Low temperature physics 2005 (enns c hunklinger S)(ISBN 3540231641)(578s)

Low-Temperature Physics Christian Enss Siegfried Hunklinger Low-Temperature Physics With 421 Figures ABC Prof Dr Christian Enss Prof Dr Siegfried Hunklinger Universität Heidelberg Kirchhoff-Institut für Physik Im Neuenheimer Feld 227 69120 Heidelberg, Germany christian.enss@physik.uni-heidelberg.de siegfried.hunklinger@physik.uni-heidelberg.de Library of Congress Control Number: 2005922610 ISBN -10 3-540-23164-1 Springer Berlin Heidelberg New York ISBN -13 978-3-540-23164-6 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors using a Springer LATEX macro package Cover design: Erich Kirchner Printed on acid-free paper SPIN: 10893031 57/3141/JVG 543210 Preface Science is often a journey to the limits of the feasible and ascertainable In low-temperature physics this journey strives towards absolute zero When Louis Cailletet on December 2nd, 1877, realized a major step in terms of the production of low temperatures, namely the first liquefaction of oxygen, he could hardly imagine the wealth of exciting physical phenomena that would be discovered in this field Despite the anticipation from everyday experience, which generally equates cold with discomfort and stiffening, condensed matter at low temperatures reveals a wide array of fascinating properties As the most prominent examples let us mention superfluidity and superconductivity, whose attraction is undiminished since their discovery With every step towards lower temperatures numerous new insights have resulted, which make the traditional subject of low-temperature physics an attractive and modern research topic The present book is based on material from lectures that both authors have given several times at the universities of Heidelberg, Bayreuth and Konstanz It is focused on the discussion of physical phenomena that become most apparent at low temperatures The book is mainly aimed at students, and provides a compact and comprehensible introduction to various topics of low-temperature physics Selection and emphasis of the material is subjective and certainly reflects our personal preferences However, we have tried to give room for as wide a spectrum of topics as possible The contents are organized in three parts, entitled quantum fluids, solids at low temperatures and principles of refrigeration and thermometry Quantum fluids, with their diverse and exotic properties, are discussed in the first five chapters of the book Here, many aspects of the extraordinary liquid, superfluid He, could only be touched upon since a thorough discussion of this topic is beyond the scope of the book Chapters six to ten cover aspects of solids at low temperatures Naturally, superconductivity has been given the largest space here Atomic tunneling systems, a topic of our own research, has also been discussed in some detail, since these degrees of freedom considerably influence the properties of many solids at low temperatures The last two chapters of the book are devoted to the common physical principles and methods of low-temperature production and thermometry In this section, we have intentionally omitted many technical details – which are admittedly often VI important for everyday work in the laboratory, but have little to with the understanding of the underlying physics The citations in the text are intended to provide references for the reader to selected important articles and reviews, and to some historically interesting articles For further studies, problems related to the material discussed are given at the end of each chapter In addition, some historic anecdotes have been included in the text to introduce some variety This book would never have appeared without the help of many colleagues and coworkers S Bandler and G Seidel have read major parts of the manuscript and have given invaluable advice We are deeply thankful for the enormous time they have committed to identify errors and shortcomings and to make this book much more readable In addition, selected topics have been read by D Einzel, A Fleischmann, R Kă uhn, H.v Lă ohneysen, D Vollhardt, M.v Schickfus, G Thummes, and V Mitrovic Their suggestions have certainly improved the quality of the book and we are most grateful for their help Special thanks go to R Weis who skillfully produced all the figures Heidelberg, February 2005 C