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Note for Mechanics - MECH By abhijith anirudhan

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2 phenomenon. temperature decreases, the number of collisions decreases and the resistance follows. A basic equation for this interaction between resistance and temperature can be seen below. R R0 R0 Fig. 2 The Meissner effect [18] Not long after this discovery Fritz London proposed that superconductivity is a quantum phenomenon. This proposal, although at the time was inconclusive, however, is now substantiated by current microscopic theory. Another important breakthrough was the discovery of the Isotope effect which points at a connection between electron and lattice vibrations in relation to superconductivity (Superconductivity in Science and Technology –Cohen). Other Important additions to the current knowledge of superconductors include the tunnel effect (which points to the likelihood of energy gap in the superconducting state), flux quantization and supercurrent flow through tunnel barrier. C. A Quest for High Temperature Until 1986, the highest critical temperature for a superconducting transition was 23.2 degrees Kelvin. The material was Nb3Ge and speculation was that there was not much more room for improvement. The focus for two European scientists then became to make an alloy in which they could enhance the electron-phonon coupling parameter. This led to a critical temperature of slightly more than 30 K. Variations and further experimentation over the past two decades have led to a current critical temperature record of 125 degrees Kelvin (significantly more than the boiling point of liquid nitrogen -77K). These recent discoveries have opened up the door for countless applications, from high powered MRIs and brain mapping (SQUIDs) to magnetic friction free trains. [3] III.  T T0 Einstein proposed a theory that the electrical resistance of the metals would fall sharply at very low (at that time unattainable) temperatures although popular opinion was that the resistances would only reach a resistance of zero at absolute zero temperature. On a microscopic level, superconductivity implies that electron pairs are forming with a spacing of hundreds of nanometers which is three times larger than the lattice spacing. These widely spaced electron pairs have been termed Cooper pairs after the scientist (Leon Cooper) involved in their discovery. Current and past understanding of “normal” electron behavior has been governed by the Pauli Exclusion Principle. This law states that no two electrons (each with half integer spin) can have the same quantum number or in other words, be in the same state simultaneously. [8] Other existing particles known as bosons carry a full integer charge. Because of this full integer spin, bosons are not governed by the Pauli Exclusion Principle and can therefore simultaneously exist in the same state. Einstein called this condensation as the bosons could move in unlimited numbers to a single ground state. [8] The first noticeable proof of Einstein’s observation was the superfluidity of Helium. This trend was found when Helium was cooled to below 2.17 degrees Kelvin, all viscosity of the liquid vanished. Viscosity, which can be termed the resistance to flow of a liquid, is yet another parallel between the waterelectricity analogies for electrical resistance. Yet another parallel between is the comparison between laminar flow and the energy band gap with tunneling effect. A diagram portraying the laminar flow of a fluid through a pipe can be seen in figure 3. MICROSCOPIC THEORY Originally, the basis for investigating the reaction of electrical resistance in relation to temperature was founded on an observable trend in the characteristics of metals. It was noted that with increasing temperatures the resistance of metals increased to a point while with decreasing temperatures, resistances decreased. The understanding of the principle of resistance varying with temperature is easily understood as resistance itself is merely the effect of collisions within a wire. Thus, as the temperature increases, atomic movement and vibration increases causing more collisions. Conversely, as Fig. 3 Laminar Flow Through a Pipe [8]

