Pauli’s exclusion principle

Physicists produce video of time crystal in action 😱

Have you heard of time crystals?

A crystal is any object whose atoms are arranged in a fixed pattern in space, with the pattern repeating itself. So what we typically know to be crystals are really space crystals. We didn’t have to bother with the prefix because space crystals were the only kind of crystals we knew until time crystals came along.

Time crystals are crystalline objects whose atoms exhibit behaviour that repeats itself in time, as periodic events. The atoms of a time crystal spin in a fixed and coordinated pattern, changing direction at fixed intervals.

Physicists sometimes prefer to quantify these spin patterns as quasiparticles to simplify their calculations. Quasiparticles are not particles per se. To understand what they are, consider a popular one called phonons. Say you strike a metal spoon on the table, producing a mild ringing sound. This sound is the result of sound waves propagating through the metal’s grid of atoms, carrying vibrational energy. You could also understand each wave to be a particle instead, carrying the same amount of energy that each sound wave carries. These quasiparticles are called phonons.

In the same way, patterns of spinning charged particles also carry some energy. Each electron in an atom, for example, generates a tiny magnetic field around itself as it spins. The directions in which the electrons in a material spin collectively determine many properties of the material’s macroscopic magnetic field. Sometimes, shifts in some electrons’ magnetic fields could set off a disturbance in the macroscopic field – like waves of magnetic energy rippling out. You could quantify these ‘spin waves’ in the form of quasiparticles called magnons. Note that magnons quantify spin waves; the waves themselves can be from electrons, ions or other charged particles.

As quasiparticles, magnons behave like a class of particles called bosons – which are nature’s force-carriers. Photons are bosons that mediate the electromagnetic force; W and Z bosons mediate the weak nuclear force responsible for radioactivity; gluons mediate the strong nuclear force, which carries the energy you see released by nuclear weapons; scientists have hypothesised the existence of gravitons, for gravity, but haven’t found them yet. Like all bosons, magnons don’t obey Pauli’s exclusion principle and they can be made to form exotic states of matter like superfluids and Bose-Einstein condensates.

Other quasiparticles include excitons and polarons (useful in the study of electronic circuits), plasmons (of plasma) and polaritons (of light-matter interactions).

Physicist Frank Wilczek proposed the existence of time crystals in 2012. One reason time crystals are interesting to physicists is that they break time-translation symmetry in their ground state.

This statement has two important parts. The first concerns time-translation symmetry-breaking. Scientists assume the laws of physics are the same in all directions – yet we still have objects like crystals, whose atoms are arranged in specific patterns that repeat themselves. Say the atoms of a crystal are arranged in a hexagonal pattern. If you kept the position of one atom fixed and rotated the atomic lattice around it or if you moved to the left or right of that atom, in both cases by an arbitrary amount, your view of the lattice will also change. This happens because crystals break spatial symmetry. Similarly, time symmetry is broken if an event repeats itself in time – like, say, a magnetic field whose structure changes between two shapes over and over.

The second part of the statement concerns the (thermodynamic) ground state – the state of any quantum mechanical system when it has its lowest possible energy. (‘Quantum mechanical system’ is a generic term for any system – like a group of electrons – in which quantum mechanical effects have the dominant influence on the system’s state and behaviour. An example of a non-quantum-mechanical system is the Solar System, where gravity dominates.) Wilczek revived interest in time crystals as objects that break time-translation symmetry in their ground states. Put another way, they are quantum mechanical systems whose constituent particles perform a periodic activity without changing the overall energy of the system.

The advent of quantum mechanics and relativity theory in the early 20th century alerted physicists to the existence of various symmetries and, through the work of Emmy Noether, their connection to different conservation laws. For example, a system in which the laws of nature were the same throughout history and will be in future – i.e. preserves time-translation symmetry – will also conserve energy. Does this mean time crystals violate the law of conservation of energy? No. The atoms’ or electrons’ spin is not the result of the electrons’ or atoms’ kinetic energy but is an inherent quantum mechanical property. This energy can’t be used to perform work the same way, say, a motor can pump water. The system’s total energy is still conserved.