Enss S Hunklinger Contents Part I Quantum Fluids Helium – General Properties 1.1 Basic Facts 1.1.1 Terrestrial Occurrence 1.1.2 Basic Atomic and Nuclear Properties 1.1.3 Van der Waals Bond 1.2 Thermodynamic Properties 1.2.1 Density 1.2.2 Specific Heat 1.2.3 Latent Heat 1.3 Phase Diagrams 1.3.1 He 10 1.3.2 He 11 Exercises 13 Superfluid He – Helium II 2.1 Experimental Observations 2.1.1 Viscosity and Superfluidity 2.1.2 Beaker Experiments 2.1.3 Thermomechanical Effect 2.1.4 Heat Transport 2.1.5 Second Sound 2.2 Two-Fluid Model 2.2.1 Two-Fluid Hydrodynamics 2.2.2 Viscosity Measurements 2.2.3 Determination of n / 2.2.4 Beaker Experiments 2.2.5 Thermomechanical Effect 2.2.6 Heat Transport 2.2.7 Momentum of the Heat Flow 2.2.8 Sound Propagation 2.3 Bose–Einstein Condensation 2.3.1 Ideal Bose Gas 2.3.2 Helium 2.3.3 Condensate Fraction in Helium II 15 15 16 18 20 21 23 24 25 27 28 29 31 32 33 35 44 44 48 50 VIII Contents 2.4 Macroscopic Quantum State 2.4.1 Wave Function of the Superfluid Component 2.4.2 Helium II Under Rotation – Quantization of Circulation 2.4.3 Josephson Effect 2.5 Excitation Spectrum of Helium II 2.5.1 Phonons and Rotons 2.5.2 Specific Heat 2.5.3 Concept of a Critical Velocity 2.5.4 Experimental Determination of the Critical Velocity 2.6 Critical Phenomena Near the Lambda Point 2.6.1 Brief Theoretical Background 2.6.2 Specific Heat 2.6.3 Order Parameter 2.6.4 Correlation Length Exercises 52 52 53 58 60 60 63 64 66 70 70 72 74 75 76 Normal-Fluid He 3.1 Ideal Fermi Gas – Comparison with Liquid He 3.1.1 Specific Heat 3.1.2 Susceptibility 3.1.3 Transport Properties 3.1.4 Quantitative Comparison: He and Ideal Fermi Gas 3.2 The Landau Fermi-Liquid Theory 3.2.1 Quasiparticle Concept 3.2.2 Interaction Function 3.2.3 Application of Landau’s Theory to Normal-Fluid He 3.3 Zero Sound 3.3.1 Longitudinal Sound Propagation 3.3.2 Transverse Sound Propagation 3.3.3 Collisionless Spin Waves 3.3.4 Final Remarks Exercises 77 77 79 81 82 86 86 86 88 89 91 92 93 94 95 96 Superfluid He 4.1 Basic Experimental Facts 4.1.1 Phase Diagram 4.1.2 Specific Heat 4.1.3 Superfluidity 4.1.4 Nuclear Magnetic Resonance (NMR) 4.2 Relevance of the Two-Fluid Model 4.2.1 Flow Experiments 4.2.2 Normal-Fluid Density 4.2.3 Viscosity 4.2.4 Heat Transport 97 98 98 100 101 102 104 104 105 106 107 Contents IX 4.3 Quantum States of Pairs of Coupled Quasiparticles 4.3.1 Spin-Triplet Pairing 4.3.2 Broken Symmetry in Superfluid He 4.3.3 Energy Gap and Superfluidity 4.4 Order-Parameter Orientation – Textures 4.4.1 Intrinsic Alignment 4.4.2 Textures in He-A 4.4.3 Surface-induced Texture – He-A in a Slab 4.4.4 Textures in He-B 4.5 Spin Dynamics – NMR Experiments 4.5.1 Leggett Equations 4.5.2 Transverse Resonance – Frequency Shift 4.5.3 Longitudinal Resonance 4.6 Macroscopic Quantum Effects 4.6.1 Superflow 4.6.2 Quantization of Circulation 4.6.3 Quantized Vortices 4.6.4 Macroscopic Quantum Interference – Josephson Effect 4.7 Normal-Fluid Density – Quasiparticle Scattering 4.7.1 Normal-Fluid Density 4.7.2 Specific Heat 4.7.3 Quasiparticle Scattering 4.8 Collective Excitations – Sound Propagation 4.8.1 Sound Propagation 4.8.2 Collective Order-Parameter Modes Exercises 107 108 110 112 114 115 116 118 119 120 120 121 122 124 124 125 127 Mixtures of He and He 5.1 Specific Heat, Phase Diagram and Solubility 5.1.1 Phase Diagram 5.1.2 Specific Heat of Dilute Solutions of He in Helium II 5.1.3 Finite Solubility of He in Liquid He at T = 5.2 Normal-Fluid Component 5.2.1 Andronikashvili Experiment 5.2.2 Osmotic Pressure 5.3 Sound Propagation 5.3.1 First Sound 5.3.2 Second Sound 5.4 Transport Properties 5.4.1 Heat Transport 5.4.2 Viscosity 5.4.3 Self-Diffusion Coefficient 5.5 Search for a Superfluid Phase of He in Mixtures Exercises 147 148 149 150 151 153 153 154 155 156 157 158 158 159 160 161 163 131 134 134 135 136 137 138 142 146 X Contents Part II Solids at Low Temperatures Phonons 6.1 Specific Heat – Debye Model 6.1.1 Significance of the Debye Temperature 6.1.2 Specific Heat of Finite-Size Systems 6.2 Heat Transport 6.2.1 Experimental Determination of the Thermal Conductivity 6.2.2 Thermal Conductivity of Dielectric Crystals 6.2.3 Phonon–Phonon Scattering 6.2.4 Defect Scattering 6.3 Significance of N-processes in Heat Transport 6.3.1 Poiseuille Flow 6.3.2 Second Sound 6.4 Ballistic Propagation of Phonons 6.4.1 Time-Resolved Measurements of Phonon