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# Superconductivity

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Superconductivity is the set of physical properties observed in certain materials, wherein electrical resistance vanishes and from which magnetic flux fields are expelled. Any material exhibiting these properties is a superconductor. Unlike an ordinary metallic conductor, whose resistance decreases gradually as its temperature is lowered even down to near absolute zero, a superconductor has a characteristic critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source. [1] [2] [3] [4]

The electrical resistance of an object is a measure of its opposition to the flow of electric current. The inverse quantity is electrical conductance, and is the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S).

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge in one or more directions. Materials made of metal are common electrical conductors. Electrical current is generated by the flow of negatively charged electrons, positively charged holes, and positive or negative ions in some cases.

## Contents

This phenomenon was discovered by Dutch physicist Heike Kamerlingh Onnes on April 8, 1911, in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum mechanical mystery. It is characterized by the Meissner effect, the complete ejection of magnetic field lines from the interior of the superconductor during its transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the idealization of perfect conductivity in classical physics.

Professor Heike Kamerlingh Onnes FRSFor HFRSE FCS was a Dutch physicist and Nobel laureate. He exploited the Hampson–Linde cycle to investigate how materials behave when cooled to nearly absolute zero and later to liquefy helium for the first time, in 1908. He was also the discoverer of superconductivity in 1911.

Leiden is a city and municipality in the province of South Holland, Netherlands. The municipality of Leiden had a population of 123,856 in August 2017, but the city forms one densely connected agglomeration with its suburbs Oegstgeest, Leiderdorp, Voorschoten and Zoeterwoude with 206,647 inhabitants. The Netherlands Central Bureau of Statistics (CBS) further includes Katwijk in the agglomeration which makes the total population of the Leiden urban agglomeration 270,879, and in the larger Leiden urban area also Teylingen, Noordwijk, and Noordwijkerhout are included with in total 348,868 inhabitants. Leiden is located on the Oude Rijn, at a distance of some 20 kilometres from The Hague to its south and some 40 km (25 mi) from Amsterdam to its north. The recreational area of the Kaag Lakes (Kagerplassen) lies just to the northeast of Leiden.

Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished. Ferromagnetism is the strongest type and is responsible for the common phenomenon of magnetism in magnets encountered in everyday life. Substances respond weakly to magnetic fields with three other types of magnetism—paramagnetism, diamagnetism, and antiferromagnetism—but the forces are usually so weak that they can only be detected by sensitive instruments in a laboratory. An everyday example of ferromagnetism is a refrigerator magnet used to hold notes on a refrigerator door. The attraction between a magnet and ferromagnetic material is "the quality of magnetism first apparent to the ancient world, and to us today".

In 1986, it was discovered that some cuprate-perovskite ceramic materials have a critical temperature above 90 K (−183 °C). [5] Such a high transition temperature is theoretically impossible for a conventional superconductor, leading the materials to be termed high-temperature superconductors. The cheaply-available coolant liquid nitrogen boils at 77 K, and thus the existence of superconductivity at higher temperatures than this facilitates many experiments and applications that are less practical at lower temperatures.

Cuprate superconductors are high temperature superconductors made of cuprates. They are layered materials, consisting of superconducting CuO2 layers separated by spacer layers.

A perovskite is any material with the same type of crystal structure as calcium titanium oxide (CaTiO3), known as the perovskite structure, or XIIA2+VIB4+X2−3 with the oxygen in the edge centers. Perovskites take their name from the mineral, which was first discovered in the Ural mountains of Russia by Gustav Rose in 1839 and is named after Russian mineralogist L. A. Perovski (1792–1856). The general chemical formula for perovskite compounds is ABX3, where 'A' and 'B' are two cations of very different sizes, and X is an anion that bonds to both. The 'A' atoms are larger than the 'B' atoms. The ideal cubic structure has the B cation in 6-fold coordination, surrounded by an octahedron of anions, and the A cation in 12-fold cuboctahedral coordination. The relative ion size requirements for stability of the cubic structure are quite stringent, so slight buckling and distortion can produce several lower-symmetry distorted versions, in which the coordination numbers of A cations, B cations or both are reduced.

A ceramic is a solid material comprising an inorganic compound of metal, non-metal or metalloid atoms primarily held in ionic and covalent bonds. Common examples are earthenware, porcelain, and brick.

## Classification

There are many criteria by which superconductors are classified. The most common are:

### Response to a magnetic field

A superconductor can be Type I , meaning it has a single critical field, above which all superconductivity is lost and below which the magnetic field is completely expelled from the superconductor; or Type II , meaning it has two critical fields, between which it allows partial penetration of the magnetic field through isolated points. These points are called vortices. Furthermore, in multicomponent superconductors it is possible to have a combination of the two behaviours. In that case the superconductor is of Type-1.5.

Superconductivity is characterized both by perfect conductivity and by the expulsion of magnetic fields. Changes in either temperature or magnetic field can cause the phase transition between normal and superconducting states. For a given temperature, the highest magnetic field under which a material remains superconducting is known as the critical field. The highest temperature under which the superconducting state is seen is known as the critical temperature. At that temperature even the smallest external magnetic field will destroy the superconducting state, so the critical field is zero. As temperature decreases, the critical field increases generally to a maximum at absolute zero.

In physics, a quantum vortex represents a quantized flux circulation of some physical quantity. In most cases quantum vortices are a type of topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was predicted by Lars Onsager in 1947 in connection with superfluid helium. Onsager also pointed out that quantum vortices describe the circulation of superfluid and conjectured that their excitations are responsible for superfluid phase transitions. These ideas of Onsager were further developed by Richard Feynman in 1955 and in 1957 were applied to describe the magnetic phase diagram of type-II superconductors by Alexei Alexeyevich Abrikosov. In 1935 Fritz London published a very closely related work on magnetic flux quantization in superconductors. London's fluxoid can also be viewed as a quantum vortex.