Now, physicists from Germany have reported that they have observed a time crystal ‘in action’ – a feat notable on three levels. First, it’s impressive that they have created a time crystal in the first place (even if they are not the first to do so). The researchers passed radio frequency waves through a strip of nickel-iron alloy a few micrometers wide. According to ScienceAlert, this ‘current’ “produced an oscillating magnetic field on the strip, with magnetic waves travelling onto it from both ends”. As a result, they “stimulated the magnons in the strip, and these moving magnons then condensed into a repeating pattern”.

Second, while quasiparticles are not actual particles per se, they exhibit some properties of particles. One of them is scattering, like two billiard balls might bounce off each other to go off in different directions at different speeds. Similarly, the researchers created more magnons and scattered them off the magnons involved in the repeating pattern. The post-scatter magnons had a shorter wavelength than they did originally, in line with expectations, and the researchers also found that they could control this wavelength by adjusting the frequency of the stimulating radio waves.

An ability to control such values often means the process could have an application. The ability to precisely manipulate systems involving the spin of electrons has evolved into a field called spintronics. Like electronics makes use of the electrical properties of subatomic particles, spintronics is expected to leverage spin-related properties and enable ultra-fast hard-drives and other technologies.

Third, the researchers were able to produce a video showing the magnons moving around. This is remarkable because the thing that makes a time crystal so unique is the result of quantum mechanical processes, which are microscopic in nature. It’s not often that you can observe their effects on the macroscopic scale. The principal reason the researchers were able achieve this is feat is the method they used to create the time crystal.

Previous efforts to create time crystals have used systems like quantum gases and Bose-Einstein condensates, both of which require sophisticated apparatuses to work with, in ultra-cold conditions, and whose behaviour researchers can track only by carefully measuring their physical and other properties. On the other hand, the current experiment works at room temperature and uses a more ‘straightforward’ setup that is also fairly large-scale – enough to be visible under an X-ray microscope.

Working this microscope is no small feat, however. Charged particles emit radiation when they’re accelerated along a circular path. An accelerator called BESSY II in Berlin uses this principle to produce X-rays. Then the microscope, called MAXYMUS, focuses the X-rays onto an extremely small spot – a few nanometers wide – and “scans across the sample”, according to its official webpage. A “variety of X-ray detectors”, including a camera, observe how the X-rays interact with the sample to produce the final images. Here’s the resulting video of the time crystal, captured at 40 billion frames per second:

I asked one of the paper’s coauthors, Joachim Gräfe, a research group leader in the department of modern magnetic systems at the Max Planck Institute for Intelligent Systems, Stuttgart, two follow-up questions. He was kind enough to reply in detail; his answers are reproduced in full below:

A time crystal represents a system that breaks time translation symmetry in its ground state. When you use radio-frequency waves to stimulate the magnons in the nickel-iron alloy, the system is no longer in its ground state – right?

The ground state debate is the interesting part of the discussion for theoreticians. Our paper is more about the experimental observation and an interaction towards a use case. It is argued that a time crystal cannot be a thermodynamic ground state. However, it is in a ground state in a periodically alternating potential, i.e. a dynamic ground state. The intriguing thing about time crystals is that they are in ground states in these periodically alternating potentials, but they do not/will not necessarily have the same periodicity as the alternating potential.
The condensation of the magnonic time crystal is a ground state of the system in the presence of the RF field (the periodically alternating potential), but it will dissipate through damping when the RF field is switched off. However, even in a system without damping, it would not form without the RF field. It really needs the periodically alternating potential. It is really a requirement to have a dynamic system to have a time crystal. I hope I have not confused you more than before my answer. Time crystals are quite mind boggling. 😵🤯

Previous experiments to observe time crystals in action have used sophisticated systems like quantum gases and Bose-Einstein condensates (BECs). Your experiment’s setup is a lot more straightforward, in a manner of speaking. Why do you think previous research teams didn’t just use your setup? Or does your setup have any particular difficulty that you overcame in the course of your study?