Propagation 6.4.2 Phonon Focusing 6.5 Thermal Conductivity of One-Dimensional Samples Exercises Conduction Electrons 7.1 Specific Heat 7.1.1 Conduction Electrons in Simple Metals 7.1.2 Heavy-Fermion Systems 7.2 Electrical Conductivity 7.2.1 Boltzmann Equation, Relaxation-Time Approximation 7.2.2 Residual Resistivity of Metals – Matthiessen’s Rule 7.2.3 Impurity Scattering 7.2.4 Electron–Phonon Scattering 7.2.5 Electron–Magnon Scattering 7.3 Thermal Conductivity of Metals 7.4 Kondo Effect 7.4.1 Localized Magnetic Moments 7.4.2 Electron Scattering by Localized Moments 7.4.3 Kondo Resistance 7.5 Heavy-Fermion Systems 7.5.1 Specific Heat 7.5.2 Susceptibility 7.5.3 Electrical Resistivity 7.5.4 Non-Fermi Liquids Exercises 167 167 172 175 178 180 181 182 185 190 190 193 195 195 198 200 203 205 205 206 208 209 210 213 214 216 218 220 224 225 228 229 233 235 237 239 241 243 Contents XI Magnetic Moments – Spins 8.1 Paramagnetic Systems – Isolated Spins 8.1.1 Magnetic Moments 8.1.2 Susceptibility 8.1.3 Specific Heat 8.2 Spin Waves – Magnons 8.2.1 Ferromagnets 8.2.2 Antiferromagnets 8.3 Spin Glasses 8.3.1 Structural Properties 8.3.2 Dynamic Behavior 8.3.3 Ageing, Rejuvenation and Memory Effects 8.4 Nuclear Magnetic Ordering 8.4.1 Strong Nucleus–Electron Coupling 8.4.2 Weak Nucleus–Electron Coupling 8.5 Negative Spin Temperatures 8.5.1 Thermodynamics at Negative Temperatures 8.5.2 Nuclear Ordering 8.5.3 Stimulated Emission Exercises 245 245 246 247 249 257 257 262 264 265 267 268 271 272 275 277 278 280 281 282 Tunneling Systems 9.1 Two-Level Tunneling Systems 9.1.1 Double-Well Potentials 9.1.2 Coupling to Electric and Elastic Fields 9.1.3 Relaxation 9.1.4 Relaxation Times 9.1.5 Resonant Interaction 9.2 Isolated Tunneling Systems in Crystals 9.2.1 Level Schemes 9.2.2 Specific Heat 9.2.3 Thermal Conductivity 9.2.4 Level Crossing 9.2.5 Dielectric Susceptibility 9.2.6 Sound Velocity 9.3 Interacting Tunneling Systems in Crystals 9.3.1 Dielectric Properties 9.3.2 Theoretical Description 9.3.3 Dielectric Susceptibility 9.4 Asymmetric Tunneling Systems in Crystals 9.4.1 Nb:O,H and Nb:O,D 9.4.2 CN− Ions in KBr:KCl 283 283 284 285 286 290 292 294 294 298 300 302 303 306 308 308 309 311 314 314 316 560 References 575 T.R Roberts, S.G Sydoriak, Phys Rev 98, 1672 (1955) 576 D.S Betts, Refrigeration and Thermometry Below One Kelvin, (Sussex Univ Press, Brighton 1976) 577 W Wiedemann, E Smolic, Proc 2nd International Cryogenics Eng Conf., Brighton, (1968), p 559 578 G Batey, V Mikheev, J Low Temp Phys 113, 933 (1998) 579 H London, Proc 2nd Int Conf on Low Temp Phys., (Oxford Univ Press, London 1951), p 157 580 H London, G.R Clarke, E Mendoza, Phys Rev 128, 1992 (1962) 581 P Das, R De Bruyn Ouboter, K.W Taconis, Proc 9th Int Conf on Low Temp Phys., (Plenum Press, London 1965), p 1253 582 D.O Edwards, E.M Ifft, R.E Sarwinski, Phys Rev 177, 380 (1969) 583 by courtesy of Oxford Instruments 584 D.J Cousins, S.N Fisher, A.M Gu´enault, R.P Haley, I.E Miller, G.R Pickett, G.N Plenderleith, P Skyba, P.Y.A Thibault, M.G Ward, J Low Temp Phys 114, 547 (1999) 585 P.L Kapitza, Zh Eksperim i Teor Fiz 11, (1941), english translation: J Phys USSR 4, 181 (1941) 586 I.M Khalatnikov, Zh Eksperim i Teor Fiz 22, 687 (1952) 587 G Frossati, J Phys 39 (C6), 1578 (1978), J Low Temp Phys 87, 595 (1992) 588 O.V Lounasmaa, Experimental Principles and Methods Below K, (Academic Press London 1974) 589 I Pomeranchuk, Zh Eksperim i Teor Fiz 20, 919 (1950) 590 Y.D Anufrier Sov Phys JETP Lett 1, 155 (1965) 591 W.F Giauque, D.P Mac Dougall, Phys Rev 43, 768 (1933) 592 W.J De Haas, E.C Wiersma, H.A Kramers, Physica 1, (1933) 593 A.H Cooke, Proc Phys Soc London, Sect A 62, 269 (1949) 594 N Kurti, F.E Simon, Proc Roy Soc Ser A 149, 152 (1935) 595 O.E Vilches, J.C Wheatley, Rev Sci Instrum 37, 819 (1966) 596 O.E Vilches, J.C Wheatley, Phys Rev 148, 509 (1966) 597 C.J Gorter, Z Phys 35, 928 (1934) 598 N Kurti, F.N Robinson, F Simon, D.A Spohr, Nature 178, 450 (1956) 599 K Gloos, P Smeibidl, C Kennedy, A Singsaas, P Sekowski, R.M Mueller, F Pobell, J Low Temp Phys 73, 101 (1988) 600 F Hund, Z Phys 42, 93 (1927) 601 K Motizuki, T Nagamiya, J Phys Soc Jpn 12, 163 (1957) 602 M Schwark, F Pobell, W.P Halperin, C Buchal, J Hanssen, M Kubota, R.M Mueller, J Low Temp Phys 53, 685 (1983) 603 M Schwark, M Kubota, R.M Mueller, F Pobell, J Low Temp Phys 58, 171 (1985) 604 W Wendler, T