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Introduction
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection0
Wed, 13 Apr 2016 23:00:00 GMT
<p>The fascinating phenomenon of superconductivity and its potential applications have attracted the attention of scientists, engineers and businessmen. Intense research has taken place to discover new superconductors, to understand the physics that underlies the properties of superconductors, and to develop new applications for these materials. In this course you will read about the history of superconductors, taking a brief look at their properties. You will also learn about modelling the properties of superconductors and the two different types of superconductor that exist today.</p><p>Superconducting electromagnets produce the large magnetic fields required in the world's largest particle accelerators, in MRI machines used for diagnostic imaging of the human body, in magnetically levitated trains (Figure 8) and in superconducting magnetic energy storage systems. But at the other extreme superconductors are used in SQUID (superconducting quantum interference device) magnetometers, which can measure the tiny magnetic fields (~10﻿<sup>−﻿13﻿</sup>T) associated with electrical activity in the brain, and there is great interest in their potential as extremely fast switches for a new generation of very powerful computers.</p><p>In this course we will focus on the macroscopic electrodynamic properties of superconductors, and particularly on some of the properties that can be explained in terms of electromagnetism concepts with which you should be familiar. A full understanding of superconductivity requires knowledge of materials science and quantum theory, and discussion of these aspects is beyond the scope of this course. We begin with a review of some of the main developments over the last hundred years, then describe in more detail some of the key electromagnetic properties. These can be modelled in a simple way without using quantum mechanics, and we shall show how this can be done. Finally, we distinguish between the type of superconductivity shown by most of the elemental superconductors, known as <i>typeI superconductivity</i>, and that shown by superconducting alloys that have commercial applications, known as <i>typeII superconductivity</i>.</p><p>This OpenLearn course is an adapted extract from the Open University course : <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www3.open.ac.uk/study/undergraduate/course/smt359.htm?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">SMT359 <i>Electromagnetism</i></a></span>.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection0
IntroductionSMT359_1<p>The fascinating phenomenon of superconductivity and its potential applications have attracted the attention of scientists, engineers and businessmen. Intense research has taken place to discover new superconductors, to understand the physics that underlies the properties of superconductors, and to develop new applications for these materials. In this course you will read about the history of superconductors, taking a brief look at their properties. You will also learn about modelling the properties of superconductors and the two different types of superconductor that exist today.</p><p>Superconducting electromagnets produce the large magnetic fields required in the world's largest particle accelerators, in MRI machines used for diagnostic imaging of the human body, in magnetically levitated trains (Figure 8) and in superconducting magnetic energy storage systems. But at the other extreme superconductors are used in SQUID (superconducting quantum interference device) magnetometers, which can measure the tiny magnetic fields (~10<sup>−13</sup>T) associated with electrical activity in the brain, and there is great interest in their potential as extremely fast switches for a new generation of very powerful computers.</p><p>In this course we will focus on the macroscopic electrodynamic properties of superconductors, and particularly on some of the properties that can be explained in terms of electromagnetism concepts with which you should be familiar. A full understanding of superconductivity requires knowledge of materials science and quantum theory, and discussion of these aspects is beyond the scope of this course. We begin with a review of some of the main developments over the last hundred years, then describe in more detail some of the key electromagnetic properties. These can be modelled in a simple way without using quantum mechanics, and we shall show how this can be done. Finally, we distinguish between the type of superconductivity shown by most of the elemental superconductors, known as <i>typeI superconductivity</i>, and that shown by superconducting alloys that have commercial applications, known as <i>typeII superconductivity</i>.</p><p>This OpenLearn course is an adapted extract from the Open University course : <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www3.open.ac.uk/study/undergraduate/course/smt359.htm?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">SMT359 <i>Electromagnetism</i></a></span>.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

Learning outcomes
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsectionlearningoutcomes
Wed, 13 Apr 2016 23:00:00 GMT
<p>After studying this course, you should be able to:</p><ul><li><p>explain the meanings of the newly defined (emboldened) terms and symbols, and use them appropriately</p></li><li><p>distinguish between perfect conduction and perfect diamagnetism, and give a qualitative description of the Meissner effect</p></li><li><p>explain how observation of a persistent current can be used to estimate an upper limit on the resistivity of a superconductor, and perform calculations related to such estimates</p></li><li><p>explain why the magnetic flux through a superconducting circuit remains constant, and describe applications of this effect</p></li><li><p>show how the London equations and Maxwell's equations lead to the prediction of the Meissner effect.</p></li></ul>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsectionlearningoutcomes
Learning outcomesSMT359_1<p>After studying this course, you should be able to:</p><ul><li><p>explain the meanings of the newly defined (emboldened) terms and symbols, and use them appropriately</p></li><li><p>distinguish between perfect conduction and perfect diamagnetism, and give a qualitative description of the Meissner effect</p></li><li><p>explain how observation of a persistent current can be used to estimate an upper limit on the resistivity of a superconductor, and perform calculations related to such estimates</p></li><li><p>explain why the magnetic flux through a superconducting circuit remains constant, and describe applications of this effect</p></li><li><p>show how the London equations and Maxwell's equations lead to the prediction of the Meissner effect.</p></li></ul>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

1 Superconductivity
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection1
Wed, 13 Apr 2016 23:00:00 GMT
<p>Superconductivity was discovered in 1911 by Heike Kamerlingh Onnes (Figure 1) as he studied the properties of metals at low temperatures. A few years earlier he had become the first person to liquefy helium, which has a boiling point of 4.2 K at atmospheric pressure, and this had opened up a new range of temperature to experimental investigation. On measuring the resistance of a small tube filled with mercury, he was astonished to observe that its resistance fell from ~﻿0.1 Ω at a temperature of 4.3 K to less than 3  × ﻿10﻿<sup>﻿−﻿6</sup> Ω at 4.1 K. His results are reproduced in Figure 2. Below 4.1 K, mercury is said to be a superconductor, and no experiment has yet detected any resistance to steady current flow in a superconducting material. The temperature below which the mercury becomes superconducting is known as its <b>critical temperature</b> <i>T</i>﻿<sub>c</sub>. Kamerlingh Onnes was awarded the Nobel Prize for Physics in 1913 ‘for his investigations on the properties of matter at low temperatures which led, <i>inter alia</i>, to the production of liquid helium’ (Nobel Prize citation).</p><div class="oucontentfigure oucontentmediamini" id="fig009_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/fa371b54/smt359_1_001i.jpg" alt="Figure 1" width="233" height="233" style="maxwidth:233px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 1 Heike Kamerlingh Onnes (left) and Johannes Van der Waals beside a helium liquefier (1908).</span></div></div></div><div class="oucontentfigure oucontentmediamini" id="fig009_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3cc5f7c0/smt359_1_002i.jpg" alt="Figure 2" width="236" height="288" style="maxwidth:236px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 2 Graph showing the resistance of a specimen of mercury versus absolute temperature.</span></div></div></div><p>Since this initial discovery, many more elements have been discovered to be superconductors. Indeed, superconductivity is by no means a rare phenomenon, as the Periodic Table in Figure 3 demonstrates. The dark pink cells indicate elements that become superconducting at atmospheric pressure, and the numbers at the bottoms of the cells are their critical temperatures, which range from 9.3 K for niobium (﻿Nb, <i>Z</i> = 41﻿) down to 3 × 10<sup>﻿−﻿4</sup> K for rhodium (Rh, <i>Z</i> = 45﻿). The orange cells are elements that become superconductors only under high pressures. The four pale pink cells are elements that are superconducting in particular forms: carbon (C, <i>Z</i> = 6﻿) in the form of nanotubes, chromium (﻿Cr, <i>Z</i> = 24﻿) as thin films, palladium (﻿Pd, <i>Z</i> = 46﻿) after irradiation with alpha particles, and platinum (﻿Pt, <i>Z</i> = 78﻿) as a compacted powder. It is worth noting that copper (﻿Cu, <i>Z</i>  = 29﻿), silver (﻿Ag, <i>Z</i> = 47﻿) and gold (﻿Au, <i>Z</i> = 79﻿), three elements that are excellent conductors at room temperature, do not become superconductors even at the lowest temperatures that are attainable.</p><div class="oucontentfigure" style="width:511px;" id="fig009_003"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3112960" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c31eeb32/smt359_1_003i.small.jpg" alt="Figure 3" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3112960">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 3 The Periodic Table showing all known elemental superconductors and their critical temperatures.</span></div></div><a id="back_thumbnailfigure_idp3112960"></a></div><p>A major advance in the understanding of superconductivity came in 1933, when Walter Meissner and Robert Ochsenfeld discovered that superconductors are more than perfect conductors of electricity. They also have the important property of excluding a magnetic field from their interior. However, the field is excluded only if it is below a certain critical field strength, which depends on the material, the temperature and the geometry of the specimen. Above this critical field strength the superconductivity disappears. Brothers Fritz and Heinz London proposed a model that described the exclusion of the field in 1935, but it was another 20 years before a microscopic explanation was developed.</p><p>The long awaited quantum theory of superconductivity was published in 1957 by three US physicists, John Bardeen, Leon Cooper and John Schrieffer, and they were awarded the Nobel Prize for Physics in 1972 ‘for their jointly developed theory of superconductivity, usually called the BCS theory’ (Nobel Prize citation). According to their theory, in the superconducting state there is an attractive interaction between electrons that is mediated by the vibrations of the ion lattice. A consequence of this interaction is that pairs of electrons are coupled together, and all of the pairs of electrons condense into a macroscopic quantum state, called the <b>condensate</b>, that extends through the superconductor. Not all of the free electrons in a superconductor are in the condensate; those that are in this state are called <b>superconducting electrons</b>, and the others are referred to as <b>normal electrons</b>. At temperatures very much lower than the critical temperature, there are very few normal electrons, but the proportion of normal electrons increases as the temperature increases, until at the critical temperature all of the electrons are normal. Because the superconducting electrons are linked in a macroscopic state, they behave coherently, and a consequence of this is that there is a characteristic distance over which their number density can change, known as the <b>coherence length</b> <i>ξ</i> (the Greek lowercase xi, pronounced ‘ksye’).</p><p>It takes a significant amount of energy to scatter an electron from the condensate – more than the thermal energy available to an electron below the critical temperature – so the superconducting electrons can flow without being scattered, that is, without any resistance. The BCS theory successfully explained many of the known properties of superconductors, but it predicted an upper bound of roughly 30 K for the critical temperature.</p><p>Another important theoretical discovery was made in 1957. Alexei Abrikosov predicted the existence of a second type of superconductor that behaved in a different way from elements like lead and tin. This new type of superconductor would expel the field from its interior when the applied field strength was low, but over a wide range of applied field strengths the superconductor would be threaded by normal metal regions through which the magnetic field could pass. The penetration of the field meant that superconductivity could exist in magnetic field strengths up to 10 T or more, which opened up the possibility of many applications. For this work, and subsequent research, Abrikosov received a Nobel Prize for Physics in 2003 ‘for pioneering contributions to the theory of superconductors and superfluids’ (Nobel Prize citation).</p><p>By the early 1960s there had been major advances in superconductor technology, with the discovery of alloys that were superconducting at temperatures higher than the critical temperatures of the elemental superconductors. In particular, alloys of niobium and titanium (NbTi, <i>T</i>﻿<sub>c</sub> = 9﻿.﻿8 K﻿) and niobium and tin (﻿Nb﻿<sub>3</sub>﻿Sn, <i>T</i>﻿<sub>c</sub> = 18﻿.﻿1 K﻿) were becoming widely used to produce highfield magnets, and a major impetus for this development was the requirement for powerful magnets for particle accelerators, like the Tevatron at Fermilab in the USA. At about the same time, Brian Josephson made an important theoretical prediction that was to have major consequences for the application of superconductivity on a very small scale. He predicted that a current could flow between two superconductors that were separated by a very thin insulating layer. The socalled Josephson tunnelling effect has been widely used for making various sensitive measurements, including the determination of fundamental physical constants and the measurement of magnetic fields that are a billion (﻿10﻿<sup>9</sup>﻿) times weaker than the Earth's field. The significance of his work was recognised when he was awarded a Nobel Prize for Physics in 1973 ‘for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects’ (Nobel Prize citation).</p><p>The hunt for superconductors with higher critical temperatures continued in the decades following publication of the BCS theory, in spite of its prediction that the upper limit for T﻿<sub>c</sub> was less than 30 K. The holy grail for scientists working in this area was a material that was superconducting at the temperature of liquid nitrogen (﻿77 K﻿), or, even better, at room temperature. This would mean that all of the technology and costs associated with use of liquid helium for cooling could be dispensed with, and applications of superconductivity would immediately become far more economically worthwhile. The breakthrough came in 1986, when Georg Bednorz and Alex Muller discovered that ceramics made of barium, lanthanum, copper and oxygen became superconducting at 30 K, the highest known critical temperature at that time. The discovery was particularly surprising because this material is an insulator at room temperature. The following year they received the Nobel Prize for Physics ‘for their important breakthrough in the discovery of superconductivity in ceramic materials’ (Nobel Prize citation), and the unprecedented rapidity with which the prize followed publication of their results reflects the importance attached to their work.</p><p>As a result of this breakthrough, a scientific bandwagon started to roll and many other scientists began to examine similar materials. In 1987, Paul Chu produced a new ceramic material by replacing lanthanum by yttrium, and found that it had a critical temperature of 90 K. This great jump in the critical temperature made it possible to use liquid nitrogen as a coolant, and with the promise of commercial viability for the new materials, a scramble ensued to find new hightemperature superconductors and to explain why they superconduct at such high temperatures. At the time of writing (2005), the highest critical temperature was 138 K, for a thalliumdoped mercuriccuprate, Hg﻿<sub>0.8</sub>﻿Tl﻿<sub>0﻿.﻿2﻿</sub>﻿Ba<sub>﻿2</sub>﻿Ca﻿<sub>2</sub>﻿Cu﻿<sub>3</sub>﻿O﻿<sub>8﻿.﻿33</sub>. Figure 4 shows the progress of the highest known superconducting critical temperature over the last century.</p><div class="oucontentfigure" style="width:511px;" id="fig009_004"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3135760" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/5c925524/smt359_1_004i.small.jpg" alt="Figure 4" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3135760">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 4 The critical temperature <i>T</i><sub>c</sub> of various superconductors plotted against their discovery date.</span></div></div><a id="back_thumbnailfigure_idp3135760"></a></div><p>In recent years, no materials with significantly higher critical temperatures have been found, but other discoveries of equal importance have been made. These include the discovery that, against conventional wisdom, several materials exhibit the coexistence of ferromagnetism and superconductivity. We have also seen the discovery of the first hightemperature superconductors that do not contain copper. Startling discoveries like these are demanding that scientists continually reexamine longstanding theories on superconductivity and consider novel combinations of elements.</p><p>Unfortunately, no superconductors have yet been found with critical temperatures above room temperature, so cryogenic cooling is still a vital part of any superconducting application. Difficulties with fabricating ceramic materials into conducting wires or strips have also slowed down the development of new applications of hightemperature superconductors. However, despite these drawbacks, the commercial use of superconductors continues to rise.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection1
1 SuperconductivitySMT359_1<p>Superconductivity was discovered in 1911 by Heike Kamerlingh Onnes (Figure 1) as he studied the properties of metals at low temperatures. A few years earlier he had become the first person to liquefy helium, which has a boiling point of 4.2 K at atmospheric pressure, and this had opened up a new range of temperature to experimental investigation. On measuring the resistance of a small tube filled with mercury, he was astonished to observe that its resistance fell from ~0.1 Ω at a temperature of 4.3 K to less than 3 × 10<sup>−6</sup> Ω at 4.1 K. His results are reproduced in Figure 2. Below 4.1 K, mercury is said to be a superconductor, and no experiment has yet detected any resistance to steady current flow in a superconducting material. The temperature below which the mercury becomes superconducting is known as its <b>critical temperature</b> <i>T</i><sub>c</sub>. Kamerlingh Onnes was awarded the Nobel Prize for Physics in 1913 ‘for his investigations on the properties of matter at low temperatures which led, <i>inter alia</i>, to the production of liquid helium’ (Nobel Prize citation).</p><div class="oucontentfigure oucontentmediamini" id="fig009_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/fa371b54/smt359_1_001i.jpg" alt="Figure 1" width="233" height="233" style="maxwidth:233px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 1 Heike Kamerlingh Onnes (left) and Johannes Van der Waals beside a helium liquefier (1908).</span></div></div></div><div class="oucontentfigure oucontentmediamini" id="fig009_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3cc5f7c0/smt359_1_002i.jpg" alt="Figure 2" width="236" height="288" style="maxwidth:236px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 2 Graph showing the resistance of a specimen of mercury versus absolute temperature.</span></div></div></div><p>Since this initial discovery, many more elements have been discovered to be superconductors. Indeed, superconductivity is by no means a rare phenomenon, as the Periodic Table in Figure 3 demonstrates. The dark pink cells indicate elements that become superconducting at atmospheric pressure, and the numbers at the bottoms of the cells are their critical temperatures, which range from 9.3 K for niobium (Nb, <i>Z</i> = 41) down to 3 × 10<sup>−4</sup> K for rhodium (Rh, <i>Z</i> = 45). The orange cells are elements that become superconductors only under high pressures. The four pale pink cells are elements that are superconducting in particular forms: carbon (C, <i>Z</i> = 6) in the form of nanotubes, chromium (Cr, <i>Z</i> = 24) as thin films, palladium (Pd, <i>Z</i> = 46) after irradiation with alpha particles, and platinum (Pt, <i>Z</i> = 78) as a compacted powder. It is worth noting that copper (Cu, <i>Z</i> = 29), silver (Ag, <i>Z</i> = 47) and gold (Au, <i>Z</i> = 79), three elements that are excellent conductors at room temperature, do not become superconductors even at the lowest temperatures that are attainable.</p><div class="oucontentfigure" style="width:511px;" id="fig009_003"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3112960" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c31eeb32/smt359_1_003i.small.jpg" alt="Figure 3" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3112960">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 3 The Periodic Table showing all known elemental superconductors and their critical temperatures.</span></div></div><a id="back_thumbnailfigure_idp3112960"></a></div><p>A major advance in the understanding of superconductivity came in 1933, when Walter Meissner and Robert Ochsenfeld discovered that superconductors are more than perfect conductors of electricity. They also have the important property of excluding a magnetic field from their interior. However, the field is excluded only if it is below a certain critical field strength, which depends on the material, the temperature and the geometry of the specimen. Above this critical field strength the superconductivity disappears. Brothers Fritz and Heinz London proposed a model that described the exclusion of the field in 1935, but it was another 20 years before a microscopic explanation was developed.</p><p>The long awaited quantum theory of superconductivity was published in 1957 by three US physicists, John Bardeen, Leon Cooper and John Schrieffer, and they were awarded the Nobel Prize for Physics in 1972 ‘for their jointly developed theory of superconductivity, usually called the BCS theory’ (Nobel Prize citation). According to their theory, in the superconducting state there is an attractive interaction between electrons that is mediated by the vibrations of the ion lattice. A consequence of this interaction is that pairs of electrons are coupled together, and all of the pairs of electrons condense into a macroscopic quantum state, called the <b>condensate</b>, that extends through the superconductor. Not all of the free electrons in a superconductor are in the condensate; those that are in this state are called <b>superconducting electrons</b>, and the others are referred to as <b>normal electrons</b>. At temperatures very much lower than the critical temperature, there are very few normal electrons, but the proportion of normal electrons increases as the temperature increases, until at the critical temperature all of the electrons are normal. Because the superconducting electrons are linked in a macroscopic state, they behave coherently, and a consequence of this is that there is a characteristic distance over which their number density can change, known as the <b>coherence length</b> <i>ξ</i> (the Greek lowercase xi, pronounced ‘ksye’).</p><p>It takes a significant amount of energy to scatter an electron from the condensate – more than the thermal energy available to an electron below the critical temperature – so the superconducting electrons can flow without being scattered, that is, without any resistance. The BCS theory successfully explained many of the known properties of superconductors, but it predicted an upper bound of roughly 30 K for the critical temperature.</p><p>Another important theoretical discovery was made in 1957. Alexei Abrikosov predicted the existence of a second type of superconductor that behaved in a different way from elements like lead and tin. This new type of superconductor would expel the field from its interior when the applied field strength was low, but over a wide range of applied field strengths the superconductor would be threaded by normal metal regions through which the magnetic field could pass. The penetration of the field meant that superconductivity could exist in magnetic field strengths up to 10 T or more, which opened up the possibility of many applications. For this work, and subsequent research, Abrikosov received a Nobel Prize for Physics in 2003 ‘for pioneering contributions to the theory of superconductors and superfluids’ (Nobel Prize citation).</p><p>By the early 1960s there had been major advances in superconductor technology, with the discovery of alloys that were superconducting at temperatures higher than the critical temperatures of the elemental superconductors. In particular, alloys of niobium and titanium (NbTi, <i>T</i><sub>c</sub> = 9.8 K) and niobium and tin (Nb<sub>3</sub>Sn, <i>T</i><sub>c</sub> = 18.1 K) were becoming widely used to produce highfield magnets, and a major impetus for this development was the requirement for powerful magnets for particle accelerators, like the Tevatron at Fermilab in the USA. At about the same time, Brian Josephson made an important theoretical prediction that was to have major consequences for the application of superconductivity on a very small scale. He predicted that a current could flow between two superconductors that were separated by a very thin insulating layer. The socalled Josephson tunnelling effect has been widely used for making various sensitive measurements, including the determination of fundamental physical constants and the measurement of magnetic fields that are a billion (10<sup>9</sup>) times weaker than the Earth's field. The significance of his work was recognised when he was awarded a Nobel Prize for Physics in 1973 ‘for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effects’ (Nobel Prize citation).</p><p>The hunt for superconductors with higher critical temperatures continued in the decades following publication of the BCS theory, in spite of its prediction that the upper limit for T<sub>c</sub> was less than 30 K. The holy grail for scientists working in this area was a material that was superconducting at the temperature of liquid nitrogen (77 K), or, even better, at room temperature. This would mean that all of the technology and costs associated with use of liquid helium for cooling could be dispensed with, and applications of superconductivity would immediately become far more economically worthwhile. The breakthrough came in 1986, when Georg Bednorz and Alex Muller discovered that ceramics made of barium, lanthanum, copper and oxygen became superconducting at 30 K, the highest known critical temperature at that time. The discovery was particularly surprising because this material is an insulator at room temperature. The following year they received the Nobel Prize for Physics ‘for their important breakthrough in the discovery of superconductivity in ceramic materials’ (Nobel Prize citation), and the unprecedented rapidity with which the prize followed publication of their results reflects the importance attached to their work.</p><p>As a result of this breakthrough, a scientific bandwagon started to roll and many other scientists began to examine similar materials. In 1987, Paul Chu produced a new ceramic material by replacing lanthanum by yttrium, and found that it had a critical temperature of 90 K. This great jump in the critical temperature made it possible to use liquid nitrogen as a coolant, and with the promise of commercial viability for the new materials, a scramble ensued to find new hightemperature superconductors and to explain why they superconduct at such high temperatures. At the time of writing (2005), the highest critical temperature was 138 K, for a thalliumdoped mercuriccuprate, Hg<sub>0.8</sub>Tl<sub>0.2</sub>Ba<sub>2</sub>Ca<sub>2</sub>Cu<sub>3</sub>O<sub>8.33</sub>. Figure 4 shows the progress of the highest known superconducting critical temperature over the last century.</p><div class="oucontentfigure" style="width:511px;" id="fig009_004"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3135760" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/5c925524/smt359_1_004i.small.jpg" alt="Figure 4" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3135760">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 4 The critical temperature <i>T</i><sub>c</sub> of various superconductors plotted against their discovery date.</span></div></div><a id="back_thumbnailfigure_idp3135760"></a></div><p>In recent years, no materials with significantly higher critical temperatures have been found, but other discoveries of equal importance have been made. These include the discovery that, against conventional wisdom, several materials exhibit the coexistence of ferromagnetism and superconductivity. We have also seen the discovery of the first hightemperature superconductors that do not contain copper. Startling discoveries like these are demanding that scientists continually reexamine longstanding theories on superconductivity and consider novel combinations of elements.</p><p>Unfortunately, no superconductors have yet been found with critical temperatures above room temperature, so cryogenic cooling is still a vital part of any superconducting application. Difficulties with fabricating ceramic materials into conducting wires or strips have also slowed down the development of new applications of hightemperature superconductors. However, despite these drawbacks, the commercial use of superconductors continues to rise.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