Interesting question. With the benefit of hindsight: our time crystal is quite obvious, why didn’t anybody else do it? Magnons only recently have emerged … as a sandbox for bosonic quantum effects (indeed, you can show BEC and superfluidity for magnons as well). So it is quite straightforward to turn towards magnons as bosons for these studies. However, our X-ray microscope (at the synchrotron light source) was probably the only instrument at the time to have the required spatial and temporal resolution with magnetic contrast to shoot a video of the space-time crystal. Most other magnon detection methods (in the lab) are indirect and don’t yield such a nice video.
On the other hand, I believe that the interesting thing about our paper is not that it was incredibly difficult to observe the space time crystal, but that it is rather simple to create one. Apparently, you can easily create a large (magnonic) space time crystal at room temperature and do something with it. Showing that it is easy to create a space time crystal opens this effect up for technological exploitation.

The awesome limits of superconductors

On June 24, a press release from CERN said that scientists and engineers working on upgrading the Large Hadron Collider (LHC) had “built and operated … the most powerful electrical transmission line … to date”. The transmission line consisted of four cables – two capable of transporting 20 kA of current and two, 7 kA.

The ‘A’ here stands for ‘ampere’, the SI unit of electric current. Twenty kilo-amperes is an extraordinary amount of current, nearly equal to the amount in a single lightning strike.

In the particulate sense: one ampere is the flow of one coulomb per second. One coulomb is equal to around 6.24 quintillion elementary charges, where each elementary charge is the charge of a single proton or electron (with opposite signs). So a cable capable of carrying a current of 20 kA can essentially transport 124.8 sextillion electrons per second.

According to the CERN press release (emphasis added):

The line is composed of cables made of magnesium diboride (MgB2), which is a superconductor and therefore presents no resistance to the flow of the current and can transmit much higher intensities than traditional non-superconducting cables. On this occasion, the line transmitted an intensity 25 times greater than could have been achieved with copper cables of a similar diameter. Magnesium diboride has the added benefit that it can be used at 25 kelvins (-248 °C), a higher temperature than is needed for conventional superconductors. This superconductor is more stable and requires less cryogenic power. The superconducting cables that make up the innovative line are inserted into a flexible cryostat, in which helium gas circulates.

The part in bold could have been more explicit and noted that superconductors, including magnesium diboride, can’t carry an arbitrarily higher amount of current than non-superconducting conductors. There is actually a limit for the same reason why there is a limit to the current-carrying capacity of a normal conductor.

This explanation wouldn’t change the impressiveness of this feat and could even interfere with readers’ impression of the most important details, so I can see why the person who drafted the statement left it out. Instead, I’ll take this matter up here.

An electric current is generated between two points when electrons move from one point to the other. The direction of current is opposite to the direction of the electrons’ movement. A metal that conducts electricity does so because its constituent atoms have one or more valence electrons that can flow throughout the metal. So if a voltage arises between two ends of the metal, the electrons can respond by flowing around, birthing an electric current.

This flow isn’t perfect, however. Sometimes, a valence electron can bump into atomic nuclei, impurities – atoms of other elements in the metallic lattice – or be thrown off course by vibrations in the lattice of atoms, produced by heat. Such disruptions across the metal collectively give rise to the metal’s resistance. And the more resistance there is, the less current the metal can carry.

These disruptions often heat the metal as well. This happens because electrons don’t just flow between the two points across which a voltage is applied. They’re accelerated. So as they’re speeding along and suddenly bump into an impurity, they’re scattered into random directions. Their kinetic energy then no longer contributes to the electric energy of the metal and instead manifests as thermal energy – or heat.

If the electrons bump into nuclei, they could impart some of their kinetic energy to the nuclei, causing the latter to vibrate more, which in turn means they heat up as well.

Copper and silver have high conductance because they have more valence electrons available to conduct electricity and these electrons are scattered to a lesser extent than in other metals. Therefore, these two also don’t heat up as quickly as other metals might, allowing them to transport a higher current for longer. Copper in particular has a higher mean free path: the average distance an electron travels before being scattered.

In superconductors, the picture is quite different because quantum physics assumes a more prominent role. There are different types of superconductors according to the theories used to understand how they conduct electricity with zero resistance and how they behave in different external conditions. The electrical behaviour of magnesium diboride, the material used to transport the 20 kA current, is described by Bardeen-Cooper-Schrieffer (BCS) theory.

