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The original interacting system can be converted to a system of non-interacting particles with a dispersion law. | 112 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 700 | 29 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In some simplest cases, the state of condensed particles can be described with a nonlinear Schrödinger equation, also known as Gross–Pitaevskii or Ginzburg–Landau equation. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments. | 316 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 701 | 30 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
This approach originates from the assumption that the state of the BEC can be described by the unique wavefunction of the condensate ψ ψ ( r → → ) . For a system of this nature, | ψ ψ ( r → → ) | 2 is interpreted as the particle density, so the total number of atoms is N = ∫ ∫ d r → → | ψ ψ ( r → → ) | 2 | 305 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 702 | 31 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Provided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using mean-field theory, the energy (E) associated with the state ψ ψ ( r → → ) is: | 207 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 703 | 32 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Minimizing this energy with respect to infinitesimal variations in ψ ψ ( r → → ) , and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a non-linear Schrödinger equation): | 208 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 704 | 33 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
where: | 6 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 705 | 34 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In the case of zero external potential, the dispersion law of interacting Bose–Einstein-condensed particles is given by so-called Bogoliubov spectrum (for T = 0 ): | 163 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 706 | 35 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The Gross-Pitaevskii equation (GPE) provides a relatively good description of the behavior of atomic BEC's. However, GPE does not take into account the temperature dependence of dynamical variables, and is therefore valid only for T = 0 . It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature. | 393 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 707 | 36 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The Gross-Pitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution and different numerical methods, such as split-step Crank-Nicolson and Fourier spectral methods, are used for its solution. There are different Fortran and C programs for its solution for contact interaction and long-range dipolar interaction which can be freely used. | 405 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 708 | 37 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The Gross–Pitaevskii model of BEC is a physical approximation valid for certain classes of BECs. By construction, the GPE uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and also neglects anomalous contributions to self-energy. These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensate wavefunction acquires the terms containing higher-order powers of the wavefunction. Moreover, for some physical systems the amount of such terms turns out to be infinite, therefore, the equation becomes essentially non-polynomial. The examples where this could happen are the Bose–Fermi composite condensates, effectively lower-dimensional condensates, and dense condensates and superfluid clusters and droplets. It is found that one has to go beyond the Gross-Pitaevskii equation. For example, the logarithmic term ψ ψ ln | ψ ψ | 2 found in the Logarithmic Schrödinger equation must be added to the Gross-Pitaevskii equation along with a Ginzburg -Sobyanin contribution to correctly determine that the speed of sound scales as the cubic root of pressure for Helium-4 at very low temperatures in close agreement with experiment. | 1,296 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 709 | 38 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
However, it is clear that in a general case the behaviour of Bose–Einstein condensate can be described by coupled evolution equations for condensate density, superfluid velocity and distribution function of elementary excitations. This problem was solved in 1977 by Peletminskii et al. in microscopical approach. The Peletminskii equations are valid for any finite temperatures below the critical point. Years after, in 1985, Kirkpatrick and Dorfman obtained similar equations using another microscopical approach. The Peletminskii equations also reproduce Khalatnikov hydrodynamical equations for superfluid as a limiting case. | 628 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 710 | 39 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The phenomena of superfluidity of a Bose gas and superconductivity of a strongly-correlated Fermi gas (a gas of Cooper pairs) are tightly connected to Bose–Einstein condensation. Under corresponding conditions, below the temperature of phase transition, these phenomena were observed in helium-4 and different classes of superconductors. In this sense, the superconductivity is often called the superfluidity of Fermi gas. In the simplest form, the origin of superfluidity can be seen from the weakly interacting bosons model. | 526 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 711 | 40 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that helium-4 became a new kind of fluid, now known as a superfluid, at temperatures less than 2.17 K (the lambda point). Superfluid helium has many unusual properties, including zero viscosity (the ability to flow without dissipating energy) and the existence of quantized vortices. It was quickly believed that the superfluidity was due to partial Bose–Einstein condensation of the liquid. In fact, many properties of superfluid helium also appear in gaseous condensates created by Cornell, Wieman and Ketterle (see below). Superfluid helium-4 is a liquid rather than a gas, which means that the interactions between the atoms are relatively strong; the original theory of Bose–Einstein condensation must be heavily modified in order to describe it. Bose–Einstein condensation remains, however, fundamental to the superfluid properties of helium-4. Note that helium-3, a fermion, also enters a superfluid phase (at a much lower temperature) which can be explained by the formation of bosonic Cooper pairs of two atoms (see also fermionic condensate). | 1,114 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 712 | 41 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and co-workers at JILA on 5 June 1995. They cooled a dilute vapor of approximately two thousand rubidium-87 atoms to below 170 nK using a combination of laser cooling (a technique that won its inventors Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips the 1997 Nobel Prize in Physics) and magnetic evaporative cooling. About four months later, an independent effort led by Wolfgang Ketterle at MIT condensed sodium-23. Ketterle's condensate had a hundred times more atoms, allowing important results such as the observation of quantum mechanical interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for their achievements. | 777 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 713 | 42 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