2.1 Zero electrical resistance
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.1
Wed, 13 Apr 2016 23:00:00 GMT
<p>In this section we shall discuss some of the most important electrical properties of superconductors, with discussion of magnetic properties to follow in the next section.</p><p>The most obvious characteristic of a superconductor is the complete disappearance of its electrical resistance below a temperature that is known as its critical temperature. Experiments have been carried out to attempt to detect whether there is any small residual resistance in the superconducting state. A sensitive test is to start a current flowing round a superconducting ring and observe whether the current decays. The current flowing in the superconducting loop clearly cannot be measured by inserting an ammeter into the loop, since this would introduce a resistance and the current would rapidly decay.</p><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq001"><div class="oucontentouterbox"><h3 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 1</h3><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Suggest a method of monitoring the current that does not involve interfering with the superconducting loop.</p></div>
<div class="oucontentsaqanswer"><h4 class="oucontenth4">Answer</h4><p>The magnetic field generated by the current in the loop could be monitored.</p></div></div></div></div><p>The magnitude of the magnetic field is directly proportional to the current circulating in the loop, and the field can be measured without drawing energy from the circuit. Experiments of this type have been carried out over periods of years, and the magnetic field – and hence the superconducting current – has always remained constant within the precision of the measuring equipment. Such a <b>persistent current</b> is characteristic of the superconducting state. From the lack of any decay of the current it has been deduced that the resistivity <i>ρ</i> of a superconductor is less than 10<sup>﻿−﻿26</sup> Ω m. This is about 18 orders of magnitude smaller than the resistivity of copper at room temperature (<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> 10﻿<sup>−﻿8</sup> Ω m﻿).</p><p>Resistivity is the reciprocal of conductivity, that is, <i>ρ</i> = σ﻿<sup>﻿−﻿1</sup>. We prefer to describe a superconductor by <i>ρ</i> = 0﻿, rather than by σ = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3be41419/infinity.jpg" alt="" width="10" height="10" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span>.</p><p>In the following Exercise you can estimate an upper limit for the resistivity of a superconductor.</p><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe009_001"><div class="oucontentouterbox"><h3 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 1</h3><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>(a) A circuit, with selfinductance <i>L</i>, has a current <i>I﻿</i><sub>0</sub> flowing in it at time <i>t</i> = 0. Assuming that the circuit has a small residual resistance <i>R</i> but contains no source of emf, what will be the current in the circuit after a time <i>T</i> has elapsed?</p><p>(b) In a classic experiment performed by Quinn and Ittner in 1962, a current was set up around the ‘squashed tube’, shown in Figure 5, which was made from two thin films of superconducting lead separated by a thin layer of insulating silicon oxide. The inductance <i>L</i> of the tube was estimated to be 1.4 × 10﻿<sup>−﻿13</sup> H. No change in the magnetic moment due to the current could be detected after 7 hours, to within the 2 per cent precision of their measurement, so the current was at least 98 per cent of its initial value. Estimate the maximum possible resistance of the tube for circulating currents.</p><p>(c) The dimensions of the tube used in the experiment are shown in Figure 5. Estimate the maximum possible resistivity <i>ρ</i><sub>max</sub> of the lead films. Compare your answer with the resistivity of pure lead at 0°C, which is 1.9  ×  10﻿<sup>–﻿7</sup>﻿Ω m.</p></div>
<div class="oucontentsaqanswer"><h4 class="oucontenth4">Answer</h4><p>(a) The current in an RL circuit (with no source of emf) decays exponentially with time:</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b4b2000f/smt359_1_ue023i.gif" alt="" width="140" height="26" style="maxwidth:140px;" class="oucontentinlinefigureimage"/></span></p><p>The time constant for the decay is <i>L/R</i>, so the smaller the resistance, the longer will be the time constant. Zero resistance means an infinite time constant – the current does not decay, but persists indefinitely (or as long as the material remains superconducting).</p><p>(b) The maximum possible resistance, <i>R</i><sub>max</sub>, of the tube corresponds to the minimum possible current, which is <i>I(T)</i> = 0.98<i>I</i><sub>0</sub>. Thus</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/63d137a2/smt359_1_ue024i.gif" alt="" width="176" height="24" style="maxwidth:176px;" class="oucontentinlinefigureimage"/></span></p><p>which can be rearranged to give</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b54d0b97/smt359_1_ue025i.gif" alt="" width="286" height="101" style="maxwidth:286px;" class="oucontentinlinefigureimage"/></span></p><p>(c) For the tube, the length of the current path, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c1d74ad2/math_l.gif" alt="" width="4" height="11" style="maxwidth:4px;" class="oucontentinlinefigureimage"/></span><i><sub>I</sub></i>, is essentially twice the width of the silicon oxide layer, and the crosssectional area perpendicular to current flow, <i>A﻿<sub>I</sub></i>, is the length of the tube times the thickness of the lead films. Since</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/876dc269/smt359_1_ue026i.gif" alt="" width="217" height="24" style="maxwidth:217px;" class="oucontentinlinefigureimage"/></span></p><p>the maximum resistivity is</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/0057505a/smt359_1_ue027i.gif" alt="" width="282" height="90" style="maxwidth:282px;" class="oucontentinlinefigureimage"/></span></p><p>The value of the resistivity of lead at 0°C is</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/5ebdb06a/smt359_1_ue028i.gif" alt="" width="237" height="24" style="maxwidth:237px;" class="oucontentinlinefigureimage"/></span></p><p>The estimate for the maximum resistivity for superconducting lead obtained from the data in this exercise is almost 18 orders of magnitude smaller than this room temperature value.</p></div></div></div></div><div class="oucontentfigure oucontentmediamini" id="fig009_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/46f2ca8d/smt359_1_005i.jpg" alt="Figure 5" width="318" height="277" style="maxwidth:318px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 5 A lead ‘tube’ used in a persistent current experiment.</span></div></div></div><p></p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.1
2.1 Zero electrical resistanceSMT359_1<p>In this section we shall discuss some of the most important electrical properties of superconductors, with discussion of magnetic properties to follow in the next section.</p><p>The most obvious characteristic of a superconductor is the complete disappearance of its electrical resistance below a temperature that is known as its critical temperature. Experiments have been carried out to attempt to detect whether there is any small residual resistance in the superconducting state. A sensitive test is to start a current flowing round a superconducting ring and observe whether the current decays. The current flowing in the superconducting loop clearly cannot be measured by inserting an ammeter into the loop, since this would introduce a resistance and the current would rapidly decay.</p><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq001"><div class="oucontentouterbox"><h3 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 1</h3><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Suggest a method of monitoring the current that does not involve interfering with the superconducting loop.</p></div>
<div class="oucontentsaqanswer"><h4 class="oucontenth4">Answer</h4><p>The magnetic field generated by the current in the loop could be monitored.</p></div></div></div></div><p>The magnitude of the magnetic field is directly proportional to the current circulating in the loop, and the field can be measured without drawing energy from the circuit. Experiments of this type have been carried out over periods of years, and the magnetic field – and hence the superconducting current – has always remained constant within the precision of the measuring equipment. Such a <b>persistent current</b> is characteristic of the superconducting state. From the lack of any decay of the current it has been deduced that the resistivity <i>ρ</i> of a superconductor is less than 10<sup>−26</sup> Ω m. This is about 18 orders of magnitude smaller than the resistivity of copper at room temperature (<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> 10<sup>−8</sup> Ω m).</p><p>Resistivity is the reciprocal of conductivity, that is, <i>ρ</i> = σ<sup>−1</sup>. We prefer to describe a superconductor by <i>ρ</i> = 0, rather than by σ = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3be41419/infinity.jpg" alt="" width="10" height="10" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span>.</p><p>In the following Exercise you can estimate an upper limit for the resistivity of a superconductor.</p><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe009_001"><div class="oucontentouterbox"><h3 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 1</h3><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>(a) A circuit, with selfinductance <i>L</i>, has a current <i>I</i><sub>0</sub> flowing in it at time <i>t</i> = 0. Assuming that the circuit has a small residual resistance <i>R</i> but contains no source of emf, what will be the current in the circuit after a time <i>T</i> has elapsed?</p><p>(b) In a classic experiment performed by Quinn and Ittner in 1962, a current was set up around the ‘squashed tube’, shown in Figure 5, which was made from two thin films of superconducting lead separated by a thin layer of insulating silicon oxide. The inductance <i>L</i> of the tube was estimated to be 1.4 × 10<sup>−13</sup> H. No change in the magnetic moment due to the current could be detected after 7 hours, to within the 2 per cent precision of their measurement, so the current was at least 98 per cent of its initial value. Estimate the maximum possible resistance of the tube for circulating currents.</p><p>(c) The dimensions of the tube used in the experiment are shown in Figure 5. Estimate the maximum possible resistivity <i>ρ</i><sub>max</sub> of the lead films. Compare your answer with the resistivity of pure lead at 0°C, which is 1.9 × 10<sup>–7</sup>Ω m.</p></div>
<div class="oucontentsaqanswer"><h4 class="oucontenth4">Answer</h4><p>(a) The current in an RL circuit (with no source of emf) decays exponentially with time:</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b4b2000f/smt359_1_ue023i.gif" alt="" width="140" height="26" style="maxwidth:140px;" class="oucontentinlinefigureimage"/></span></p><p>The time constant for the decay is <i>L/R</i>, so the smaller the resistance, the longer will be the time constant. Zero resistance means an infinite time constant – the current does not decay, but persists indefinitely (or as long as the material remains superconducting).</p><p>(b) The maximum possible resistance, <i>R</i><sub>max</sub>, of the tube corresponds to the minimum possible current, which is <i>I(T)</i> = 0.98<i>I</i><sub>0</sub>. Thus</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/63d137a2/smt359_1_ue024i.gif" alt="" width="176" height="24" style="maxwidth:176px;" class="oucontentinlinefigureimage"/></span></p><p>which can be rearranged to give</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b54d0b97/smt359_1_ue025i.gif" alt="" width="286" height="101" style="maxwidth:286px;" class="oucontentinlinefigureimage"/></span></p><p>(c) For the tube, the length of the current path, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c1d74ad2/math_l.gif" alt="" width="4" height="11" style="maxwidth:4px;" class="oucontentinlinefigureimage"/></span><i><sub>I</sub></i>, is essentially twice the width of the silicon oxide layer, and the crosssectional area perpendicular to current flow, <i>A<sub>I</sub></i>, is the length of the tube times the thickness of the lead films. Since</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/876dc269/smt359_1_ue026i.gif" alt="" width="217" height="24" style="maxwidth:217px;" class="oucontentinlinefigureimage"/></span></p><p>the maximum resistivity is</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/0057505a/smt359_1_ue027i.gif" alt="" width="282" height="90" style="maxwidth:282px;" class="oucontentinlinefigureimage"/></span></p><p>The value of the resistivity of lead at 0°C is</p><p><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/5ebdb06a/smt359_1_ue028i.gif" alt="" width="237" height="24" style="maxwidth:237px;" class="oucontentinlinefigureimage"/></span></p><p>The estimate for the maximum resistivity for superconducting lead obtained from the data in this exercise is almost 18 orders of magnitude smaller than this room temperature value.</p></div></div></div></div><div class="oucontentfigure oucontentmediamini" id="fig009_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/46f2ca8d/smt359_1_005i.jpg" alt="Figure 5" width="318" height="277" style="maxwidth:318px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 5 A lead ‘tube’ used in a persistent current experiment.</span></div></div></div><p></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