Type-1.5 superconductors are multicomponent superconductors characterized by two or more coherence lengths, at least one of which is shorter than the magnetic field penetration length , and at least one of which is longer. This is in contrast to single-component superconductors, where there is only one coherence length and the superconductor is necessarily either type 1 or type 2. When placed in magnetic field, type-1.5 superconductors should form quantum vortices: magnetic-flux-carrying excitations. They allow magnetic field to pass through superconductors due to a vortex-like circulation of superconducting particles. In type-1.5 superconductors these vortices have long-range attractive, short-range repulsive interaction. As a consequence a type-1.5 superconductor in a magnetic field can form a phase separation into domains with expelled magnetic field and clusters of quantum vortices which are bound together by attractive intervortex forces. The domains of the Meissner state retain the two-component superconductivity, while in the vortex clusters one of the superconducting components is suppressed. Thus such materials should allow coexistence of various properties of type-I and type-II superconductors.

### By theory of operation

It is conventional if it can be explained by the BCS theory or its derivatives, or unconventional , otherwise. [6]

Conventional superconductors are materials that display superconductivity as described by BCS theory or its extensions. This is in contrast to unconventional superconductors, which do not. Conventional superconductors can be either type-I or type-II.

BCS theory or Bardeen–Cooper–Schrieffer theory is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes superconductivity as a microscopic effect caused by a condensation of Cooper pairs into a boson-like state. The theory is also used in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus.

Unconventional superconductors are materials that display superconductivity which does not conform to either the conventional BCS theory or Nikolay Bogolyubov's theory or its extensions.

### By critical temperature

A superconductor is generally considered high-temperature if it reaches a superconducting state above a temperature of 30 K [7] ; as in the initial discovery by Georg Bednorz and K. Alex Müller [5] . It may also reference materials that transition to superconductivity when cooled using liquid nitrogen – that is, at only Tc > 77 K, although this is generally used only to emphasize that liquid nitrogen coolant is sufficient. Low temperature superconductors refer to materials with a critical temperature below 30 K. One exception to this rule is the iron pnictide group of superconductors which display behaviour and properties typical of high temperature superconductors, yet some of the group have critical temperatures below 30K.

High-temperature superconductors are materials that behave as superconductors at unusually high temperatures. The first high-Tc superconductor was discovered in 1986 by IBM researchers Georg Bednorz and K. Alex Müller, who were awarded the 1987 Nobel Prize in Physics "for their important break-through in the discovery of superconductivity in ceramic materials".

Johannes Georg Bednorz is a German physicist who, together with K. Alex Müller, discovered high-temperature superconductivity in ceramics, for which they shared the 1987 Nobel Prize in Physics.

Karl Alexander Müller is a Swiss physicist and Nobel laureate. He received the Nobel Prize in Physics in 1987 with Georg Bednorz for their work in superconductivity in ceramic materials.

### By material

Superconductor material classes include chemical elements (e.g. mercury or lead), alloys (such as niobium-titanium, germanium-niobium, and niobium nitride), ceramics (YBCO and magnesium diboride), superconducting pnictides (like fluorine-doped LaOFeAs) or organic superconductors (fullerenes and carbon nanotubes; though perhaps these examples should be included among the chemical elements, as they are composed entirely of carbon).

## Elementary properties of superconductors

Most of the physical properties of superconductors vary from material to material, such as the heat capacity and the critical temperature, critical field, and critical current density at which superconductivity is destroyed.

On the other hand, there is a class of properties that are independent of the underlying material. For instance, all superconductors have exactly zero resistivity to low applied currents when there is no magnetic field present or if the applied field does not exceed a critical value. The existence of these "universal" properties implies that superconductivity is a thermodynamic phase, and thus possesses certain distinguishing properties which are largely independent of microscopic details.

### Zero electrical DC resistance

The simplest method to measure the electrical resistance of a sample of some material is to place it in an electrical circuit in series with a current source I and measure the resulting voltage V across the sample. The resistance of the sample is given by Ohm's law as R = V / I. If the voltage is zero, this means that the resistance is zero.

Superconductors are also able to maintain a current with no applied voltage whatsoever, a property exploited in superconducting electromagnets such as those found in MRI machines. Experiments have demonstrated that currents in superconducting coils can persist for years without any measurable degradation. Experimental evidence points to a current lifetime of at least 100,000 years. Theoretical estimates for the lifetime of a persistent current can exceed the estimated lifetime of the universe, depending on the wire geometry and the temperature. [3] In practice, currents injected in superconducting coils have persisted for more than 23 years (as in August 2018) in superconducting gravimeters. [8] [9] In such instruments, the measurement principle is based on the monitoring of the levitation of a superconducting niobium sphere with a mass of 4 grams.

In a normal conductor, an electric current may be visualized as a fluid of electrons moving across a heavy ionic lattice. The electrons are constantly colliding with the ions in the lattice, and during each collision some of the energy carried by the current is absorbed by the lattice and converted into heat, which is essentially the vibrational kinetic energy of the lattice ions. As a result, the energy carried by the current is constantly being dissipated. This is the phenomenon of electrical resistance and Joule heating.

The situation is different in a superconductor. In a conventional superconductor, the electronic fluid cannot be resolved into individual electrons. Instead, it consists of bound pairs of electrons known as Cooper pairs. This pairing is caused by an attractive force between electrons from the exchange of phonons. Due to quantum mechanics, the energy spectrum of this Cooper pair fluid possesses an energy gap , meaning there is a minimum amount of energy ΔE that must be supplied in order to excite the fluid. Therefore, if ΔE is larger than the thermal energy of the lattice, given by kT, where k is Boltzmann's constant and T is the temperature, the fluid will not be scattered by the lattice. The Cooper pair fluid is thus a superfluid, meaning it can flow without energy dissipation.