A group led by Randall Hulet at Rice University announced a condensate of lithium atoms only one month following the JILA work. Lithium has attractive interactions, causing the condensate to be unstable and collapse for all but a few atoms. Hulet's team subsequently showed the condensate could be stabilized by confinement quantum pressure for up to about 1000 atoms. Various isotopes have since been condensed. | 412 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 714 | 43 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In the image accompanying this article, the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle : spatially confined atoms have a minimum width velocity distribution. This width is given by the curvature of the magnetic potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotropy of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left. This graph served as the cover design for the 1999 textbook Thermal Physics by Ralph Baierlein. | 910 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 715 | 44 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Bose–Einstein condensation also applies to quasiparticles in solids. Magnons, excitons, and polaritons have integer spin which means they are bosons that can form condensates. | 175 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 716 | 45 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Magnons, electron spin waves, can be controlled by a magnetic field. Densities from the limit of a dilute gas to a strongly interacting Bose liquid are possible. Magnetic ordering is the analog of superfluidity. In 1999 condensation was demonstrated in antiferromagnetic Tl Cu Cl 3, at temperatures as great as 14 K. The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. In 2006, condensation in a ferromagnetic yttrium-iron-garnet thin film was seen even at room temperature, with optical pumping. | 596 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 717 | 46 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Excitons, electron-hole pairs, were predicted to condense at low temperature and high density by Boer et al., in 1961. Bilayer system experiments first demonstrated condensation in 2003, by Hall voltage disappearance. Fast optical exciton creation was used to form condensates in sub-kelvin Cu 2 O in 2005 on. | 309 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 718 | 47 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Polariton condensation was first detected for exciton-polaritons in a quantum well microcavity kept at 5 K. | 107 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 719 | 48 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In June 2020, the Cold Atom Laboratory experiment on board the International Space Station successfully created a BEC of rubidium atoms and observed them for over a second in free-fall. Although initially just a proof of function, early results showed that, in the microgravity environment of the ISS, about half of the atoms formed into a magnetically insensitive halo-like cloud around the main body of the BEC. | 413 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 720 | 49 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
As in many other systems, vortices can exist in BECs. Vortices can be created, for example, by "stirring" the condensate with lasers, rotating the confining trap, or by rapid cooling across the phase transition. The vortex created will be a quantum vortex with core shape determined by the interactions. Fluid circulation around any point is quantized due to the single-valued nature of the order BEC order parameter or wavefunction, that can be written in the form ψ ψ ( r → → ) = ϕ ϕ ( ρ ρ , z ) e i ℓ ℓ θ θ where ρ ρ , z and θ θ are as in the cylindrical coordinate system, and ℓ ℓ is the angular quantum number (a.k.a. the "charge" of the vortex). Since the energy of a vortex is proportional to the square of its angular momentum, in trivial topology only ℓ ℓ = 1 vortices can exist in the steady state ; Higher-charge vortices will have a tendency to split into ℓ ℓ = 1 vortices, if allowed by the topology of the geometry. | 929 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 721 | 50 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
An axially symmetric (for instance, harmonic) confining potential is commonly used for the study of vortices in BEC. To determine ϕ ϕ ( ρ ρ , z ) , the energy of ψ ψ ( r → → ) must be minimized, according to the constraint ψ ψ ( r → → ) = ϕ ϕ ( ρ ρ , z ) e i ℓ ℓ θ θ . This is usually done computationally, however, in a uniform medium, the following analytic form demonstrates the correct behavior, and is a good approximation: | 428 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 722 | 51 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Here, n is the density far from the vortex and x = ρ ρ / ( ℓ ℓ ξ ξ ) , where ξ ξ = 1 / 8 π π a s n 0 is the healing length of the condensate. | 141 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 723 | 52 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
A singly charged vortex ( ℓ ℓ = 1 ) is in the ground state, with its energy ϵ ϵ v given by | 90 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 724 | 53 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
where b is the farthest distance from the vortices considered.(To obtain an energy which is well defined it is necessary to include this boundary b .) | 150 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 725 | 54 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
For multiply charged vortices ( ℓ ℓ > 1 ) the energy is approximated by | 71 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 726 | 55 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
which is greater than that of ℓ ℓ singly charged vortices, indicating that these multiply charged vortices are unstable to decay. Research has, however, indicated they are metastable states, so may have relatively long lifetimes. | 229 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 727 | 56 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Closely related to the creation of vortices in BECs is the generation of so-called dark solitons in one-dimensional BECs. These topological objects feature a phase gradient across their nodal plane, which stabilizes their shape even in propagation and interaction. Although solitons carry no charge and are thus prone to decay, relatively long-lived dark solitons have been produced and studied extensively. | 407 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 728 | 57 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Experiments led by Randall Hulet at Rice University from 1995 through 2000 showed that lithium condensates with attractive interactions could stably exist up to a critical atom number. Quench cooling the gas, they observed the condensate to grow, then subsequently collapse as the attraction overwhelmed the zero-point energy of the confining potential, in a burst reminiscent of a supernova, with an explosion preceded by an implosion. | 436 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 729 | 58 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Further work on attractive condensates was performed in 2000 by the JILA team, of Cornell, Wieman and coworkers. Their instrumentation now had better control so they used naturally attracting atoms of rubidium-85 (having negative atom–atom scattering length). Through Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, they lowered the characteristic, discrete energies at which rubidium bonds, making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among wave-like condensate atoms. | 616 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 730 | 59 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