Herrmannsdă orfer, S Rehmann, F Pobell, Euro Phys Lett 38, 619 (1997) 605 R.M Mueller, C Buchal, T Oversluizen, F Pobell, Rev Sci Instrum 49, 515 (1978) 606 R.M Mueller, C Buchal, H.R Folle, M Kubota, F Pobell, Cryogenics 20, 395 (1980) 607 F Pobell, Physica B, C 109 & 110, 1485 (1982) References 561 Chapter 12 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 W Thomson (later Lord Kelvin), Philos Mag 33, 313 (1848) J.P Joule, W Thomson (later Lord Kelvin), Philos Trans 144, 321 (1854) A Celsius, Kgl Svenska Vetenskaps akad Handl 4, 197 (1742) R.L Rusby, M Durieux, A.L Reesink, R.P Hudson, G Schuster, M Kă uhne, W.E Fogle, R.J Soulen, E.D Adams, J Low Temp Phys 126, 633 (2002) G.K White, P.J Meeson, Experimental Techniques in Low-Temperature Physics, (Oxford Univ Press, Oxford 2002) H Neumann, K Stecker, Temperaturmessung (Akademie-Verlag, Berlin 1983) O Reynolds, Philos Trans R Soc London, B170, 727 (1880) D.S Greywall, P.A Busch, Rev Sci Instrum 51, 509 (1980) V Steinberg, G Ahlers, J Low Temp Phys 53, 255 (1983) H Preston-Thomas, Metrologia 27, (1990) B.W Magnum, J Res Nat Inst Stand Technol 95, 69 (1990) B.W Magnum, G.T Furukawa, Nat Inst Stand Technol., Technical Note 1265 (1990) R.P Hudson, H Marshak, R.J Soulen Jr., D.B Utton, J Low Temp Phys 20, (1975) D.S Greywall, Phys Rev 33, 7520 (1986) G Schuster, D Hechtfischer, B Fellmuth, Rep Prog Phys 57, 187 (1994) J.B Johnson, Nature 119, 50 (1927); Phys Rev 29, 367 (1927) and 32, 97 (1928) H Nyquist, Phys Rev 29 614 (1927) and 32, 110 (1928) S Menkel, D Drung, Y.S Greensberg, T Schurig, J Low Temp Phys 120, 382 (2000) C.P Lusher, J Li, V.A Maidanov, M.E Digby, H Dyball, A Casey, J Ny’e ki, V.V Dmitriev, B.P Cowan, J Saunders, Meas Sci Technol 12, (2001) A Netsch, E Hassinger, A Fleischmann, C Enss, to be published R.J Soulen Jr., R.B Dove, Standard Reference Materials, SRM 768, Temperature Reference Standard for use Below 0.5 K, National Bureau of Standards, US Department of Commerce, Special Publication, (1979), p 260 J.H Colwell, W.E Fogle, R.J Soulen Jr., Proc 17th International Conf on Low Temp Phys (E Eckern, A Schmid, W Weber, H Wă uhl eds.) (NorthHolland, Amsterdam 1984), p 395 J.F Schooley, R.J Soulen Jr., Temperature, its Measurement and Control in Science and Industry 6, (J.F Schooley ed.), (AIP, New York 1992), p 251 A.J Storm, W.A Bosch, M.J de Groot, R Jochemsen, F Mathu, G.J Nieuwenhuys, Physica B 284–288, 2008 (2000) W.A Bosch, J Flokstra, G.E de Groot, M.J de Groot, R Jochemsen, F Mathu, P Peruzzi, D Veldhuis, in Temperature: Its Measurement and Control in Sience and Industry (D.C Ripple ed.), (AIP, New York 2003), p 155 S Schă ottl, R Rusby, H Godfrin, M Meschke, V Goudon, S Trqueneaux, A Peruzzi, M.J de Groot, R Jochemsen, W.A Bosch, Y Hermier, L Pitre, C Rives, B Fellmuth, J Engert, Proc Quantum Fluids Solids 2004 H Marshak, J Res NBS 88, 175 (1983) H Marshak, Low-Temperature Nuclear Orientation, (N.J Stone, H Postma, eds.), (North-Holland, Amsterdam 1986), p 769 562 References 636 J.P Pekola, K.P Hirvi, J.P Kauppinen, M.A Paalanen, Phys Rev Lett 73, 2903 (1994) 637 J.P Kauppinen, K.T Loberg, A.J Manninen, J.P Pekola, R.A Voutilaninen, Rev Sci Instrum 69, 4166 (1998) 638 Single Charge Tunneling, Coulomb Blockade Phenomena in Nanostructures, (H Grabert, M.H Devoret eds.), (Plenum Press, New York, 1992) 639 K.P Hirvi, J.P Kauppinen, A.N Korotkov, M.A Paalanen, J.P Pekola, Appl Phys Lett 67, 2096 (1995) 640 J.P Pekola, J.J Toppari, J.P Kauppinen, K.M Kinnunen, A.J Manninen, A.G.M Jansen, J Appl Phys 83, 5582 (1998) 641 N.F Mott, Philos Mag 19, 835 (1969) 642 W Schoepe, Z Phys B 71, 455 (1988) 643 B.I Shklovskii, A.L Efros, Electronic Properties of Doped Semiconductors, (Springer, Heidelberg 1984) 644 H Fritsche, M Cuevas, Phys Rev 119, 1238 (1960) 645 E.E Haller, Infrared Phys 25, 263 (1985) 646 E.E Haller, J Appl Phys 77, 2857 (1995) 647 W.C Black, W.R Roach, J.C Wheatley, Rev Sci Instrum 35, 587 (1964) 648 K Neumeier, G Eska, Cryogenics 23, 84 (1983) 649 R.W Willekers, F Mathu, H.C Meijer, H Postma, Cryogenics 30, 351 (1990) 650 H Armbră uster, W.P Kirk, Physica B 107, 335 (1981) 651 D.J Bradley, A.M Gu´enault, V Keith, G.R Pickett, W.P Pratt Jr., J Low Temp Phys 45, 357 (1981) 652 P Strehlow, M Wohlfahrt, A.G.M Jansen, R Haueisen, G Weiss, C Enss, S Hunklinger, Phys Rev Lett 84, 1938 (2000) 653 S.A.J Wiegers, R Jochemsen, C.C