In a class of superconductors known as type II superconductors, including all known high-temperature superconductors, an extremely low but nonzero resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field, which may be caused by the electric current. This is due to the motion of magnetic vortices in the electronic superfluid, which dissipates some of the energy carried by the current. If the current is sufficiently small, the vortices are stationary, and the resistivity vanishes. The resistance due to this effect is tiny compared with that of non-superconducting materials, but must be taken into account in sensitive experiments. However, as the temperature decreases far enough below the nominal superconducting transition, these vortices can become frozen into a disordered but stationary phase known as a "vortex glass". Below this vortex glass transition temperature, the resistance of the material becomes truly zero.

### Phase transition

In superconducting materials, the characteristics of superconductivity appear when the temperature T is lowered below a critical temperature Tc. The value of this critical temperature varies from material to material. Conventional superconductors usually have critical temperatures ranging from around 20  K to less than 1 K. Solid mercury, for example, has a critical temperature of 4.2 K. As of 2015, the highest critical temperature found for a conventional superconductor is 203K for H2S, although high pressures of approximately 90 gigapascals were required. [10] Cuprate superconductors can have much higher critical temperatures: YBa2Cu3O7, one of the first cuprate superconductors to be discovered, has a critical temperature above 90 K, and mercury-based cuprates have been found with critical temperatures in excess of 130 K. The explanation for these high critical temperatures remains unknown. Electron pairing due to phonon exchanges explains superconductivity in conventional superconductors, but it does not explain superconductivity in the newer superconductors that have a very high critical temperature.

Similarly, at a fixed temperature below the critical temperature, superconducting materials cease to superconduct when an external magnetic field is applied which is greater than the critical magnetic field. This is because the Gibbs free energy of the superconducting phase increases quadratically with the magnetic field while the free energy of the normal phase is roughly independent of the magnetic field. If the material superconducts in the absence of a field, then the superconducting phase free energy is lower than that of the normal phase and so for some finite value of the magnetic field (proportional to the square root of the difference of the free energies at zero magnetic field) the two free energies will be equal and a phase transition to the normal phase will occur. More generally, a higher temperature and a stronger magnetic field lead to a smaller fraction of electrons that are superconducting and consequently to a longer London penetration depth of external magnetic fields and currents. The penetration depth becomes infinite at the phase transition.

The onset of superconductivity is accompanied by abrupt changes in various physical properties, which is the hallmark of a phase transition. For example, the electronic heat capacity is proportional to the temperature in the normal (non-superconducting) regime. At the superconducting transition, it suffers a discontinuous jump and thereafter ceases to be linear. At low temperatures, it varies instead as e−α/T for some constant, α. This exponential behavior is one of the pieces of evidence for the existence of the energy gap.

The order of the superconducting phase transition was long a matter of debate. Experiments indicate that the transition is second-order, meaning there is no latent heat. However, in the presence of an external magnetic field there is latent heat, because the superconducting phase has a lower entropy below the critical temperature than the normal phase. It has been experimentally demonstrated [11] that, as a consequence, when the magnetic field is increased beyond the critical field, the resulting phase transition leads to a decrease in the temperature of the superconducting material.

Calculations in the 1970s suggested that it may actually be weakly first-order due to the effect of long-range fluctuations in the electromagnetic field. In the 1980s it was shown theoretically with the help of a disorder field theory, in which the vortex lines of the superconductor play a major role, that the transition is of second order within the type II regime and of first order (i.e., latent heat) within the type I regime, and that the two regions are separated by a tricritical point. [12] The results were strongly supported by Monte Carlo computer simulations. [13]

### Meissner effect

When a superconductor is placed in a weak external magnetic field H, and cooled below its transition temperature, the magnetic field is ejected. The Meissner effect does not cause the field to be completely ejected but instead the field penetrates the superconductor but only to a very small distance, characterized by a parameter λ, called the London penetration depth, decaying exponentially to zero within the bulk of the material. The Meissner effect is a defining characteristic of superconductivity. For most superconductors, the London penetration depth is on the order of 100 nm.

The Meissner effect is sometimes confused with the kind of diamagnetism one would expect in a perfect electrical conductor: according to Lenz's law, when a changing magnetic field is applied to a conductor, it will induce an electric current in the conductor that creates an opposing magnetic field. In a perfect conductor, an arbitrarily large current can be induced, and the resulting magnetic field exactly cancels the applied field.

The Meissner effect is distinct from this—it is the spontaneous expulsion which occurs during transition to superconductivity. Suppose we have a material in its normal state, containing a constant internal magnetic field. When the material is cooled below the critical temperature, we would observe the abrupt expulsion of the internal magnetic field, which we would not expect based on Lenz's law.

The Meissner effect was given a phenomenological explanation by the brothers Fritz and Heinz London, who showed that the electromagnetic free energy in a superconductor is minimized provided

${\displaystyle \nabla ^{2}\mathbf {H} =\lambda ^{-2}\mathbf {H} \,}$

where H is the magnetic field and λ is the London penetration depth.

This equation, which is known as the London equation, predicts that the magnetic field in a superconductor decays exponentially from whatever value it possesses at the surface.

A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value Hc. Depending on the geometry of the sample, one may obtain an intermediate state [14] consisting of a baroque pattern [15] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value Hc1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength Hc2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.

### London moment

Conversely, a spinning superconductor generates a magnetic field, precisely aligned with the spin axis. The effect, the London moment, was put to good use in Gravity Probe B. This experiment measured the magnetic fields of four superconducting gyroscopes to determine their spin axes. This was critical to the experiment since it is one of the few ways to accurately determine the spin axis of an otherwise featureless sphere.

## History of superconductivity

Superconductivity was discovered on April 8, 1911 by Heike Kamerlingh Onnes, who was studying the resistance of solid mercury at cryogenic temperatures using the recently produced liquid helium as a refrigerant. At the temperature of 4.2 K, he observed that the resistance abruptly disappeared. [16] In the same experiment, he also observed the superfluid transition of helium at 2.2 K, without recognizing its significance. The precise date and circumstances of the discovery were only reconstructed a century later, when Onnes's notebook was found. [17] In subsequent decades, superconductivity was observed in several other materials. In 1913, lead was found to superconduct at 7 K, and in 1941 niobium nitride was found to superconduct at 16 K.

