Vitaly Ginzburg and Lev Landau formulated a theory that incorporated a mathematical function to clarify the interplay between superconductivity and magnetism. The theory was intended for the first type of superconductivity, but it enabled a theory for the second type of superconductivity.”
MLA style: Vitaly L. Ginzburg – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/2003/ginzburg/facts/
2003 Physics Nobel Prize awarded to Leggett.
WORK (Leggett): “When certain substances are cooled to extremely low temperatures, they become superconductors, conducting electrical current entirely without resistance. This applies to helium-4, the most common form of helium, but for a long time the superfluidity of helium-3 was in dispute. The different types of helium are described by different quantum mechanical rules and equations under which helium-4 has a whole-number spin while helium-3 has a half-number spin. After it was discovered that at extremely low temperatures helium-3 also becomes superconducting, Anthony Leggett formulated a theory that explained this.”
MLA style: Anthony J. Leggett – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/2003/leggett/facts/
Ginzburg mentions in his Nobel Lecture On Superconductivity and Superfluidity that “Superconductivity is, if you please, the superfluidity of a charged liquid or, equivalently, superfluidity is the superconductivity of a noncharged liquid.”
In 1952 Alexei Abrikosov formulated a theory for the second type of superconductor and introduced a mathematical function that described vortexes that forms in the conductor when an external magnetic field is applied.
In 1957 a microscopic theory of superconductivity was proposed by Bardeen, Cooper and Schrieffer (BCS). In this theory the superconducting current is consider as a superfluid of Cooper pairs of electrons that interact through the exchange of phonons, the quantized vibrations of the crystal lattice.
Although in metals individual electrons behave as fermions, the strong coupling between the Cooper-pairs produces a collective pattern characterized by the absence of electric resistance (see Figure 12.2).
MLA style: Leon N. Cooper – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sun. 18 Jun 2023. https://www.nobelprize.org/prizes/physics/1972/cooper/lecture/
(Source; taken from Cooper´s Nobel Lecture Microscopic Quantum Interference Effects in the Theory of Superconductivity.)
Figure 12.2. The concept of Cooper pairs.
For detailed explanations see also the other 1972 Nobel Lectures: Electron-Phonon Interactions and Superconductivity by Bardeen, and Macroscopic Quantum Phenomena from Pairing in Superconductors by Schrieffer.
MLA style: John Bardeen – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Tue. 6 Feb 2024. https://www.nobelprize.org/prizes/physics/1972/bardeen/lecture/
MLA style: Robert Schrieffer – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Tue. 6 Feb 2024. https://www.nobelprize.org/prizes/physics/1972/schrieffer/lecture/
1972 Physics Nobel Prize awarded to Bardeen, Cooper and Schrieffer.
WORK: “When certain metals are cooled to extremely low temperatures, they become superconductors, conducting electrical current entirely without resistance. Based on quantum mechanics, John Bardeen, Leon Cooper, and Robert Schrieffer formulated a theory for the phenomenon in 1957. At extremely low temperatures, the interaction between electrons and atoms in the metals’ crystalline structure causes the electrons to pair up with one another. As a result, their movement becomes orderly, unlike the random movement at normal temperatures, and electrical resistance disappears.”
MLA style: John Bardeen – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/1972/bardeen/facts/
In 1962 Brian Josephson (1940) predicted that a supercurrent can flow through a tunnel barrier when between two pieces of a superconductor are separated by a thin layer of insulator. See his 1973 Nobel Lecture The Discovery of Tunnelling Supercurrents.
MLA style: Brian D. Josephson – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Tue. 6 Feb 2024. https://www.nobelprize.org/prizes/physics/1973/josephson/lecture/
1973 Physics Nobel Prize awarded to Josephson.
WORK: “In quantum physics matter is described as both waves and particles. One result of this is the tunneling phenomenon, which means that particles can pass through barriers that they should not be able to squeeze through according to classic physics. In 1962 Brian Josephson predicted unexpected results with superconductors, material that at low temperatures lacks electrical resistance. Without superimposed voltage, a current can result between two superconductors that are separated by a thin insulator. If a rectified voltage is added, on the other hand, an alternating current can result.”