Kranenburg, G Frossati, Rev Sci Instrum 57, 1413 (1986) 654 T.C.P Chui, D.R Swanson, M.J Adriaans, J.A Nissen, J.A Lipa, Phys Rev Lett 69, 3005 (1992) 655 M Jutzler, B Schră oder, K Gloos, F Pobell, Z Phys B 64, 115 (1986) 656 R Kă onig, T Herrmannsdă orfer, C Enss, private communication 657 R.A Buhrmann, W.P Halperin, S.W Schwenterly, J Reppy, R.C Richardson, W.W Webb, Proc 12th International Conf Low Temp Phys., (E Kanda ed.) (Academic Press Japan, Tokyo 1971), p 831 658 D.J Meredith, G.R Pickett, O.G Symko, J Low Temp Phys 13, 607 (1973) 659 A.I Ahonen, M Krusius, M.A Paalanen, J Low Temp Phys 25, 421 (1976) 660 A.I Ahonen, P.M Berglund, M.T Haikala, M Krusius, O.V Lounasmaa, M.A Paalanen, Cryogenics 16, 521 (1976) Index Abrikosov lattice 57, 358, 376 adiabatic demagnetization 485–505 cooling capacity 488 cooling mechanism 486 electron spins 489–490 nuclear spins 491–505 ageing 270–272 Andreev reflection 139 Andronikashvili’s experiment 28 superfluid He 108 angle-resolved photoemission spectroscopy 441 Arrhenius law 292 asymmetric tunneling systems in crystals 316–319 atomic tunneling 285–342 atomic volume 45 ballistic propagation of phonons 198–202 bath cryostats 466–469 BCS theory 378–402 Cooper pair 378–383 finite temperatures 389 ground state 383 beaker experiments 18, 29 Bloch equations 336, 540 Bloch’s T 3/2 law 262 Bogoliubov quasiparticles 115, 136, 387 boiling temperature 452 He and He Boltzmann equation 212 Bose–Einstein condensation 44–51 Bose–Einstein statistics 45 Brillouin function 249 broken symmetry 112 capacitive thermometers 534 Casimir regime 187, 327 chemical potential 45 Chevrel-compound superconductor 420 circulation 54 clapping mode 144 Claude cycle 461 Clausius–Clapeyron equation 10, 469 coherence length 129, 134, 371, 403 collective excitations helium II 50 collision term 212 collisionless spin waves 96 complex susceptibility tunneling systems 290 condensate 48 condensate concentration 51 condensation energy 361, 386 condensation temperature 47 conduction electrons 207 effective mass 210 electrical conductivity 211–242 heat transport 221–226 simple metals 208 specific heat 208–211 cooling capacity 488 cooling media 500 cooling power dilution refrigerator 480 evaporation cryostats 470 Pomeranchuk cooling 484 cooling techniques 451–505 adiabatic demagnetization 485–505 bath cryostats 466–469 closed-cycle refrigerators 462–466 cryocoolers 462–466 dilution refrigerators 472–482 564 Index electron spin demagnetization 489–490 evaporation cryostats 469–472 nuclear spin demagnetization 491–505 Pomeranchuk cooling 482–485 Cooper pair 378–383 corner SQUID interferometer 443 correlation length 56, 75 Coulomb-blockade thermometers 522 Coulomb gap 528 critical current density 401 critical field 402 critical intensity 330 critical magnetic field 352, 437 critical pressure 452 critical temperature 345, 452 He and He critical velocity helium II 23, 65 superfluid He 107 cryocoolers 462–466 Gifford–McMahon coolers 462 pulse-tube coolers 466 cryogenic liquids 452 crystal field 248, 249 cuprates 434 Curie approximation 249 Curie law 249 Debye density of states 170 Debye frequency 170 Debye model 169–181 Debye relaxator 291 Debye temperature 171, 174, 175 Debye velocity 170 deformation potential 288 density He and He density of states free particles 47 phonons 170, 178 quasiparticles in superconductors 388 tunneling systems in glasses 321 Dewar vessel 466 dielectric constant 291 glasses 332–335, 535 dielectric loss 291 glasses 327–331 dielectric susceptibility crystals with tunneling systems 305–307 dilute solutions of He in superfluid He 152 dilution refrigerators 472–482 construction 474 cooling power 480 heat exchanger 479 principle of operation 473 dipole gap 335 direct process 230 dirty limit 369 disclination 130 dislocations thermal conductivity 190 dispersion curve helium II 62 normalfluid He 97 phonons 171 spin waves antiferromagnets 264 ferromagnets 220, 260 dominant phonon approximation 181 double-well potential 286 drift velocity of ions in helium II 67 droplet theory spin glasses 267 Dulong–Petit law 172 echo experiments 336–341 eddy-current heating 494 effective mass 208, 210 elastic compliance 292 electrical conductivity 211–242 heavy-fermion systems 241 Kondo alloys 231 electrical resistivity 211–242 electromagnetic coherence length 368 electron spin demagnetization 489–490 electron