Great efforts have been devoted to finding out how and why superconductivity works; the important step occurred in 1933, when Meissner and Ochsenfeld discovered that superconductors expelled applied magnetic fields, a phenomenon which has come to be known as the Meissner effect. [18] In 1935, Fritz and Heinz London showed that the Meissner effect was a consequence of the minimization of the electromagnetic free energy carried by superconducting current. [19]

### London theory

The first phenomenological theory of superconductivity was London theory. It was put forward by the brothers Fritz and Heinz London in 1935, shortly after the discovery that magnetic fields are expelled from superconductors. A major triumph of the equations of this theory is their ability to explain the Meissner effect, [18] wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold. By using the London equation, one can obtain the dependence of the magnetic field inside the superconductor on the distance to the surface. [20]

There are two London equations:

${\displaystyle {\frac {\partial \mathbf {j} _{s}}{\partial t}}={\frac {n_{s}e^{2}}{m}}\mathbf {E} ,\qquad \mathbf {\nabla } \times \mathbf {j} _{s}=-{\frac {n_{s}e^{2}}{m}}\mathbf {B} .}$

The first equation follows from Newton's second law for superconducting electrons.

### Conventional theories (1950s)

During the 1950s, theoretical condensed matter physicists arrived at an understanding of "conventional" superconductivity, through a pair of remarkable and important theories: the phenomenological Ginzburg-Landau theory (1950) and the microscopic BCS theory (1957). [21] [22]

In 1950, the phenomenological Ginzburg-Landau theory of superconductivity was devised by Landau and Ginzburg. [23] This theory, which combined Landau's theory of second-order phase transitions with a Schrödinger-like wave equation, had great success in explaining the macroscopic properties of superconductors. In particular, Abrikosov showed that Ginzburg-Landau theory predicts the division of superconductors into the two categories now referred to as Type I and Type II. Abrikosov and Ginzburg were awarded the 2003 Nobel Prize for their work (Landau had received the 1962 Nobel Prize for other work, and died in 1968). The four-dimensional extension of the Ginzburg-Landau theory, the Coleman-Weinberg model, is important in quantum field theory and cosmology.

Also in 1950, Maxwell and Reynolds et al. found that the critical temperature of a superconductor depends on the isotopic mass of the constituent element. [24] [25] This important discovery pointed to the electron-phonon interaction as the microscopic mechanism responsible for superconductivity.

The complete microscopic theory of superconductivity was finally proposed in 1957 by Bardeen, Cooper and Schrieffer. [22] This BCS theory explained the superconducting current as a superfluid of Cooper pairs, pairs of electrons interacting through the exchange of phonons. For this work, the authors were awarded the Nobel Prize in 1972.

The BCS theory was set on a firmer footing in 1958, when N. N. Bogolyubov showed that the BCS wavefunction, which had originally been derived from a variational argument, could be obtained using a canonical transformation of the electronic Hamiltonian. [26] In 1959, Lev Gor'kov showed that the BCS theory reduced to the Ginzburg-Landau theory close to the critical temperature. [27] [28]

Generalizations of BCS theory for conventional superconductors form the basis for understanding of the phenomenon of superfluidity, because they fall into the lambda transition universality class. The extent to which such generalizations can be applied to unconventional superconductors is still controversial.

### Further history

The first practical application of superconductivity was developed in 1954 with Dudley Allen Buck's invention of the cryotron. [29] Two superconductors with greatly different values of critical magnetic field are combined to produce a fast, simple switch for computer elements.

Soon after discovering superconductivity in 1911, Kamerlingh Onnes attempted to make an electromagnet with superconducting windings but found that relatively low magnetic fields destroyed superconductivity in the materials he investigated. Much later, in 1955, G.B. Yntema [30] succeeded in constructing a small 0.7-tesla iron-core electromagnet with superconducting niobium wire windings. Then, in 1961, J.E. Kunzler, E. Buehler, F.S.L. Hsu, and J.H. Wernick [31] made the startling discovery that, at 4.2 kelvin, a compound consisting of three parts niobium and one part tin, was capable of supporting a current density of more than 100,000 amperes per square centimeter in a magnetic field of 8.8 tesla. Despite being brittle and difficult to fabricate, niobium-tin has since proved extremely useful in supermagnets generating magnetic fields as high as 20 tesla. In 1962 T.G. Berlincourt and R.R. Hake [32] [33] discovered that alloys of niobium and titanium are suitable for applications up to 10 tesla. Promptly thereafter, commercial production of niobium-titanium supermagnet wire commenced at Westinghouse Electric Corporation and at Wah Chang Corporation. Although niobium-titanium boasts less-impressive superconducting properties than those of niobium-tin, niobium-titanium has, nevertheless, become the most widely used “workhorse” supermagnet material, in large measure a consequence of its very-high ductility and ease of fabrication. However, both niobium-tin and niobium-titanium find wide application in MRI medical imagers, bending and focusing magnets for enormous high-energy-particle accelerators, and a host of other applications. Conectus, a European superconductivity consortium, estimated that in 2014, global economic activity for which superconductivity was indispensable amounted to about five billion euros, with MRI systems accounting for about 80% of that total.

In 1962, Josephson made the important theoretical prediction that a supercurrent can flow between two pieces of superconductor separated by a thin layer of insulator. [34] This phenomenon, now called the Josephson effect, is exploited by superconducting devices such as SQUIDs. It is used in the most accurate available measurements of the magnetic flux quantum Φ0 = h/(2e), where h is the Planck constant. Coupled with the quantum Hall resistivity, this leads to a precise measurement of the Planck constant. Josephson was awarded the Nobel Prize for this work in 1973.

