5. Phase Transitions
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Lecture Notes: BCS Theory of Superconductivity
Lecture Notes: BCS theory of superconductivity Prof. Rafael M. Fernandes Here we will discuss a new ground state of the interacting electron gas: the superconducting state. In this macroscopic quantum state, the electrons form coherent bound states called Cooper pairs, which dramatically change the macroscopic properties of the system, giving rise to perfect conductivity and perfect diamagnetism. We will mostly focus on conventional superconductors, where the Cooper pairs originate from a small attractive electron-electron interaction mediated by phonons. However, in the so- called unconventional superconductors - a topic of intense research in current solid state physics - the pairing can originate even from purely repulsive interactions. 1 Phenomenology Superconductivity was discovered by Kamerlingh-Onnes in 1911, when he was studying the transport properties of Hg (mercury) at low temperatures. He found that below the liquifying temperature of helium, at around 4:2 K, the resistivity of Hg would suddenly drop to zero. Although at the time there was not a well established model for the low-temperature behavior of transport in metals, the result was quite surprising, as the expectations were that the resistivity would either go to zero or diverge at T = 0, but not vanish at a finite temperature. In a metal the resistivity at low temperatures has a constant contribution from impurity scattering, a T 2 contribution from electron-electron scattering, and a T 5 contribution from phonon scattering. Thus, the vanishing of the resistivity at low temperatures is a clear indication of a new ground state. Another key property of the superconductor was discovered in 1933 by Meissner. -
Phase-Transition Phenomena in Colloidal Systems with Attractive and Repulsive Particle Interactions Agienus Vrij," Marcel H
Faraday Discuss. Chem. SOC.,1990,90, 31-40 Phase-transition Phenomena in Colloidal Systems with Attractive and Repulsive Particle Interactions Agienus Vrij," Marcel H. G. M. Penders, Piet W. ROUW,Cornelis G. de Kruif, Jan K. G. Dhont, Carla Smits and Henk N. W. Lekkerkerker Van't Hof laboratory, University of Utrecht, Padualaan 8, 3584 CH Utrecht, The Netherlands We discuss certain aspects of phase transitions in colloidal systems with attractive or repulsive particle interactions. The colloidal systems studied are dispersions of spherical particles consisting of an amorphous silica core, coated with a variety of stabilizing layers, in organic solvents. The interaction may be varied from (steeply) repulsive to (deeply) attractive, by an appropri- ate choice of the stabilizing coating, the temperature and the solvent. In systems with an attractive interaction potential, a separation into two liquid- like phases which differ in concentration is observed. The location of the spinodal associated with this demining process is measured with pulse- induced critical light scattering. If the interaction potential is repulsive, crystallization is observed. The rate of formation of crystallites as a function of the concentration of the colloidal particles is studied by means of time- resolved light scattering. Colloidal systems exhibit phase transitions which are also known for molecular/ atomic systems. In systems consisting of spherical Brownian particles, liquid-liquid phase separation and crystallization may occur. Also gel and glass transitions are found. Moreover, in systems containing rod-like Brownian particles, nematic, smectic and crystalline phases are observed. A major advantage for the experimental study of phase equilibria and phase-separation kinetics in colloidal systems over molecular systems is the length- and time-scales that are involved. -
Supercritical Fluid Extraction of Positron-Emitting Radioisotopes from Solid Target Matrices