According to this theory, when certain materials are cooled below a certain temperature, the residual vibrations of their atomic lattice encourages their valence electrons to overcome their mutual repulsion and become correlated, especially in terms of their movement. That is, the electrons pair up.

While individual electrons belong to a class of particles called fermions, these electron pairs – a.k.a. Cooper pairs – belong to another class called bosons. One difference between these two classes is that bosons don’t obey Pauli’s exclusion principle: that no two fermions in the same quantum system (like an atom) can have the same set of quantum numbers at the same time.

As a result, all the electron pairs in the material are now free to occupy the same quantum state – which they will when the material is supercooled. When they do, the pairs collectively make up an exotic state of matter called a Bose-Einstein condensate: the electron pairs now flow through the material as if they were one cohesive liquid.

In this state, even if one pair gets scattered by an impurity, the current doesn’t experience resistance because the condensate’s overall flow isn’t affected. In fact, given that breaking up one pair will cause all other pairs to break up as well, the energy required to break up one pair is roughly equal to the energy required to break up all pairs. This feature affords the condensate a measure of robustness.

But while current can keep flowing through a BCS superconductor with zero resistance, the superconducting state itself doesn’t have infinite persistence. It can break if it stops being cooled below a specific temperature, called the critical temperature; if the material is too impure, contributing to a sufficient number of collisions to ‘kick’ all electrons pairs out of their condensate reverie; or if the current density crosses a particular threshold.

At the LHC, the magnesium diboride cables will be wrapped around electromagnets. When a large current flows through the cables, the electromagnets will produce a magnetic field. The LHC uses a circular arrangement of such magnetic fields to bend the beam of protons it will accelerate into a circular path. The more powerful the magnetic field, the more accurate the bending. The current operational field strength is 8.36 tesla, about 128,000-times more powerful than Earth’s magnetic field. The cables will be insulated but they will still be exposed to a large magnetic field.

Type I superconductors completely expel an external magnetic field when they transition to their superconducting state. That is, the magnetic field can’t penetrate the material’s surface and enter the bulk. Type II superconductors are slightly more complicated. Below one critical temperature and one critical magnetic field strength, they behave like type I superconductors. Below the same temperature but a slightly stronger magnetic field, they are superconducting and allow the fields to penetrate their bulk to a certain extent. This is called the mixed state.

A hand-drawn phase diagram showing the conditions in which a mixed-state type II superconductor exists. Credit: Frederic Bouquet/Wikimedia Commons, CC BY-SA 3.0

Say a uniform magnetic field is applied over a mixed-state superconductor. The field will plunge into the material’s bulk in the form of vortices. All these vortices will have the same magnetic flux – the number of magnetic field lines per unit area – and will repel each other, settling down in a triangular pattern equidistant from each other.

An annotated image of vortices in a type II superconductor. The scale is specified at the bottom right. Source: A set of slides entitled ‘Superconductors and Vortices at Radio Frequency Magnetic Fields’ by Ernst Helmut Brandt, Max Planck Institute for Metals Research, October 2010.

When an electric current passes through this material, the vortices are slightly displaced, and also begin to experience a force proportional to how closely they’re packed together and their pattern of displacement. As a result, to quote from this technical (yet lucid) paper by Praveen Chaddah:

This force on each vortex … will cause the vortices to move. The vortex motion produces an electric field¹ parallel to [the direction of the existing current], thus causing a resistance, and this is called the flux-flow resistance. The resistance is much smaller than the normal state resistance, but the material no longer [has] infinite conductivity.

1. According to Maxwell’s equations of electromagnetism, a changing magnetic field produces an electric field.

Since the vortices’ displacement depends on the current density: the greater the number of electrons being transported, the more flux-flow resistance there is. So the magnesium diboride cables can’t simply carry more and more current. At some point, setting aside other sources of resistance, the flux-flow resistance itself will damage the cable.