temperature 274 electron–magnon scattering 220 electron–phonon interaction 380 electron–phonon scattering 218 Eliashberg theory 391 energy gap high-Tc superconductors 439 Index 565 superconductors 391–401 experimental determination 392–401 suprafluid He 114 energy of vortex rings 68 entropy He in superfluid He 474 helium II 25 liquid and solid He 11 normal-fluid He 83 nuclear spins 277, 499 paramagnetic salts 486, 489–490 solid He 483 spin systems 252–255 superconductors 362 two-level systems 254 Euler equation 25 evaporation cryostats 469–472 exchange interaction ferromagnets 220 He 82 exchange process 230 excitation spectrum helium II 59–66 expansion engines 453–459 superfluid He 106 flux line 356 flux quantization 403–405, 442 flux through narrow capillaries helium II 16 fountain effect 20 fourth sound helium II 42 superfluid He 143 free induction decay 337 Friedel oscillation 229 frustration spin glasses 266 fullerides 419 Fermi gas 88 magnetic susceptibility 83 self-diffusion coefficient 86 specific heat 81 thermal conductivity 87 viscosity 85 Fermi liquid 88–98, 236 Landau parameter 92 magnetic susceptibility 92 quasiparticle concept 88 sound velocity 92 specific heat 92 zero sound 93–96 Fermi-liquid theory application to liquid He 91 Fermi temperature 81, 153 field cooling 269 finite-size systems 178–181 first sound He/4 He mixtures 158 helium II 37 flow through thin capillaries helium II 27 Hagen–Poiseuille law 32, 106, 482 Hamaker constant 30 He/4 He mixtures 149–164 heat transport 160 normal-fluid component 156 osmotic pressure 156 phase diagram 150 self-diffusion coefficient 162 specific heat 150 viscosity 161 He melting-curve thermometers 510 healing length 56, 74, 75 heat exchanger 453, 454, 479 heat release 323 heat switch 501 heat transport conduction electrons 221 He/4 He mixtures 160 helium II 21, 32–34 momentum of heat flux in helium II 33 phonons 181–197 superfluid He 109 gas thermometer 508 Gifford–McMahon coolers 462 Ginzburg–Landau coherence length 371 Ginzburg–Landau theory 369–376 Gorter model 356 grain boundaries influence on thermal conductivity 192 group velocity surface 200 566 Index heavy-fermion systems 235 electrical resistivity 241 magnetic susceptibility 239 specific heat 210, 237 superconductivity 430 Heisenberg model of ferromagnets 259 He-A collective order-parameter modes 144 energy gap 114 fourth sound 143 mass flow 113 NMR experiments 104, 122–125 normal-fluid density 136 order parameter 116 quantization of circulation 127 quantized vortices 129 second sound 140 specific heat 101 spin dynamics 122–125 superflow 127 superfluidity 103 textures 116–122 wave function 111 He-A1 second sound 140 specific heat 103 wave function 111 He-B collective order-parameter modes 145 energy gap 115 fourth sound 143 heat transport 109 Josephson effects 133 NMR experiments 104, 122–125 normal-fluid density 107, 136 quantization of circulation 128 quantized vortices 131 second sound 140 specific heat 101 spin dynamics 122–125 superflow 126 superfluidity 103 textures 117, 121 third sound 142 viscosity 108 wave function 112 helium films 39, 178 helium monolayers 178 helium II beaker experiments 18, 29 Bose–Einstein condensation 44–51 critical velocity 23, 65 excitation spectrum 59–66 fountain effect 20 heat transport 21, 32–34 Josephson effects 52, 58 macroscopic wave function 52–59 quantization of circulation 53–58 sound propagation 35–44 specific heat 63–64 thermomechanical effect 20, 31 two-fluid model 24–44 under rotation 53–58 vortices with quantized circulation 55 high-Tc superconductors 434–448 anisotropy 436 critical magnetic field 436, 437 energy gap 439 flux quantization 442 pseudogap 447 specific resistivity 436 structure 434 hysteresis parameter 413 ideal diamagnet 352 ideal Fermi gas 79–88 imaginary squashing mode 147 impurity scattering conduction electrons 215 interaction between tunneling systems 310–315 intermediate state 353 internal friction 292 inversion temperature 452 ions in helium II 67 iron group magnetic moment 248 Ising model 266 isotope effect 378 Josephson effects 406–411, 513 ac current 407 dc current 406 helium II 52, 58 Index superfluid He 133 Josephson junction 411 Joule–Thomson effect 459–462 Kapitza resistance 476 Kondo effect 227–235 Kondo lattice 241 Kondo resistance 231 Kondo temperature 233 Korringa constant 274, 492 Korringa relation 540 Korringa