XA0101188 11. United States of America Supercritical Fluid Extraction of Positron-Emitting Radioisotopes From Solid Target Matrices D. Schlyer, Brookhaven National Laboratory, Chemistry Department, Upton, Bldg. 901, New York 11973-5000, USA Project Description Supercritical fluids are attractive as media for both chemical reactions, as well as process extraction since their physical properties can be manipulated by small changes in pressure and temperature near the critical point of the fluid. What is a supercritical fluid? Above a certain temperature, a vapor can no longer be liquefied regardless of pressure critical temperature - Tc supercritical fluid r«gi on solid a u & temperature Fig. 1. Phase diagram depicting regions of solid, liquid, gas and supercritical fluid behavior. The critical point is defined by a critical pressure (Pc) and critical temperature (Tc) for a particular substance. Such changes can result in drastic effects on density-dependent properties such as solubility, refractive index, dielectric constant, viscosity and diffusivity of the fluid[l,2,3]. This suggests that pressure tuning of a pure supercritical fluid may be a useful means to manipulate chemical reactions on the basis of a thermodynamic solvent effect. It also means that the solvation properties of the fluid can be precisely controlled to enable selective component extraction from a matrix. In recent years there has been a growing interest in applying supercritical fluid extraction to the selective removal of trace metals from solid samples [4-10]. Much of the work has been done on simple systems comprised of inert matrices such as silica or cellulose. Recently, this process as been expanded to environmental samples as well [11,12]. -
Phase Transitions in Quantum Condensed Matter
Diss. ETH No. 15104 Phase Transitions in Quantum Condensed Matter A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH¨ (ETH Zuric¨ h) for the degree of Doctor of Natural Science presented by HANS PETER BUCHLER¨ Dipl. Phys. ETH born December 5, 1973 Swiss citizien accepted on the recommendation of Prof. Dr. J. W. Blatter, examiner Prof. Dr. W. Zwerger, co-examiner PD. Dr. V. B. Geshkenbein, co-examiner 2003 Abstract In this thesis, phase transitions in superconducting metals and ultra-cold atomic gases (Bose-Einstein condensates) are studied. Both systems are examples of quantum condensed matter, where quantum effects operate on a macroscopic level. Their main characteristics are the condensation of a macroscopic number of particles into the same quantum state and their ability to sustain a particle current at a constant velocity without any driving force. Pushing these materials to extreme conditions, such as reducing their dimensionality or enhancing the interactions between the particles, thermal and quantum fluctuations start to play a crucial role and entail a rich phase diagram. It is the subject of this thesis to study some of the most intriguing phase transitions in these systems. Reducing the dimensionality of a superconductor one finds that fluctuations and disorder strongly influence the superconducting transition temperature and eventually drive a superconductor to insulator quantum phase transition. In one-dimensional wires, the fluctuations of Cooper pairs appearing below the mean-field critical temperature Tc0 define a finite resistance via the nucleation of thermally activated phase slips, removing the finite temperature phase tran- sition. Superconductivity possibly survives only at zero temperature. -
The Superconductor-Metal Quantum Phase Transition in Ultra-Narrow Wires
The superconductor-metal quantum phase transition in ultra-narrow wires Adissertationpresented by Adrian Giuseppe Del Maestro to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2008 c 2008 - Adrian Giuseppe Del Maestro ! All rights reserved. Thesis advisor Author Subir Sachdev Adrian Giuseppe Del Maestro The superconductor-metal quantum phase transition in ultra- narrow wires Abstract We present a complete description of a zero temperature phasetransitionbetween superconducting and diffusive metallic states in very thin wires due to a Cooper pair breaking mechanism originating from a number of possible sources. These include impurities localized to the surface of the wire, a magnetic field orientated parallel to the wire or, disorder in an unconventional superconductor. The order parameter describing pairing is strongly overdamped by its coupling toaneffectivelyinfinite bath of unpaired electrons imagined to reside in the transverse conduction channels of the wire. The dissipative critical theory thus contains current