When the JILA team raised the magnetic field strength further, the condensate suddenly reverted to attraction, imploded and shrank beyond detection, then exploded, expelling about two-thirds of its 10,000 atoms. About half of the atoms in the condensate seemed to have disappeared from the experiment altogether, not seen in the cold remnant or expanding gas cloud. Carl Wieman explained that under current atomic theory this characteristic of Bose–Einstein condensate could not be explained because the energy state of an atom near absolute zero should not be enough to cause an implosion; however, subsequent mean-field theories have been proposed to explain it. Most likely they formed molecules of two rubidium atoms; energy gained by this bond imparts velocity sufficient to leave the trap without being detected. | 818 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 731 | 60 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The process of creation of molecular Bose condensate during the sweep of the magnetic field throughout the Feshbach resonance, as well as the reverse process, are described by the exactly solvable model that can explain many experimental observations. | 251 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 732 | 61 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile. The slightest interaction with the external environment can be enough to warm them past the condensation threshold, eliminating their interesting properties and forming a normal gas. | 287 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 733 | 62 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Nevertheless, they have proven useful in exploring a wide range of questions in fundamental physics, and the years since the initial discoveries by the JILA and MIT groups have seen an increase in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave–particle duality, the study of superfluidity and quantized vortices, the creation of bright matter wave solitons from Bose condensates confined to one dimension, and the slowing of light pulses to very low speeds using electromagnetically induced transparency. Vortices in Bose–Einstein condensates are also currently the subject of analogue gravity research, studying the possibility of modeling black holes and their related phenomena in such environments in the laboratory. Experimenters have also realized " optical lattices ", where the interference pattern from overlapping lasers provides a periodic potential. These have been used to explore the transition between a superfluid and a Mott insulator, and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks–Girardeau gas. Further, the sensitivity of the pinning transition of strongly interacting bosons confined in a shallow one-dimensional optical lattice originally observed by Haller has been explored via a tweaking of the primary optical lattice by a secondary weaker one. Thus for a resulting weak bichromatic optical lattice, it has been found that the pinning transition is robust against the introduction of the weaker secondary optical lattice. Studies of vortices in nonuniform Bose–Einstein condensates as well as excitations of these systems by the application of moving repulsive or attractive obstacles, have also been undertaken. Within this context, the conditions for order and chaos in the dynamics of a trapped Bose–Einstein condensate have been explored by the application of moving blue and red- detuned laser beams (hitting frequencies slightly above and below the resonance frequency, respectively) via the time-dependent Gross-Pitaevskii equation. | 2,118 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 734 | 63 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Bose–Einstein condensates composed of a wide range of isotopes have been produced. | 82 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 735 | 64 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Cooling fermions to extremely low temperatures has created degenerate gases, subject to the Pauli exclusion principle. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form bosonic compound particles (e.g. molecules or Cooper pairs). The first molecular condensates were created in November 2003 by the groups of Rudolf Grimm at the University of Innsbruck, Deborah S. Jin at the University of Colorado at Boulder and Wolfgang Ketterle at MIT. Jin quickly went on to create the first fermionic condensate, working with the same system but outside the molecular regime. | 589 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 736 | 65 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 1999, Danish physicist Lene Hau led a team from Harvard University which slowed a beam of light to about 17 meters per second using a superfluid. Hau and her associates have since made a group of condensate atoms recoil from a light pulse such that they recorded the light's phase and amplitude, recovered by a second nearby condensate, in what they term "slow-light-mediated atomic matter-wave amplification" using Bose–Einstein condensates. | 445 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 737 | 66 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Another current research interest is the creation of Bose–Einstein condensates in microgravity in order to use its properties for high precision atom interferometry. The first demonstration of a BEC in weightlessness was achieved in 2008 at a drop tower in Bremen, Germany by a consortium of researchers led by Ernst M. Rasel from Leibniz University Hannover. The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space and it is also the subject of two upcoming experiments on the International Space Station. | 543 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 738 | 67 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Researchers in the new field of atomtronics use the properties of Bose–Einstein condensates in the emerging quantum technology of matter-wave circuits. | 151 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 739 | 68 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 1970, BECs were proposed by Emmanuel David Tannenbaum for anti- stealth technology. | 86 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 740 | 69 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 2020, researchers reported the development of superconducting BEC and that there appears to be a "smooth transition between" BEC and Bardeen–Cooper–Shrieffer regimes. | 169 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 741 | 70 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Limitations of evaporative cooling have restricted atomic BECs to "pulsed" operation, involving a highly inefficient duty cycle that discards more than 99% of atoms to reach BEC. Achieving continuous BEC has been a major open problem of experimental BEC research, driven by the same motivations as continuous optical laser development: high flux, high coherence matter waves produced continuously would enable new sensing applications. | 435 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 742 | 71 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Continuous BEC was achieved for the first time in 2022. | 55 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 743 | 72 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
