Superconductivity and its applications

15 Mar

“If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is that…all things are made of atoms.”
Richard P. Feynman, winner of the 1965 Nobel Prize in Physics

PART 1

History

The idea of the absolute zero of temperature was advanced by J. A. C. Charles and J. L. Gay-Lussac (Fig.1). In fact, they have shown that by extrapolating their measurements of the volume of any gas as a function of the temperature (at constant pressure), their volume would tends toward a zero value at a temperature near -273 degree Celsius (see fig.2).

Gay-Lussac. Image credits: chemistryland.com

By extrapolation of the curve of volume versus temperature, Charles and Gay-Lussac predicted the existence of the absolute zero. Im. credit: chempaths.chemeddl.org

In 1911, Kamerlingh Onnes discovered that mercury suddenly lose its resisitivity  when the temperature dropped to -260 degree Celsius. Omnes started investigating the field of low-temperature physics in 1908, when he succeeded in liquifying helium. Whith these findings we may say that the field of superconductivity is open, bringing to us the compreension of astounding phenomena, characterized by the manifestation of the wave nature of particles at a macroscopic scale.

It is not always fundamental to reduce friction to put the body at zero degrees Kelvin. In fact, russian scientists succeeded that achievement with a simple experiment. They launched a sphere of steel on a surface made of molybdenium and sulphur, while at the same time thay beam a cloud of electrons on the surface. They observe a strong reduction of friction from a coefficient 0.9 to 0.0015!

Fundamental Properties of Superconductors
superconducting or superfluid phase is considered a different thermodynamical phase when compared to the normal state that exists when the temperature T>T_c, where T_c means the critical temperature below which the superconducting or superfuid state appears. For example, liquid 4He under its vapor pressure becomes superfluid at T_c=2.17 K. So, the passage from one state to the other is called a phase transition, and it is accompanied by a strong increase of the specifi heat when the temperature is near T_c, with a strong release of entropy. This phenomena, experimentally seen, leads to the formation of a more ordered state of matter (see also precedent video). Fundamentally, a superconductor is a conductor that has undergone a phase transition to a lower energy state below the critical temperature T_c, characterized by the appearance of groups of paired electrons, the so called Cooper pairs, carrying electrical current without any resistance and responsible, among other properties, of perfect diamagnetism.

We may distinguish two types of superconductors. A Type I supercondcutor, is characterized by the following properties [1]:

  • zero electrical resistance and perfect diamagnetism at a temperature below T_c. Normally, at temperatures above T_c this material is a normal metal, although not a very good conductor;
  • perfect diamagnetism, also called Meissner effect, the magnetic field stays outside the material, cann’t penetrate the material. Curiosly enough, if you appply an external magnetic field  B_app above a given critial magnetic field, B_c, the material suffers a transition from superconductor to normal state. An approximate functional dependence on temperature for this critical magnetic field is given by

And what is the diamagnetic property of matter? This property  is a kind of negative magnetism. This effect was studied in the framework of  classical mechanical by Paul Langevin in 1905 [2], using previous and revolutionary ideas proposed formerly by André-Marie Ampère[3]  and Wilhelm Weber [4] (nowadays we barely talk about these two great men of science that really use their minds for the advancement of science and the progress of mankind…). Langevin found, in the classic framework provided by Ampère and Weber, that N electrons moving in orbits around the nucleus at an average distance <r²>, such that the (constant and negative) magnetic susceptibility χ is given by

The magnetic susceptibility is the ratio of M/H, where M is the magnetization field and H the magnetic field. The susceptibility χ is slightly negative for diamagnets, but acquires small positive values for paramagnetic substances (e.g., ), and is strongly positive for ferromagnetics substances (e.g., Fe). It can be shown that a material constitute of paramagnetic ions with magnetic moment μ, obeys the Curie-Weiss law:

where n means the concentration of paramagnetic ions, Θ is the Curie-Weiss constant () . when perfect diamagnetism is achieved, χ=-1, that is, the magnetization M is directed opposite to the H field, cancelling it, M=-H. For example, when a superconductor with spherical form is placed nearby the poles of a magnet, it results a superposition of the applied magnetic field B_app, and the resulting dipole field (Fig.1a) , giving a curvature of the magnetic field lines of the form as shown in Fig.1b.

The dipole result when a uniform permanent magnetization M is parallel to the axis Oz (see Section 5.10 in the textbook of J. D. Jackson [4]).

Besides perfect diamagnetism, the other important property of the superconducting state is its zero resistance. In ideal conditions, an electric current established in a loop of supercuncting wire will last indefenitely. The surface resistance of the material with a current flowing along a film of thickness d, must satisfy the condition

where ρ is the electric resistivity, h is the Planck constant and e is the absolute charge of the electron.

Methods of Quantum field Theory

In quantum field theory (QFT), the Lagrangian plays a central role in working out mathematically the physical problems and from which the dynamic field equations are generated. In the Hamiltonian formulation of classical mechanics, the equations of motion of a system of particles can be obtained from a special function called the Action. Usually, the Lagrangian is dependent on the positions and velocities of the particles:

Hamilton’s minimum action principle states that Nature prefer to follow movements that  extremize this quantity called action.

In QFT, this formulation can be translated, replacing que classical quantities by a “field”, in the manner of the Table 1 below:

In field theory, the fields φ(x,t) play the role of the generalized coordinates, {qi(t)}, where the discrete index i represents the number of discrete coordinates of the system. In local field theories the Lagrangian may be written as an integral over another function called the Lagrangian density, L, such as

depending on the set of a possible fields present on the given point of space and their first derivatives:

When seeking to extremize the functional called Action, it is obtained the so called Euler-Lagrange equations for any field:The Fig. below represents several possible “trajectories”, or “stories” associated to a given dynamical field. But Nature prefer only one of them, the one “story” which is represented by the above Euler-Lagrange equations…

As written by Pierre Fermat in a letter of 1662 to M. de la Chambre:

Natura operatur per modos faciliores et expeditiones (Nature works by the easiest and readiest means).

Various possible "stories" that a physical system may undergo. However, Nature prefer the one that extremize the Action.

Maupertius argued that the principle of least action (was the first to enunciate it), showed the wisdom of the Creator. Maupertuis believed that the vis viva (which today is twice the kinetic energy) should be minimal.}: “ The action is proportional to the product of mass and velocity and space. Now here this principle, so wise, so worthy of the Supreme Being: once a change occurs in nature, the amount of action employed for this variation is always as small as possible. ” Pierre Louis Moreau de Maupertuis was a French mathematician and astronomer. Born July 7, 1698 in Saint-Malo and died in the Bale July 27, 1759. Interestingly, he was the son of a pirate.

Let us take as an example (although for non-mathematicians, a little bit too complex problem that can however be grasped as an example of the methods in QFT), the Lagrangian of a physical system composed by matter + electromagnetic field, such as described below, and we will recover essential properties of the superconductor state.
                                                            matter field                                                                             EM field                   Current

PART 2.

To follow

REFERENCES:

[1] Handbook of Superconductivity, Charles P. Poole, Jr. (Academic Press, San Diego, 2000)

[2] Paul Langevin

[3] André-Marie Ampère, Essay sur la Philosophie des Sciences

[4] J. D. Jackson, Classical Electrodynamics

[FN1] Annales de Chimie et Physique, 5 (1905); see paper by Paul Langevin at p.70-127

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PART 2 – To follow soon…

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