relaxation 274 Kramers–Kronig relation 290 lambda point 8, 73 lambda transition 16 Landau theory of Fermi liquids 88 Landau velocity 68 Larmor frequency 540 latent heat 9, 452 Leggett angle 113 Leggett equations 122 level crossing 304 Linde liquefier 461 linearized Boltzmann equation 213 liquefaction of gases 451–462 localization length 527 London equation 32, 364 London limit 368 London model 355 London penetration depth 365 Lorenz number 222 macroscopic wave function helium II 52–59 magnetic flux inside a Josephson junction 411 magnetic moment free ions 248 iron group 248 rare-earth elements 248 magnetic phase diagram 268 magnetic superconductor 420 magnetic susceptibility He 83 heavy-fermion systems 239 nuclear spins 277 paramagnetic salts 250 magnetic thermometers 535–539 567 electron spins 537 nuclear spins 537 magnetically mediated superconductivity 424 magnetization spin waves ferromagnets 262 magnons 259–266 antiferromagnets 264–266 dispersion curve 264 specific heat 265 dispersion relation 220 ferromagnets 220, 259–263 dispersion curve 260 magnetization 262 specific heat 263 magnetization 259 specific heat 259 Magnus force 54 Matthiessen’s rule 215 maxons 61 mean free path conduction electrons 215, 225 phonons 181 Meissner–Ochsenfeld effect 351 melting curve He 12 memory effect 270, 334 metallic glasses 335–336 metals thermal conductivity 221 metals with heavy electrons 235 electrical resistivity 241 magnetic susceptibility 239 specific heat 210, 237 mode-coupling theory 313 Mă ossbauer-eect thermometers 521 momentum of heat flux helium II 33 monolayers of helium 178 near-re-entrant superconductivity negative temperatures 279–284 nuclear ordering 282 stimulated emission 283 thermodynamics 280 NMR experiments He 104, 122 421 568 Index metals at negative temperatures 283 NMR thermometers 539–543 nodal lines 417 nodal points 417 nodes 417 noise thermometers 512 nonequilibrium phenomena amorphous dielectrics 334 non-Fermi liquid 237 Nordheim’s rule 217 normal process 185, 186, 193 normal-flapping mode 144 normal-fluid He collisionless spin waves 96 magnetic susceptibility 83 self-diffusion coefficient 86 specific heat 81 thermal conductivity 87 viscosity 85 zero sound 93 normal-fluid density He/4 He mixtures 156 helium II 28 nuclear magnetic ordering 273 nuclear magnetic resonance He 104, 122 longitudinal rf field 106 transversal rf-field 105, 123 nuclear orientation thermometers 518–521 nuclear quadrupole moment 341 nuclear spin demagnetization 491–505 cooling media 500 cooling process 503 heat leaks 493 heat switch 501 technical realization 499–503 Van-Vleck paramagnets 500 nuclear spin resonance thermometers 539–543 pulsed method 541 stationary method 540 nuclear spin temperature 274 nuclear spins entropy 277 magnetic susceptibility 277 specific heat 257, 275 spontaneous magnetic order 279 spontaneous nuclear magnetic ordering 273 nuclei–electron coupling 491 Nyquist theorem 512 one-phonon process 293, 294 order parameter helium II 74 superconductor 369, 431 superfluid He 116 organic superconductor 417 ortho-hydrogen 496 ortho-para conversion of H2 495 oscillating disc viscosimeter helium II 27 osmotic pressure He/4 He mixtures 156 osmotic pressure thermometer 524 para-hydrogen 496 paramagnetic salts demagnetization 489–490 magnetic susceptibility 250 specific heat 255 paramagnetic systems 247–259 magnetic moment 248 magnetic susceptibility 249–251 paramagnons 82, 425 partition function 252 penetration depth 365 persistent current experiments helium II 16 persistent flow experiments superfluid He 103 phase-boundary energy 373 phase diagram He 11, 100 He/4 He mixtures 150 He 10 nitrogen-oxygen mixtures 458 spin glasses 267 phase memory time 340 phonon exchange 380 phonon focusing 200 phonon generation 401 phonon–defect scattering dislocations 190 grain boundaries 192 Index point defects 188 surfaces 187 phonon–phonon scattering normal process 186, 193–196 three-phonon process 184 umklapp process 186–187, 193 phonons 169 ballistic propagation 198–202 density of states 170 dispersion relation 170 fine powders 179 finite-size systems 178–181 group velocity surface 200 heat transport 181–197 low-dimensional systems 178 Poiseuille flow 193 second sound 196 specific heat 169–181 pinning centers 377 Pippard equation 368 Pippard limit 369 Poiseuille flow phonons 193 Pomeranchuk cooling 12, 482–485 