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Long-Range Order and Superconductivity Alexander Gabovich, KPI, Lecture 1

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Density matrix in quantum mechanics If one has a large closed quantum-mechanical system with co-ordinates q and a subsystem with co-ordinates x, its wave function Ψ(q,x) generally speaking does not decompose into two ones, each dependent on q and x. If f is a physical quantity, its mean value is given by

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Density matrix in quantum mechanics In the pure case, when the system concerned is described by the wave function one has

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Density matrix in quantum mechanics Another kind of the long-range order is the following:

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Off-diagonal long-range order

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Long-range orders below critical lines of phase transitions (4He)

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Phase transitions

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MICHAEL FARADAY, THE PRECURSOR OF LIQUEFACTION Michael Faraday, 1791-1867 He liquefied all gases known to him except O2, N2, CO, NO, CH4, H2. Permanent gases? – NO! COLD WAR OF LIQUEFACTION: O2 – Louis-Paul Cailletet (France) and Raoul-Pierre Pictet (Switzerland) [1877]; N2, Ar – Zygmund Wróblewski and Karol Olszewski (Poland) [1883]

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JAMES DEWAR, THE COMPETITOR – A MAN, WHO LIQUEFIED HYDROGEN IN 1898 A Dewar flask in the hands of the inventor. James Dewar’s laboratory in the basement of the Royal Institution in London appears as the background.

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KAMERLINGH-ONNES, THE WINNER – PHYSICIST AND ENGINEER (Nobel Prize in Physics, 1913) Heike Kamerlingh Onnes (right) in his Cryogenic Laboratory at Leiden University, with his assistant Gerrit Jan Flim, around the time of the discovery of superconductivity: 1911

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LOW TEMPERATURE STUDIES USING LIQUID HELIUM LED TO NEW DISCOVERIES: NOT ONLY SUPERCONDUCTIVITY!

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Superconducting phenomenology

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SUPERCONDUCTIVITY AMONG ELEMENTS

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SUPERCONDUCTIVITY, A MIRACLE FOUND BY KAMERLINGH-ONNES

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ANNIVERSARIES OF key discoveries 1908-2008 (100) Helium liquefying 1911-2011 (100) Superconductivity 1933-2013 (70) Meissner-Ochsenfeld effect 1956-2011 (55) Cooper pairing concept 1962-2012 (50) Josephson effect 1971-2011 (40) Superfluidity of 3He 1986-2011 (25) High-Tc oxide superconductivity 2001-2011 (10) MgB2 with Tc = 39 K 2008-2013 (5) Iron-based superconductors with Tc = 75 K (in single layers of FeSe)

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PHENOMENOLOGY. NORMAL METALS

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Superconducting phenomenology

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Magnetic field, magnetic induction, and magnetization

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Superconducting phenomenology

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Superconducting phenomenology

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Superconducting phenomenology

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Superconducting phenomenology

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Superconducting phenomenology

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Creators of the type II superconductors

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Superconducting phenomenology

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Superconducting phenomenology

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Superconducting phenomenology: London equation

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Superconducting phenomenology: London equation

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Superconducting phenomenology: London equation Let us consider the second Newton law mdv/dt = eE. This equations means that there is no resistance! (The main point! – infinite conductivity). The current density j = nsev. Then d(Λj)/dt = E (*), where Λ=m/(nse2). One knows that the full and partial time derivative are connected by the equation d/dt = / t + v. Since real current velocities v in metals are small in comparison with the Fermi velocity vF, one can replace the full derivative by the partial one. Then (Λj)/t = E (i). We have the Maxwell equation (Faraday electromagnetic induction equation): rot E = − c-1H/t (**). Let us apply a rotor operation to the equation (i). Then (Λ rot j)/t = rot E (***).

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Superconducting phenomenology: London equation From (**) and (***) one obtains (Λ rot j)/t = − c-1H/t (***). Or /t(rot Λj + c-1H ) =0 (****). It means that the quantity in the parentheses of Eq. (****) is conserved in time. Now, it is another main step, that takes into account the superconductivity itself! Specifically, in the bulk of the superconductor both j = 0 And H = 0. It simply reflects the Meissner effect! Then rot Λj + c-1H = 0 (*****). Equations (*****) and (i) constitute the basis of the London theory.

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Superconducting phenomenology: London equation Equation (*****) and the Maxwell equation rot H = 4πj/c leads to the characteristic result of London electrodynamics. Below, we shall write relevant equations in the SI unit system.

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Superconducting phenomenology: London equation

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Superconducting phenomenology: London equation

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Superconducting phenomenology: London equation Eq. (3.46) can be transformed and solved to obtain Eq. (3.52). Namely, one knows the vector identity rot rot B =  div B – Δ B, where B is an arbitrary vector. However, div B = 0, because there are no magnetic charges. Therefore, Δ B = B/2. Now, for the special geometry of Fig. 3.12 one has