2.2 Persistent currents lead to constant magnetic flux
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.2
Wed, 13 Apr 2016 23:00:00 GMT
<p>An important consequence of the persistent currents that flow in materials with zero resistance is that the magnetic flux that passes through a continuous loop of such a material remains constant. To see how this comes about, consider a ring of metal, enclosing a fixed area <i>A</i>, as shown in Figure 6a. An initial magnetic field <b>B</b>﻿<sub>0</sub> is applied perpendicular to the plane of the ring when the temperature is above the critical temperature of the material from which the ring is made. The magnetic flux Φ through the ring is <i>B</i>﻿<sub>0</sub>﻿<i>A</i>, and if the ring is cooled below its critical temperature while in this applied field, then the flux passing through it is unchanged. If we now change the applied field, then a current will be induced in the ring, and according to Lenz's law the direction of this current will be such that the magnetic flux it generates compensates for the flux change due to the change in the applied field. From Faraday's law, the induced emf in the ring is −﻿d﻿Φ﻿/﻿d﻿<i>t</i> = −﻿<i>A</i>﻿d(<i>B</i> − <i>B</i>﻿<sub>0</sub>)﻿/﻿d﻿<i>t</i>, and this generates an induced current <i>I</i> given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8639ffc1/smt359_1_ue001i.gif" alt=""/></div><p>where <i>L</i> is the selfinductance of the ring. Note that there is no ohmic term, <i>IR</i>, on the lefthand side of this equation, because we are assuming that <i>R</i> = 0. Integrating this equation, we obtain</p><div class="oucontentquote oucontentsbox" id="quo001"><blockquote><p><i>LI</i> + <i>BA</i> = constant.</p></blockquote></div><div class="oucontentfigure" style="width:511px;" id="fig009_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/0fc0d267/smt359_1_006i.jpg" alt="Figure 6" width="511" height="232" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 6 (a) A ring cooled below its critical temperature in an applied field B<sub>0</sub>. (b) When the applied field is removed, a superconducting current maintains the flux through the ring at the same value.</span></div></div></div><p>But <i>LI</i> is the amount of flux passing through the ring generated by the current <i>I</i> flowing in the ring – this is just the definition of selfinductance <i>L</i> – so (﻿<i>LI</i>﻿ + <i>BA</i>) is the total magnetic flux through the ring. The total flux threading a circuit with zero resistance must therefore remain constant – it cannot change. If the applied magnetic field is changed, an induced current is set up that creates a flux to compensate exactly for the change in the flux from the applied magnetic field. Because the circuit has no resistance, the induced current can flow indefinitely, and the original amount of flux through the ring can be maintained indefinitely. This is true even if the external field is removed altogether; the flux through the ring is maintained by a persistent induced current, as in Figure 6b. However, note that constant flux through the ring does not mean that the magnetic field is unchanged. In Figure 6a there is a uniform field within the ring, whereas in Figure 6b the field is produced by a current flowing in the ring and will be much larger close to the ring than at its centre.</p><p>An important application of the constant flux through a superconducting circuit is shown in Figure 7. A superconducting solenoid, used to produce large magnetic fields, is connected to a power supply that can be adjusted to provide the appropriate current to generate the required field. For some applications it is important for the field to remain constant to a higher precision than the stability of the power supply would allow. A stable field is achieved by including a superconducting switch in parallel with the solenoid. This is not a mechanical switch, but a length of superconducting wire that is heated to above its critical temperature to ‘open’ the switch, and cooled below the critical temperature to ‘close’ it. With the switch open, the current from the power supply is set to give the required field strength. The switch is then closed to produce a completely superconducting circuit that includes the solenoid, the switch and the leads connecting them. The flux through this circuit must remain constant in time, so the field inside the solenoid will also remain constant in time. An added bonus is that the power supply can now be disconnected, which means that no energy is being dissipated while maintaining the field.</p><div class="oucontentfigure oucontentmediamini" id="fig009_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/caf7d6d6/smt359_1_007i.jpg" alt="Figure 7" width="279" height="327" style="maxwidth:279px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 7 A superconducting solenoid with a superconducting switch that allows it to operate in a persistent current mode.</span></div></div></div><p>Superconducting coils with persistent currents can be used in highspeed magneticallylevitated trains. In the system used on the Yamanashi Maglev Test Line in Japan (Figure 8), superconducting coils mounted on the sides of the train induce currents in coils mounted in the walls of a guideway, and the attractive and repulsive forces between the superconducting magnets and the trackmounted coils both levitate the train and provide lateral guidance. The train is propelled forwards by attractive and repulsive forces between the superconducting magnets and propulsion coils located on the walls of the guideway that are energised by a threephase alternating current that creates a shifting magnetic field along the guideway. In 2003, a train reached the recordbreaking speed of 581 km h﻿<sup>−﻿1</sup> on this track.</p><div class="oucontentfigure" style="width:511px;" id="fig009_008"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3229488" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/1b06e8b0/smt359_1_008i.small.jpg" alt="Figure 8" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3229488">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 8 (a) A train that uses superconducting coils for magnetic levitation. (b) The guideway for the train, showing the coils used for levitation, guidance and propulsion.</span></div></div><a id="back_thumbnailfigure_idp3229488"></a></div><p>A second application is the use of a superconducting tube to screen sensitive components from magnetic fields, as shown in Figure 9. The tube is cooled below its critical temperature in a very small magnetic field. If a magnetic field is subsequently applied in the region of the tube, screening currents will be induced that generate fields which cancel out the applied field within the tube. However, note that effective screening requires a long tube, because only this geometry will generate a uniform magnetic field in the middle of the tube.</p><div class="oucontentfigure oucontentmediamini" id="fig009_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c81efc3a/smt359_1_009i.jpg" alt="Figure 9" width="238" height="305" style="maxwidth:238px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 9 A long superconducting tube screens the region inside from externally applied magnetic fields.</span></div></div></div><p></p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.2
2.2 Persistent currents lead to constant magnetic fluxSMT359_1<p>An important consequence of the persistent currents that flow in materials with zero resistance is that the magnetic flux that passes through a continuous loop of such a material remains constant. To see how this comes about, consider a ring of metal, enclosing a fixed area <i>A</i>, as shown in Figure 6a. An initial magnetic field <b>B</b><sub>0</sub> is applied perpendicular to the plane of the ring when the temperature is above the critical temperature of the material from which the ring is made. The magnetic flux Φ through the ring is <i>B</i><sub>0</sub><i>A</i>, and if the ring is cooled below its critical temperature while in this applied field, then the flux passing through it is unchanged. If we now change the applied field, then a current will be induced in the ring, and according to Lenz's law the direction of this current will be such that the magnetic flux it generates compensates for the flux change due to the change in the applied field. From Faraday's law, the induced emf in the ring is −dΦ/d<i>t</i> = −<i>A</i>d(<i>B</i> − <i>B</i><sub>0</sub>)/d<i>t</i>, and this generates an induced current <i>I</i> given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8639ffc1/smt359_1_ue001i.gif" alt=""/></div><p>where <i>L</i> is the selfinductance of the ring. Note that there is no ohmic term, <i>IR</i>, on the lefthand side of this equation, because we are assuming that <i>R</i> = 0. Integrating this equation, we obtain</p><div class="oucontentquote oucontentsbox" id="quo001"><blockquote><p><i>LI</i> + <i>BA</i> = constant.</p></blockquote></div><div class="oucontentfigure" style="width:511px;" id="fig009_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/0fc0d267/smt359_1_006i.jpg" alt="Figure 6" width="511" height="232" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 6 (a) A ring cooled below its critical temperature in an applied field B<sub>0</sub>. (b) When the applied field is removed, a superconducting current maintains the flux through the ring at the same value.</span></div></div></div><p>But <i>LI</i> is the amount of flux passing through the ring generated by the current <i>I</i> flowing in the ring – this is just the definition of selfinductance <i>L</i> – so (<i>LI</i> + <i>BA</i>) is the total magnetic flux through the ring. The total flux threading a circuit with zero resistance must therefore remain constant – it cannot change. If the applied magnetic field is changed, an induced current is set up that creates a flux to compensate exactly for the change in the flux from the applied magnetic field. Because the circuit has no resistance, the induced current can flow indefinitely, and the original amount of flux through the ring can be maintained indefinitely. This is true even if the external field is removed altogether; the flux through the ring is maintained by a persistent induced current, as in Figure 6b. However, note that constant flux through the ring does not mean that the magnetic field is unchanged. In Figure 6a there is a uniform field within the ring, whereas in Figure 6b the field is produced by a current flowing in the ring and will be much larger close to the ring than at its centre.</p><p>An important application of the constant flux through a superconducting circuit is shown in Figure 7. A superconducting solenoid, used to produce large magnetic fields, is connected to a power supply that can be adjusted to provide the appropriate current to generate the required field. For some applications it is important for the field to remain constant to a higher precision than the stability of the power supply would allow. A stable field is achieved by including a superconducting switch in parallel with the solenoid. This is not a mechanical switch, but a length of superconducting wire that is heated to above its critical temperature to ‘open’ the switch, and cooled below the critical temperature to ‘close’ it. With the switch open, the current from the power supply is set to give the required field strength. The switch is then closed to produce a completely superconducting circuit that includes the solenoid, the switch and the leads connecting them. The flux through this circuit must remain constant in time, so the field inside the solenoid will also remain constant in time. An added bonus is that the power supply can now be disconnected, which means that no energy is being dissipated while maintaining the field.</p><div class="oucontentfigure oucontentmediamini" id="fig009_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/caf7d6d6/smt359_1_007i.jpg" alt="Figure 7" width="279" height="327" style="maxwidth:279px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 7 A superconducting solenoid with a superconducting switch that allows it to operate in a persistent current mode.</span></div></div></div><p>Superconducting coils with persistent currents can be used in highspeed magneticallylevitated trains. In the system used on the Yamanashi Maglev Test Line in Japan (Figure 8), superconducting coils mounted on the sides of the train induce currents in coils mounted in the walls of a guideway, and the attractive and repulsive forces between the superconducting magnets and the trackmounted coils both levitate the train and provide lateral guidance. The train is propelled forwards by attractive and repulsive forces between the superconducting magnets and propulsion coils located on the walls of the guideway that are energised by a threephase alternating current that creates a shifting magnetic field along the guideway. In 2003, a train reached the recordbreaking speed of 581 km h<sup>−1</sup> on this track.</p><div class="oucontentfigure" style="width:511px;" id="fig009_008"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3229488" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/1b06e8b0/smt359_1_008i.small.jpg" alt="Figure 8" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3229488">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 8 (a) A train that uses superconducting coils for magnetic levitation. (b) The guideway for the train, showing the coils used for levitation, guidance and propulsion.</span></div></div><a id="back_thumbnailfigure_idp3229488"></a></div><p>A second application is the use of a superconducting tube to screen sensitive components from magnetic fields, as shown in Figure 9. The tube is cooled below its critical temperature in a very small magnetic field. If a magnetic field is subsequently applied in the region of the tube, screening currents will be induced that generate fields which cancel out the applied field within the tube. However, note that effective screening requires a long tube, because only this geometry will generate a uniform magnetic field in the middle of the tube.</p><div class="oucontentfigure oucontentmediamini" id="fig009_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c81efc3a/smt359_1_009i.jpg" alt="Figure 9" width="238" height="305" style="maxwidth:238px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 9 A long superconducting tube screens the region inside from externally applied magnetic fields.</span></div></div></div><p></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

2.3 The Meissner effect
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.3
Wed, 13 Apr 2016 23:00:00 GMT
<p>The second defining characteristic of a superconducting material is much less obvious than its zero electrical resistance. It was over 20 years after the discovery of superconductivity that Meissner and Ochsenfeld published a paper describing this second characteristic. They discovered that when a magnetic field is applied to a sample of tin, say, in the superconducting state, the applied field is excluded, so that <i>B</i> = 0 throughout its interior. This property of the superconducting state is known as the <b>Meissner effect</b>.</p><p>The exclusion of the magnetic field from a superconductor takes place regardless of whether the sample becomes superconducting before or after the external magnetic field is applied. In the steady state, the external magnetic field is cancelled in the interior of the superconductor by opposing magnetic fields produced by a steady screening current that flows on the surface of the superconductor.</p><p>It is important to recognise that the exclusion of the magnetic field from inside a superconductor cannot be predicted by applying Maxwell's equations to a material that has zero electrical resistance. We shall refer to a material that has zero resistance but does not exhibit the Meissner effect as a <b>perfect conductor</b>, and we shall show that a superconductor has additional properties besides those that can be predicted from its zero resistance.</p><p>Consider first the behaviour of a perfect conductor. We showed in the previous subsection that the flux enclosed by a continuous path through zero resistance material – a perfect conductor – remains constant, and this must be true for <i>any</i> path within the material, whatever its size or orientation. This means that the magnetic field throughout the material must remain constant, that is, ∂﻿<b>B</b>/∂﻿<i>t</i> = <b>0</b>. The consequences of this are shown in Figure 10 parts (a) and (b).</p><div class="oucontentfigure" style="width:512px;" id="fig009_010"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3247072" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/cae95e49/smt359_1_010i.small.jpg" alt="Figure 10" style="maxwidth:512px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3247072">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 10 A comparison of the response of a perfect conductor, (a) and (b), and a superconductor, (c) and (d), to an applied magnetic field.</span></div></div><a id="back_thumbnailfigure_idp3247072"></a></div><p>In part (a) of this figure, a perfect conductor is cooled in zero magnetic field to below the temperature at which its resistance becomes zero. When a magnetic field is applied, screening currents are induced in the surface to maintain the field at zero within the material, and when the field is removed, the field within the material stays at zero. In contrast, part (b) shows that cooling a perfect conductor to below its critical temperature in a uniform magnetic field leads to a situation where the uniform field is maintained within the material. If the applied field is then removed, the field within the conductor remains uniform, and continuity of magnetic field lines means there is a field in the region around the perfect conductor. Clearly, the magnetisation state of the perfect conductor depends not just on temperature and magnetic field, but also on the previous history of the material.</p><p>Contrast this with the behaviour of a superconductor, shown in Figure 10 parts (c) and (d). Whether a material is cooled below its superconducting critical temperature in zero field, (c), or in a finite field, (d), the magnetic field within a superconducting material is always zero. The magnetic field is always expelled from a superconductor. This is achieved spontaneously by producing currents on the surface of the superconductor. The direction of the currents is such as to create a magnetic field that exactly cancels the applied field in the superconductor. It is this active exclusion of magnetic field – the Meissner effect – that distinguishes a superconductor from a perfect conductor, a material that merely has zero resistance. Thus we can regard zero resistance and zero magnetic field as the two key characteristics of superconductivity.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.3
2.3 The Meissner effectSMT359_1<p>The second defining characteristic of a superconducting material is much less obvious than its zero electrical resistance. It was over 20 years after the discovery of superconductivity that Meissner and Ochsenfeld published a paper describing this second characteristic. They discovered that when a magnetic field is applied to a sample of tin, say, in the superconducting state, the applied field is excluded, so that <i>B</i> = 0 throughout its interior. This property of the superconducting state is known as the <b>Meissner effect</b>.</p><p>The exclusion of the magnetic field from a superconductor takes place regardless of whether the sample becomes superconducting before or after the external magnetic field is applied. In the steady state, the external magnetic field is cancelled in the interior of the superconductor by opposing magnetic fields produced by a steady screening current that flows on the surface of the superconductor.</p><p>It is important to recognise that the exclusion of the magnetic field from inside a superconductor cannot be predicted by applying Maxwell's equations to a material that has zero electrical resistance. We shall refer to a material that has zero resistance but does not exhibit the Meissner effect as a <b>perfect conductor</b>, and we shall show that a superconductor has additional properties besides those that can be predicted from its zero resistance.</p><p>Consider first the behaviour of a perfect conductor. We showed in the previous subsection that the flux enclosed by a continuous path through zero resistance material – a perfect conductor – remains constant, and this must be true for <i>any</i> path within the material, whatever its size or orientation. This means that the magnetic field throughout the material must remain constant, that is, ∂<b>B</b>/∂<i>t</i> = <b>0</b>. The consequences of this are shown in Figure 10 parts (a) and (b).</p><div class="oucontentfigure" style="width:512px;" id="fig009_010"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3247072" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/cae95e49/smt359_1_010i.small.jpg" alt="Figure 10" style="maxwidth:512px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3247072">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 10 A comparison of the response of a perfect conductor, (a) and (b), and a superconductor, (c) and (d), to an applied magnetic field.</span></div></div><a id="back_thumbnailfigure_idp3247072"></a></div><p>In part (a) of this figure, a perfect conductor is cooled in zero magnetic field to below the temperature at which its resistance becomes zero. When a magnetic field is applied, screening currents are induced in the surface to maintain the field at zero within the material, and when the field is removed, the field within the material stays at zero. In contrast, part (b) shows that cooling a perfect conductor to below its critical temperature in a uniform magnetic field leads to a situation where the uniform field is maintained within the material. If the applied field is then removed, the field within the conductor remains uniform, and continuity of magnetic field lines means there is a field in the region around the perfect conductor. Clearly, the magnetisation state of the perfect conductor depends not just on temperature and magnetic field, but also on the previous history of the material.</p><p>Contrast this with the behaviour of a superconductor, shown in Figure 10 parts (c) and (d). Whether a material is cooled below its superconducting critical temperature in zero field, (c), or in a finite field, (d), the magnetic field within a superconducting material is always zero. The magnetic field is always expelled from a superconductor. This is achieved spontaneously by producing currents on the surface of the superconductor. The direction of the currents is such as to create a magnetic field that exactly cancels the applied field in the superconductor. It is this active exclusion of magnetic field – the Meissner effect – that distinguishes a superconductor from a perfect conductor, a material that merely has zero resistance. Thus we can regard zero resistance and zero magnetic field as the two key characteristics of superconductivity.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

2.3.1 Perfect diamagnetism
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.3.1
Wed, 13 Apr 2016 23:00:00 GMT
<p>Diamagnetism is due to currents induced in atomic orbitals by an applied magnetic field. The induced currents produce a magnetisation within the diamagnetic material that opposes the applied field, and the magnetisation disappears when the applied field is removed. However, this effect is very small: the magnetisation generally reduces the applied field by less than one part in 10﻿<sup>5</sup> within the material. In diamagnetic material, <b>B</b> = <i>﻿μ﻿μ﻿</i><sub>0</sub>﻿<b>H</b>, with the relative permeability <i>μ</i> slightly less than unity.</p><p>Superconductors take the diamagnetic effect to the extreme, since in a superconductor the field <b>B</b> is zero – the field is completely screened from the interior of the material. Thus the relative permeability of a superconductor is zero.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.3.1
2.3.1 Perfect diamagnetismSMT359_1<p>Diamagnetism is due to currents induced in atomic orbitals by an applied magnetic field. The induced currents produce a magnetisation within the diamagnetic material that opposes the applied field, and the magnetisation disappears when the applied field is removed. However, this effect is very small: the magnetisation generally reduces the applied field by less than one part in 10<sup>5</sup> within the material. In diamagnetic material, <b>B</b> = <i>μμ</i><sub>0</sub><b>H</b>, with the relative permeability <i>μ</i> slightly less than unity.</p><p>Superconductors take the diamagnetic effect to the extreme, since in a superconductor the field <b>B</b> is zero – the field is completely screened from the interior of the material. Thus the relative permeability of a superconductor is zero.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