In 2008, it was proposed that the same mechanism that produces superconductivity could produce a superinsulator state in some materials, with almost infinite electrical resistance. [35]

## High-temperature superconductivity

Until 1986, physicists had believed that BCS theory forbade superconductivity at temperatures above about 30 K. In that year, Bednorz and Müller discovered superconductivity in a lanthanum-based cuprate perovskite material, which had a transition temperature of 35 K (Nobel Prize in Physics, 1987). [5] It was soon found that replacing the lanthanum with yttrium (i.e., making YBCO) raised the critical temperature above 90 K. [36]

This temperature jump is particularly significant, since it allows liquid nitrogen as a refrigerant, replacing liquid helium. [36] This can be important commercially because liquid nitrogen can be produced relatively cheaply, even on-site. Also, the higher temperatures help avoid some of the problems that arise at liquid helium temperatures, such as the formation of plugs of frozen air that can block cryogenic lines and cause unanticipated and potentially hazardous pressure buildup. [37] [38]

Many other cuprate superconductors have since been discovered, and the theory of superconductivity in these materials is one of the major outstanding challenges of theoretical condensed matter physics. [39] There are currently two main hypotheses – the resonating-valence-bond theory, and spin fluctuation which has the most support in the research community. [40] The second hypothesis proposed that electron pairing in high-temperature superconductors is mediated by short-range spin waves known as paramagnons. [41] [42] [ dubious ]

In 2008, holographic superconductivity, which uses holographic duality or AdS/CFT correspondence theory, was proposed by Gubser, Hartnoll, Herzog, and Horowitz, as a possible explanation of high-temperature superconductivity in certain materials. [43]

Since about 1993, the highest-temperature superconductor has been a ceramic material consisting of mercury, barium, calcium, copper and oxygen (HgBa2Ca2Cu3O8+δ) with Tc = 133–138 K. [44] [45] The latter experiment (138 K) still awaits experimental confirmation, however.

In February 2008, an iron-based family of high-temperature superconductors was discovered. [46] [47] Hideo Hosono, of the Tokyo Institute of Technology, and colleagues found lanthanum oxygen fluorine iron arsenide (LaO1−xFxFeAs), an oxypnictide that superconducts below 26 K. Replacing the lanthanum in LaO1−xFxFeAs with samarium leads to superconductors that work at 55 K. [48]

In May 2014, hydrogen sulfide (H
2
S
) was predicted to be a high-temperature superconductor with a transition temperature of 80 K at 160 gigapascals of pressure. [49] In 2015, H
2
S
has been observed to exhibit superconductivity at below 203 K but at extremely high pressures — around 150 gigapascals. [50]

In 2018, a research team from the Department of Physics, Massachusetts Institute of Technology, discovered superconductivity in bilayer graphene with one layer twisted at an angle of approximately 1.1 degrees with cooling and applying a small electric charge. Even if the experiments were not carried out in a high-temperature environment, the results are correlated less to classical but high temperature superconductors, given that no foreign atoms need to be introduced. [51]

## Applications

Superconducting magnets are some of the most powerful electromagnets known. They are used in MRI/NMR machines, mass spectrometers, the beam-steering magnets used in particle accelerators and plasma confining magnets in some tokamaks. They can also be used for magnetic separation, where weakly magnetic particles are extracted from a background of less or non-magnetic particles, as in the pigment industries.

In the 1950s and 1960s, superconductors were used to build experimental digital computers using cryotron switches. More recently, superconductors have been used to make digital circuits based on rapid single flux quantum technology and RF and microwave filters for mobile phone base stations.

Superconductors are used to build Josephson junctions which are the building blocks of SQUIDs (superconducting quantum interference devices), the most sensitive magnetometers known. SQUIDs are used in scanning SQUID microscopes and magnetoencephalography. Series of Josephson devices are used to realize the SI volt. Depending on the particular mode of operation, a superconductor-insulator-superconductor Josephson junction can be used as a photon detector or as a mixer. The large resistance change at the transition from the normal- to the superconducting state is used to build thermometers in cryogenic micro-calorimeter photon detectors. The same effect is used in ultrasensitive bolometers made from superconducting materials.

Other early markets are arising where the relative efficiency, size and weight advantages of devices based on high-temperature superconductivity outweigh the additional costs involved. For example, in wind turbines the lower weight and volume of superconducting generators could lead to savings in construction and tower costs, offsetting the higher costs for the generator and lowering the total LCOE. [52]

Promising future applications include high-performance smart grid, electric power transmission, transformers, power storage devices, electric motors (e.g. for vehicle propulsion, as in vactrains or maglev trains), magnetic levitation devices, fault current limiters, enhancing spintronic devices with superconducting materials, [53] and superconducting magnetic refrigeration. However, superconductivity is sensitive to moving magnetic fields so applications that use alternating current (e.g. transformers) will be more difficult to develop than those that rely upon direct current. Compared to traditional power lines superconducting transmission lines are more efficient and require only a fraction of the space, which would not only lead to a better environmental performance but could also improve public acceptance for expansion of the electric grid. [54]

## Related Research Articles

Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the constituents are strong. The most familiar examples of condensed phases are solids and liquids, which arise from the electromagnetic forces between atoms. Condensed matter physicists seek to understand the behavior of these phases by using physical laws. In particular, they include the laws of quantum mechanics, electromagnetism and statistical mechanics.

The Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. The German physicists Walther Meissner and Robert Ochsenfeld discovered this phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin and lead samples. The samples, in the presence of an applied magnetic field, were cooled below their superconducting transition temperature, whereupon the samples cancelled nearly all interior magnetic fields. They detected this effect only indirectly because the magnetic flux is conserved by a superconductor: when the interior field decreases, the exterior field increases. The experiment demonstrated for the first time that superconductors were more than just perfect conductors and provided a uniquely defining property of the superconductor state. The ability for the expulsion effect is determined by the nature of equilibrium formed by the neutralization within the unit cell of a superconductor.

Magnesium diboride is the inorganic compound with the formula MgB2. It is a dark gray, water-insoluble solid. The compound has attracted attention because it becomes superconducting at Tc = 39K. In terms of its composition, MgB2 differs strikingly from most superconductors of comparable Tc, which feature transition metals.

