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.