There are ways to minimise this resistance. For example, the material can be doped with impurities that will ‘pin’ the vortices to fixed locations and prevent them from moving around. However, optimising these solutions for a given magnetic field and other conditions involves complex calculations that we don’t need to get into.

The point is that superconductors have their limits too. And knowing these limits could improve our appreciation for the feats of physics and engineering that underlie achievements like cables being able to transport 124.8 sextillion electrons per second with zero resistance. In fact, according to the CERN press release,

The [line] that is currently being tested is the forerunner of the final version that will be installed in the accelerator. It is composed of 19 cables that supply the various magnet circuits and could transmit intensities of up to 120 kA!

While writing this post, I was frequently tempted to quote from Lisa Randall‘s excellent book-length introduction to the LHC, Knocking on Heaven’s Door (2011). Here’s a short excerpt:

One of the most impressive objects I saw when I visited CERN was a prototype of LHC’s gigantic cylindrical dipole magnets. Event with 1,232 such magnets, each of them is an impressive 15 metres long and weighs 30 tonnes. … Each of these magnets cost EUR 700,000, making the ned cost of the LHC magnets alone more than a billion dollars.
The narrow pipes that hold the proton beams extend inside the dipoles, which are strung together end to end so that they wind through the extent of the LHC tunnel’s interior. They produce a magnetic field that can be as strong as 8.3 tesla, about a thousand times the field of the average refrigerator magnet. As the energy of the proton beams increases from 450 GeV to 7 TeV, the magnetic field increases from 0.54 to 8.3 teslas, in order to keep guiding the increasingly energetic protons around.
The field these magnets produce is so enormous that it would displace the magnets themselves if no restraints were in place. This force is alleviated through the geometry of the coils, but the magnets are ultimately kept in place through specially constructed collars made of four-centimetre thick steel.
… Each LHC dipole contains coils of niobium-titanium superconducting cables, each of which contains stranded filaments a mere six microns thick – much smaller than a human hair. The LHC contains 1,200 tonnes of these remarkable filaments. If you unwrapped them, they would be long enough to encircle the orbit of Mars.
When operating, the dipoles need to be extremely cold, since they work only when the temperature is sufficiently low. The superconducting wires are maintained at 1.9 degrees above absolute zero … This temperature is even lower than the 2.7-degree cosmic microwave background radiation in outer space. The LHC tunnel houses the coldest extended region in the universe – at least that we know of. The magnets are known as cryodipoles to take into account their special refrigerated nature.
In addition to the impressive filament technology used for the magnets, the refrigeration (cryogenic) system is also an imposing accomplishment meriting its own superlatives. The system is in fact the world’s largest. Flowing helium maintains the extremely low temperature. A casing of approximately 97 metric tonnes of liquid helium surrounds the magnets to cool the cables. It is not ordinary helium gas, but helium with the necessary pressure to keep it in a superfluid phase. Superfluid helium is not subject to the viscosity of ordinary materials, so it can dissipate any heat produced in the dipole system with great efficiency: 10,000 metric tonnes of liquid nitrogen are first cooled, and this in turn cools the 130 metric tonnes of helium that circulate in the dipoles.

Featured image: A view of the experimental MgB2 transmission line at the LHC. Credit: CERN.

Physicists could have to wait 66,000 yottayears to see an electron decay

The longest coherently described span of time I’ve encountered is from Hindu cosmology. It concerns the age of Brahma, one of Hinduism’s principal deities, who is described as being 51 years old (with 49 more to go). But these are no simple years. Each day in Brahma’s life lasts for a period called the kalpa: 4.32 billion Earth-years. In 51 years, he will actually have lived for almost 80 trillion Earth-years. In a 100, he will have lived 157 trillion Earth-years.

157,000,000,000,000. That’s stupidly huge. Forget astronomy – I doubt even economic crises have use for such numbers.

On December 3, scientists announced that we’ve all known something that will live for even longer: the electron.

Yup, the same tiny lepton that zips around inside atoms with gay abandon, that’s swimming through the power lines in your home, has been found to be stable for at least 66,000 yottayears – yotta- being the largest available prefix in the decimal system.