MLA style: Brian D. Josephson – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/1973/josephson/facts/
In 1966 Anthony Legett considered that a two-band superconductor should show a sort of “internal Josephson effect” corresponding to fluctuations of the relative number of electrons in the two bands and of the relative phase of the Cooper pairs in them.
In 1970 Lee, Osheroff and Richardson discovered superfluidity in He-3. In the superfluid phase of He-3 the atoms themselves present a coupling pairing interaction which is mediated by quantum spin fluctuations rather than by the exchange of the vibratory energy corresponding to phonons as in superconductors.
When two atoms with odd numbers of nucleons pair up with each other, they collectively possess an even number of nucleons and behave like bosons, condensing together into a superfluid state. Fermions such as atomic He-3 do not condensate, therefore it is not possible that they present superfluidity like the boson atomic He-4. However, fermions in liquid He-3 form boson pairs and behave as a superfluid. Other experiments reported in Lee´s Lecture show that the specific heat measurements in superfluid He-3 present a discontinuity at the transition temperature like the one that characterizes phase transitions in superconductors (see Figure 12.3).
MLA style: David M. Lee – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sun. 18 Jun 2023. https://www.nobelprize.org/prizes/physics/1996/lee/lecture/
(Source: MLA style: David M. Lee – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sun. 18 Jun 2023. https://www.nobelprize.org/prizes/physics/1996/lee/lecture/)
Figure 12.3. Experimental behavior in superfluid He-3 near the critical temperature.
1996 Physics Nobel Prize awarded to Lee, Osheroff and Richardson.
WORK: “When certain substances are cooled to extremely low temperatures, they become superfluid, flowing without any friction. This applies to helium-4, the most common form of helium, but for a long time the superfluidity of helium-3 was in dispute. The different types of helium are described by different quantum mechanical rules and equations under which helium-4 has a whole-number spin while helium-3 has a half-number spin. In 1972 David Lee, Douglas Osheroff, and Robert Richardson verified that helium-3 also becomes superfluid at extremely low temperatures.”
MLA style: David M. Lee – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/1996/lee/facts/
In 1986 J. Georg Bednorz (1950) and K. Alexander Müller (1927-2023) discovered superconductivity in ceramic materials at temperatures above 77°K (−196.2 °C; −321.1 °F). See their joint 1987 Nobel Lecture Perovskite-Type Oxides – The New Approach to High- \(T_c\) Superconductivity. (See Figure 12.4).
MLA style: J. Georg Bednorz – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sun. 18 Jun 2023. https://www.nobelprize.org/prizes/physics/1987/bednorz/lecture/
1987 Physics Nobel Prize awarded to Bednorz and Müller.
WORK: “When certain metals are cooled to extremely low temperatures, they become superconductors, conducting electrical current entirely without resistance. However, very low temperatures, just a few degrees above absolute zero, are required for this phenomenon to occur. In 1986 Georg Bednorz and Alex Müller discovered that a material composed of copper oxide with lantanum and barium additives became superconducting at a significantly higher temperature than previously tested materials. This sparked extensive research into similar materials.”
MLA style: J. Georg Bednorz – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/1987/bednorz/facts/
(Source: MLA style: J. Georg Bednorz – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sun. 18 Jun 2023. https://www.nobelprize.org/prizes/physics/1987/bednorz/lecture/)
Figure 12.4. Increasing values of the superconducting transition temperatures.
In 1995 Eric Cornell and Carl Wieman Cornell produced a pure condensate of about 2,000 rubidium atoms at 20 nK (\(2 \times 10^{-8}\) degrees above absolute zero). Independently, Ketterle performed experiments with sodium atoms.
2001 Physics Nobel Prize awarded to Cornell and Wieman.
WORK: “One of the fundamental numbers in the world of quantum mechanics is the spin quantum number. Particles and atoms that have whole-number spin are described by other rules and equations than those that have half-number spin. Satyendra Nath Bose and Albert Einstein predicted in 1924 that at very low temperatures atoms with whole-number spin would be able to concentrate themselves in the lowest energy state and form a Bose-Einstein condensate. In 1995 Eric Cornell and Carl Wieman succeeded in proving the phenomenon in a rarefied gas of rubidium atoms at an extremely low temperature.”