P. Sikivie and Q. Yang showed that cold dark matter axions would form a Bose–Einstein condensate by thermalisation because of gravitational self-interactions. Axions have not yet been confirmed to exist. However the important search for them has been greatly enhanced with the completion of upgrades to the Axion Dark Matter Experiment (ADMX) at the University of Washington in early 2018. | 389 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 744 | 73 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 2014, a potential dibaryon was detected at the Jülich Research Center at about 2380 MeV. The center claimed that the measurements confirm results from 2011, via a more replicable method. The particle existed for 10 seconds and was named d*(2380). This particle is hypothesized to consist of three up and three down quarks. It is theorized that groups of d* (d-stars) could form Bose–Einstein condensates due to prevailing low temperatures in the early universe, and that BECs made of such hexaquarks with trapped electrons could behave like dark matter. | 556 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 745 | 74 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The effect has mainly been observed on alkaline atoms which have nuclear properties particularly suitable for working with traps. As of 2012, using ultra-low temperatures of 10 − − 7 K or below, Bose–Einstein condensates had been obtained for a multitude of isotopes, mainly of alkali metal, alkaline earth metal, and lanthanide atoms (Li, Na, K, K, Rb, Rb, Cs, Cr, Ca, Sr, Sr, Sr, Yb, Dy, and Er). Research was finally successful in hydrogen with the aid of the newly developed method of 'evaporative cooling'. In contrast, the superfluid state of He below 2.17 K is not a good example, because the interaction between the atoms is too strong. Only 8% of atoms are in the ground state of the trap near absolute zero, rather than the 100% of a true condensate. | 760 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 746 | 75 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The bosonic behavior of some of these alkaline gases appears odd at first sight, because their nuclei have half-integer total spin. It arises from a subtle interplay of electronic and nuclear spins: at ultra-low temperatures and corresponding excitation energies, the half-integer total spin of the electronic shell and half-integer total spin of the nucleus are coupled by a very weak hyperfine interaction. The total spin of the atom, arising from this coupling, is an integer lower value. The chemistry of systems at room temperature is determined by the electronic properties, which is essentially fermionic, since room temperature thermal excitations have typical energies much higher than the hyperfine values. | 716 | Bose–Einstein_condensate | https://en.wikipedia.org/wiki/Bose–Einstein_condensate | 747 | 76 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The Beer–Lambert law is commonly applied to chemical analysis measurements to determine the concentration of chemical species that absorb light. It is often referred to as Beer's law. In physics, the Bouguer–Lambert law is an empirical law which relates the extinction or attenuation of light to the properties of the material through which the light is travelling. It had its first use in astronomical extinction. The fundamental law of extinction (the process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant) is sometimes called the Beer–Bouguer–Lambert law or the Bouguer–Beer–Lambert law or merely the extinction law. The extinction law is also used in understanding attenuation in physical optics, for photons, neutrons, or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation. | 903 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 748 | 0 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Bouguer–Lambert law: This law is based on observations made by Pierre Bouguer before 1729. It is often attributed to Johann Heinrich Lambert, who cited Bouguer's Essai d'optique sur la gradation de la lumière (Claude Jombert, Paris, 1729) – and even quoted from it – in his Photometria in 1760. Lambert expressed the law, which states that the loss of light intensity when it propagates in a medium is directly proportional to intensity and path length, in the mathematical form used today. | 490 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 749 | 1 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Lambert began by assuming that the intensity I of light traveling into an absorbing body would be given by the differential equation: − − d I = μ μ I d x , which is compatible with Bouguer's observations. The constant of proportionality μ was often termed the "optical density" of the body. Integrating to find the intensity at a distance d into the body, one obtains: ln ( I 0 / I ) = ∫ ∫ 0 d μ μ d x . For a homogeneous medium, this reduces to: ln ( I 0 / I ) = μ μ d , from which follows the exponential attenuation law: I = I 0 e − − μ μ d . | 553 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 750 | 2 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Beer's law: Much later, in 1852, the German scientist August Beer studied another attenuation relation. In the introduction to his classic paper, he wrote: "The absorption of light during the irradiation of a colored substance has often been the object of experiment; but attention has always been directed to the relative diminution of the various colors or, in the case of crystalline bodies, the relation between the absorption and the direction of polarization. Concerning the absolute magnitude of the absorption that a particular ray of light suffers during its propagation through an absorbing medium, there is no information available." By studying absorption of red light in colored aqueous solutions of various salts, he concluded that "the transmittance of a concentrated solution can be derived from a measurement of the transmittance of a dilute solution". It is clear that he understood the exponential relationship, as he wrote: "If λ λ is the coefficient (fraction) of diminution, then this coefficient (fraction) will have the value λ λ 2 for double this thickness." Furthermore Beer stated: "We shall take the absorption coefficient to be the coefficient giving the diminution in amplitude suffered by a light ray as it passes through a unit length of an absorbing material. We then have, according to theory, and as I have found verified by experiment, λ λ = μ μ D where μ μ is the absorption coefficient and D the length of the absorbing material traversed in the experiment." This is the relationship that might properly be called Beer's law. There is no evidence that Beer saw concentration and path length as symmetrical variables in an equation in the manner of the Beer-Lambert law. | 1,707 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 751 | 3 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Beer–Lambert law: The modern formulation of the Beer–Lambert law combines the observations of Bouguer and Beer into the mathematical form of Lambert. It correlates the absorbance, most often expressed as the negative decadic logarithm of the transmittance, to both the concentrations of the attenuating species and the thickness of the material sample. An early, possibly the first, modern formulation was given by Robert Luther and Andreas Nikolopulos in 1913. | 461 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 752 | 4 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