The Little–Parks effect was discovered in 1962 by William A. Little and Roland D. Parks in experiments with empty and thin-walled superconducting cylinders subjected to a parallel magnetic field. It was one of the first experiments that indicate the importance of Cooper-pairing principle in BCS theory.

Superconductivity is the phenomenon of certain materials exhibiting zero electrical resistance and the expulsion of magnetic fields below a characteristic temperature. The history of superconductivity began with Dutch physicist Heike Kamerlingh Onnes's discovery of superconductivity in mercury in 1911. Since then, many other superconducting materials have been discovered and the theory of superconductivity has been developed. These subjects remain active areas of study in the field of condensed matter physics.

Cryogenic particle detectors operate at very low temperature, typically only a few degrees above absolute zero. These sensors interact with an energetic elementary particle and deliver a signal that can be related to the type of particle and the nature of the interaction. While many types of particle detectors might be operated with improved performance at cryogenic temperatures, this term generally refers to types that take advantage of special effects or properties occurring only at low temperature.

In superconductivity, a type-II superconductor is characterized by the formation of magnetic vortices in an applied magnetic field. This occurs above a certain critical field strength Hc1. The vortex density increases with increasing field strength. At a higher critical field Hc2, superconductivity is destroyed. Type-II superconductors do not exhibit a complete Meissner effect.

A Josephson junction is a quantum mechanical device, which is made of two superconducting electrodes separated by a barrier. A π Josephson junction is a Josephson junction in which the Josephson phase φ equals π in the ground state, i.e. when no external current or magnetic field is applied.

A charge density wave (CDW) is an ordered quantum fluid of electrons in a linear chain compound or layered crystal. The electrons within a CDW form a standing wave pattern and sometimes collectively carry an electric current. The electrons in such a CDW, like those in a superconductor, can flow through a linear chain compound en masse, in a highly correlated fashion. Unlike a superconductor, however, the electric CDW current often flows in a jerky fashion, much like water dripping from a faucet due to its electrostatic properties. In a CDW, the combined effects of pinning and electrostatic interactions likely play critical roles in the CDW current's jerky behavior, as discussed in sections 4 & 5 below.

Superconductors can be classified in accordance with several criteria that depend on physical properties, current understanding, how expensive is cooling them or their material.

Scanning SQUID microscopy is a technique where a superconducting quantum interference device (SQUID) is used to image surface magnetic field strength with micrometre scale resolution. A tiny SQUID is mounted onto a tip which is then rastered near the surface of the sample to be measured. As the SQUID is the most sensitive detector of magnetic fields available and can be constructed at submicrometre widths via lithography, the scanning SQUID microscope allows magnetic fields to be measured with unparalleled resolution and sensitivity. The first scanning SQUID microscope was built in 1992 by Black et al. Since then the technique has been used to confirm unconventional superconductity in several high-temperature superconductors including YBCO and BSCCO compounds.

The 122 iron arsenide unconventional superconductors are part of a new class of iron-based superconductors. They form in the tetragonal I4/mmm, ThCr2Si2 type, crystal structure. The shorthand name "122" comes from their stoichiometry; the 122s have the chemical formula AEFe2Pn2, where AE stands for alkaline earth metal (Ca, Ba, Sr or Eu) and Pn is pnictide (As, P, etc.). These materials become superconducting under pressure and also upon doping. The maximum superconducting transition temperature found to date is 38 K in the Ba0.6K0.4Fe2As2. The microscopic description of superconductivity in the 122s is yet unclear.

Flux pumping is a method for magnetising superconductors to fields in excess of 15 teslas. The method can be applied to any type II superconductor and exploits a fundamental property of superconductors. That is their ability to support and maintain currents on the length scale of the superconductor. Conventional magnetic materials are magnetised on a molecular scale which means that superconductors can maintain a flux density orders of magnitude bigger than conventional materials. Flux pumping is especially significant when one bears in mind that all other methods of magnetising superconductors require application of a magnetic flux density at least as high as the final required field. This is not true of flux pumping.

Macroscopic quantum phenomena refer to processes showing quantum behavior at the macroscopic scale, rather than at the atomic scale where quantum effects are prevalent. The best-known examples of macroscopic quantum phenomena are superfluidity and superconductivity; other examples include the quantum Hall effect. Since 2000 there has been extensive experimental work on quantum gases, particularly Bose–Einstein Condensates.

Several hundred metals, compounds, alloys and ceramics possess the property of superconductivity at low temperatures. The SU(2) color quark matter adjoins the list of superconducting systems. Although it is a mathematical abstraction, its properties are believed to be closely related to the SU(3) color quark matter, which exists in nature when ordinary matter is compressed at supranuclear densities above ~ 0.5 1039 nucleon/cm3.

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text:
# Properties

 Modulus 640 Structure $$C_{32}\times C_{4}\times C_{2}$$ Order 256

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(640)

pari: g = idealstar(,640,2)

## Character group

 sage: G.order()  pari: g.no Order = 256 sage: H.invariants()  pari: g.cyc Structure = $$C_{32}\times C_{4}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{640}(261,\cdot)$, $\chi_{640}(257,\cdot)$, $\chi_{640}(511,\cdot)$