In stupidly huge terms, that’s 66,000,000,000,000,000,000,000,000,000 (66,000 trillion trillion) years. Brahma just slipped to second place among the mortals.

But why were scientists making this measurement in the first place?

Because they’re desperately trying to disprove a prevailing theory in physics. Called the Standard Model, it describes how fundamental particles interact with each other. Though it was meticulously studied and built over a period of more than 30 years to explain a variety of phenomena, the Standard Model hasn’t been able to answer few of the more important questions. For example, why is gravity among the four fundamental forces so much weaker than the rest? Or why is there more matter than antimatter in the universe? Or why does the Higgs boson not weigh more than it does? Or what is dark matter?

Silence.

The electron belongs to a class of particles called leptons, which in turn is well described by the Standard Model. So if physicists are able to find that an electron is less stable the model predicts, it’d be a breakthrough. But despite multiple attempts to find an equally freak event, physicists haven’t succeeded – not even with the LHC (though hopeful rumours are doing the rounds that that could change soon).

The measurement of 66,000 yottayears was published in the journal Physical Review Letters on December 3 (a preprint copy is available on the arXiv server dated November 11). It was made at the Borexino neutrino experiment buried under the Gran Sasso mountain in Italy. The value itself is hinged on a simple idea: the conservation of charge.

If an electron becomes unstable and has to break down, it’ll break down into a photon and a neutrino. There are almost no other options because the electron is the lightest charged particle and whatever it breaks down into has to be even lighter. However, neither the photon nor the neutrino has an electric charge so the breaking-down would violate a fundamental law of nature – and definitely overturn the Standard Model.

The Borexino experiment is actually a solar neutrino detector, using 300 tonnes of a petroleum-based liquid to detect and study neutrinos streaming in from the Sun. When a neutrino strikes the liquid, it knocks out an electron in a tiny flash of energy. Some 2,210 photomultiplier tubes surrounding the tank amplify this flash for examination. The energy released is about 256 keV (by the mass-energy equivalence, corresponding to about a 4,000th the mass of a proton).

However, the innards of the mountain where the detector is located also produce photons thanks to the radioactive decay of bismuth and polonium in it. So the team making the measurement used a simulator to calculate how often photons of 256 keV are logged by the detector against the ‘background’ of all the photons striking the detector. Kinda like a filter. They used data logged over 408 days (January 2012 to May 2013).

The answer: once every 66,000 yotta-years (that’s 420 trillion Brahma-years).

Physics World reports that if photons from the ‘background’ radiation could be eliminated further, the electron’s lifetime could probably be increased by a thousand times. But there’s historical precedent that to some extent encourages stronger probes of the humble electron’s properties.

In 2006, another experiment situated under the Gran Sasso mountain tried to measure the rate at which electrons violated a defining rule in particle physics called Pauli’s exclusion principle. All electrons can be described by four distinct attibutes called their quantum numbers, and the principle holds that no two electrons can have the same four numbers at any given time.

The experiment was called DEAR (DAΦNE Exotic Atom Research). It energised electrons and then measured how much of it was released when the particles returned to a lower-energy state. After three years of data-taking, its team announced in 2009 that the principle was being violated once every 570 trillion trillion measurements (another stupidly large number).

That’s a violation 0.0000000000000000000000001% of the time – but it’s still something. And it could amount to more when compared to the Borexino measurement of an electron’s stability. In March 2013, the team that worked DEAR submitted a proposal for building an instrument that improve the measurement by a 100-times, and in May 2015, reported that such an instrument was under construction.

Here’s hoping they don’t find what they were looking for?

Superconductivity: From Feshbach to Fermi

(This post is continued from this one.)

After a bit of searching on Wikipedia, I found that the fundamental philosophical underpinnings of superconductivity were to be found in a statistical concept called the Feshbach resonance. If I had to teach superconductivity to those who only knew of the phenomenon superfluously, that’s where I’d begin. So.

Imagine a group of students who have gathered in a room to study together for a paper the next day. Usually, there is that one guy among them who will be hell-bent on gossiping more than studying, affecting the performance of the rest of the group. In fact, given sufficient time, the entire group’s interest will gradually shift in the direction of the gossip and away from its syllabus. The way to get the entire group back on track is to introduce a Feshbach resonance: cut the bond between the group’s interest and the entity causing the disruption. If done properly, the group will turn coherent in its interest and to focusing on studying for the paper.