reducing fluctuations in the guise of both quantum and thermally activated phase slips. A full cross-over phase diagram is computed via an expansion in the inverse number of complex com- ponents of the superconducting order parameter (equal to oneinthephysicalcase). The fluctuation corrections to the electrical and thermal conductivities are deter- mined, and we find that the zero frequency electrical transport has a non-monotonic temperature dependence when moving from the quantum critical to low tempera- ture metallic phase, which may be consistent with recent experimental results on ultra-narrow MoGe wires. Near criticality, the ratio of the thermal to electrical con- ductivity displays a linear temperature dependence and thustheWiedemann-Franz law is obeyed. -
VISCOSITY of a GAS -Dr S P Singh Department of Chemistry, a N College, Patna
Lecture Note on VISCOSITY OF A GAS -Dr S P Singh Department of Chemistry, A N College, Patna A sketchy summary of the main points Viscosity of gases, relation between mean free path and coefficient of viscosity, temperature and pressure dependence of viscosity, calculation of collision diameter from the coefficient of viscosity Viscosity is the property of a fluid which implies resistance to flow. Viscosity arises from jump of molecules from one layer to another in case of a gas. There is a transfer of momentum of molecules from faster layer to slower layer or vice-versa. Let us consider a gas having laminar flow over a horizontal surface OX with a velocity smaller than the thermal velocity of the molecule. The velocity of the gaseous layer in contact with the surface is zero which goes on increasing upon increasing the distance from OX towards OY (the direction perpendicular to OX) at a uniform rate . Suppose a layer ‘B’ of the gas is at a certain distance from the fixed surface OX having velocity ‘v’. Two layers ‘A’ and ‘C’ above and below are taken into consideration at a distance ‘l’ (mean free path of the gaseous molecules) so that the molecules moving vertically up and down can’t collide while moving between the two layers. Thus, the velocity of a gas in the layer ‘A’ ---------- (i) = + Likely, the velocity of the gas in the layer ‘C’ ---------- (ii) The gaseous molecules are moving in all directions due= to −thermal velocity; therefore, it may be supposed that of the gaseous molecules are moving along the three Cartesian coordinates each. -
Viscosity of Gases References
VISCOSITY OF GASES Marcia L. Huber and Allan H. Harvey The following table gives the viscosity of some common gases generally less than 2% . Uncertainties for the viscosities of gases in as a function of temperature . Unless otherwise noted, the viscosity this table are generally less than 3%; uncertainty information on values refer to a pressure of 100 kPa (1 bar) . The notation P = 0 specific fluids can be found in the references . Viscosity is given in indicates that the low-pressure limiting value is given . The dif- units of μPa s; note that 1 μPa s = 10–5 poise . Substances are listed ference between the viscosity at 100 kPa and the limiting value is in the modified Hill order (see Introduction) . Viscosity in μPa s 100 K 200 K 300 K 400 K 500 K 600 K Ref. Air 7 .1 13 .3 18 .5 23 .1 27 .1 30 .8 1 Ar Argon (P = 0) 8 .1 15 .9 22 .7 28 .6 33 .9 38 .8 2, 3*, 4* BF3 Boron trifluoride 12 .3 17 .1 21 .7 26 .1 30 .2 5 ClH Hydrogen chloride 14 .6 19 .7 24 .3 5 F6S Sulfur hexafluoride (P = 0) 15 .3 19 .7 23 .8 27 .6 6 H2 Normal hydrogen (P = 0) 4 .1 6 .8 8 .9 10 .9 12 .8 14 .5 3*, 7 D2 Deuterium (P = 0) 5 .9 9 .6 12 .6 15 .4 17 .9 20 .3 8 H2O Water (P = 0) 9 .8 13 .4 17 .3 21 .4 9 D2O Deuterium oxide (P = 0) 10 .2 13 .7 17 .8 22 .0 10 H2S Hydrogen sulfide 12 .5 16 .9 21 .2 25 .4 11 H3N Ammonia 10 .2 14 .0 17 .9 21 .7 12 He Helium (P = 0) 9 .6 15 .1 19 .9 24 .3 28 .3 32 .2 13 Kr Krypton (P = 0) 17 .4 25 .5 32 .9 39 .6 45 .8 14 NO Nitric oxide 13 .8 19 .2 23 .8 28 .0 31 .9 5 N2 Nitrogen 7 .0 12 .9 17 .9 22 .2 26 .1 29 .6 1, 15* N2O Nitrous -
Equation of State and Phase Transitions in the Nuclear