## First 32 of 256 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

orbit label order primitive -1 1 3 7 9 11 13 17 19 21 23 27
$$\chi_{640}(1,\cdot)$$ 640.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{640}(3,\cdot)$$ 640.bl 32 yes $$1$$ $$1$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$
$$\chi_{640}(7,\cdot)$$ 640.bj 16 no $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$
$$\chi_{640}(9,\cdot)$$ 640.bf 16 no $$1$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{640}(11,\cdot)$$ 640.bp 32 no $$-1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$
$$\chi_{640}(13,\cdot)$$ 640.bk 32 yes $$-1$$ $$1$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{31}{32}\right)$$
$$\chi_{640}(17,\cdot)$$ 640.bb 8 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{640}(19,\cdot)$$ 640.bn 32 yes $$-1$$ $$1$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$
$$\chi_{640}(21,\cdot)$$ 640.bm 32 no $$1$$ $$1$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$
$$\chi_{640}(23,\cdot)$$ 640.bj 16 no $$1$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{640}(27,\cdot)$$ 640.bl 32 yes $$1$$ $$1$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$
$$\chi_{640}(29,\cdot)$$ 640.bo 32 yes $$1$$ $$1$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$
$$\chi_{640}(31,\cdot)$$ 640.r 4 no $$-1$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$-i$$ $$1$$ $$i$$ $$i$$ $$1$$ $$-i$$
$$\chi_{640}(33,\cdot)$$ 640.i 4 no $$-1$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$-i$$ $$-i$$ $$-1$$
$$\chi_{640}(37,\cdot)$$ 640.bk 32 yes $$-1$$ $$1$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$
$$\chi_{640}(39,\cdot)$$ 640.bh 16 no $$-1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{640}(41,\cdot)$$ 640.be 16 no $$1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$
$$\chi_{640}(43,\cdot)$$ 640.br 32 yes $$1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$
$$\chi_{640}(47,\cdot)$$ 640.u 8 no $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{640}(49,\cdot)$$ 640.z 8 no $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{640}(51,\cdot)$$ 640.bp 32 no $$-1$$ $$1$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{32}\right)$$
$$\chi_{640}(53,\cdot)$$ 640.bq 32 yes $$-1$$ $$1$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{32}\right)$$
$$\chi_{640}(57,\cdot)$$ 640.bi 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$
$$\chi_{640}(59,\cdot)$$ 640.bn 32 yes $$-1$$ $$1$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$
$$\chi_{640}(61,\cdot)$$ 640.bm 32 no $$1$$ $$1$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{32}\right)$$
$$\chi_{640}(63,\cdot)$$ 640.o 4 no $$1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$-i$$
$$\chi_{640}(67,\cdot)$$ 640.br 32 yes $$1$$ $$1$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$
$$\chi_{640}(69,\cdot)$$ 640.bo 32 yes $$1$$ $$1$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$
$$\chi_{640}(71,\cdot)$$ 640.bg 16 no $$-1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{640}(73,\cdot)$$ 640.bi 16 no $$-1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$
$$\chi_{640}(77,\cdot)$$ 640.bq 32 yes $$-1$$ $$1$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{31}{32}\right)$$
$$\chi_{640}(79,\cdot)$$ 640.y 8 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$

format: plain


text:
Speaker: Ji Suk So, UNIST
When: June 2nd, Wendesday, 2021 16:00--18:00
Where: Building 110, Room N104

Abstract: The aim of this talk is to explain how to construct $p$-adic $L$-function and introduce basic theories related p-adic number theory. Using these theory, we studied some theorem about p-adic zeta function. Finally, we give some problems about p-adic zeta function for our future work.


text:
The open source CFD toolbox
Go to the documentation of this file.
1/*---------------------------------------------------------------------------*\
2 ========= |
3 \\ / F ield | OpenFOAM: The Open Source CFD Toolbox
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31// * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * * * //
32
33template<class Type>
35(
36 const dictionary& spec
37)
38{
39 const word readerType = spec.getOrDefault<word>
40 (
42 "openFoam"
43 );
44
46
47 if (!ctorPtr)
48 {
50 (
51 spec,
54 *dictionaryConstructorTablePtr_
55 ) << exit(FatalIOError);
56 }
57
59}
60
61
62// * * * * * * * * * * * * * * * * Constructors * * * * * * * * * * * * * * //
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69// * * * * * * * * * * * * * * * Member Functions * * * * * * * * * * * * * //
70
71template<class Type>
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An Ostream is an abstract base class for all output systems (streams, files, token lists,...
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Construct (dummy) Time - no functionObjects or libraries.
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Write the output fields.