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3 In a superconductor, the Cooper pairs, mentioned earlier, act as bosons. This means that the pairs can exist simultaneously in ultra-low energy states. These pairs form closely to the top of the collection of energy levels (also called the Fermi level) by interacting with the crystal lattice. The slight lattice vibrations attract these Cooper pairs, leaving an energy gap. Because the pairs are not subject to Pauli Exclusion Principle due to their combined integer spin, they can simultaneously occupy the same energy state. This attraction is termed the phonon interaction. The energy gap is then the sort of “tunnel” through which resistance free traveling can occur in the case that the thermal energy is less than the band gap. Where in normal circumstances, collisions would occur causing normal resistivity, at low temperatures, the thermal energy drops to a value less than the band gap, leaving the band gap open for tunneling. This theory of functionality for superconducting principles is known as the Bardeen-CoorperSchrieffer Theory (or BCS theory). [7] [5] The BCS theory not only substantiates the microscopic theory of superconductivity, but also predicts a bandgap based on the critical temperature. The equation for this prediction can be seen below where Eg is the predicted bandgap and Tc is the critical temperature. The graph in figure 4 displays some superconductors and their respective bandgaps and critical temperatures. Fig. 5 Superconductors Energy Gap Reduction. [8] The energy gap reduction then provides the means through which a superconducting state is produced in that when the thermal energy is less than the energy gap the resistance of the material drops to 0. IV. SUPERCONDUCTIVE MATERIALS Currently two basic types of superconductors are recognized. These types are aptly termed type I and type II superconductors. In order to understand applications of superconductors it is first necessary to understand these two types as they relate specific characteristics of materials to certain attainable results. A. Type I Superconductors Contrary to intuition, the best normal conductors (i.e. gold, silver, and copper) are not superconductors at all due to their small lattice vibrations. Metals listed in figure 2 are all type 1 superconductors as their lattice vibrations are an attractive force to the Cooper pairs. These types of superconductors fall under the BCS theory mentioned earlier. Typical metals in this category posses “softer” characteristics. They do not maintain their superconductivity at higher temperatures and exhibit lower temperature magnetic fields than type II. Fig. 4 Superconductors and Relational Bangaps. [8] As the superconductor approaches its critical temperature, the energy gap decreases exponentially. A plot of this relationship can be seen in figure 5.

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4 Mat. Tc Mat. Tc Be 0 Gd* 1.1 Rh 0 Al 1.2 W 0.015 Pa 1.4 Ir 0.1 Th 1.4 Lu 0.1 Re 1.4 Hf 0.1 Tl 2.39 Ru 0.5 In 3.408 Os 0.7 Sn 3.722 Mo 0.92 Hg 4.153 Zr 0.546 Ta 4.47 Cd 0.56 V 5.38 U 0.2 La 6.00 Ti 0.39 Pb 7.193 Zn 0.85 Tc 7.77 Ga 1.083 Nb 9.46 flexibility reasons. A table of type II superconductors can be seen in figure 8. Fig. 8 Table of Type II Superconductors [8] Type II superconductors possess not only harder characteristics, but also retain a higher critical temperature and the ability to produce high powered magnetic fields. An alloy of niobium and titanium is currently used in the making of the MRI and other supermagnets utilized by Fermilab. In addition to harder characteristics and a higher critical temperature, type II superconductors possess a higher threshold tolerance to magnetic fields so that they retain their superconducting properties even when in contact with a stronger magnetic field. Overall, some of the differences between type I and type II superconductors can be seen in Figure 9. Fig. 6 Table of Type I Superconductors [8] In addition to having a lower critical temperature than type II superconductors, type I superconductors also have a lower threshold for magnetic field tolerance. This means that when a magnetic field greater than the threshold is applied to a type I superconductor, its superconducting state ceases. Fig. 7 Reaction of Superconductivity to Magnetic Fields [8] A. Type II Superconductors Type II superconductors are often referred to as the “hard” superconductors. They are composed of alloys of ceramics and metal oxides. An example of a newer type II superconductor is a material known as BSCCO which is an oxide composed of bismuth, strontium, calcium and copper. [4] In addition to these materials, silver was added for Fig. 9 Differences in Trend between Type I and Type II [18]