National Academy of Sciences of Ukraine Bogolyubov Institute for Theoretical Physics Has the rights of a manuscript Bugaev Kyrill Alekseevich UDC: 532.51; 533.77; 539.125/126; 544.586.6 Equation of State and Phase Transitions in the Nuclear and Hadronic Systems Speciality 01.04.02 - theoretical physics DISSERTATION to receive a scientific degree of the Doctor of Science in physics and mathematics arXiv:1012.3400v1 [nucl-th] 15 Dec 2010 Kiev - 2009 2 Abstract An investigation of strongly interacting matter equation of state remains one of the major tasks of modern high energy nuclear physics for almost a quarter of century. The present work is my doctor of science thesis which contains my contribution (42 works) to this field made between 1993 and 2008. Inhere I mainly discuss the common physical and mathematical features of several exactly solvable statistical models which describe the nuclear liquid-gas phase transition and the deconfinement phase transition. Luckily, in some cases it was possible to rigorously extend the solutions found in thermodynamic limit to finite volumes and to formulate the finite volume analogs of phases directly from the grand canonical partition. It turns out that finite volume (surface) of a system generates also the temporal constraints, i.e. the finite formation/decay time of possible states in this finite system. Among other results I would like to mention the calculation of upper and lower bounds for the surface entropy of physical clusters within the Hills and Dales model; evaluation of the second virial coefficient which accounts for the Lorentz contraction of the hard core repulsing potential between hadrons; inclusion of large width of heavy quark-gluon bags into statistical description. -
Qualitative Picture of Scaling in the Entropy Formalism
Entropy 2008, 10, 224-239; DOI: 10.3390/e10030224 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.org/entropy Article Qualitative Picture of Scaling in the Entropy Formalism Hans Behringer Faculty of Physics, University of Bielefeld, D-33615 Bielefeld, Germany E-mail: [email protected] Received: 30 June 2008; in revised form: 27 August 2008 / Accepted: 2 September 2008 / Published: 5 September 2008 Abstract: The properties of an infinite system at a continuous phase transition are charac- terised by non-trivial critical exponents. These non-trivial exponents are related to scaling relations of the thermodynamic potential. The scaling properties of the singular part of the specific entropy of infinite systems are deduced starting from the well-established scaling re- lations of the Gibbs free energy. Moreover, it turns out that the corrections to scaling are suppressed in the microcanonical ensemble compared to the corresponding corrections in the canonical ensemble. Keywords: Critical phenomena, microcanonical entropy, scaling relations, Gaussian and spherical model 1. Introduction In a mathematically idealised way continuous phase transitions are described in infinite systems al- though physical systems in nature or in a computer experiment are finite. The singular behaviour of physical quantities such as the response functions is related to scaling properties of the thermodynamic potential which is used for the formulation [1, 2]. Normally the Gibbs free energy is chosen for the description although other potentials also contain the full information about an equilibrium phase transi- tion. The Hankey-Stanley theorem relates the scaling properties of the Gibbs free energy to the scaling properties of any other energy-like potential that is obtained from the Gibbs free energy by a Legendre transformation [3]. -
Guide to Understanding Condensation
Guide to Understanding Condensation The complete Andersen® Owner-To-Owner™ limited warranty is available at: www.andersenwindows.com. “Andersen” is a registered trademark of Andersen Corporation. All other marks where denoted are marks of Andersen Corporation. © 2007 Andersen Corporation. All rights reserved. 