While the observations of Bouguer and Beer have a similar form in the Beer–Lambert law, their areas of observation were very different. For both experimenters, the incident beam was well collimated, with a light sensor which preferentially detected directly transmitted light. | 276 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 753 | 5 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Beer specifically looked at solutions. Solutions are homogeneous and do not scatter light (Ultraviolet, visible, Infrared) of wavelengths commonly used in analytical spectroscopy (except upon entry and exit). The attenuation of a beam of light within a solution is assumed to be only due to absorption. In order to approximate the conditions required for the Beer Lambert law to hold, often the intensity of transmitted light through a reference sample ( I R ) consisting of pure solvent is measured, and compared to the intensity of light transmitted through a sample ( I S ) , with the absorbance of the sample taken as: log 10 ( I R / I S ) . It is for this case that the common mathematical formulation (see below) applies: log 10 ( I R / I S ) = A = ε ε ℓ ℓ c | 772 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 754 | 6 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Bouguer looked at astronomical phenomena where the size of a detector is very small compared to the distance traveled by the light. In this case, any light that is scattered by a particle, either in the forward or backward direction, will not strike the detector. The loss of intensity to the detector will be due to both absorption and scatter. Consequently, the total loss is called attenuation (rather than absorption). A single measurement cannot separate the two, but conceptually the contribution of each can be separated in the attenuation coefficient. If I 0 is the intensity of the light at the beginning of the travel and I d is the intensity of the light detected after travel of a distance d , the fraction transmitted, T , is given by: T = I d I 0 = exp ( − − μ μ d ) , where μ μ is called an attenuation constant or coefficient. The amount of light transmitted is falling off exponentially with distance. Taking the natural logarithm in the above equation, we get: − − ln ( T ) = ln I 0 I d = μ μ d . For scattering media, the constant is often divided into two parts, μ μ = μ μ s + μ μ a , separating it into a scattering coefficient, μ μ s , and an absorption coefficient, μ μ a . | 1,208 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 755 | 7 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The fundamental law of extinction states that the extinction process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant. (Neither concentration or length are fundamental parameters.) There are two factors that determine the degree to which a medium containing particles will attenuate a light beam: the number of particles encountered by the light beam, and the degree to which each particle extinguishes the light. | 495 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 756 | 8 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
For the case of absorption (Beer), this later quantity is called the absorptivity [ ϵ ϵ ], which is defined as "the property of a body that determines the fraction of incident radiation absorbed by the body". The Beer–Lambert law [ log 10 ( I 0 / I ) = A = ϵ ϵ ℓ ℓ c ] uses concentration and length in order to determine the number of particles the beam encounters. For a collimated beam (directed radiation) of cross-sectional area S , the number of particles encountered over a distance ℓ ℓ is N A c σ σ ℓ ℓ S , where N A is the Avogadro constant, c the molar concentration (in mol/m), and σ σ the particle cross section. | 627 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 757 | 9 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
There must be a large number of particles that are uniformly distributed for this relationship to hold. In practice, the beam area is thought of as a constant, and since the fraction [ I / I 0 ] has the area in both the numerator and denominator, the beam area cancels in the calculation of the absorbance. The units of the absorptivity must match the units in which the sample is described. For example, if the sample is described by mass concentration (g/L) and length (cm), then the units on the absorptivity would be [ L g cm ], so that the absorbance has no units. | 569 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 758 | 10 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
For the case of " extinction " (Bouguer), the sum of absorption and scatter, the terms absorption, scattering, and extinction cross-sections are often used. The fraction of light extinguished by the sample may be described by the extinction cross section (fraction extinguished per particle). the number of particles in a unit distance and the distance in those units. For example: [ (fraction extinguished / particle) (# particles / meter) (# meters / sample) = fraction extinguished / sample ] | 495 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 759 | 11 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
A common and practical expression of the Beer–Lambert law relates the optical attenuation of a physical material containing a single attenuating species of uniform concentration to the optical path length through the sample and absorptivity of the species. This expression is: log 10 ( I 0 / I ) = A = ε ε ℓ ℓ c where | 321 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 760 | 12 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
A more general form of the Beer–Lambert law states that, for N attenuating species in the material sample, T = exp ( − − ∑ ∑ i = 1 N σ σ i ∫ ∫ 0 ℓ ℓ n i ( z ) d z ) = 10 ∧ ∧ ( − − ∑ ∑ i = 1 N ε ε i ∫ ∫ 0 ℓ ℓ c i ( z ) d z ) , or equivalently that τ τ = ∑ ∑ i = 1 N τ τ i = ∑ ∑ i = 1 N σ σ i ∫ ∫ 0 ℓ ℓ n i ( z ) d z , A = ∑ ∑ i = 1 N A i = ∑ ∑ i = 1 N ε ε i ∫ ∫ 0 ℓ ℓ c i ( z ) d z , where | 392 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 761 | 13 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In the above equations, the transmittance T of material sample is related to its optical depth τ and to its absorbance A by the following definition T = Φ Φ e t Φ Φ e i = e − − τ τ = 10 − − A , where | 199 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 762 | 14 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Attenuation cross section and molar attenuation coefficient are related by ε ε i = N A ln ( 10 ) σ σ i , and number density and amount concentration by τ τ = ℓ ℓ ∑ ∑ i = 1 N σ σ i n i , A = ℓ ℓ ∑ ∑ i = 1 N ε ε i c i . c i = n i N A , where N A is the Avogadro constant. In case of uniform attenuation, these relations become T = exp ( − − ℓ ℓ ∑ ∑ i = 1 N σ σ i n i ) = 10 ∧ ∧ ( − − ℓ ℓ ∑ ∑ i = 1 N ε ε i c i ) , or equivalently τ τ = ℓ ℓ ∑ ∑ i = 1 N σ σ i n i , A = ℓ ℓ ∑ ∑ i = 1 N ε ε i c i . τ τ = ℓ ℓ ∑ ∑ i = 1 N σ σ i n i , A = ℓ ℓ ∑ ∑ i = 1 N ε ε i c i . Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance. | 687 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 763 | 15 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The law tends to break down at very high concentrations, especially if the material is highly scattering. Absorbance within range of 0.2 to 0.5 is ideal to maintain linearity in the Beer–Lambert law. If the radiation is especially intense, nonlinear optical processes can also cause variances. The main reason, however, is that the concentration dependence is in general non-linear and Beer's law is valid only under certain conditions as shown by derivation below. For strong oscillators and at high concentrations the deviations are stronger. If the molecules are closer to each other interactions can set in. These interactions can be roughly divided into physical and chemical interactions. Physical interaction do not alter the polarizability of the molecules as long as the interaction is not so strong that light and molecular quantum state intermix (strong coupling), but cause the attenuation cross sections to be non-additive via electromagnetic coupling. Chemical interactions in contrast change the polarizability and thus absorption. | 1,046 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 764 | 16 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The law can be expressed in terms of attenuation coefficient, but in this case is better called the Bouguer-Lambert's law. The (Napierian) attenuation coefficient μ μ and the decadic attenuation coefficient μ μ 10 = μ μ ln 10 of a material sample are related to its number densities and amount concentrations as μ μ ( z ) = ∑ ∑ i = 1 N μ μ i ( z ) = ∑ ∑ i = 1 N σ σ i n i ( z ) , μ μ 10 ( z ) = ∑ ∑ i = 1 N μ μ 10 , i ( z ) = ∑ ∑ i = 1 N ε ε i c i ( z ) respectively, by definition of attenuation cross section and molar attenuation coefficient. Then the law becomes T = exp ( − − ∫ ∫ 0 ℓ ℓ μ μ ( z ) d z ) = 10 ∧ ∧ ( − − ∫ ∫ 0 ℓ ℓ μ μ 10 ( z ) d z ) , and τ τ = ∫ ∫ 0 ℓ ℓ μ μ ( z ) d z , A = ∫ ∫ 0 ℓ ℓ μ μ 10 ( z ) d z . In case of uniform attenuation, these relations become T = e − − μ μ ℓ ℓ = 10 − − μ μ 10 ℓ ℓ , or equivalently τ τ = μ μ ℓ ℓ , A = μ μ 10 ℓ ℓ . | 873 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 765 | 17 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In many cases, the attenuation coefficient does not vary with z , in which case one does not have to perform an integral and can express the law as: I ( z ) = I 0 e − − μ μ z where the attenuation is usually an addition of absorption coefficient α α (creation of electron-hole pairs) or scattering (for example Rayleigh scattering if the scattering centers are much smaller than the incident wavelength). Also note that for some systems we can put 1 / λ λ (1 over inelastic mean free path) in place of μ μ . | 507 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 766 | 18 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Assume that a beam of light enters a material sample. Define z as an axis parallel to the direction of the beam. Divide the material sample into thin slices, perpendicular to the beam of light, with thickness d z sufficiently small that one particle in a slice cannot obscure another particle in the same slice when viewed along the z direction. The radiant flux of the light that emerges from a slice is reduced, compared to that of the light that entered, by d Φ Φ e ( z ) = − − μ μ ( z ) Φ Φ e ( z ) d z , where μ is the (Napierian) attenuation coefficient, which yields the following first-order linear, ordinary differential equation : d Φ Φ e ( z ) d z = − − μ μ ( z ) Φ Φ e ( z ) . The attenuation is caused by the photons that did not make it to the other side of the slice because of scattering or absorption. The solution to this differential equation is obtained by multiplying the integrating factor exp ( ∫ ∫ 0 z μ μ ( z ′ ) d z ′ ) throughout to obtain d Φ Φ e ( z ) d z exp ( ∫ ∫ 0 z μ μ ( z ′ ) d z ′ ) + μ μ ( z ) Φ Φ e ( z ) exp ( ∫ ∫ 0 z μ μ ( z ′ ) d z ′ ) = 0 , which simplifies due to the product rule (applied backwards) to d d z [ Φ Φ e ( z ) exp ( ∫ ∫ 0 z μ μ ( z ′ ) d z ′ ) ] = 0. | 1,223 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 767 | 19 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Integrating both sides and solving for Φ e for a material of real thickness ℓ, with the incident radiant flux upon the slice Φ Φ e i = Φ Φ e ( 0 ) and the transmitted radiant flux Φ Φ e t = Φ Φ e ( ℓ ℓ ) gives Φ Φ e t = Φ Φ e i exp ( − − ∫ ∫ 0 ℓ ℓ μ μ ( z ) d z ) , and finally T = Φ Φ e t Φ Φ e i = exp ( − − ∫ ∫ 0 ℓ ℓ μ μ ( z ) d z ) . | 345 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 768 | 20 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Since the decadic attenuation coefficient μ 10 is related to the (Napierian) attenuation coefficient by μ μ 10 = μ μ ln 10 , we also have T = exp ( − − ∫ ∫ 0 ℓ ℓ ln ( 10 ) μ μ 10 ( z ) d z ) = 10 ∧ ∧ ( − − ∫ ∫ 0 ℓ ℓ μ μ 10 ( z ) d z ) . | 248 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 769 | 21 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
To describe the attenuation coefficient in a way independent of the number densities n i of the N attenuating species of the material sample, one introduces the attenuation cross section σ σ i = μ μ i ( z ) n i ( z ) . σ i has the dimension of an area; it expresses the likelihood of interaction between the particles of the beam and the particles of the species i in the material sample: T = exp ( − − ∑ ∑ i = 1 N σ σ i ∫ ∫ 0 ℓ ℓ n i ( z ) d z ) . | 452 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 770 | 22 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
One can also use the molar attenuation coefficients ε ε i = N A ln 10 σ σ i , where N A is the Avogadro constant, to describe the attenuation coefficient in a way independent of the amount concentrations c i ( z ) = n i z N A of the attenuating species of the material sample: T = exp ( − − ∑ ∑ i = 1 N ln ( 10 ) N A ε ε i ∫ ∫ 0 ℓ ℓ n i ( z ) d z ) = exp ( − − ∑ ∑ i = 1 N ε ε i ∫ ∫ 0 ℓ ℓ n i ( z ) N A d z ) ln ( 10 ) = 10 ∧ ∧ ( − − ∑ ∑ i = 1 N ε ε i ∫ ∫ 0 ℓ ℓ c i ( z ) d z ) . | 499 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 771 | 23 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Under certain conditions the Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte. These deviations are classified into three categories: | 190 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 772 | 24 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
There are at least six conditions that need to be fulfilled in order for the Beer–Lambert law to be valid. These are: | 117 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 773 | 25 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
If any of these conditions are not fulfilled, there will be deviations from the Beer–Lambert law. | 97 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 774 | 26 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The Beer–Lambert law can be applied to the analysis of a mixture by spectrophotometry, without the need for extensive pre-processing of the sample. An example is the determination of bilirubin in blood plasma samples. The spectrum of pure bilirubin is known, so the molar attenuation coefficient ε is known. Measurements of decadic attenuation coefficient μ 10 are made at one wavelength λ that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences. The amount concentration c is then given by c = μ μ 10 ( λ λ ) ε ε ( λ λ ) . | 577 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 775 | 27 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
For a more complicated example, consider a mixture in solution containing two species at amount concentrations c 1 and c 2. The decadic attenuation coefficient at any wavelength λ is, given by μ μ 10 ( λ λ ) = ε ε 1 ( λ λ ) c 1 + ε ε 2 ( λ λ ) c 2 . | 249 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 776 | 28 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations c 1 and c 2 as long as the molar attenuation coefficients of the two components, ε 1 and ε 2 are known at both wavelengths. This two system equation can be solved using Cramer's rule. In practice it is better to use linear least squares to determine the two amount concentrations from measurements made at more than two wavelengths. Mixtures containing more than two components can be analyzed in the same way, using a minimum of N wavelengths for a mixture containing N components. | 618 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 777 | 29 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The law is used widely in infra-red spectroscopy and near-infrared spectroscopy for analysis of polymer degradation and oxidation (also in biological tissue) as well as to measure the concentration of various compounds in different food samples. The carbonyl group attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the polymer calculated. | 380 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 778 | 30 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The Bouguer–Lambert law may be applied to describe the attenuation of solar or stellar radiation as it travels through the atmosphere. In this case, there is scattering of radiation as well as absorption. The optical depth for a slant path is τ ′ = mτ, where τ refers to a vertical path, m is called the relative airmass, and for a plane-parallel atmosphere it is determined as m = sec θ where θ is the zenith angle corresponding to the given path. The Bouguer-Lambert law for the atmosphere is usually written T = exp ( − − m ( τ τ a + τ τ g + τ τ R S + τ τ N O 2 + τ τ w + τ τ O 3 + τ τ r + ⋯ ⋯ ) ) , where each τ x is the optical depth whose subscript identifies the source of the absorption or scattering it describes: | 726 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 779 | 31 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