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5 V. APPLICATIONS OF SUPERCONDUCTORS Knowing the basic principles and methodology of superconductors, it becomes easier and more intuitive to understand current and future applications of superconductors. General applications include both DC and AC possibilities. A. AC Applications One AC application of superconductors is simply the idea of replacing existing high voltage power lines with superconducting power lines. The idea of driving a high power line over miles and miles of terrain with no loss due to resistance is appealing to both the public and power companies. This option is becoming more viable as critical temperatures of type II superconductors are nearing the boiling point of nitrogen. Current superconducting wires use liquid helium as the cooling agent as critical temperatures are lower than the range nitrogen can provide. The cost of cooling with liquid helium, however, is not feasible for such a large scale application so it is likely that before action is taken in this region a new breed of high-temperature superconductors will need to emerge. Another AC application of superconductors is the magnetically levitated or “maglev” train. This application is works on principles utilizing the ultra-strong magnetic fields produced by superconductors. Strong opposition of magnetic fields keep the train “levitated” so that it never comes into contact with the track. This method of transportation provides clear benefits over other forms of transportation. First of all, the train never comes into contact with the rail so theoretically there would need to be no maintenance as there are no moving parts. Secondly, there is no friction between the train and railway because it is levitated. As a result of this the train can travel at higher speeds although there is still friction due to air. In addition to this the ride in the maglev train is smoother as there no bumps due to railways. Rather than being propelled by conventional methods, the train is moved through the use of magnetic forces. The diagram in figure 10 depicts these forces and how they affect the train. [19] Fig. 10 Maglev Train Propulsion Forces [19] Many disadvantages exist, however, in that the track utilizing superconducting currents to produce the magnetic fields is expensive to construct. In addition, the current maglev trains do not possess the ability to travel on regular tracks so their application would be simply high speed transit between large cities. In addition, the largest forces at high speeds are due to wind resistance so although it may be “friction free”, these forces still affect the efficiency of the train. Overall, whether the train will truly be implemented remains to be seen from test tracks built in China and Germany. One other AC application in particular is in relation to high speed particle collision research. Much of this research is done at fermilab where the world’s largest high speed particle accelerator is located. This accelerator brings protons and antiprotons to extremely high speeds at which they collide together using the charged characteristics of the particles. Through the use superconductors, high-powered magnetic fields keep the particles in place while the charged plates are used to accelerate the particles. Through these experiments, scientists hope to discover the basic properties of particles and reasons behind the basic forces governing matter. B. DC Applications The DC applications of superconductors in many ways are conducive to further DC technology. One example of this is Josephson Junctions which lead to the possibility of superconducting quantum interference devices (SQUIDs). In addition to this, high power multiplexers are attaining the ability to decrease in size and yet maintain their power levels. The first items of interest in regard to the DC applications of superconductors are then Josephson Junctions. While he was researching superconductors, a British physicist named Brian David Josephson investigated the properties of a junction between two superconductors. Josephson observed that when no voltage was applied to either superconductor, current would flow through a thin insulating layer. When a voltage was applied, the current would cease to flow and oscillate at a high frequency. [12] The reason for the flowing of current through the insulating layer was discovered to be the effect of ambient magnetic fields. It was discovered that because of the Meissner Effect in a superconductor, the magnetic field would externally induce current flow inside the superconductor. The Meissner Effect, in essence, is the characteristic of a superconductor that when in superconducting state, acts as a perfect diamagnet, excluding any magnetic field that would normally flow through it by inducing current loops internally for cancellation. The ability of the Josephson junction to induce this current was then found to be far more sensitive to magnetic fields than the normal superconductor which allows for the capability to sense very small magnetic fields in a vicinity. These junctions are found to be particularly applicable in the areas of highly sensitive microwave detectors and SQUIDs. Consequently, the discovery of the Josephson junction has led to the creation of the superconducting quantum interference device. The purpose of these mechanisms is then to measure extremely weak electromagnetic signals, particularly those exhibited by the human brain. By employing Josephson Junctions, SQUIDs have the ability to measure signals 100 billion times weaker than the electromagnetic energy required to move a compass needle. The actual structure of the SQUID consists of extremely small loops of superconductors which use Josephson Junctions to employ superposition.

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