text: THE METHOD OF SECTIONS In the method of sections, a truss is divided into two parts by taking an imaginary “cut” (shown here as a-a) through the truss. trailer << /Size 266 /Info 244 0 R /Root 247 0 R /Prev 419168 /ID[<3b35815937c66a182b685eab7871658a>] >> startxref 0 %%EOF 247 0 obj << /Type /Catalog /Pages 242 0 R /Metadata 245 0 R /Outlines 73 0 R /OpenAction [ 249 0 R /Fit ] /PageMode /UseNone /PageLayout /SinglePage /PageLabels 240 0 R /StructTreeRoot 248 0 R /PieceInfo << /MarkedPDF << /LastModified (D:20030923161743)>> >> /LastModified (D:20030923161743) /MarkInfo << /Marked true /LetterspaceFlags 0 >> >> endobj 248 0 obj << /Type /StructTreeRoot /ParentTree 92 0 R /ParentTreeNextKey 17 /K [ 96 0 R 104 0 R 111 0 R 118 0 R 126 0 R 133 0 R 140 0 R 149 0 R 158 0 R 166 0 R 172 0 R 180 0 R 190 0 R 200 0 R 211 0 R 222 0 R 233 0 R ] /RoleMap 238 0 R >> endobj 264 0 obj << /S 448 /O 525 /L 541 /C 557 /Filter /FlateDecode /Length 265 0 R >> stream 0000002277 00000 n Problems and Solutions in Real and Complex Analysis, Integration, Functional Equations and Inequalities by Willi-Hans Steeb International School for Scienti c Computing Preface The purpose of this book is to supply a collection of Demand engt’s utility function is U(x1, x2)= x1 + ln x 2 x 1 - stamps x 2 - beer Bengts budget p1 x 1 The Simplex Method A-5 The Simplex Method Finally, consider an example wheres 1 0 and s 2 0. 0000020469 00000 n P-428. Chapter 6 : Surface Integrals Here are a set of practice problems for the Surface Integrals chapter of the Calculus III notes. This will give us the boundary conditions we need to progress in solving the structure. Read moreabout In this unit, you will again use some of Most sections should have a range of difficulty levels in the problems although this will vary from section to section. $F_{BE} = 150.78 \, \text{ kN tension}$           answer, $F_{BC} = 120 \, \text{ kN compression}$           answer, $F_{DE} = 64 \, \text{ kN tension}$           answer, Method of Sections | Analysis of Simple Trusses, Method of Joints | Analysis of Simple Trusses, Problem 417 - Roof Truss by Method of Sections, Problem 418 - Warren Truss by Method of Sections, Problem 419 - Warren Truss by Method of Sections, Problem 420 - Howe Truss by Method of Sections, Problem 421 - Cantilever Truss by Method of Sections, Problem 422 - Right-triangular Truss by Method of Sections, Problem 423 - Howe Roof Truss by Method of Sections, Problem 424 - Method of Joints Checked by Method of Sections, Problem 425 - Fink Truss by Method of Sections, Problem 426 - Fink Truss by Method of Sections, Problem 427 - Interior Members of Nacelle Truss by Method of Sections, Problem 428 - Howe Truss by Method of Sections, Problem 429 - Cantilever Truss by Method of Sections, Problem 430 - Parker Truss by Method of Sections, Problem 431 - Members in the Third Panel of a Parker Truss, Problem 432 - Force in Members of a Truss by Method of Sections, Problem 433 - Scissors Truss by Method of Sections, Problem 434 - Scissors Truss by Method of Sections, Problem 435 - Transmission Tower by Method of Sections, Problem 436 - Howe Truss With Counter Braces, Problem 437 - Truss With Counter Diagonals, Problem 438 - Truss With Redundant Members, Method of Members | Frames Containing Three-Force Members. KINETICS Practice Problems and Solutions Determining rate law from Initial Rates. (Use the ratio of initial rates to get the orders). 500 lb. Mathematical Economics Practice Problems and Solutions – Second Edition – G. Stolyarov II 1 MatheMatical econoMics Practice ProbleMs and solutions Second Edition G. Stolyarov II, ASA, ACAS, MAAA, CPCU, ARe, ARC 2. minimization problems with any combination of ≤, ≥, = constraints 2 Example Maximize P = 2x 1 + x 2 subject to x 1 + x 2 < 10 –x 1 + x 2 > 2 0 3 x 1, x 2 > To form an equation out of the first inequality, we introduce a s 1x 1 + 0000002307 00000 n CS Topics covered : Greedy … Problem 003-ms C++ Solved programs, problems/Examples with solutions. Consider the table of initial rates for the reaction: 2ClO 2 + 2OH 1- … Useful solutions for standard problems Preface Modelling is a key part of design. 0000023226 00000 n Open Digital Education.Data for CBSE, GCSE, ICSE and Indian state boards. MethodofSections The Method of Sections involves analytically cutting the truss into sections and solving for static equilibrium for each section. 0000001891 00000 n 6.4 THE METHOD OF SECTIONS In the method of sections, a truss is divided into two parts by taking an imaginary “cut” (shown here as a-a) through the truss. This engineering statics tutorial goes over a method of sections example problem for truss analysis. on scalar ordinary and partial we focus onlinearrL Example problem using method of sections for truss analysis - statics and structural analysis. Problems with solutions, Intermediate microeconomics, part 1 Niklas Jakobsson, nja@nova.no Katarina.Katz@kau.se Problem 1. In the early stage, approximate modelling establishes whether the … Principles of Statics Equilibrium of Force System 800 lb. 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text:
# How do you simplify m+m-m+m?

##### 1 Answer
Mar 23, 2018

The answer is $2 m$.

#### Explanation:

$m + m - m + m$

As we know, $m - m$ is equal to zero, so

$m + m = 2 m$


text:
trans_dsp16_twiddle_factors.c File Reference
#include "trans_dsp16_twiddle_factors.h"

## Macros

#define DSP16_Q_CONVERT(x)   DSP16_Q((x)/2.)
This macro is used to convert the raw data according to the algorithm's optimization we need. More...

## Variables

1024 elements twiddle factors table
A_ALIGNED
TWIDDLE_FACTORS_PREFIX_TAB
dsp16_t
dsp16_twiddle_factors [DSP16_N_TWIDDLE_FACTORS/2+2]
This table has been generated using the following algorithm: More...

A_ALIGNED
TWIDDLE_FACTORS_PREFIX_TAB
dsp16_t
dsp16_twiddle_factors2 [DSP16_N_TWIDDLE_FACTORS]
This table has been generated using the following algorithm: More...

 #define DSP16_Q_CONVERT ( x ) DSP16_Q((x)/2.)

This macro is used to convert the raw data according to the algorithm's optimization we need.

 A_ALIGNED TWIDDLE_FACTORS_PREFIX_TAB dsp16_t dsp16_twiddle_factors[DSP16_N_TWIDDLE_FACTORS/2+2]

This table has been generated using the following algorithm:

w = exp(-2*PI*%i*k);
end;

It is a one dimensional array containing the real parts (even parts) and imaginary parts (odd parts) of the w value.

Referenced by dsp16_trans_complexfft(), dsp16_trans_complexifft(), and dsp16_trans_realcomplexfft().

 A_ALIGNED TWIDDLE_FACTORS_PREFIX_TAB dsp16_t dsp16_twiddle_factors2[DSP16_N_TWIDDLE_FACTORS]

This table has been generated using the following algorithm:

w2 = exp(-2*PI*%i*k*2);
w3 = exp(-2*PI*%i*k*3);
end;

It is a one dimensional array containing the real parts (4*i parts) and imaginary parts (4*i+1 parts) of the w2 value and the real parts (4*i+2 parts) and imaginary parts (4*i+3 parts) of the w3 value.

Referenced by dsp16_trans_complexfft(), dsp16_trans_complexifft(), and dsp16_trans_realcomplexfft().