In multi-body systems, such as a conductor harboring electrons, the presence of a Feshbach resonance renders an internal degree of freedom independent of those coordinates “along” which dissociation is most like to occur. And in a superconductor, a Feshbach resonance results in each electron pairing up with another (i.e., electron-vibrations are quelled by eliminating thermal excitation) owing to both being influenced by an attractive potential that arises out of the electron’s interaction with the vibrating lattice.

Feshbach resonance & BCS theory

For particulate considerations, the lattice-vibrations are quantized in the form of hypothetical particles called phonons. As for why the Feshbach resonance must occur the way it does in a superconductor: that is the conclusion, rather implication, of the BCS theory formulated in 1957 by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer.

The BCS theory essentially treats electrons like rebellious, teenage kids (I must be getting old). As negatively charged electrons pass through the crystal lattice, they draw the positively charged nuclei toward themselves, creating an increase in the positive charge density in their vicinity that attracts more electrons in turn. The resulting electrostatic pull is stronger near nuclei and very weak at larger distances. The BCS theory states that two electrons that would otherwise repel each other will pair up in the face of such a unifying electrostatic potential, howsoever weak it is.

This is something like rebellious teens who, in the face of a common enemy, will unite with each other no matter what the differences between them earlier were.

Since electrons are fermions, they bow down to Pauli’s exclusion principle, which states that no two fermions may occupy the same quantum state. As each quantum state is defined by some specific combination of state variables called quantum numbers, at least one quantum number must differ between the two co-paired electrons.

prof_wolfgang_pauli — Prof. Wolfgang Pauli (1900-1958)

In the case of superconductors, this is particle spin: the electrons in the member-pair will have opposite spins. Further, once such unions have been achieved between different pairs of electrons, each pair becomes indistinguishable from the other, even in principle. Imagine: they are all electron-pairs with two opposing spins but with the same values for all other quantum numbers. Each pair, called a Cooper pair, is just the same as the next!

Bose-Einstein condensates

This unification results in the sea of electrons displaying many properties normally associated with Bose-Einstein condensates (BECs). In a BEC, the particles that attain the state of indistinguishability are bosons (particles with integer spin), not fermions (particles with half-integer spin). The phenomenon occurs at temperatures close to absolute zero and in the presence of an external confining potential, such as an electric field.

Since bosons don’t follow Pauli’s exclusion principle, a major fraction of the indistinguishable entities in the condensate may and do occupy the same quantum state. This causes quantum mechanical effects to become apparent on a macroscopic scale.

By extension, the formulation and conclusions of the BCS theory, alongside its success in supporting associated phenomena, imply that superconductivity may be a quantum phenomenon manifesting in a macroscopic scale.

Note: If even one Cooper pair is “broken”, the superconducting state will be lost as the passage of electric current will be disrupted, and the condensate will dissolve into individual electrons, which means the energy required to break one Cooper pair is the same as the energy required to break the composition of the condensate. So thermal vibrations of the crystal lattice, usually weak, become insufficient to interrupt the flow of Cooper pairs, which is the flow of electrons.

The Meissner effect

In this context, the Meissner effect is simply an extrapolation of Lenz’s law but with zero electrical resistance.

Lenz’s law states that the electromotive force (EMF) because of a current in a conductor acts in a direction that always resists a change in the magnetic flux that causes the EMF. In the absence of resistance, the magnetic fields due to electric currents at the surface of a superconductor cancel all magnetic fields inside the bulk of the material, effectively pushing magnetic field lines of an external magnetic potential outward. However, the Meissner effect manifests only when the externally applied field is weaker than a certain critical threshold: if it is stronger, then the superconductor returns to its conducting state.

Now, there are a class of materials called Type II superconductors – as opposed to the Type I class described earlier – that only push some of the magnetic field outward, the rest remaining conserved inside the material in filaments while being surrounded by supercurrents. This state is called the vortex state, and its occurrence means the material can withstand much stronger magnetic fields and continue to remain superconducting while also exhibiting the hybrid Meissner effect.