2.4 Critical magnetic field
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.4
Wed, 13 Apr 2016 23:00:00 GMT
<p>An important characteristic of a superconductor is that its normal resistance is restored if a sufficiently large magnetic field is applied. The nature of this transition to the normal state depends on the shape of the superconductor and the orientation of the magnetic field, and it is also different for pure elements and for alloys. In this subsection we describe the behaviour in the simplest situation; we shall discuss other more complex behaviour in Section 4.</p><p>If an increasing magnetic field is applied parallel to a long thin cylinder of tin at a constant temperature below the critical temperature, then the cylinder will make a transition from the superconducting state to the normal state when the field reaches a welldefined strength. This field at which the superconductivity is destroyed is known as the <b>critical magnetic field strength</b>, <i>B</i>﻿<sub>c</sub>. If the field is reduced, with the temperature held constant, the tin cylinder returns to the superconducting state at the same critical field strength <i>B</i>﻿<sub>c</sub>.</p><p>Experiments indicate that the critical magnetic field strength depends on temperature, and the form of this temperature dependence is shown in Figure 11 for several elements. At very low temperatures, the critical field strength is essentially independent of temperature, but as the temperature increases, the critical field strength drops, and becomes zero at the critical temperature. At temperatures just below the critical temperature it requires only a very small magnetic field to destroy the superconductivity.</p><p>The temperature dependence of the critical field strength is approximately parabolic:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/24a90791/smt359_1_e001i.gif" alt=""/></div><p>where <i>B</i><sub>c</sub>(0) is the extrapolated value of the critical field strength at absolute zero and <i>T</i>﻿<sub>c</sub> is the critical temperature. The curves in Figure 11 indicate that a superconductor with a high critical temperature <i>T</i>﻿<sub>c</sub> has a high critical field strength <i>B</i>﻿<sub>c</sub> at <i>T</i> = 0 K, and Table 1 confirms this correlation for a larger number of superconducting elements.</p><div class="oucontentfigure oucontentmediamini" id="fig009_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b0c52cac/smt359_1_011i.jpg" alt="Figure 11" width="256" height="255" style="maxwidth:256px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 11 The temperature dependences of the critical magnetic field strengths of mercury, tin, indium and thallium.</span></div></div></div><p></p><div class="oucontenttable oucontentsnormal oucontentsbox" id="tbl001"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Table 1 The critical temperatures <i>T</i><sub>c</sub> and critical magnetic field strengths <i>B</i><sub>c</sub>(0) for various superconducting elements.</h2><div class="oucontenttablewrapper"><table><tr><th scope="col"></th><th scope="col"><i>T</i>﻿<sub>c</sub>﻿/﻿K</th><th scope="col"><i>B</i>﻿<sub>c</sub>﻿(﻿0﻿)﻿/﻿mT</th></tr><tr><td>aluminium</td><td>1.2</td><td>10</td></tr><tr><td>cadmium</td><td>0.52</td><td>2.8</td></tr><tr><td>indium</td><td>3.4</td><td>28</td></tr><tr><td>lead</td><td>7.2</td><td>80</td></tr><tr><td>mercury</td><td>4.2</td><td>41</td></tr><tr><td>tantalum</td><td>4.5</td><td>83</td></tr><tr><td>thallium</td><td>2.4</td><td>18</td></tr><tr><td>tin</td><td>3.7</td><td>31</td></tr><tr><td>titanium</td><td>0.40</td><td>5.6</td></tr><tr><td>zinc</td><td>0.85</td><td>5.4</td></tr></table></div><div class="oucontentsourcereference"></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe009_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 2</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Estimate the magnetic field strength necessary to destroy superconductivity in a sample of lead at 4.2 K.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From Table 1, for lead <i>T</i>﻿<sub>c</sub> = 7.2 K and <i>B</i>﻿<sub>c</sub>﻿(﻿0﻿)  =  0.080 T. From Equation 1,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_029"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/0b584702/smt359_1_ue029i.gif" alt=""/></div></div></div></div></div><p>It is interesting to compare the magnetic behaviour of a superconducting element with typical curves for diamagnetic, paramagnetic and ferromagnetic materials. The magnetic behaviour of magnetic materials can be represented by <i>B</i> versus <i>H</i> graphs. Figure 12a shows the behaviour of typical diamagnetic and paramagnetic materials. Note that we have plotted <i>μ</i><sub>0</sub><i>H</i> on the horizontal axis rather than <i>H</i>, so that both axes use the same unit (tesla). The straight lines plotted correspond to the relationship <i>B</i> = <i>μ﻿μ</i>﻿<sub>0</sub>﻿<i>H</i>, with <i>μ</i> slightly smaller than unity for the diamagnetic material and slightly greater than unity for the paramagnetic material. The behaviour of a ferromagnet, shown in Figure 12b, is quite different, with <i>B</i> ≫ <i>μ</i>﻿<sub>0</sub>﻿<i>H</i>, and a highly nonlinear and irreversible curve until the magnetisation saturates, after which <i>B</i> increases linearly with <i>H</i>.</p><div class="oucontentfigure" style="width:511px;" id="fig009_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/84239698/smt359_1_012i.jpg" alt="Figure 12" width="511" height="262" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 12 Graphs showing <i>B</i> versus <i>μ﻿</i><sub>0</sub>﻿<i>H</i> for (a) diamagnetic and paramagnetic materials, and (b) a ferromagnetic material. Note the different scales for the <i>μ﻿</i><sub>0</sub>﻿<i>H</i>axes in the two graphs.</span></div></div></div><p>Compare these graphs with Figure 13, which shows the <i>B﻿﻿H</i> curve for a superconducting cylinder of tin, with the field parallel to its axis. The field strength <i>B</i> within the superconductor is zero when <i>μ</i><sub>0</sub>﻿<i>H</i> is less than the critical field strength <i>B</i>﻿<sub>c</sub><i>;</i> the superconductor behaves like a perfect diamagnetic material and completely excludes the field from its interior. But then <i>B</i> jumps abruptly to a value <i>B</i>﻿<sub>c</sub>, and at higher fields the tin cylinder obeys the relationship <i>B</i> <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> <i>﻿μ</i><sub>0</sub>﻿<i>H</i>, since the material is weakly diamagnetic in its normal state, with <i>μ</i> = 0.9998. The linear graphs in Figure 12a are similar to those for a superconductor above the critical field strength.</p><div class="oucontentfigure oucontentmediamini" id="fig009_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/11d3fd63/smt359_1_013i.jpg" alt="Figure 13" width="256" height="213" style="maxwidth:256px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 13 Graph showing <i>B</i> versus <i>μ</i>﻿<sub>0</sub>﻿<i>H</i> for a tin cylinder, aligned parallel to the field.</span></div></div></div><p></p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.4
2.4 Critical magnetic fieldSMT359_1<p>An important characteristic of a superconductor is that its normal resistance is restored if a sufficiently large magnetic field is applied. The nature of this transition to the normal state depends on the shape of the superconductor and the orientation of the magnetic field, and it is also different for pure elements and for alloys. In this subsection we describe the behaviour in the simplest situation; we shall discuss other more complex behaviour in Section 4.</p><p>If an increasing magnetic field is applied parallel to a long thin cylinder of tin at a constant temperature below the critical temperature, then the cylinder will make a transition from the superconducting state to the normal state when the field reaches a welldefined strength. This field at which the superconductivity is destroyed is known as the <b>critical magnetic field strength</b>, <i>B</i><sub>c</sub>. If the field is reduced, with the temperature held constant, the tin cylinder returns to the superconducting state at the same critical field strength <i>B</i><sub>c</sub>.</p><p>Experiments indicate that the critical magnetic field strength depends on temperature, and the form of this temperature dependence is shown in Figure 11 for several elements. At very low temperatures, the critical field strength is essentially independent of temperature, but as the temperature increases, the critical field strength drops, and becomes zero at the critical temperature. At temperatures just below the critical temperature it requires only a very small magnetic field to destroy the superconductivity.</p><p>The temperature dependence of the critical field strength is approximately parabolic:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/24a90791/smt359_1_e001i.gif" alt=""/></div><p>where <i>B</i><sub>c</sub>(0) is the extrapolated value of the critical field strength at absolute zero and <i>T</i><sub>c</sub> is the critical temperature. The curves in Figure 11 indicate that a superconductor with a high critical temperature <i>T</i><sub>c</sub> has a high critical field strength <i>B</i><sub>c</sub> at <i>T</i> = 0 K, and Table 1 confirms this correlation for a larger number of superconducting elements.</p><div class="oucontentfigure oucontentmediamini" id="fig009_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b0c52cac/smt359_1_011i.jpg" alt="Figure 11" width="256" height="255" style="maxwidth:256px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 11 The temperature dependences of the critical magnetic field strengths of mercury, tin, indium and thallium.</span></div></div></div><p></p><div class="oucontenttable oucontentsnormal oucontentsbox" id="tbl001"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Table 1 The critical temperatures <i>T</i><sub>c</sub> and critical magnetic field strengths <i>B</i><sub>c</sub>(0) for various superconducting elements.</h2><div class="oucontenttablewrapper"><table><tr><th scope="col"></th><th scope="col"><i>T</i><sub>c</sub>/K</th><th scope="col"><i>B</i><sub>c</sub>(0)/mT</th></tr><tr><td>aluminium</td><td>1.2</td><td>10</td></tr><tr><td>cadmium</td><td>0.52</td><td>2.8</td></tr><tr><td>indium</td><td>3.4</td><td>28</td></tr><tr><td>lead</td><td>7.2</td><td>80</td></tr><tr><td>mercury</td><td>4.2</td><td>41</td></tr><tr><td>tantalum</td><td>4.5</td><td>83</td></tr><tr><td>thallium</td><td>2.4</td><td>18</td></tr><tr><td>tin</td><td>3.7</td><td>31</td></tr><tr><td>titanium</td><td>0.40</td><td>5.6</td></tr><tr><td>zinc</td><td>0.85</td><td>5.4</td></tr></table></div><div class="oucontentsourcereference"></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe009_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 2</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Estimate the magnetic field strength necessary to destroy superconductivity in a sample of lead at 4.2 K.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From Table 1, for lead <i>T</i><sub>c</sub> = 7.2 K and <i>B</i><sub>c</sub>(0) = 0.080 T. From Equation 1,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_029"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/0b584702/smt359_1_ue029i.gif" alt=""/></div></div></div></div></div><p>It is interesting to compare the magnetic behaviour of a superconducting element with typical curves for diamagnetic, paramagnetic and ferromagnetic materials. The magnetic behaviour of magnetic materials can be represented by <i>B</i> versus <i>H</i> graphs. Figure 12a shows the behaviour of typical diamagnetic and paramagnetic materials. Note that we have plotted <i>μ</i><sub>0</sub><i>H</i> on the horizontal axis rather than <i>H</i>, so that both axes use the same unit (tesla). The straight lines plotted correspond to the relationship <i>B</i> = <i>μμ</i><sub>0</sub><i>H</i>, with <i>μ</i> slightly smaller than unity for the diamagnetic material and slightly greater than unity for the paramagnetic material. The behaviour of a ferromagnet, shown in Figure 12b, is quite different, with <i>B</i> ≫ <i>μ</i><sub>0</sub><i>H</i>, and a highly nonlinear and irreversible curve until the magnetisation saturates, after which <i>B</i> increases linearly with <i>H</i>.</p><div class="oucontentfigure" style="width:511px;" id="fig009_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/84239698/smt359_1_012i.jpg" alt="Figure 12" width="511" height="262" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 12 Graphs showing <i>B</i> versus <i>μ</i><sub>0</sub><i>H</i> for (a) diamagnetic and paramagnetic materials, and (b) a ferromagnetic material. Note the different scales for the <i>μ</i><sub>0</sub><i>H</i>axes in the two graphs.</span></div></div></div><p>Compare these graphs with Figure 13, which shows the <i>BH</i> curve for a superconducting cylinder of tin, with the field parallel to its axis. The field strength <i>B</i> within the superconductor is zero when <i>μ</i><sub>0</sub><i>H</i> is less than the critical field strength <i>B</i><sub>c</sub><i>;</i> the superconductor behaves like a perfect diamagnetic material and completely excludes the field from its interior. But then <i>B</i> jumps abruptly to a value <i>B</i><sub>c</sub>, and at higher fields the tin cylinder obeys the relationship <i>B</i> <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> <i>μ</i><sub>0</sub><i>H</i>, since the material is weakly diamagnetic in its normal state, with <i>μ</i> = 0.9998. The linear graphs in Figure 12a are similar to those for a superconductor above the critical field strength.</p><div class="oucontentfigure oucontentmediamini" id="fig009_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/11d3fd63/smt359_1_013i.jpg" alt="Figure 13" width="256" height="213" style="maxwidth:256px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 13 Graph showing <i>B</i> versus <i>μ</i><sub>0</sub><i>H</i> for a tin cylinder, aligned parallel to the field.</span></div></div></div><p></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

2.5 Critical current
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.5
Wed, 13 Apr 2016 23:00:00 GMT
<p>The current density for a steady current flowing along a wire in its normal state is essentially uniform over its crosssection. A consequence of this is that the magnetic field strength <i>B</i> within a wire of radius <i>a</i>, carrying current <i>I</i>, increases linearly with distance from the centre of the wire, and reaches a maximum value of <i>μ</i>﻿<sub>0</sub><i>I / </i>2﻿<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>﻿<i>a</i> at the surface of the wire (see Exercise 3.6). Within a superconductor, however, the magnetic field <b>B</b> is zero.</p><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 2</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>What can you deduce about the current flow in a superconducting wire from the fact that <b>B</b> = <b>0</b> within a superconductor?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The current density within the bulk of the wire must be zero, since Ampère's law (﻿curl <b>B</b> = <i>﻿μ﻿</i><sub>0</sub>﻿<b>J</b>﻿) indicates that a nonzero current density would produce a magnetic field. The current must therefore flow in the surface of the wire.</p></div></div></div></div><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 3</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>How does the magnetic field just outside the surface of a superconducting wire, radius <i>a</i>, carrying current <i>I</i>, compare with the field just outside the surface of a normal wire with the same radius, carrying the same current?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The fields just outside the surface are identical. The currents in both wires are axially symmetric, so the integral version of Ampère's law indicates that the fields just outside the surfaces of the wires are the same.</p></div></div></div></div><p>The magnetic field strength <i>B</i> just outside the surface of the wire is <i>μ</i>﻿<sub>0</sub>﻿<i>I</i> / 2﻿<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span><i>﻿a</i>. It follows that if the current flowing in a superconducting wire is increased, eventually the field strength at the surface of the wire will exceed <i>B﻿<sub>c</sub></i> and the sample will revert to its normal state. The maximum current that a wire can carry with zero resistance is known as its <b>critical current</b>, and for a long straight wire the critical current <i>I<sub>﻿c</sub></i> is given by <i>I<sub>﻿c</sub></i> = 2﻿<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span><i>﻿aB﻿</i><sub>c</sub> / <i>μ</i>﻿<sub>0</sub>. A current greater than <i>I<sub>﻿c</sub></i> will cause the wire to revert to its normal state. This critical current is proportional to the radius of the wire.</p><p>In the previous subsection you saw that the critical field strength is dependent on temperature, decreasing to zero as the temperature is increased to the critical temperature. This means that the superconducting current that a wire can carry will also decrease as the temperature gets closer to the critical temperature. Because of this, in real applications superconductors generally operate at temperatures less than half of the critical temperature, where the critical field strength, and therefore the critical current, is greater than 75 per cent of the maximum value.</p><p>Now, the current carried by a superconducting wire actually flows in a thin layer at the surface; it cannot be restricted to an infinitesimal layer, because that would lead to an infinite current density. As you will see in Section 3, this means that the magnetic field penetrates into this thin layer, and we derive there relationships between the field and the current density. But in the present context, the point to note is that the transition to the normal state takes place when the magnetic field strength at the surface corresponds to the critical field strength, and this occurs when the current density at the surface reaches a critical current density. This critical current density is much greater than <i>I﻿<sub>c</sub></i> / <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>﻿<i>a</i>﻿<sup>2</sup> because the current flows only in a thin surface layer.</p><p>The magnetic field at the surface of a superconductor may have a contribution from an external source of magnetic field, as well as from the field produced by the current in the wire. This external field will set up screening currents in the surface layer of the material. The transition to the normal state then occurs when the vector sum of the current densities at the surface due to the current in the wire and due to the screening current exceeds the critical current density, or, equivalently, when the magnitude of the vector sum of the magnetic fields that are present at the surface of the wire exceeds the critical field strength.</p><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe009_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 3</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Tin has <i>T</i>﻿<sub>c</sub> = 3.7 K and <i>B</i>﻿<sub>c</sub> = 31 mT at <i>T</i> = 0 K. What is the minimum radius required for tin wire if it is to carry a current of 200 A at <i>T</i> = 2.0 K﻿?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From Ampère's law, the field strength <i>B</i> at the surface of a wire of radius <i>R</i> carrying a current <i>I</i> is <i>B</i> = <i>μ</i>﻿<sub>0</sub>﻿<i>I</i> / 2﻿<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>﻿<i>R</i>. From Equation 1, we also have that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_030"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/d155739a/smt359_1_ue030i.gif" alt=""/></div><p>so the radius <i>R</i> required is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_031"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/1d227cdf/smt359_1_ue031i.gif" alt=""/></div></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection2.5
2.5 Critical currentSMT359_1<p>The current density for a steady current flowing along a wire in its normal state is essentially uniform over its crosssection. A consequence of this is that the magnetic field strength <i>B</i> within a wire of radius <i>a</i>, carrying current <i>I</i>, increases linearly with distance from the centre of the wire, and reaches a maximum value of <i>μ</i><sub>0</sub><i>I / </i>2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span><i>a</i> at the surface of the wire (see Exercise 3.6). Within a superconductor, however, the magnetic field <b>B</b> is zero.</p><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 2</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>What can you deduce about the current flow in a superconducting wire from the fact that <b>B</b> = <b>0</b> within a superconductor?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The current density within the bulk of the wire must be zero, since Ampère's law (curl <b>B</b> = <i>μ</i><sub>0</sub><b>J</b>) indicates that a nonzero current density would produce a magnetic field. The current must therefore flow in the surface of the wire.</p></div></div></div></div><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 3</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>How does the magnetic field just outside the surface of a superconducting wire, radius <i>a</i>, carrying current <i>I</i>, compare with the field just outside the surface of a normal wire with the same radius, carrying the same current?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The fields just outside the surface are identical. The currents in both wires are axially symmetric, so the integral version of Ampère's law indicates that the fields just outside the surfaces of the wires are the same.</p></div></div></div></div><p>The magnetic field strength <i>B</i> just outside the surface of the wire is <i>μ</i><sub>0</sub><i>I</i> / 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span><i>a</i>. It follows that if the current flowing in a superconducting wire is increased, eventually the field strength at the surface of the wire will exceed <i>B<sub>c</sub></i> and the sample will revert to its normal state. The maximum current that a wire can carry with zero resistance is known as its <b>critical current</b>, and for a long straight wire the critical current <i>I<sub>c</sub></i> is given by <i>I<sub>c</sub></i> = 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span><i>aB</i><sub>c</sub> / <i>μ</i><sub>0</sub>. A current greater than <i>I<sub>c</sub></i> will cause the wire to revert to its normal state. This critical current is proportional to the radius of the wire.</p><p>In the previous subsection you saw that the critical field strength is dependent on temperature, decreasing to zero as the temperature is increased to the critical temperature. This means that the superconducting current that a wire can carry will also decrease as the temperature gets closer to the critical temperature. Because of this, in real applications superconductors generally operate at temperatures less than half of the critical temperature, where the critical field strength, and therefore the critical current, is greater than 75 per cent of the maximum value.</p><p>Now, the current carried by a superconducting wire actually flows in a thin layer at the surface; it cannot be restricted to an infinitesimal layer, because that would lead to an infinite current density. As you will see in Section 3, this means that the magnetic field penetrates into this thin layer, and we derive there relationships between the field and the current density. But in the present context, the point to note is that the transition to the normal state takes place when the magnetic field strength at the surface corresponds to the critical field strength, and this occurs when the current density at the surface reaches a critical current density. This critical current density is much greater than <i>I<sub>c</sub></i> / <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span><i>a</i><sup>2</sup> because the current flows only in a thin surface layer.</p><p>The magnetic field at the surface of a superconductor may have a contribution from an external source of magnetic field, as well as from the field produced by the current in the wire. This external field will set up screening currents in the surface layer of the material. The transition to the normal state then occurs when the vector sum of the current densities at the surface due to the current in the wire and due to the screening current exceeds the critical current density, or, equivalently, when the magnitude of the vector sum of the magnetic fields that are present at the surface of the wire exceeds the critical field strength.</p><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe009_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 3</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Tin has <i>T</i><sub>c</sub> = 3.7 K and <i>B</i><sub>c</sub> = 31 mT at <i>T</i> = 0 K. What is the minimum radius required for tin wire if it is to carry a current of 200 A at <i>T</i> = 2.0 K?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From Ampère's law, the field strength <i>B</i> at the surface of a wire of radius <i>R</i> carrying a current <i>I</i> is <i>B</i> = <i>μ</i><sub>0</sub><i>I</i> / 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span><i>R</i>. From Equation 1, we also have that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_030"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/d155739a/smt359_1_ue030i.gif" alt=""/></div><p>so the radius <i>R</i> required is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_031"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/1d227cdf/smt359_1_ue031i.gif" alt=""/></div></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

3.1 A twofluid model
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.1
Wed, 13 Apr 2016 23:00:00 GMT
<p>As was mentioned earlier, a substantial dose of quantum mechanics would be required to provide a full explanation of the properties of superconductors. This would take us too far away from electromagnetism, and we shall therefore restrict our discussion to aspects that can be discussed using classical concepts of electromagnetism.</p><p>We shall model the free electrons within a superconductor as two fluids. According to this <b>twofluid model</b>, one fluid consists of ‘normal’ electrons, number density <i>n</i>﻿<sub>n</sub>, and these behave in exactly the same way as the free electrons in a normal metal. They are accelerated by an electric field <b>E</b>, but are frequently scattered by impurities and defects in the ion lattice and by thermal vibrations of the lattice. The scattering limits the speed of the electrons, and they attain a mean drift velocity <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/e03856f3/vn.gif" alt="" width="28" height="18" style="maxwidth:28px;" class="oucontentinlinefigureimage"/></span> = −﻿e﻿<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/2c539de0/tau.gif" alt="" width="5" height="7" style="maxwidth:5px;" class="oucontentinlinefigureimage"/></span>﻿<b>E</b> / <i>m</i>, where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/2c539de0/tau.gif" alt="" width="5" height="7" style="maxwidth:5px;" class="oucontentinlinefigureimage"/></span> is the mean time between scattering events for the electrons and <i>m</i> is the electron mass. The current density <b>J</b><sub>n</sub> due to flow of these electrons is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/6138d25a/smt359_1_e002i.gif" alt=""/></div><p>Interspersed with the normal electrons are what we shall call the superconducting electrons, or superelectrons, which form a fluid with number density <i>n</i><sub>s</sub>. The superconducting electrons are not scattered by impurities, defects or thermal vibrations, so they are freely accelerated by an electric field. If the velocity of a superconducting electron is <b>v</b><sub>s</sub>, then its equation of motion is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c5f82f75/smt359_1_ue003i.gif" alt=""/></div><p>Combining this with the expression for the current density, <b>J</b>﻿<sub>s</sub> = −﻿<i>n</i>﻿<sub>s</sub>﻿<i>e</i>﻿<b>v</b>﻿<sub>s</sub>, we find that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/eff7eb98/smt359_1_e003i.gif" alt=""/></div><p>Compare this with Equation 2, which relates current density and electric field in a normal conductor. Scattering of the normal electrons leads to a constant current in a constant electric field, whereas the absence of scattering of the electrons in a superconductor means that the current density would increase steadily in a constant electric field. However, if we consider a constant current flowing in the superconductor, then ∂﻿<b>J</b>﻿<sub>s</sub>/∂﻿<i>t</i> = 0, so <b>E</b> = <b>0</b>. Therefore the normal current density must be zero – all of the steady current in a superconductor is carried by the superconducting electrons. Of course, with no electric field within the superconductor, there will be no potential difference across it, and so it has zero resistance.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.1
3.1 A twofluid modelSMT359_1<p>As was mentioned earlier, a substantial dose of quantum mechanics would be required to provide a full explanation of the properties of superconductors. This would take us too far away from electromagnetism, and we shall therefore restrict our discussion to aspects that can be discussed using classical concepts of electromagnetism.</p><p>We shall model the free electrons within a superconductor as two fluids. According to this <b>twofluid model</b>, one fluid consists of ‘normal’ electrons, number density <i>n</i><sub>n</sub>, and these behave in exactly the same way as the free electrons in a normal metal. They are accelerated by an electric field <b>E</b>, but are frequently scattered by impurities and defects in the ion lattice and by thermal vibrations of the lattice. The scattering limits the speed of the electrons, and they attain a mean drift velocity <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/e03856f3/vn.gif" alt="" width="28" height="18" style="maxwidth:28px;" class="oucontentinlinefigureimage"/></span> = −e<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/2c539de0/tau.gif" alt="" width="5" height="7" style="maxwidth:5px;" class="oucontentinlinefigureimage"/></span><b>E</b> / <i>m</i>, where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/2c539de0/tau.gif" alt="" width="5" height="7" style="maxwidth:5px;" class="oucontentinlinefigureimage"/></span> is the mean time between scattering events for the electrons and <i>m</i> is the electron mass. The current density <b>J</b><sub>n</sub> due to flow of these electrons is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/6138d25a/smt359_1_e002i.gif" alt=""/></div><p>Interspersed with the normal electrons are what we shall call the superconducting electrons, or superelectrons, which form a fluid with number density <i>n</i><sub>s</sub>. The superconducting electrons are not scattered by impurities, defects or thermal vibrations, so they are freely accelerated by an electric field. If the velocity of a superconducting electron is <b>v</b><sub>s</sub>, then its equation of motion is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c5f82f75/smt359_1_ue003i.gif" alt=""/></div><p>Combining this with the expression for the current density, <b>J</b><sub>s</sub> = −<i>n</i><sub>s</sub><i>e</i><b>v</b><sub>s</sub>, we find that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/eff7eb98/smt359_1_e003i.gif" alt=""/></div><p>Compare this with Equation 2, which relates current density and electric field in a normal conductor. Scattering of the normal electrons leads to a constant current in a constant electric field, whereas the absence of scattering of the electrons in a superconductor means that the current density would increase steadily in a constant electric field. However, if we consider a constant current flowing in the superconductor, then ∂<b>J</b><sub>s</sub>/∂<i>t</i> = 0, so <b>E</b> = <b>0</b>. Therefore the normal current density must be zero – all of the steady current in a superconductor is carried by the superconducting electrons. Of course, with no electric field within the superconductor, there will be no potential difference across it, and so it has zero resistance.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

3.2 Magnetic field in a perfect conductor
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.2
Wed, 13 Apr 2016 23:00:00 GMT
<p>When discussing the Meissner effect in Subsection 2.3, we argued qualitatively that a material that just had the property of zero resistance – a perfect conductor rather than a superconductor – would maintain a constant magnetic field in its interior, and would not expel any field that was present when the material became superconducting. We shall now show how that conclusion follows from an application of Maxwell's equations to a perfect conductor. We can then see what additional assumptions are needed to account for the Meissner effect in a superconductor.</p><p>We assume that the electrons in a perfect conductor (or a proportion of them) are not scattered, and therefore the current density is governed by Equation 3. However, we shall use the subscript ‘pc’ (for perfect conductor) here to indicate that we are not dealing with a superconductor. We are interested in the magnetic field in a perfect conductor, so we shall apply Maxwell's equations to this situation. Faraday's law is valid in all situations,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/04dab8c4/smt359_1_ue004i.gif" alt=""/></div><p>and if we substitute for <b>E</b> using Equation 3, we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/f01b40d7/smt359_1_e004i.gif" alt=""/></div><p>Looking now at the AmpèreMaxwell law, curl <b>H</b> = <b>J</b>﻿<sub>f</sub> + ∂﻿<b>D</b>/∂﻿<i>t</i>, we shall assume that our perfect conductor is either weakly diamagnetic or weakly paramagnetic, so that</p><p><i>μ</i> <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> 1 and <b>H</b> <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> <b>B</b>/<i>μ</i><sub>0</sub> are very good approximations.</p><p>We shall also omit the Maxwell term, ∂﻿<b>D</b>/∂﻿<i>t</i>, since this is negligible for the static, or slowlyvarying, fields that we shall be considering. With these approximations, the AmpèreMaxwell law simplifies to Ampère's law,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/e0dc08b0/smt359_1_e005i.gif" alt=""/></div><p>where use of the subscript pc for the current density reminds us that the free current <b>J</b>﻿<sub>f</sub> is carried by the perfectlyconducting electrons. We now use this expression to eliminate <b>J</b>﻿<sub>pc</sub> from Equation 4:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/23b8ba07/smt359_1_e006i.gif" alt=""/></div><p>We can use a standard vector identity from inside the back cover to rewrite the lefthand side of this equation:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/f97124e8/smt359_1_ue005i.gif" alt=""/></div><p>The nomonopole law, div <b>B</b> = 0, means that the first term on the righthand side of this equation is zero, so Equation 6 can be rewritten as</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/dcce2e15/smt359_1_e007i.gif" alt=""/></div><p>This equation determines how ∂﻿<b>B</b><i>﻿/﻿</i>∂﻿<i>t</i> varies in a perfect conductor.</p><p>We shall look for the solution to Equation 7 for the simple geometry shown in Figure 14; a conductor has a boundary corresponding to the plane <i>z</i> = 0, and occupies the region <i>z</i> > 0, with a uniform field outside the conductor given by <b>B</b>﻿<sub>0</sub> = <i>B</i>﻿<sub>0</sub>﻿<b>e</b>﻿<sub><i>x</i></sub>.</p><div class="oucontentfigure oucontentmediamini" id="fig009_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/fa9f8f3b/smt359_1_014i.jpg" alt="Figure 14" width="248" height="278" style="maxwidth:248px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 14 A plane boundary between a conductor and air.</span></div></div></div><p>The uniform external field in the <i>x</i>direction means that the field inside the conductor will also be in the <i>x</i>direction, and its strength will depend only on <i>z</i>. So Equation 7 reduces to the onedimensional form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/2e7ec8ea/smt359_1_ue006i.gif" alt=""/></div><p>where we have simplified the equation, for reasons that will soon become clear, by writing </p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="x009_011ie"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/eb59ea79/smt359_1_ie011i.gif" alt=""/></div><p>The general solution of this equation is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/d0586a95/smt359_1_ue007i.gif" alt=""/></div><p>where <i>a</i> and <i>b</i> are independent of position. The second term on the righthand side corresponds to a rate of change of field strength that continues to increase exponentially with distance from the boundary; since this is unphysical, we set <i>b</i> = 0. The boundary condition for the field parallel to the boundary is that <i>H</i><sub>∥</sub> is continuous, and since we are assuming that <i>μ</i> <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> 1 in both the air and the conductor, this is equivalent to <i>B</i>﻿<sub>∥</sub> being the same on either side of the boundary at all times. This means that ∂﻿<i>B</i>/∂﻿<i>t</i> is the same on either side of the boundary, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="x009_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/871e940f/smt359_1_ie014i.gif" alt=""/></div><p>and the field within the perfect conductor satisfies the equation</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/7cd2ac06/smt359_1_e008i.gif" alt=""/></div><p>This indicates that any <i>changes</i> in the external magnetic field are attenuated exponentially with distance below the surface of the perfect conductor. If the distance λ<sub>pc</sub> is very small, then the field will not change within the bulk of the perfect conductor, and this is the behaviour that we described qualitatively in Subsection 2.3. Note that this does not mean the magnetic field <i>must</i> be expelled: flux expulsion <i>requires B</i> = 0, rather than just ∂﻿<i>B</i>/∂﻿<i>t</i> = 0. So how do we modify the description that we have given of a perfect conductor so that it describes a superconductor and leads to a prediction that <i>B</i> = 0?</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.2
3.2 Magnetic field in a perfect conductorSMT359_1<p>When discussing the Meissner effect in Subsection 2.3, we argued qualitatively that a material that just had the property of zero resistance – a perfect conductor rather than a superconductor – would maintain a constant magnetic field in its interior, and would not expel any field that was present when the material became superconducting. We shall now show how that conclusion follows from an application of Maxwell's equations to a perfect conductor. We can then see what additional assumptions are needed to account for the Meissner effect in a superconductor.</p><p>We assume that the electrons in a perfect conductor (or a proportion of them) are not scattered, and therefore the current density is governed by Equation 3. However, we shall use the subscript ‘pc’ (for perfect conductor) here to indicate that we are not dealing with a superconductor. We are interested in the magnetic field in a perfect conductor, so we shall apply Maxwell's equations to this situation. Faraday's law is valid in all situations,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/04dab8c4/smt359_1_ue004i.gif" alt=""/></div><p>and if we substitute for <b>E</b> using Equation 3, we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/f01b40d7/smt359_1_e004i.gif" alt=""/></div><p>Looking now at the AmpèreMaxwell law, curl <b>H</b> = <b>J</b><sub>f</sub> + ∂<b>D</b>/∂<i>t</i>, we shall assume that our perfect conductor is either weakly diamagnetic or weakly paramagnetic, so that</p><p><i>μ</i> <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> 1 and <b>H</b> <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> <b>B</b>/<i>μ</i><sub>0</sub> are very good approximations.</p><p>We shall also omit the Maxwell term, ∂<b>D</b>/∂<i>t</i>, since this is negligible for the static, or slowlyvarying, fields that we shall be considering. With these approximations, the AmpèreMaxwell law simplifies to Ampère's law,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/e0dc08b0/smt359_1_e005i.gif" alt=""/></div><p>where use of the subscript pc for the current density reminds us that the free current <b>J</b><sub>f</sub> is carried by the perfectlyconducting electrons. We now use this expression to eliminate <b>J</b><sub>pc</sub> from Equation 4:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/23b8ba07/smt359_1_e006i.gif" alt=""/></div><p>We can use a standard vector identity from inside the back cover to rewrite the lefthand side of this equation:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/f97124e8/smt359_1_ue005i.gif" alt=""/></div><p>The nomonopole law, div <b>B</b> = 0, means that the first term on the righthand side of this equation is zero, so Equation 6 can be rewritten as</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/dcce2e15/smt359_1_e007i.gif" alt=""/></div><p>This equation determines how ∂<b>B</b><i>/</i>∂<i>t</i> varies in a perfect conductor.</p><p>We shall look for the solution to Equation 7 for the simple geometry shown in Figure 14; a conductor has a boundary corresponding to the plane <i>z</i> = 0, and occupies the region <i>z</i> > 0, with a uniform field outside the conductor given by <b>B</b><sub>0</sub> = <i>B</i><sub>0</sub><b>e</b><sub><i>x</i></sub>.</p><div class="oucontentfigure oucontentmediamini" id="fig009_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/fa9f8f3b/smt359_1_014i.jpg" alt="Figure 14" width="248" height="278" style="maxwidth:248px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 14 A plane boundary between a conductor and air.</span></div></div></div><p>The uniform external field in the <i>x</i>direction means that the field inside the conductor will also be in the <i>x</i>direction, and its strength will depend only on <i>z</i>. So Equation 7 reduces to the onedimensional form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/2e7ec8ea/smt359_1_ue006i.gif" alt=""/></div><p>where we have simplified the equation, for reasons that will soon become clear, by writing </p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="x009_011ie"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/eb59ea79/smt359_1_ie011i.gif" alt=""/></div><p>The general solution of this equation is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/d0586a95/smt359_1_ue007i.gif" alt=""/></div><p>where <i>a</i> and <i>b</i> are independent of position. The second term on the righthand side corresponds to a rate of change of field strength that continues to increase exponentially with distance from the boundary; since this is unphysical, we set <i>b</i> = 0. The boundary condition for the field parallel to the boundary is that <i>H</i><sub>∥</sub> is continuous, and since we are assuming that <i>μ</i> <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8ddb333f/equal_to.gif" alt="" width="10" height="5" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> 1 in both the air and the conductor, this is equivalent to <i>B</i><sub>∥</sub> being the same on either side of the boundary at all times. This means that ∂<i>B</i>/∂<i>t</i> is the same on either side of the boundary, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="x009_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/871e940f/smt359_1_ie014i.gif" alt=""/></div><p>and the field within the perfect conductor satisfies the equation</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/7cd2ac06/smt359_1_e008i.gif" alt=""/></div><p>This indicates that any <i>changes</i> in the external magnetic field are attenuated exponentially with distance below the surface of the perfect conductor. If the distance λ<sub>pc</sub> is very small, then the field will not change within the bulk of the perfect conductor, and this is the behaviour that we described qualitatively in Subsection 2.3. Note that this does not mean the magnetic field <i>must</i> be expelled: flux expulsion <i>requires B</i> = 0, rather than just ∂<i>B</i>/∂<i>t</i> = 0. So how do we modify the description that we have given of a perfect conductor so that it describes a superconductor and leads to a prediction that <i>B</i> = 0?</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

3.3 The London equations
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.3
Wed, 13 Apr 2016 23:00:00 GMT
<p>A simple but useful description of the electrodynamics of superconductivity was put forward by the brothers Fritz and Heinz London in 1935, shortly after the discovery that magnetic fields are expelled from superconductors. Their proposed equations are consistent with the Meissner effect and can be used with Maxwell's equations to predict how the magnetic field and surface current vary with distance from the surface of a superconductor.</p><p>In order to account for the Meissner effect, the London brothers proposed that in a superconductor, Equation 4 is replaced by the more restrictive relationship</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8bb2f5b6/smt359_1_ue008i.gif" alt=""/></div><p>This equation, and Equation 3 which relates the rate of change of current to the electric field, are now known as the <b>London equations</b>.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box009_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">London equations</h2><div class="oucontentinnerbox"><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/76257161/smt359_1_e009i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_009a"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/ed9fc7ac/smt359_1_e010i.gif" alt=""/></div></div></div></div><p>It is important to note that these equations are not an explanation of superconductivity. They were introduced as a restriction on Maxwell's equations so that the behaviour of superconductors deduced from the equations was consistent with experimental observations, and in particular with the Meissner effect. Their status is somewhat similar to Ohm's law, which is a useful description of the behaviour of many normal metals, but which does not provide any explanation for the conduction process at the microscopic level.</p><p>To demonstrate how the London equations lead to the Meissner effect, we proceed in the same way as for the perfect conductor. First we use Ampère's law, curl <b>B</b> = <i>μ﻿</i><sub>0</sub>﻿<b>J</b>﻿<sub>s</sub>, to substitute for <b>J</b>﻿<sub>s</sub> in Equation 9, and we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/1cd4d4f9/smt359_1_ue009i.gif" alt=""/></div><p>where</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/fbf49eaf/smt359_1_e011i.gif" alt=""/></div><p>But curl﻿(curl <b>B</b>﻿) = grad﻿(﻿div <b>B</b>﻿) − ∇﻿<sup>2</sup>﻿<b>B</b> = −﻿∇﻿<sup>2</sup>﻿<b>B</b>, since div <b>B</b> = 0. So</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/88450cd3/smt359_1_e012i.gif" alt=""/></div><p>This equation is similar to Equation 7, but ∂﻿<i>B</i>/∂﻿<i>t</i> has been replaced by <b>B</b>. The important point to note about this equation is that the only solution that corresponds to a spatially uniform field (for which ∇﻿<sup>2</sup>﻿<b>B</b> = 0﻿) is the field that is identically zero everywhere. If <b>B</b> were not equal to zero, then ∇﻿<sup>2</sup>﻿<b>B</b> would not be zero, so <b>B</b> would depend on position. Thus, a uniform magnetic field like that shown in Figure 10b cannot exist in a superconductor.</p><p>If we consider again the simple onedimensional geometry shown in Figure 14, then we obtain the solution to Equation 11 by simply replacing the partial time derivatives of the fields in the solution for the perfect conductor (Equation 8) by the fields themselves, that is,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/79f7a57f/smt359_1_e013i.gif" alt=""/></div><p>Therefore, the London equations lead to the prediction of an exponential decay of the magnetic field within the superconductor, as shown in Figure 15.</p><div class="oucontentfigure oucontentmediamini" id="fig009_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/1337a269/smt359_1_015i.jpg" alt="Figure 15" width="288" height="334" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 15 The penetration of a magnetic field into a superconducting material, showing the penetration depth, λ.</span></div></div></div><p></p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.3
3.3 The London equationsSMT359_1<p>A simple but useful description of the electrodynamics of superconductivity was put forward by the brothers Fritz and Heinz London in 1935, shortly after the discovery that magnetic fields are expelled from superconductors. Their proposed equations are consistent with the Meissner effect and can be used with Maxwell's equations to predict how the magnetic field and surface current vary with distance from the surface of a superconductor.</p><p>In order to account for the Meissner effect, the London brothers proposed that in a superconductor, Equation 4 is replaced by the more restrictive relationship</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8bb2f5b6/smt359_1_ue008i.gif" alt=""/></div><p>This equation, and Equation 3 which relates the rate of change of current to the electric field, are now known as the <b>London equations</b>.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box009_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">London equations</h2><div class="oucontentinnerbox"><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/76257161/smt359_1_e009i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_009a"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/ed9fc7ac/smt359_1_e010i.gif" alt=""/></div></div></div></div><p>It is important to note that these equations are not an explanation of superconductivity. They were introduced as a restriction on Maxwell's equations so that the behaviour of superconductors deduced from the equations was consistent with experimental observations, and in particular with the Meissner effect. Their status is somewhat similar to Ohm's law, which is a useful description of the behaviour of many normal metals, but which does not provide any explanation for the conduction process at the microscopic level.</p><p>To demonstrate how the London equations lead to the Meissner effect, we proceed in the same way as for the perfect conductor. First we use Ampère's law, curl <b>B</b> = <i>μ</i><sub>0</sub><b>J</b><sub>s</sub>, to substitute for <b>J</b><sub>s</sub> in Equation 9, and we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/1cd4d4f9/smt359_1_ue009i.gif" alt=""/></div><p>where</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/fbf49eaf/smt359_1_e011i.gif" alt=""/></div><p>But curl(curl <b>B</b>) = grad(div <b>B</b>) − ∇<sup>2</sup><b>B</b> = −∇<sup>2</sup><b>B</b>, since div <b>B</b> = 0. So</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/88450cd3/smt359_1_e012i.gif" alt=""/></div><p>This equation is similar to Equation 7, but ∂<i>B</i>/∂<i>t</i> has been replaced by <b>B</b>. The important point to note about this equation is that the only solution that corresponds to a spatially uniform field (for which ∇<sup>2</sup><b>B</b> = 0) is the field that is identically zero everywhere. If <b>B</b> were not equal to zero, then ∇<sup>2</sup><b>B</b> would not be zero, so <b>B</b> would depend on position. Thus, a uniform magnetic field like that shown in Figure 10b cannot exist in a superconductor.</p><p>If we consider again the simple onedimensional geometry shown in Figure 14, then we obtain the solution to Equation 11 by simply replacing the partial time derivatives of the fields in the solution for the perfect conductor (Equation 8) by the fields themselves, that is,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/79f7a57f/smt359_1_e013i.gif" alt=""/></div><p>Therefore, the London equations lead to the prediction of an exponential decay of the magnetic field within the superconductor, as shown in Figure 15.</p><div class="oucontentfigure oucontentmediamini" id="fig009_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/1337a269/smt359_1_015i.jpg" alt="Figure 15" width="288" height="334" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 15 The penetration of a magnetic field into a superconducting material, showing the penetration depth, λ.</span></div></div></div><p></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