7/07 INTRODUCTION 2 The moisture that suddenly appears in cold weather on the interior We have created this brochure to answer questions you may have or exterior of window and patio door glass can block the view, drip about condensation, indoor humidity and exterior condensation. on the floor or freeze on the glass. It can be an annoying problem. We’ll start with the basics and offer solutions and alternatives While it may seem natural to blame the windows or doors, interior along the way. condensation is really an indication of excess humidity in the home. Exterior condensation, on the other hand, is a form of dew — the Should you run into problems or situations not covered in the glass simply provides a surface on which the moisture can condense. following pages, please contact your Andersen retailer. The important thing to realize is that if excessive humidity is Visit the Andersen website: www.andersenwindows.com causing window condensation, it may also be causing problems elsewhere in your home. Here are some other signs of excess The Andersen customer service toll-free number: 1-888-888-7020. humidity: • A “damp feeling” in the home. • Staining or discoloration of interior surfaces. • Mold or mildew on surfaces or a “musty smell.” • Warped wooden surfaces. • Cracking, peeling or blistering interior or exterior paint. -
Ionic Liquid and Supercritical Fluid Hyphenated Techniques for Dissolution and Separation of Lanthanides, Actinides, and Fission Products
Project No. 09-805 Ionic Liquid and Supercritical Fluid Hyphenated Techniques for Dissolution and Separation of Lanthanides, Actinides, and Fission Products ItIntegrat tdUied Universit itPy Programs Dr. Chien Wai University of Idaho In collaboration with: Idaho National Laboratory Jack Law, Technical POC James Bresee, Federal POC 1 Ionic Liquid and Supercritical Fluid Hyphenated Techniques For Dissolution and Separation of Lanthanides and Actinides DOE-NEUP Project (TO 00058) Final Technical Report Principal Investigator: Chien M. Wai Department of Chemistry, University of Idaho, Moscow, Idaho 83844 Date: December 1, 2012 2 Table of Contents Project Summary 3 Publications Derived from the Project 5 Chapter I. Introduction 6 Chapter II. Uranium Dioxide in Ionic Liquid with a TP-HNO3 Complex – Dissolution and Coordination Environment 9 1. Dissolution of UO2 in Ionic Liquid with TBP(HNO3)1.8(H2O)0.6 2. Raman Spectra of Dissolved Uranyl Species in IL 13 3. Transferring Uranium from IL Phase to sc-CO2 15 Chapter III. Kinetic Study on Dissolution of Uranium Dioxide and Neodymium Sesquioxide in Ionic Liquid 19 1. Rate of Dissolution of UO2 and Nd2O3 in RTIL 19 2. Temperature Effect on Dissolution of UO2 and Nd2O3 24 3. Viscosity Effect on Dissolution of UO2 in IL with TBP(HNO3)1.8(H2O)0.6 Chapter IV. Separation of UO2(NO3)2(TBP)2 and Nd(NO3)3(TBP)3 in Ionic Liquid Using Diglycolamide and Supercritical CO2 Extraction 30 1. Complexation of Uranyl with Diglycolamide TBDGA in Ionic Liquid 31 2. Complexation of Neodymium(III) with TBDGA in Ionic Liquid 35 3. Solubility and Distribution Ratio of UO2(NO3)2(TBP)2 and Nd(NO3)3(TBP)3 in Supercritical CO2 Phase 38 4. -
Chapter 3 Bose-Einstein Condensation of an Ideal
Chapter 3 Bose-Einstein Condensation of An Ideal Gas An ideal gas consisting of non-interacting Bose particles is a ¯ctitious system since every realistic Bose gas shows some level of particle-particle interaction. Nevertheless, such a mathematical model provides the simplest example for the realization of Bose-Einstein condensation. This simple model, ¯rst studied by A. Einstein [1], correctly describes important basic properties of actual non-ideal (interacting) Bose gas. In particular, such basic concepts as BEC critical temperature Tc (or critical particle density nc), condensate fraction N0=N and the dimensionality issue will be obtained. 3.1 The ideal Bose gas in the canonical and grand canonical ensemble Suppose an ideal gas of non-interacting particles with ¯xed particle number N is trapped in a box with a volume V and at equilibrium temperature T . We assume a particle system somehow establishes an equilibrium temperature in spite of the absence of interaction. Such a system can be characterized by the thermodynamic partition function of canonical ensemble X Z = e¡¯ER ; (3.1) R where R stands for a macroscopic state of the gas and is uniquely speci¯ed by the occupa- tion number ni of each single particle state i: fn0; n1; ¢ ¢ ¢ ¢ ¢ ¢g. ¯ = 1=kBT is a temperature parameter. Then, the total energy of a macroscopic state R is given by only the kinetic energy: X ER = "ini; (3.2) i where "i is the eigen-energy of the single particle state i and the occupation number ni satis¯es the normalization condition X N = ni: (3.3) i 1 The probability