Temperature & superconductivity

There are also a host of other effects that only superconductors can exhibit, including Cooper-pair tunneling, flux quantization, and the isotope effect, and it was by studying them that a strong relationship was observed between temperature and superconductivity in various forms.

Bardeen_Cooper_Schrieffer — (L to R) John Bardeen, Leon Cooper, and John Schrieffer

In fact, Bardeen, Cooper, and Schrieffer hit upon their eponymous theory after observing a band gap in the electronic spectra of superconductors. The electrons in any conductor can exist at specific energies, each well-defined. Electrons above a certain energy, usually in the valence band, become free to pass through the entire material instead of staying in motion around the nuclei, and are responsible for conduction.

The trio observed that upon cooling the material to closer and closer to absolute zero, there was a curious gap in the energies at which electrons could be found in the material at a particular temperature. This meant that, at that temperature, the electrons were jumping from existing at one energy to existing at some other lower energy. The observation indicated that some form of condensation was occurring. However, a BEC was ruled out because of Pauli’s exclusion principle. At the same time, a BEC-like state had to have been achieved by the electrons.

This temperature is called the transition temperature, and is the temperature below which a conductor transitions into its superconducting state, and Cooper pairs form, leading to the drop in the energy of each electron. Also, the differences in various properties of the material on either side of this threshold are also attributed to this temperature, including an important notion called the Fermi energy: it is the potential energy that any system possesses when all its thermal energy has been removed from it. This is a significant idea because it defines both the kind and amount of energy that a superconductor has to offer for an externally applied electric current.

enrico_fermi — Enrico Fermi, along with Paul Dirac, defined the Fermi-Dirac statistics that governs the behavior all identical particles that obey Pauli’s exclusion principle (i.e., fermions). Fermi level and Fermi energy are concepts named for him; however, as long as we’re discussing eponymy, Fermilab overshadows them all.

In simple terms, the density of various energy states of the electrons at the Fermi energy of a given material dictates the “breadth” of the band gap if the electron-phonon interaction energy were to be held fixed at some value: a direct proportionality. Thus, the value of the energy gap at absolute zero should be a fixed multiple of the value of the energy gap at the superconducting transition temperature (the multiplication factor was found to be 3.5 universally, irrespective of the material).

Similarly, because of the suppression of thermal excitation (because of the low temperature), the heat capacity of the material reduces drastically at low temperatures, and vanishes below the transition temperature. However, just before hitting zero at the threshold, the heat capacity balloons up to beyond its original value, and then pops. It was found that the ballooned value was always 2.5 times the material’s normal heat capacity value… again, universally, irrespective of the material!

The temperature-dependence of superconductors gains further importance with respect to applications and industrial deployment in the context of its possible occurring at higher temperatures. The low temperatures currently necessary eliminate thermal excitations, in the form of vibrations, of nuclei and almost entirely counter the possibility of electrons, or Cooper pairs, colliding into them.The low temperatures also assist in the flow of Cooper pairs as a superfluid apart from allowing for the energy of the superfluid being higher than the phononic energy of the lattice.

However, to achieve all these states in order to turn a conductor into a superconductor at a higher temperature, a more definitive theory of superconductivity is required. One that allows for the conception of superconductivity that requires only certain internal conditions to prevail while the ambient temperature soars. The 1986-discovery of high-temperature superconductors in ceramics by Bednorz and Muller was the turning point. It started to displace the BCS theory which, physicists realized, doesn’t contain the necessary mechanisms for superconductivity to manifest itself in ceramics – insulators at room temperature – at temperatures as high as 125 K.

A firmer description of superconductivity, therefore, still remains elusive. Its construction should not only pave the for one of the few phenomena that hardly appears in nature and natural processes to be fully understood, but also for its substitution against standard conductors that are responsible for lossy transmission and other such undesirable effects. After all, superconductors are the creation of humankind, and only by its hand while they ever be fully worked.

Stages of superconductivity: Researchers’ insight on ‘pseudogap’ an important advance (phys.org)

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