3.4 Penetration depth
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.4
Wed, 13 Apr 2016 23:00:00 GMT
<p>The characteristic length, λ, associated with the decay of the magnetic field at the surface of a superconductor is known as the <b>penetration depth</b>, and it depends on the number density <i>n</i>﻿<sub>s</sub> of superconducting electrons.</p><p>We can estimate a value for λ by assuming that all of the free electrons are superconducting. If we set <i>n</i>﻿<sub>s</sub> = 10﻿<sup>29</sup> m<sup>﻿−﻿3</sup>, a typical free electron density in a metal, then we find that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/a264229e/smt359_1_ue010i.gif" alt=""/></div><p>The small size of λ indicates that the magnetic field is effectively excluded from the interior of macroscopic specimens of superconductors, in agreement with the experimentally observed Meissner effect.</p><p>The small scale of the field penetration means that carefullydesigned experiments are needed to measure the value of λ. Many experiments have been done with samples that have a large surface to volume ratio to make the penetration effect of the field appreciable. Thin films, thin wires and colloidal particles of superconductors have all been used for this purpose. But it is also possible to use large specimens if the measurement is sensitive to the amount of magnetic flux passing through the superconductor's surface, and not to the ratio of flux excluded by the superconductor to flux through the normal material, which is close to unity.</p><p>In a classic experiment performed in the 1950s, Schawlow and Devlin measured the selfinductance of a solenoid within which they inserted a long singlecrystal cylinder of superconducting tin, 7.4 mm in diameter. They minimised the space between the coil and the tin cylinder, and since no magnetic flux passed through the bulk of the superconductor, the flux was essentially restricted to a thin cylindrical shell of thickness λ at the surface of the cylinder. The inductance of the solenoid was therefore determined mainly by the magnitude of the penetration depth. To measure the inductance, a capacitor was connected in parallel with the solenoid, and the natural angular frequency, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/7b2688a4/small_omega.gif" alt="" width="14" height="9" style="maxwidth:14px;" class="oucontentinlinefigureimage"/></span><sub>n</sub> = 1/<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/bcb13e7a/smt359_1_ie017i.gif" alt="" width="26" height="19" style="maxwidth:26px;" class="oucontentinlinefigureimage"/></span>, of the LC circuit was measured. The precision of the frequency measurement was about one part in 10﻿<sup>6</sup>, which corresponded to a precision of 0.4 nm in the value of the penetration depth. The result that they obtained for the penetration depth of tin for temperatures much lower than the critical temperature was 52 nm.</p><p>The number density of superconducting electrons depends on temperature, so the penetration depth is temperature dependent. For <i>T</i> ≪ <i>T</i>﻿<sub>c</sub>, all of the free electrons are superconducting, but the number density falls steadily with increasing temperature until it reaches zero at the critical temperature. Since λ <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/cab49858/proportional.gif" alt="" width="10" height="7" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> <i>n</i><sub>s</sub><sup>1/2</sup> according to the London model, the penetration depth increases as the temperature approaches the critical temperature, becoming effectively infinite – corresponding to a uniform field in the material – at and above the critical temperature. Figure 16 shows this temperature dependence for tin, which is well represented by the expression</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/fbea0d2d/smt359_1_ue011i.gif" alt=""/></div><p>where λ(0) is the value of the penetration depth at <i>T</i> = 0 K.</p><div class="oucontentfigure oucontentmediamini" id="fig009_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/59f50a4c/smt359_1_016i.jpg" alt="Figure 16" width="256" height="210" style="maxwidth:256px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 16 The penetration depth λ as a function of temperature for tin.</span></div></div></div><p></p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.4
3.4 Penetration depthSMT359_1<p>The characteristic length, λ, associated with the decay of the magnetic field at the surface of a superconductor is known as the <b>penetration depth</b>, and it depends on the number density <i>n</i><sub>s</sub> of superconducting electrons.</p><p>We can estimate a value for λ by assuming that all of the free electrons are superconducting. If we set <i>n</i><sub>s</sub> = 10<sup>29</sup> m<sup>−3</sup>, a typical free electron density in a metal, then we find that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/a264229e/smt359_1_ue010i.gif" alt=""/></div><p>The small size of λ indicates that the magnetic field is effectively excluded from the interior of macroscopic specimens of superconductors, in agreement with the experimentally observed Meissner effect.</p><p>The small scale of the field penetration means that carefullydesigned experiments are needed to measure the value of λ. Many experiments have been done with samples that have a large surface to volume ratio to make the penetration effect of the field appreciable. Thin films, thin wires and colloidal particles of superconductors have all been used for this purpose. But it is also possible to use large specimens if the measurement is sensitive to the amount of magnetic flux passing through the superconductor's surface, and not to the ratio of flux excluded by the superconductor to flux through the normal material, which is close to unity.</p><p>In a classic experiment performed in the 1950s, Schawlow and Devlin measured the selfinductance of a solenoid within which they inserted a long singlecrystal cylinder of superconducting tin, 7.4 mm in diameter. They minimised the space between the coil and the tin cylinder, and since no magnetic flux passed through the bulk of the superconductor, the flux was essentially restricted to a thin cylindrical shell of thickness λ at the surface of the cylinder. The inductance of the solenoid was therefore determined mainly by the magnitude of the penetration depth. To measure the inductance, a capacitor was connected in parallel with the solenoid, and the natural angular frequency, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/7b2688a4/small_omega.gif" alt="" width="14" height="9" style="maxwidth:14px;" class="oucontentinlinefigureimage"/></span><sub>n</sub> = 1/<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/bcb13e7a/smt359_1_ie017i.gif" alt="" width="26" height="19" style="maxwidth:26px;" class="oucontentinlinefigureimage"/></span>, of the LC circuit was measured. The precision of the frequency measurement was about one part in 10<sup>6</sup>, which corresponded to a precision of 0.4 nm in the value of the penetration depth. The result that they obtained for the penetration depth of tin for temperatures much lower than the critical temperature was 52 nm.</p><p>The number density of superconducting electrons depends on temperature, so the penetration depth is temperature dependent. For <i>T</i> ≪ <i>T</i><sub>c</sub>, all of the free electrons are superconducting, but the number density falls steadily with increasing temperature until it reaches zero at the critical temperature. Since λ <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/cab49858/proportional.gif" alt="" width="10" height="7" style="maxwidth:10px;" class="oucontentinlinefigureimage"/></span> <i>n</i><sub>s</sub><sup>1/2</sup> according to the London model, the penetration depth increases as the temperature approaches the critical temperature, becoming effectively infinite – corresponding to a uniform field in the material – at and above the critical temperature. Figure 16 shows this temperature dependence for tin, which is well represented by the expression</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/fbea0d2d/smt359_1_ue011i.gif" alt=""/></div><p>where λ(0) is the value of the penetration depth at <i>T</i> = 0 K.</p><div class="oucontentfigure oucontentmediamini" id="fig009_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/59f50a4c/smt359_1_016i.jpg" alt="Figure 16" width="256" height="210" style="maxwidth:256px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 16 The penetration depth λ as a function of temperature for tin.</span></div></div></div><p></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

3.5 The screening current
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.5
Wed, 13 Apr 2016 23:00:00 GMT
<p>The London equations relate the magnetic field in a superconductor to the superconducting current density, and we derived the dependence of field on position by eliminating the current density. However, if we eliminate the magnetic field instead, we can derive the following equation for the current density:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/670f84b6/smt359_1_e014i.gif" alt=""/></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe009_004"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 4</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Derive Equation 13 by taking the curl of both sides of Equation 9 and then using Ampère's law to eliminate curl <b>B</b>. Assume that the currents are steady.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>Taking the curl of both sides of Equation 9 and using Ampère's law, curl <b>B</b> = <i>μ﻿</i><sub>0</sub>﻿<b>J</b>﻿<sub>s</sub>, to eliminate <b>B</b>, we find that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_032"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c1252b2f/smt359_1_ue032i.gif" alt=""/></div><p>We now use a standard vector identity to rewrite the curl(curl <b>J</b><sub>s</sub>) term:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_033"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/58609725/smt359_1_ue033i.gif" alt=""/></div><p>For our steadystate situation, where ∂﻿<i>ρ</i><sub>s</sub>/∂﻿<i>t</i> = 0, the equation of continuity, ∂﻿<i>ρ</i><sub>s</sub>/∂<i>t</i> + div <b>J</b><sub>s</sub> = 0, reduces to div <b>J</b>﻿<sub>s</sub> = 0, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_034"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c0b37a25/smt359_1_ue034i.gif" alt=""/></div></div></div></div></div><p>Equation 13 has exactly the same form as Equation 11. So for the planar symmetry that we discussed earlier – superconducting material occupying the region <i>z</i> > 0 (Figure 14) – the solution for the current density will have the same form as Equation 12, that is,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/13f23fff/smt359_1_e015i.gif" alt=""/></div><p>This equation gives no indication of the absolute magnitude or direction of current flow, but we can deduce this by using Ampère's law, curl <b>B</b> = <i>μ﻿</i><sub>0</sub>﻿<b>J</b>﻿<sub>s</sub>. In the planar situation that we are considering, <b>B</b> = <i>B<sub>x</sub></i> (<i>z</i>) <b>e</b><sub><i>x</i></sub>, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/92489e37/smt359_1_ue012i.gif" alt=""/></div><p>Then</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/9abe522b/smt359_1_ue013i.gif" alt=""/></div><p>where</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b4ba5615/smt359_1_ue014i.gif" alt=""/></div><p>But we know that <i>B</i><i><sub>x</sub></i>(<i>z</i>) = <i>B</i>﻿<sub>0</sub>﻿e<sup>﻿−﻿<i>z</i>﻿/﻿λ</sup> (Equation 12), so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/6057bfd9/smt359_1_ue015i.gif" alt=""/></div><p>Thus the current that screens the interior of the superconductor from an applied field flows within a thin surface layer, which has a thickness characterised by the penetration depth λ, and the current flows parallel to the surface and in a direction perpendicular to the magnetic field, as shown in Figure 17.</p><div class="oucontentfigure oucontentmediamini" id="fig009_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b827d174/smt359_1_017i.jpg" alt="Figure17" width="288" height="314" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 17 The magnetic field and current density vectors in the surface layer of a superconductor.</span></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe009_005"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 5</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>The number density of free electrons in tin is 1.5 × 10﻿<sup>29</sup> m﻿<sup>−﻿3</sup>. Calculate the penetration depth predicted by the London model, assuming that all of the free electrons are superconducting, and compare the result with the value measured by Schawlow and Devlin.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From Equation 10, the penetration depth is given by λ = (<i>m</i>/<i>μ</i><sub>0</sub><i>n</i><sub>s</sub><i>e</i><sup>2</sup>)<sup>1/2</sup>, so (in metres)</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_035"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/2b3515b6/smt359_1_ue035i.gif" alt=""/></div><p>This value, predicted by the London model, is about a quarter of the measured value.</p></div></div></div></div><p>The numerical discrepancy between the London model prediction for the penetration depth of tin and the experimentally measured value indicates that this model has limitations. One limitation is that the model is essentially a <i>local</i> model, relating current density and magnetic field at each point. Superconductivity, though, is a nonlocal phenomenon, involving coherent behaviour of the superconducting electrons that are condensed into a macroscopic quantum state. The characteristic distance over which the behaviour of the superconducting electrons is linked is known as the coherence length, ξ, introduced in Section 1. This distance represents the distance over which the number density of the superconducting electrons changes, and is a measure of the intrinsic nonlocal nature of the superconducting state. The London local model is a good description if ξ ≪ λ, that is, the coherence length is much shorter than the distance λ over which the fields and current density are changing. Since the penetration depth increases sharply as the temperature approaches the critical temperature (Figure 16), the London model becomes a good approximation in this limit. More importantly, the coherence length of superconductors decreases as the critical temperature increases and as the scattering time for normal electrons decreases. Both of these effects mean that the coherence length is short compared with the penetration depth in alloy and ceramic superconductors, so the London local model is a good approximation in these cases too, and predicted and experimental results for the penetration depth are in good agreement.</p><p>For pure elements, well below their critical temperatures, the penetration depth is generally much shorter than the coherence length, so a local model is not appropriate. In this limit, the number density of superconducting electrons does not reach the bulk value until a distance of the order of ξ, which is greater than λ, from the surface, and the reduced value of <i>n﻿<sub>s</sub></i> accounts for the discrepancy between the predicted and experimental results for the penetration depth of tin discussed earlier.</p><p>The ratio of the penetration depth to the coherence length is an important parameter for a superconductor, and we shall return to this subject in Subsection 4.2.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection3.5
3.5 The screening currentSMT359_1<p>The London equations relate the magnetic field in a superconductor to the superconducting current density, and we derived the dependence of field on position by eliminating the current density. However, if we eliminate the magnetic field instead, we can derive the following equation for the current density:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/670f84b6/smt359_1_e014i.gif" alt=""/></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe009_004"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 4</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Derive Equation 13 by taking the curl of both sides of Equation 9 and then using Ampère's law to eliminate curl <b>B</b>. Assume that the currents are steady.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>Taking the curl of both sides of Equation 9 and using Ampère's law, curl <b>B</b> = <i>μ</i><sub>0</sub><b>J</b><sub>s</sub>, to eliminate <b>B</b>, we find that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_032"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c1252b2f/smt359_1_ue032i.gif" alt=""/></div><p>We now use a standard vector identity to rewrite the curl(curl <b>J</b><sub>s</sub>) term:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_033"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/58609725/smt359_1_ue033i.gif" alt=""/></div><p>For our steadystate situation, where ∂<i>ρ</i><sub>s</sub>/∂<i>t</i> = 0, the equation of continuity, ∂<i>ρ</i><sub>s</sub>/∂<i>t</i> + div <b>J</b><sub>s</sub> = 0, reduces to div <b>J</b><sub>s</sub> = 0, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_034"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c0b37a25/smt359_1_ue034i.gif" alt=""/></div></div></div></div></div><p>Equation 13 has exactly the same form as Equation 11. So for the planar symmetry that we discussed earlier – superconducting material occupying the region <i>z</i> > 0 (Figure 14) – the solution for the current density will have the same form as Equation 12, that is,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn009_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/13f23fff/smt359_1_e015i.gif" alt=""/></div><p>This equation gives no indication of the absolute magnitude or direction of current flow, but we can deduce this by using Ampère's law, curl <b>B</b> = <i>μ</i><sub>0</sub><b>J</b><sub>s</sub>. In the planar situation that we are considering, <b>B</b> = <i>B<sub>x</sub></i> (<i>z</i>) <b>e</b><sub><i>x</i></sub>, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/92489e37/smt359_1_ue012i.gif" alt=""/></div><p>Then</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/9abe522b/smt359_1_ue013i.gif" alt=""/></div><p>where</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b4ba5615/smt359_1_ue014i.gif" alt=""/></div><p>But we know that <i>B</i><i><sub>x</sub></i>(<i>z</i>) = <i>B</i><sub>0</sub>e<sup>−<i>z</i>/λ</sup> (Equation 12), so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/6057bfd9/smt359_1_ue015i.gif" alt=""/></div><p>Thus the current that screens the interior of the superconductor from an applied field flows within a thin surface layer, which has a thickness characterised by the penetration depth λ, and the current flows parallel to the surface and in a direction perpendicular to the magnetic field, as shown in Figure 17.</p><div class="oucontentfigure oucontentmediamini" id="fig009_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/b827d174/smt359_1_017i.jpg" alt="Figure17" width="288" height="314" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 17 The magnetic field and current density vectors in the surface layer of a superconductor.</span></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe009_005"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 5</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>The number density of free electrons in tin is 1.5 × 10<sup>29</sup> m<sup>−3</sup>. Calculate the penetration depth predicted by the London model, assuming that all of the free electrons are superconducting, and compare the result with the value measured by Schawlow and Devlin.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From Equation 10, the penetration depth is given by λ = (<i>m</i>/<i>μ</i><sub>0</sub><i>n</i><sub>s</sub><i>e</i><sup>2</sup>)<sup>1/2</sup>, so (in metres)</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_035"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/2b3515b6/smt359_1_ue035i.gif" alt=""/></div><p>This value, predicted by the London model, is about a quarter of the measured value.</p></div></div></div></div><p>The numerical discrepancy between the London model prediction for the penetration depth of tin and the experimentally measured value indicates that this model has limitations. One limitation is that the model is essentially a <i>local</i> model, relating current density and magnetic field at each point. Superconductivity, though, is a nonlocal phenomenon, involving coherent behaviour of the superconducting electrons that are condensed into a macroscopic quantum state. The characteristic distance over which the behaviour of the superconducting electrons is linked is known as the coherence length, ξ, introduced in Section 1. This distance represents the distance over which the number density of the superconducting electrons changes, and is a measure of the intrinsic nonlocal nature of the superconducting state. The London local model is a good description if ξ ≪ λ, that is, the coherence length is much shorter than the distance λ over which the fields and current density are changing. Since the penetration depth increases sharply as the temperature approaches the critical temperature (Figure 16), the London model becomes a good approximation in this limit. More importantly, the coherence length of superconductors decreases as the critical temperature increases and as the scattering time for normal electrons decreases. Both of these effects mean that the coherence length is short compared with the penetration depth in alloy and ceramic superconductors, so the London local model is a good approximation in these cases too, and predicted and experimental results for the penetration depth are in good agreement.</p><p>For pure elements, well below their critical temperatures, the penetration depth is generally much shorter than the coherence length, so a local model is not appropriate. In this limit, the number density of superconducting electrons does not reach the bulk value until a distance of the order of ξ, which is greater than λ, from the surface, and the reduced value of <i>n<sub>s</sub></i> accounts for the discrepancy between the predicted and experimental results for the penetration depth of tin discussed earlier.</p><p>The ratio of the penetration depth to the coherence length is an important parameter for a superconductor, and we shall return to this subject in Subsection 4.2.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