MLA style: Eric A. Cornell – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/2001/cornell/facts/
NOBEL LECTURE: Bose-Einstein Condensation in a Dilute Gas; The First 70 Years and Some Recent Experiments by Cornell and Wieman.
MLA style: Eric A. Cornell – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Thu. 13 Jun 2024. https://www.nobelprize.org/prizes/physics/2001/cornell/lecture/
"2001 Physics Nobel Prize awarded to Ketterle.
WORK: “One of the fundamental numbers in the world of quantum mechanics is the spin quantum number. Particles and atoms that have whole-number spin are described by other rules and equations than those that have half-number spin. Satyendra Nath Bose and Albert Einstein predicted in 1924 that at very low temperatures atoms with whole-number spin would be able to concentrate themselves in the lowest energy state and form a Bose-Einstein condensate. In 1995 Wolfgang Ketterle succeeded in proving the phenomenon in a rarefied gas of sodium atoms at an extremely low temperature.”
MLA style: Wolfgang Ketterle – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 8 Jun 2024. https://www.nobelprize.org/prizes/physics/2001/ketterle/facts/
Two methods have been used to force the atoms to attain their lowest energy levels: laser cooling and evaporative cooling. In the first method when a photon bounces off the atom, the electron in the atom that absorbs the photon jumps to a higher energy level and quickly jumps back to its original level, ejecting the photon again which produces a temperature decrease. In the second method the atoms are located inside a magnetic trap from where the most energetic atoms escape but the slowest atoms remain.
As an application of Heisenberg’s uncertainty principle to the condensation effect, we next quote the explanation given by Ketterle in his Nobel Lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser.
MLA style: Wolfgang Ketterle – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sun. 18 Jun 2023. https://www.nobelprize.org/prizes/physics/2001/ketterle/lecture/
"When a gas of bosonic atoms is cooled below a critical temperature \(T_c\), a large fraction of the atoms condenses in the lowest quantum state. Atoms at temperature T and with mass m can be regarded as quantum-mechanical wavepackets that have a spatial extent on the order of a thermal de Broglie wavelength \(λ_{dB}= (2πħ^2/mk_BT)^{1/2}\). The value of \(λ_{dB}\) is the position uncertainty associated with the thermal momentum distribution and increases with decreasing temperature. When atoms are cooled to the point where \(λ_{dB}\) is comparable to the interatomic separation, the atomic wavepackets “overlap” and the gas starts to become a “quantum soup” of indistinguishable particles. Bosonic atoms undergo a quantum-mechanical phase transition and form a Bose-Einstein condensate, a cloud of atoms all occupying the same quantum mechanical state at a precise temperature which, for an ideal gas, is related to the peak atomic density n by \(n(λ_{dB})^3 = 2.612\). If the atoms are fermions, cooling gradually brings the gas closer to being a “Fermi sea” in which exactly one atom occupies each low-energy state".
Note: the reduced Planck constant or Dirac constant is \(ħ =h⁄(2π)\).
Next three Figures 12.5, 12.6 and 12.7 come from Ketterle´s Nobel Lecture: When Atoms Behave as Waves: Bose-Einstein Condensation and the Atom Laser
Formato MLA
Figure 12.5. Conditions for obtaining Bose-Einstein condensation.
Figure 12.6. Experimental setup for producing Bose-Einstein condensate.
Figure 12.7. Observation of Bose-Einstein condensate.
For more details see the Nobel Lecture: Bose-Einstein Condensation in a Dilute Gas; The First 70 Years and Some Recent Experiments by E.A. Cornell and C.E Wieman.
MLA style: Eric A. Cornell – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Thu. 13 Jun 2024. https://www.nobelprize.org/prizes/physics/2001/cornell/lecture/
See also Advanced information:* Bose-Einstein Condensation in Alkali Gases*.
MLA style: Advanced information. NobelPrize.org. Nobel Prize Outreach AB 2023. Mon. 19 Jun 2023. https://www.nobelprize.org/prizes/physics/2001/advanced-information/
In 2018 Pablo Jarillo-Herrero (1976) and collaborators reported the discovery of superconductivity on twisted bilayer graphene. This is one example of Twistronics, the study of how the angle (the twist) between layers of two-dimensional materials can change their electrical properties from conductors to superconductors.