m is the optical mass or airmass factor, a term approximately equal (for small and moderate values of θ) to 1 cos θ θ , where θ is the observed object's zenith angle (the angle measured from the direction perpendicular to the Earth's surface at the observation site). This equation can be used to retrieve τ a, the aerosol optical thickness, which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate. | 477 | Beer–Lambert_law | https://en.wikipedia.org/wiki/Beer–Lambert_law | 780 | 32 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
A black hole is a region of spacetime where gravity is so strong that nothing, not even light and other electromagnetic waves, is capable of possessing enough energy to escape it. Einstein 's theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of no escape is called the event horizon. A black hole has a great effect on the fate and circumstances of an object crossing it, but it has no locally detectable features according to general relativity. In many ways, a black hole acts like an ideal black body, as it reflects no light. Quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is of the order of billionths of a kelvin for stellar black holes, making it essentially impossible to observe directly. | 930 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 781 | 0 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. In 1916, Karl Schwarzschild found the first modern solution of general relativity that would characterize a black hole. David Finkelstein, in 1958, first published the interpretation of "black hole" as a region of space from which nothing can escape. Black holes were long considered a mathematical curiosity; it was not until the 1960s that theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars by Jocelyn Bell Burnell in 1967 sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality. The first black hole known was Cygnus X-1, identified by several researchers independently in 1971. | 835 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 782 | 1 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Black holes of stellar mass form when massive stars collapse at the end of their life cycle. After a black hole has formed, it can grow by absorbing mass from its surroundings. Supermassive black holes of millions of solar masses (M ☉) may form by absorbing other stars and merging with other black holes, or via direct collapse of gas clouds. There is consensus that supermassive black holes exist in the centres of most galaxies. | 431 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 783 | 2 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light. Any matter that falls toward a black hole can form an external accretion disk heated by friction, forming quasars, some of the brightest objects in the universe. Stars passing too close to a supermassive black hole can be shredded into streamers that shine very brightly before being "swallowed." If other stars are orbiting a black hole, their orbits can be used to determine the black hole's mass and location. Such observations can be used to exclude possible alternatives such as neutron stars. In this way, astronomers have identified numerous stellar black hole candidates in binary systems and established that the radio source known as Sagittarius A*, at the core of the Milky Way galaxy, contains a supermassive black hole of about 4.3 million solar masses. | 911 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 784 | 3 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The idea of a body so big that even light could not escape was briefly proposed by English astronomical pioneer and clergyman John Michell in a letter published in November 1784. Michell's simplistic calculations assumed such a body might have the same density as the Sun, and concluded that one would form when a star's diameter exceeds the Sun's by a factor of 500, and its surface escape velocity exceeds the usual speed of light. Michell correctly noted that such supermassive but non-radiating bodies might be detectable through their gravitational effects on nearby visible bodies. Scholars of the time were initially excited by the proposal that giant but invisible 'dark stars' might be hiding in plain view, but enthusiasm dampened when the wavelike nature of light became apparent in the early nineteenth century, as if light were a wave rather than a particle, it was unclear what, if any, influence gravity would have on escaping light waves. | 954 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 785 | 4 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
The modern theory of gravity, general relativity, discredits Michell's notion of a light ray shooting directly from the surface of a supermassive star, being slowed down by the star's gravity, stopping, and then free-falling back to the star's surface. Instead, spacetime itself is curved such that the geodesic that light travels on never leaves the surface of the "star" (black hole). | 386 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 786 | 5 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 1915, Albert Einstein developed his theory of general relativity, having earlier shown that gravity does influence light's motion. Only a few months later, Karl Schwarzschild found a solution to the Einstein field equations that describes the gravitational field of a point mass and a spherical mass. A few months after Schwarzschild, Johannes Droste, a student of Hendrik Lorentz, independently gave the same solution for the point mass and wrote more extensively about its properties. This solution had a peculiar behaviour at what is now called the Schwarzschild radius, where it became singular, meaning that some of the terms in the Einstein equations became infinite. The nature of this surface was not quite understood at the time. | 741 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 787 | 6 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 1924, Arthur Eddington showed that the singularity disappeared after a change of coordinates. In 1933, Georges Lemaître realized that this meant the singularity at the Schwarzschild radius was a non-physical coordinate singularity. Arthur Eddington commented on the possibility of a star with mass compressed to the Schwarzschild radius in a 1926 book, noting that Einstein's theory allows us to rule out overly large densities for visible stars like Betelgeuse because "a star of 250 million km radius could not possibly have so high a density as the Sun. Firstly, the force of gravitation would be so great that light would be unable to escape from it, the rays falling back to the star like a stone to the earth. Secondly, the red shift of the spectral lines would be so great that the spectrum would be shifted out of existence. Thirdly, the mass would produce so much curvature of the spacetime metric that space would close up around the star, leaving us outside (i.e., nowhere)." | 989 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 788 | 7 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 1931, Subrahmanyan Chandrasekhar calculated, using special relativity, that a non-rotating body of electron-degenerate matter above a certain limiting mass (now called the