Preamble
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection4.1
Wed, 13 Apr 2016 23:00:00 GMT
<p>The two main types of superconducting materials are known as <b>typeI</b> and <b>typeII superconductors</b>, and their properties will be discussed in the remainder of this course. All of the pure elemental superconductors are typeI, with the exception of niobium, vanadium and technetium. The discussion of the effects of magnetic fields and currents on superconductors earlier in this course has been confined to thin cylinders of typeI materials like lead or tin in a <i>parallel</i> magnetic field. In Subsection 4.1 we shall discuss what happens when the magnetic field is <i>perpendicular</i> to cylinders made of these materials.</p><p>Superconducting alloys and high critical temperature ceramics are all typeII, and these are the materials that are used in most practical applications. In Subsection 4.2, we shall consider the response to a magnetic field of this type of superconductor. Such materials behave quite differently from lead and tin, and this is the reason that they are widely used.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection4.1
PreambleSMT359_1<p>The two main types of superconducting materials are known as <b>typeI</b> and <b>typeII superconductors</b>, and their properties will be discussed in the remainder of this course. All of the pure elemental superconductors are typeI, with the exception of niobium, vanadium and technetium. The discussion of the effects of magnetic fields and currents on superconductors earlier in this course has been confined to thin cylinders of typeI materials like lead or tin in a <i>parallel</i> magnetic field. In Subsection 4.1 we shall discuss what happens when the magnetic field is <i>perpendicular</i> to cylinders made of these materials.</p><p>Superconducting alloys and high critical temperature ceramics are all typeII, and these are the materials that are used in most practical applications. In Subsection 4.2, we shall consider the response to a magnetic field of this type of superconductor. Such materials behave quite differently from lead and tin, and this is the reason that they are widely used.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

4.2 TypeI superconductors
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection4.2
Wed, 13 Apr 2016 23:00:00 GMT
<p>You saw in Subsection 2.4 that superconductivity in a tin cylinder is destroyed when an applied field with strength <i>B</i>﻿<sub>0</sub>  > <i>B</i>﻿<sub>c</sub> is applied parallel to the cylinder. However, when the field is applied perpendicular to the cylinder, as shown in Figure 18, the field strength at points A and C is substantially greater than the strength of the applied field at a distance from the cylinder, and this is indicated by the increased concentration of the field lines shown near these points. In fact, it can be shown that the field strength at these points is a factor of two greater than the applied field strength. This means that as the applied field <b>B</b>﻿<sub>0</sub> is increased, the field at points A and C will reach the critical field strength <i>B</i><sub>c</sub> when <i>B</i><sub>0</sub> = <i>B</i>﻿<sub>c</sub>﻿/﻿2.</p><div class="oucontentfigure oucontentmediamini" id="fig009_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8d60bf9d/smt359_1_018i.jpg" alt="Figure 18" width="288" height="266" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 18 The magnetic field B in a plane perpendicular to the axis of a superconducting cylinder (shown in crosssection) for an applied field with <i>B</i>﻿<sub>0</sub> < <i>B</i>﻿<sub>c</sub>﻿/﻿2.</span></div></div></div><p>You might think that superconductivity in the cylinder would be completely destroyed at this lower field strength. However, were this to be the case, then the material would be in the normal state with a field strength in its interior of less than <i>B</i>﻿<sub>c</sub>, which is not possible. Instead, for applied field strengths <i>B</i>﻿<sub>0</sub> in the range <i>B</i>﻿<sub>c</sub>﻿/﻿2 < <i>B</i>﻿<sub>0</sub> < <i>B</i>﻿<sub>c</sub>, the cylinder splits up into small slices of normal and superconducting material that run parallel to the applied field. This state in which regions of normal and superconducting material coexist in a typeI superconductor is known as the <b>intermediate state</b>, and it is shown schematically in Figure 19. Within the normal regions, <i>B</i> = <i>B</i>﻿<sub>c</sub>, and in the superconducting regions, <i>B</i> decreases rapidly and is confined to a thin layer, the width of which is determined by the penetration depth, as shown in Figure 20. The number density of superconducting electrons <i>n</i>﻿<sub>s</sub> increases from zero at the boundary to the bulk value over the coherence length ξ. Note that since <i>n﻿<sub>s</sub></i> is not constant in the superconducting regions, the magnetic field strength does not fall exponentially. The proportion of the material in the normal state increases from zero for <i>B</i><sub>﻿0</sub> = <i>B</i><sub>﻿c</sub>﻿/﻿2 to 100 per cent for <i>B</i>﻿<sub>0</sub> = <i>B</i>﻿<sub>c</sub>.</p><div class="oucontentfigure oucontentmediamini" id="fig009_019"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8b097b21/smt359_1_019i.jpg" alt="Figure 19" width="208" height="198" style="maxwidth:208px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 19 Schematic representation of the intermediate state for a cylinder of a typeI superconductor aligned perpendicular to the magnetic field.</span></div></div></div><div class="oucontentfigure oucontentmediamini" id="fig009_020"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/cbf2801f/smt359_1_020i.jpg" alt="Figure 20" width="238" height="159" style="maxwidth:238px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 20 Variation of magnetic field strength and number density of superconducting electrons in the region of a boundary between normal and superconducting regions in the intermediate state of a typeI superconductor.</span></div></div></div><p>The lowest applied field strength at which the intermediate state appears depends on the shape of the specimen and the orientation of the field. Essentially, it is determined by the extent to which the field is deviated by the superconductor, or equivalently, by how much the field strength is enhanced at the edges of the superconductor. For the thin cylinder shown in Figure 21a, the field just outside is essentially the same as the applied field, so there is a direct transition from superconducting to normal state, without the intervening intermediate state. Contrast this with the thin plate oriented perpendicular to the applied field shown in Figure 21b, where the field strength would be greatly enhanced outside the plate's edges if it could not penetrate the plate. Samples like this enter the intermediate state when the applied field strength is a very small fraction of the critical field strength (Figure 21c).</p><div class="oucontentfigure" style="width:511px;" id="fig009_021"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3692416" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/f856dd5d/smt359_1_021i.small.jpg" alt="Figure 21" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3692416">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 21 (a) A thin superconducting cylinder hardly distorts a uniform applied field. (b) Highly distorted field that would result if a magnetic field did not penetrate a thin plate. (c) Intermediate state in a thin plate.</span></div></div><a id="back_thumbnailfigure_idp3692416"></a></div><p>Figure 22 shows the pattern of normal and superconducting regions for an aluminium plate in the intermediate state, with the field perpendicular to the surface.</p><div class="oucontentfigure oucontentmediamini" id="fig009_022"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/e44290a0/smt359_1_022i.jpg" alt="Figure 22" width="288" height="189" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 22 Intermediate state in an aluminium plate. Tin powder was deposited on the plate and collected in the superconducting regions – the dark areas of the image – where the magnetic field was low. The light normal regions are about 1 mm wide.</span></div></div></div><p></p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection4.2
4.2 TypeI superconductorsSMT359_1<p>You saw in Subsection 2.4 that superconductivity in a tin cylinder is destroyed when an applied field with strength <i>B</i><sub>0</sub> > <i>B</i><sub>c</sub> is applied parallel to the cylinder. However, when the field is applied perpendicular to the cylinder, as shown in Figure 18, the field strength at points A and C is substantially greater than the strength of the applied field at a distance from the cylinder, and this is indicated by the increased concentration of the field lines shown near these points. In fact, it can be shown that the field strength at these points is a factor of two greater than the applied field strength. This means that as the applied field <b>B</b><sub>0</sub> is increased, the field at points A and C will reach the critical field strength <i>B</i><sub>c</sub> when <i>B</i><sub>0</sub> = <i>B</i><sub>c</sub>/2.</p><div class="oucontentfigure oucontentmediamini" id="fig009_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8d60bf9d/smt359_1_018i.jpg" alt="Figure 18" width="288" height="266" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 18 The magnetic field B in a plane perpendicular to the axis of a superconducting cylinder (shown in crosssection) for an applied field with <i>B</i><sub>0</sub> < <i>B</i><sub>c</sub>/2.</span></div></div></div><p>You might think that superconductivity in the cylinder would be completely destroyed at this lower field strength. However, were this to be the case, then the material would be in the normal state with a field strength in its interior of less than <i>B</i><sub>c</sub>, which is not possible. Instead, for applied field strengths <i>B</i><sub>0</sub> in the range <i>B</i><sub>c</sub>/2 < <i>B</i><sub>0</sub> < <i>B</i><sub>c</sub>, the cylinder splits up into small slices of normal and superconducting material that run parallel to the applied field. This state in which regions of normal and superconducting material coexist in a typeI superconductor is known as the <b>intermediate state</b>, and it is shown schematically in Figure 19. Within the normal regions, <i>B</i> = <i>B</i><sub>c</sub>, and in the superconducting regions, <i>B</i> decreases rapidly and is confined to a thin layer, the width of which is determined by the penetration depth, as shown in Figure 20. The number density of superconducting electrons <i>n</i><sub>s</sub> increases from zero at the boundary to the bulk value over the coherence length ξ. Note that since <i>n<sub>s</sub></i> is not constant in the superconducting regions, the magnetic field strength does not fall exponentially. The proportion of the material in the normal state increases from zero for <i>B</i><sub>0</sub> = <i>B</i><sub>c</sub>/2 to 100 per cent for <i>B</i><sub>0</sub> = <i>B</i><sub>c</sub>.</p><div class="oucontentfigure oucontentmediamini" id="fig009_019"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/8b097b21/smt359_1_019i.jpg" alt="Figure 19" width="208" height="198" style="maxwidth:208px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 19 Schematic representation of the intermediate state for a cylinder of a typeI superconductor aligned perpendicular to the magnetic field.</span></div></div></div><div class="oucontentfigure oucontentmediamini" id="fig009_020"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/cbf2801f/smt359_1_020i.jpg" alt="Figure 20" width="238" height="159" style="maxwidth:238px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 20 Variation of magnetic field strength and number density of superconducting electrons in the region of a boundary between normal and superconducting regions in the intermediate state of a typeI superconductor.</span></div></div></div><p>The lowest applied field strength at which the intermediate state appears depends on the shape of the specimen and the orientation of the field. Essentially, it is determined by the extent to which the field is deviated by the superconductor, or equivalently, by how much the field strength is enhanced at the edges of the superconductor. For the thin cylinder shown in Figure 21a, the field just outside is essentially the same as the applied field, so there is a direct transition from superconducting to normal state, without the intervening intermediate state. Contrast this with the thin plate oriented perpendicular to the applied field shown in Figure 21b, where the field strength would be greatly enhanced outside the plate's edges if it could not penetrate the plate. Samples like this enter the intermediate state when the applied field strength is a very small fraction of the critical field strength (Figure 21c).</p><div class="oucontentfigure" style="width:511px;" id="fig009_021"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3692416" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/f856dd5d/smt359_1_021i.small.jpg" alt="Figure 21" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2685&extra=thumbnailfigure_idp3692416">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 21 (a) A thin superconducting cylinder hardly distorts a uniform applied field. (b) Highly distorted field that would result if a magnetic field did not penetrate a thin plate. (c) Intermediate state in a thin plate.</span></div></div><a id="back_thumbnailfigure_idp3692416"></a></div><p>Figure 22 shows the pattern of normal and superconducting regions for an aluminium plate in the intermediate state, with the field perpendicular to the surface.</p><div class="oucontentfigure oucontentmediamini" id="fig009_022"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/e44290a0/smt359_1_022i.jpg" alt="Figure 22" width="288" height="189" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 22 Intermediate state in an aluminium plate. Tin powder was deposited on the plate and collected in the superconducting regions – the dark areas of the image – where the magnetic field was low. The light normal regions are about 1 mm wide.</span></div></div></div><p></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