Chandrasekhar limit at 1.4 M ☉) has no stable solutions. His arguments were opposed by many of his contemporaries like Eddington and Lev Landau, who argued that some yet unknown mechanism would stop the collapse. They were partly correct: a white dwarf slightly more massive than the Chandrasekhar limit will collapse into a neutron star, which is itself stable. | 537 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 789 | 8 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 1939, Robert Oppenheimer and others predicted that neutron stars above another limit, the Tolman–Oppenheimer–Volkoff limit, would collapse further for the reasons presented by Chandrasekhar, and concluded that no law of physics was likely to intervene and stop at least some stars from collapsing to black holes. Their original calculations, based on the Pauli exclusion principle, gave it as 0.7 M ☉. Subsequent consideration of neutron-neutron repulsion mediated by the strong force raised the estimate to approximately 1.5 M ☉ to 3.0 M ☉. Observations of the neutron star merger GW170817, which is thought to have generated a black hole shortly afterward, have refined the TOV limit estimate to ~2.17 M ☉. | 711 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 790 | 9 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Oppenheimer and his co-authors interpreted the singularity at the boundary of the Schwarzschild radius as indicating that this was the boundary of a bubble in which time stopped. This is a valid point of view for external observers, but not for infalling observers. The hypothetical collapsed stars were called "frozen stars", because an outside observer would see the surface of the star frozen in time at the instant where its collapse takes it to the Schwarzschild radius. | 475 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 791 | 10 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Also in 1939, Einstein attempted to prove that black holes were impossible in his publication "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses", using his theory of general relativity to defend his argument. Months later, Oppenheimer and his student Hartland Snyder provided the Oppenheimer–Snyder model in their paper "On Continued Gravitational Contraction", which predicted the existence of black holes. In the paper, which made no reference to Einstein's recent publication, Oppenheimer and Snyder used Einstein's own theory of general relativity to show the conditions on how a black hole could develop, for the first time in contemporary physics. | 689 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 792 | 11 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In 1958, David Finkelstein identified the Schwarzschild surface as an event horizon, "a perfect unidirectional membrane: causal influences can cross it in only one direction". This did not strictly contradict Oppenheimer's results, but extended them to include the point of view of infalling observers. Finkelstein's solution extended the Schwarzschild solution for the future of observers falling into a black hole. A complete extension had already been found by Martin Kruskal, who was urged to publish it. | 508 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 793 | 12 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
These results came at the beginning of the golden age of general relativity, which was marked by general relativity and black holes becoming mainstream subjects of research. This process was helped by the discovery of pulsars by Jocelyn Bell Burnell in 1967, which, by 1969, were shown to be rapidly rotating neutron stars. Until that time, neutron stars, like black holes, were regarded as just theoretical curiosities; but the discovery of pulsars showed their physical relevance and spurred a further interest in all types of compact objects that might be formed by gravitational collapse. | 592 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 794 | 13 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
In this period more general black hole solutions were found. In 1963, Roy Kerr found the exact solution for a rotating black hole. Two years later, Ezra Newman found the axisymmetric solution for a black hole that is both rotating and electrically charged. Through the work of Werner Israel, Brandon Carter, and David Robinson the no-hair theorem emerged, stating that a stationary black hole solution is completely described by the three parameters of the Kerr–Newman metric : mass, angular momentum, and electric charge. | 522 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 795 | 14 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
At first, it was suspected that the strange features of the black hole solutions were pathological artifacts from the symmetry conditions imposed, and that the singularities would not appear in generic situations. This view was held in particular by Vladimir Belinsky, Isaak Khalatnikov, and Evgeny Lifshitz, who tried to prove that no singularities appear in generic solutions. However, in the late 1960s Roger Penrose and Stephen Hawking used global techniques to prove that singularities appear generically. For this work, Penrose received half of the 2020 Nobel Prize in Physics, Hawking having died in 2018. Based on observations in Greenwich and Toronto in the early 1970s, Cygnus X-1, a galactic X-ray source discovered in 1964, became the first astronomical object commonly accepted to be a black hole. | 810 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 796 | 15 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
Work by James Bardeen, Jacob Bekenstein, Carter, and Hawking in the early 1970s led to the formulation of black hole thermodynamics. These laws describe the behaviour of a black hole in close analogy to the laws of thermodynamics by relating mass to energy, area to entropy, and surface gravity to temperature. The analogy was completed when Hawking, in 1974, showed that quantum field theory implies that black holes should radiate like a black body with a temperature proportional to the surface gravity of the black hole, predicting the effect now known as Hawking radiation. | 578 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 797 | 16 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
On 11 February 2016, the LIGO Scientific Collaboration and the Virgo collaboration announced the first direct detection of gravitational waves, representing the first observation of a black hole merger. On 10 April 2019, the first direct image of a black hole and its vicinity was published, following observations made by the Event Horizon Telescope (EHT) in 2017 of the supermassive black hole in Messier 87 's galactic centre. As of 2023, the nearest known body thought to be a black hole, Gaia BH1, is around 1,560 light-years (480 parsecs) away. Though only a couple dozen black holes have been found so far in the Milky Way, there are thought to be hundreds of millions, most of which are solitary and do not cause emission of radiation. Therefore, they would only be detectable by gravitational lensing. | 810 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 798 | 17 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 | |
John Michell used the term "dark star" in a November 1783 letter to Henry Cavendish, and in the early 20th century, physicists used the term "gravitationally collapsed object". Science writer Marcia Bartusiak traces the term "black hole" to physicist Robert H. Dicke, who in the early 1960s reportedly compared the phenomenon to the Black Hole of Calcutta, notorious as a prison where people entered but never left alive. | 421 | Black_hole | https://en.wikipedia.org/wiki/Black_hole | 799 | 18 | 2,024 | 8 | 10 | 0 | 31 | 1 | 0 |