4.3 TypeII superconductors
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection4.3
Wed, 13 Apr 2016 23:00:00 GMT
<p>For decades it was assumed that all superconductors, elements and alloys, behaved in similar ways, and that any differences could be attributed to impurities or defects in the materials. However, in 1957, Abrikosov predicted the existence of a different sort of superconductor, and Figure 23 shows direct evidence for the existence of what are now known as typeII superconductors. A comparison of Figures 23 and 22 indicates that the effect of an applied field on a typeII superconductor is rather different from that for typeI superconductors.</p><div class="oucontentfigure oucontentmediamini" id="fig009_023"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/d4a2868c/smt359_1_023i.jpg" alt="Figure 23" width="276" height="177" style="maxwidth:276px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 23 Surface of a superconducting alloy that had a magnetic field applied perpendicular to the surface. The dark regions were normal and the light regions superconducting. In this case, small ferromagnetic particles were applied to the surface, and collected where the field strength was largest. The particles remained in position when the specimen warmed up to room temperature, and the surface was then imaged with an electron microscope.</span></div></div></div><p>For simplicity, we shall consider first a long cylindrical specimen of typeII material, and apply a field parallel to its axis. Below a certain critical field strength, known as the <b>lower critical field strength</b> and denoted by the symbol <i>B</i>﻿<sub>c1</sub>, the applied magnetic field is excluded from the bulk of the material, penetrating into only a thin layer at the surface, just as for typeI materials. But above <i>B</i>﻿<sub>c1</sub>, the material does not make a sudden transition to the normal state. Instead, very thin cylindrical regions of normal material appear, passing through the specimen parallel to its axis. We shall refer to such a normal region as a <b>normal core</b>. The normal cores are arranged on a triangular lattice, as shown in Figure 23, and as the applied field is increased, more normal cores appear and they become more and more closely packed together. Eventually, a second critical field strength, the <b>upper critical field strength</b> <i>B</i>﻿<sub>c2</sub>, is reached, above which the material reverts to the normal state. The state that exists between the lower and upper critical field strengths, in which a typeII superconductor is threaded by normal cores, is known as the <b>mixed state</b>. As Figure 24 shows, both the upper and lower critical field strengths depend on temperature in a similar way to the critical field strength for a typeI material (Figure 11).</p><div class="oucontentfigure oucontentmediamini" id="fig009_024"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/42f3ae24/smt359_1_024i.jpg" alt="Figure 24" width="238" height="296" style="maxwidth:238px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 24 Temperature dependence of the lower critical field strength (<i>B</i><sub>c1</sub>) and upper critical field strength (﻿<i>B</i>﻿<sub>c2</sub>﻿) for a typeII superconductor.</span></div></div></div><p>The normal cores that exist in typeII superconductors in the mixed state are not sharply delineated. Figure 25 shows how the number density of superelectrons and the magnetic field strength vary along a line passing through the axes of three neighbouring cores. The value of <i>n</i>﻿<sub>s</sub> is zero at the centres of the cores and rises over a characteristic distance ξ, the coherence length. The magnetic field associated with each normal core is spread over a region with a diameter of 2λ, and each normal core is surrounded by a vortex of circulating current.</p><div class="oucontentfigure oucontentmediamini" id="fig009_025"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/92f9b763/smt359_1_025i.jpg" alt="Figure 25" width="288" height="232" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 25 Number density of superelectrons <i>n﻿<sub>s</sub></i> and magnetic field strength <i>B</i> around normal cores in a typeII superconductor.</span></div></div></div><p>You can see from Figure 25 that the coherence length ξ, the characteristic distance for changes in <i>n<sub>﻿s</sub></i>, is shorter than the penetration depth λ, the characteristic distance for changes in the magnetic field in a superconductor. This is generally true for typeII superconductors, whereas for typeI superconductors, ξ > λ (Figure 20). For a pure typeI superconductor, typical values of the characteristic lengths are ξ ~ 1 <i>μ</i>﻿m and λ = 50 nm. Contrast this with the values for a widelyused typeII alloy of niobium and tin, Nb﻿<sub>3</sub>Sn, for which ξ ~ 3.5 nm and λ = 80 nm.</p><p>The reason that the relative magnitude of the coherence length and the penetration depth is so important is that when ξ > λ, the surface energy associated with the boundary between superconducting and normal regions is positive, whereas when ξ < λ, this surface energy is negative. Justifying this statement would involve a discussion of the thermodynamics of superconductors, but for our purposes it is sufficient to just look at the consequences. For a positive surface energy, the system will prefer few boundaries and we expect relatively thick normal and superconducting regions, as observed in the intermediate state in typeI materials. Conversely, a negative surface energy favours formation of as much boundary between normal and superconducting regions as possible, and this is what happens in the mixed state in typeII materials with many narrow normal cores. The lower limit for the diameter of the cores is 2ξ, as shown in Figure 25, because ξ is the distance over which <i>n</i><sub>s</sub> can vary.</p><p>This energy argument does not explicitly indicate how much magnetic flux passes through each of the normal cores. However, quantum mechanical arguments show that the magnetic flux linking any superconducting loop must be quantised, and that the quantum of magnetic flux is</p><div class="oucontentquote oucontentsbox" id="quo002"><blockquote><p><i>h</i> / 2<i>e</i> = 2.07 × 10<sup>﻿−﻿15</sup>﻿T m﻿<sup>2</sup></p></blockquote></div><p>where <i>h</i> = 6.63 × 10﻿<sup>−﻿34﻿</sup>J s is Planck's constant. In fact, each of the normal cores shown in Figure 23 contains just one quantum of flux, since this is more favorable energetically than having two or more quanta of flux in a core.</p><p>The quantisation of flux in a superconductor is of particular importance in SQUIDs, the superconducting quantum interference devices that are at the heart of the magnetometers used for measuring the magnetic fields produced by currents in the brain. A SQUID contains a small loop of superconductor with a weakly superconducting link in it, and the quantisation of flux in the loop causes its electrical properties to depend on the flux applied to it, with a periodicity equal to the flux quantum. The very small magnitude of the flux quantum is responsible for the sensitivity of the device to very small magnetic fields.</p><p>A final point is worth noting about the quantum of flux: the factor of 2 in the denominator of the expression <i>h</i> / 2﻿<i>e</i> is a consequence of the coupling of pairs of electrons in a superconductor and their condensation into a superconducting ground state. There is charge −﻿2﻿<i>e</i> associated with each of these electron pairs.</p><p><b>Critical currents in typeII superconductors</b></p><p>The high values of the upper critical field strength <i>B</i><sub>c2</sub> of many typeII superconducting alloys make them very attractive for winding coils for generating high magnetic fields. For example, alloys of niobium and titanium (﻿NbTi﻿<sub>2</sub>﻿) and of niobium and tin (﻿Nb﻿<sub>3</sub>﻿Sn﻿) have values of <i>B</i>﻿<sub>c2</sub> of about 10 T and 20 T, respectively, compared with 0.08 T for lead, a typeI superconductor. However, for typeII materials to be usable for this purpose, they must also have high critical currents at high field strengths, and this requires some help from metallurgists to overcome a significant problem.</p><p>This problem is related to the interaction between the current flowing through a typeII superconductor in the mixed state and the ‘tubes’ of magnetic flux that thread through the normal cores. The electrons will experience a Lorentz force, perpendicular to both the current density and the magnetic field. We can regard this as a mutual interaction between the electrons and the flux in the normal cores, as a result of which each normal core experiences a force that is in the opposite direction to the Lorentz force on the electrons. The directions of the magnetic field, current and forces are shown in Figure 26.</p><div class="oucontentfigure oucontentmediamini" id="fig009_026"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/e619efd9/smt359_1_026i.jpg" alt="Figure 26" width="288" height="214" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 26 Electrons and normal cores experience forces perpendicular to the current and to the magnetic field, but in opposite directions.</span></div></div></div><p>This Lorentz force can cause the cores and their associated magnetic flux to move, and the flux motion will induce an emf that drives a current through the normal cores, somewhat like an eddy current. Energy is therefore dissipated in the normal cores, and this energy must come from the power supply. The energy dissipation means that the flow of electrons is impeded, and therefore there is a resistance to the flow of the current.</p><p>Flux motion is therefore undesirable in typeII superconductors, and the aim of the metallurgists who develop processes for manufacturing wire for magnets is to make flux motion as difficult as possible. This is done by introducing defects into the crystalline structure, particularly by preparing the material in such a way that it comprises many small crystalline grains with different orientations and small precipitates of different composition. These defects effectively pin the normal cores in position – they provide a potential barrier to motion of the cores, so that the force on the cores must exceed a certain value before the cores can move. The more of these flux pinning centres that are present, and the greater the potential barrier they provide, the greater will be the current required to set them in motion, i.e. the greater the critical current. So, unlike a normal conductor, for which improving the purity and reducing imperfections in the crystal structure lead to better conductivity, with typeII superconductors the inclusion of impurities and defects in the crystal structure can improve the critical current and make the material more suitable for use in electromagnets.</p><p>Undoubtedly the largest use of superconducting material for a single project is in the Large Hadron Collider at CERN, due for commissioning in 2007. The 27 km﻿﻿long accelerator tunnel contains 1232 superconducting magnets that are responsible for steering the particle beams around their circular paths. Each of these magnets is 15 m long, has a mass of 35 tonnes and produces a magnetic field strength of 8.5 T. The coils in each magnet are made from about 6 km of niobiumtitanium cable, with a mass of about a tonne, and will be maintained at a temperature of 1.9 K using liquid helium. To construct an accelerator with a similar specification using nonsuperconducting magnets would have required a 120 km tunnel and phenomenal amounts of power to operate.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection4.3
4.3 TypeII superconductorsSMT359_1<p>For decades it was assumed that all superconductors, elements and alloys, behaved in similar ways, and that any differences could be attributed to impurities or defects in the materials. However, in 1957, Abrikosov predicted the existence of a different sort of superconductor, and Figure 23 shows direct evidence for the existence of what are now known as typeII superconductors. A comparison of Figures 23 and 22 indicates that the effect of an applied field on a typeII superconductor is rather different from that for typeI superconductors.</p><div class="oucontentfigure oucontentmediamini" id="fig009_023"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/d4a2868c/smt359_1_023i.jpg" alt="Figure 23" width="276" height="177" style="maxwidth:276px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 23 Surface of a superconducting alloy that had a magnetic field applied perpendicular to the surface. The dark regions were normal and the light regions superconducting. In this case, small ferromagnetic particles were applied to the surface, and collected where the field strength was largest. The particles remained in position when the specimen warmed up to room temperature, and the surface was then imaged with an electron microscope.</span></div></div></div><p>For simplicity, we shall consider first a long cylindrical specimen of typeII material, and apply a field parallel to its axis. Below a certain critical field strength, known as the <b>lower critical field strength</b> and denoted by the symbol <i>B</i><sub>c1</sub>, the applied magnetic field is excluded from the bulk of the material, penetrating into only a thin layer at the surface, just as for typeI materials. But above <i>B</i><sub>c1</sub>, the material does not make a sudden transition to the normal state. Instead, very thin cylindrical regions of normal material appear, passing through the specimen parallel to its axis. We shall refer to such a normal region as a <b>normal core</b>. The normal cores are arranged on a triangular lattice, as shown in Figure 23, and as the applied field is increased, more normal cores appear and they become more and more closely packed together. Eventually, a second critical field strength, the <b>upper critical field strength</b> <i>B</i><sub>c2</sub>, is reached, above which the material reverts to the normal state. The state that exists between the lower and upper critical field strengths, in which a typeII superconductor is threaded by normal cores, is known as the <b>mixed state</b>. As Figure 24 shows, both the upper and lower critical field strengths depend on temperature in a similar way to the critical field strength for a typeI material (Figure 11).</p><div class="oucontentfigure oucontentmediamini" id="fig009_024"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/42f3ae24/smt359_1_024i.jpg" alt="Figure 24" width="238" height="296" style="maxwidth:238px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 24 Temperature dependence of the lower critical field strength (<i>B</i><sub>c1</sub>) and upper critical field strength (<i>B</i><sub>c2</sub>) for a typeII superconductor.</span></div></div></div><p>The normal cores that exist in typeII superconductors in the mixed state are not sharply delineated. Figure 25 shows how the number density of superelectrons and the magnetic field strength vary along a line passing through the axes of three neighbouring cores. The value of <i>n</i><sub>s</sub> is zero at the centres of the cores and rises over a characteristic distance ξ, the coherence length. The magnetic field associated with each normal core is spread over a region with a diameter of 2λ, and each normal core is surrounded by a vortex of circulating current.</p><div class="oucontentfigure oucontentmediamini" id="fig009_025"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/92f9b763/smt359_1_025i.jpg" alt="Figure 25" width="288" height="232" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 25 Number density of superelectrons <i>n<sub>s</sub></i> and magnetic field strength <i>B</i> around normal cores in a typeII superconductor.</span></div></div></div><p>You can see from Figure 25 that the coherence length ξ, the characteristic distance for changes in <i>n<sub>s</sub></i>, is shorter than the penetration depth λ, the characteristic distance for changes in the magnetic field in a superconductor. This is generally true for typeII superconductors, whereas for typeI superconductors, ξ > λ (Figure 20). For a pure typeI superconductor, typical values of the characteristic lengths are ξ ~ 1 <i>μ</i>m and λ = 50 nm. Contrast this with the values for a widelyused typeII alloy of niobium and tin, Nb<sub>3</sub>Sn, for which ξ ~ 3.5 nm and λ = 80 nm.</p><p>The reason that the relative magnitude of the coherence length and the penetration depth is so important is that when ξ > λ, the surface energy associated with the boundary between superconducting and normal regions is positive, whereas when ξ < λ, this surface energy is negative. Justifying this statement would involve a discussion of the thermodynamics of superconductors, but for our purposes it is sufficient to just look at the consequences. For a positive surface energy, the system will prefer few boundaries and we expect relatively thick normal and superconducting regions, as observed in the intermediate state in typeI materials. Conversely, a negative surface energy favours formation of as much boundary between normal and superconducting regions as possible, and this is what happens in the mixed state in typeII materials with many narrow normal cores. The lower limit for the diameter of the cores is 2ξ, as shown in Figure 25, because ξ is the distance over which <i>n</i><sub>s</sub> can vary.</p><p>This energy argument does not explicitly indicate how much magnetic flux passes through each of the normal cores. However, quantum mechanical arguments show that the magnetic flux linking any superconducting loop must be quantised, and that the quantum of magnetic flux is</p><div class="oucontentquote oucontentsbox" id="quo002"><blockquote><p><i>h</i> / 2<i>e</i> = 2.07 × 10<sup>−15</sup>T m<sup>2</sup></p></blockquote></div><p>where <i>h</i> = 6.63 × 10<sup>−34</sup>J s is Planck's constant. In fact, each of the normal cores shown in Figure 23 contains just one quantum of flux, since this is more favorable energetically than having two or more quanta of flux in a core.</p><p>The quantisation of flux in a superconductor is of particular importance in SQUIDs, the superconducting quantum interference devices that are at the heart of the magnetometers used for measuring the magnetic fields produced by currents in the brain. A SQUID contains a small loop of superconductor with a weakly superconducting link in it, and the quantisation of flux in the loop causes its electrical properties to depend on the flux applied to it, with a periodicity equal to the flux quantum. The very small magnitude of the flux quantum is responsible for the sensitivity of the device to very small magnetic fields.</p><p>A final point is worth noting about the quantum of flux: the factor of 2 in the denominator of the expression <i>h</i> / 2<i>e</i> is a consequence of the coupling of pairs of electrons in a superconductor and their condensation into a superconducting ground state. There is charge −2<i>e</i> associated with each of these electron pairs.</p><p><b>Critical currents in typeII superconductors</b></p><p>The high values of the upper critical field strength <i>B</i><sub>c2</sub> of many typeII superconducting alloys make them very attractive for winding coils for generating high magnetic fields. For example, alloys of niobium and titanium (NbTi<sub>2</sub>) and of niobium and tin (Nb<sub>3</sub>Sn) have values of <i>B</i><sub>c2</sub> of about 10 T and 20 T, respectively, compared with 0.08 T for lead, a typeI superconductor. However, for typeII materials to be usable for this purpose, they must also have high critical currents at high field strengths, and this requires some help from metallurgists to overcome a significant problem.</p><p>This problem is related to the interaction between the current flowing through a typeII superconductor in the mixed state and the ‘tubes’ of magnetic flux that thread through the normal cores. The electrons will experience a Lorentz force, perpendicular to both the current density and the magnetic field. We can regard this as a mutual interaction between the electrons and the flux in the normal cores, as a result of which each normal core experiences a force that is in the opposite direction to the Lorentz force on the electrons. The directions of the magnetic field, current and forces are shown in Figure 26.</p><div class="oucontentfigure oucontentmediamini" id="fig009_026"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/e619efd9/smt359_1_026i.jpg" alt="Figure 26" width="288" height="214" style="maxwidth:288px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 26 Electrons and normal cores experience forces perpendicular to the current and to the magnetic field, but in opposite directions.</span></div></div></div><p>This Lorentz force can cause the cores and their associated magnetic flux to move, and the flux motion will induce an emf that drives a current through the normal cores, somewhat like an eddy current. Energy is therefore dissipated in the normal cores, and this energy must come from the power supply. The energy dissipation means that the flow of electrons is impeded, and therefore there is a resistance to the flow of the current.</p><p>Flux motion is therefore undesirable in typeII superconductors, and the aim of the metallurgists who develop processes for manufacturing wire for magnets is to make flux motion as difficult as possible. This is done by introducing defects into the crystalline structure, particularly by preparing the material in such a way that it comprises many small crystalline grains with different orientations and small precipitates of different composition. These defects effectively pin the normal cores in position – they provide a potential barrier to motion of the cores, so that the force on the cores must exceed a certain value before the cores can move. The more of these flux pinning centres that are present, and the greater the potential barrier they provide, the greater will be the current required to set them in motion, i.e. the greater the critical current. So, unlike a normal conductor, for which improving the purity and reducing imperfections in the crystal structure lead to better conductivity, with typeII superconductors the inclusion of impurities and defects in the crystal structure can improve the critical current and make the material more suitable for use in electromagnets.</p><p>Undoubtedly the largest use of superconducting material for a single project is in the Large Hadron Collider at CERN, due for commissioning in 2007. The 27 kmlong accelerator tunnel contains 1232 superconducting magnets that are responsible for steering the particle beams around their circular paths. Each of these magnets is 15 m long, has a mass of 35 tonnes and produces a magnetic field strength of 8.5 T. The coils in each magnet are made from about 6 km of niobiumtitanium cable, with a mass of about a tonne, and will be maintained at a temperature of 1.9 K using liquid helium. To construct an accelerator with a similar specification using nonsuperconducting magnets would have required a 120 km tunnel and phenomenal amounts of power to operate.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

Conclusion
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection5
Wed, 13 Apr 2016 23:00:00 GMT
<p><b>Section 1</b> Superconductivity was discovered in 1911, and in the century since then there have been many developments in knowledge of the properties of superconductors and the materials that become superconducting, in the theoretical understanding of superconductivity, and in the applications of superconductors.</p><p><b>Section 2</b> A superconductor has zero resistance to flow of electric current, and can sustain a current indefinitely. The magnetic flux remains constant in a completely superconducting circuit, since changes in the flux from the field applied to the circuit are balanced by changes to (persistent) currents induced in the circuit. For each superconductor there is a critical temperature <i>T</i>﻿<sub>c</sub> below which the material is superconducting.</p><p>Superconductors also exhibit perfect diamagnetism, with <b>B</b> = <b>0</b> in the bulk of the material. The exclusion of magnetic field is known as the Meissner effect. An external magnetic field penetrates for a short distance into the surface of a superconducting material, and a current flows in the surface layer to screen the interior of the material from the applied field. Superconductivity is destroyed when the magnetic field strength exceeds a critical value for the material. The critical field strength falls to zero as the temperature is raised to the critical temperature. A superconducting specimen will have a critical current <i>I</i>﻿<sub>c</sub> above which the material reverts to the normal state. This critical current corresponds to the field strength exceeding the critical field strength in some region of the specimen.</p><p><b>Section 3</b> The twofluid model of a superconductor regards some of the conduction electrons as behaving like normal electrons and some like superconducting electrons. For <i>T</i> ≪ <i>T</i>﻿<sub>c</sub>, all of the conduction electrons are superconducting electrons, but the proportion of superconducting electrons drops to zero at the critical temperature.</p><p>For a perfect conductor (which has <i>R</i> = 0), Maxwell's equations predict that the magnetic field cannot change, except in a thin surface layer. This does not predict the Meissner effect. The London equations are relationships between current density and magnetic field that are consistent with the Meissner effect:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c5b4eb15/smt359_1_ue017i.gif" alt=""/></div><p>When combined with Maxwell's equations, they lead to the prediction that the magnetic field strength and the surface current decrease exponentially below the surface of a superconductor, over a characteristic distance called the penetration depth λ, which is typically tens of nanometres. The London equations are local relationships and therefore are strictly valid only when λ ≫ ξ, where the coherence length ξ is the characteristic distance over which <i>n</i>﻿<sub>s</sub> varies.</p><p><b>Section 4</b> There are two types of superconductors, typeI and typeII. For a typeI material in the form of a thin specimen parallel to the field, there is an abrupt transition to the normal state at the critical field strength <i>B</i>﻿<sub>c</sub>. When the field is inclined to the surface of a typeI material, the material exists in the intermediate state over a range of field strengths below <i>B</i>﻿<sub>c</sub>. In this state there are thin layers of normal and superconducting material, with the proportion of normal material rising to unity at field strength <i>B</i>﻿<sub>c</sub>. In typeI materials, the coherence length ξ is greater than the penetration depth λ, and the surface energy of the boundary between superconducting and normal material is positive, which favours a course subdivision into regions of normal and superconducting material.</p><p>A typeII superconductor has two critical field strengths, <i>B</i>﻿<sub>c1</sub> and <i>B</i>﻿<sub>c2</sub>, between which the material is in the mixed state. In this state the superconductor is threaded by thin cores of normal material, through which the magnetic field passes. The coherence length ξ is shorter than the penetration depth λ, and the surface energy of the boundary between superconducting and normal material is negative, which favours a fine subdivision into regions of normal and superconducting material. To take advantage of the high values of <i>B</i><sub>c2</sub> to produce high magnetic fields with superconducting magnets, it is essential to pin the normal cores to inhibit their motion.</p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsection5
ConclusionSMT359_1<p><b>Section 1</b> Superconductivity was discovered in 1911, and in the century since then there have been many developments in knowledge of the properties of superconductors and the materials that become superconducting, in the theoretical understanding of superconductivity, and in the applications of superconductors.</p><p><b>Section 2</b> A superconductor has zero resistance to flow of electric current, and can sustain a current indefinitely. The magnetic flux remains constant in a completely superconducting circuit, since changes in the flux from the field applied to the circuit are balanced by changes to (persistent) currents induced in the circuit. For each superconductor there is a critical temperature <i>T</i><sub>c</sub> below which the material is superconducting.</p><p>Superconductors also exhibit perfect diamagnetism, with <b>B</b> = <b>0</b> in the bulk of the material. The exclusion of magnetic field is known as the Meissner effect. An external magnetic field penetrates for a short distance into the surface of a superconducting material, and a current flows in the surface layer to screen the interior of the material from the applied field. Superconductivity is destroyed when the magnetic field strength exceeds a critical value for the material. The critical field strength falls to zero as the temperature is raised to the critical temperature. A superconducting specimen will have a critical current <i>I</i><sub>c</sub> above which the material reverts to the normal state. This critical current corresponds to the field strength exceeding the critical field strength in some region of the specimen.</p><p><b>Section 3</b> The twofluid model of a superconductor regards some of the conduction electrons as behaving like normal electrons and some like superconducting electrons. For <i>T</i> ≪ <i>T</i><sub>c</sub>, all of the conduction electrons are superconducting electrons, but the proportion of superconducting electrons drops to zero at the critical temperature.</p><p>For a perfect conductor (which has <i>R</i> = 0), Maxwell's equations predict that the magnetic field cannot change, except in a thin surface layer. This does not predict the Meissner effect. The London equations are relationships between current density and magnetic field that are consistent with the Meissner effect:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn009_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69548/mod_oucontent/oucontent/514/cbca6394/c5b4eb15/smt359_1_ue017i.gif" alt=""/></div><p>When combined with Maxwell's equations, they lead to the prediction that the magnetic field strength and the surface current decrease exponentially below the surface of a superconductor, over a characteristic distance called the penetration depth λ, which is typically tens of nanometres. The London equations are local relationships and therefore are strictly valid only when λ ≫ ξ, where the coherence length ξ is the characteristic distance over which <i>n</i><sub>s</sub> varies.</p><p><b>Section 4</b> There are two types of superconductors, typeI and typeII. For a typeI material in the form of a thin specimen parallel to the field, there is an abrupt transition to the normal state at the critical field strength <i>B</i><sub>c</sub>. When the field is inclined to the surface of a typeI material, the material exists in the intermediate state over a range of field strengths below <i>B</i><sub>c</sub>. In this state there are thin layers of normal and superconducting material, with the proportion of normal material rising to unity at field strength <i>B</i><sub>c</sub>. In typeI materials, the coherence length ξ is greater than the penetration depth λ, and the surface energy of the boundary between superconducting and normal material is positive, which favours a course subdivision into regions of normal and superconducting material.</p><p>A typeII superconductor has two critical field strengths, <i>B</i><sub>c1</sub> and <i>B</i><sub>c2</sub>, between which the material is in the mixed state. In this state the superconductor is threaded by thin cores of normal material, through which the magnetic field passes. The coherence length ξ is shorter than the penetration depth λ, and the surface energy of the boundary between superconducting and normal material is negative, which favours a fine subdivision into regions of normal and superconducting material. To take advantage of the high values of <i>B</i><sub>c2</sub> to produce high magnetic fields with superconducting magnets, it is essential to pin the normal cores to inhibit their motion.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University

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Acknowledgements
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Wed, 13 Apr 2016 23:00:00 GMT
<p>The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.</p><p>Grateful acknowledgement is made to the following sources for permission to reproduce material in this course:</p><p>Course image: <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.flickr.com/photos/criminalintent/">Lars Plougmann</a></span> in Flickr made available under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/bysa/2.0/legalcode">Creative Commons AttributionShareAlike 2.0 Licence</a>.</p><p>Figure 1 Science Photo Library;</p><p>Figure 8a This photograph has been provided by Railway Technical Research Institute in Japan;</p><p>Figure 22 Proceedings of the Royal Society A248 464. The Royal Society;</p><p>Figure 23 V Essmann and H Trauble, Max Planck Institut fiir Metallforschung.</p><p>This photograph has been provided by Railway Technical Research Institute in Japan;</p><p><b>Don't miss out:</b></p><p>If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University  <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a></p>
https://www.open.edu/openlearn/sciencemathstechnology/engineeringandtechnology/engineering/superconductivity/contentsectionacknowledgements
AcknowledgementsSMT359_1<p>The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.</p><p>Grateful acknowledgement is made to the following sources for permission to reproduce material in this course:</p><p>Course image: <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.flickr.com/photos/criminalintent/">Lars Plougmann</a></span> in Flickr made available under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/bysa/2.0/legalcode">Creative Commons AttributionShareAlike 2.0 Licence</a>.</p><p>Figure 1 Science Photo Library;</p><p>Figure 8a This photograph has been provided by Railway Technical Research Institute in Japan;</p><p>Figure 22 Proceedings of the Royal Society A248 464. The Royal Society;</p><p>Figure 23 V Essmann and H Trauble, Max Planck Institut fiir Metallforschung.</p><p>This photograph has been provided by Railway Technical Research Institute in Japan;</p><p><b>Don't miss out:</b></p><p>If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University  <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBSuperconductivity  SMT359_1Copyright © 2016 The Open University