cooling power 484 technical realization 483 primary thermometers 508–524 Coulomb-blockade thermometers 522 gas thermometers 508 He melting-curve thermometers 510 Mă ossbauer-eect thermometers 521 noise thermometers 512 nuclear-orientation thermometers 518–521 osmotic-pressure thermometers 524 superconducting fixed-point thermometers 516–517 vapor-pressure thermometers 508 pulse-tube coolers 466 quantization of circulation helium II 53–58 superfluid He 127 quantized vortices superfluid He 129 quantum critical point 425 quantum interference 410 569 quantum phase transition 425 quantum states of superfluid He 110 quantum volume 45 quasiparticle scattering superfluid He 136–139 quasiparticle tunneling 396 quasiparticles 220 density of states 387 heavy-fermion systems 239 normal-fluid He 89 superconductors 387, 396 superfluid He 136–139 unconventional superconductors 416 Rabi frequency 338 Raman process 293 rare-earth elements magnetic moment 248 Rayleigh disc 109 real squashing mode 147 rectifier 458 re-entrant superconductivity 421 regenerator 455 rejuvenation 270 relaxation absorption tunneling systems in glasses 327–328 relaxation processes tunneling systems 288–294 relaxation-time approximation 213, 289 relaxation times 292 residual resistivity 215, 226 resistance thermometers 524–532 carbon resistors 530 doped semiconductors 526 metals 525 RuO2 thick-film resistors 532 resistive mixed state 378 resonant absorption tunneling systems in glasses 329–331 saturation 330 resonant interaction tunneling systems 294–295 RKKY interaction 230 spin glasses 266 rotary viscosimeter 570 Index helium II 27 rotating helium II 53–58 rotons 60, 62 Ruderman–Kittel interaction 274 scattering of conduction electrons impurities 215 localized magnetic moments 230 magnons 220 phonons 218 scattering term 212 Schottky anomaly 253 second sound He/4 He mixtures 159 helium II 37 phonons 196 superfluid He 140 secondary thermometers 524–543 capacitive thermometers 534 magnetic thermometers 535–539 resistance thermometers 524–532 thermoelectric elements 533 Seebeck effect 533 self-diffusion coefficient He/4 He mixtures 162 normal-fluid He 86 Shubnikov phase 356 SK-model 267 slip phenomena 109 Snell’s law of refraction 476 Sommerfeld coefficient 208, 242 sound propagation in He/4 He mixtures first sound 158 second sound 159 sound propagation in helium II 35–44 first sound 37 second sound 23, 37 third sound 39 fourth sound 42 fifth sound 43 sound propagation in normal-fluid He 93–96 longitudinal zero sound 94 transverse zero sound 95 sound propagation in superfluid He 139–143 second sound 140 third sound 142 fourth sound 143 sound velocity crystals with tunneling systems 308–310 glasses 332–335 specific heat adsorbed He on graphite 178 conduction electrons 208 glasses 320–325 He/4 He mixtures 150 He 7, 81, 101, 137 He 7, 72 heavy-fermion systems 210, 237 He-A1 103 helium II 63 magnons 259 nuclear spins 257, 275 paramagnetic systems 251–259 phonons 169–181 spin waves antiferromagnets 265 ferromagnets 263 superconductor 362 tunneling systems in crystals 300–302 two-dimensional phonon gas 178 two-level systems 252–255 spectral diffusion 341 spin dynamics superfluid He 122–125 spin-glass temperature 268 spin glasses 266–272 ageing 270 droplet theory 267 dynamical behavior 268 field cooling 269 frustration 266 memory effect 270 phase diagram 267 rejuvenation 270 RKKY exchange interaction 266 SK-model 267 stuctural properties 267 zero-field cooling 269 spin-singlet state 383 spin waves 259–266 antiferromagnets 264–266 dispersion curve 264 specific heat 265 Index ferromagnets 220, 259–263 dispersion curve 220, 260 magnetization 262 specific heat 263 normalfluid He 96 spins 247–284 spontaneous echo 337 SQUID 412–415 static susceptibility 289 Stirling cycle 455 super-flapping mode 144 superconducting fixed-point thermometers 516–517 superconducting itinerant ferromagnets 426 superconductivity 345 critical temperature 345 BCS theory 378–402 coherent length 368 condensation energy 361, 386 Cooper pair 378–383 Eliashberg theory 391 energy gap 391 flux quantization 403–405 Ginzburg–Landau theory 369–376 Gorter model 356 ideal diamagnet 352 Josephson effects 406–411 London equation 364 London model 355 macroscopic wave function 403–411 Meissner–Ochsenfeld effect 351 persistent current 345 phase-boundary energy 373 phenomenological description 359 Pippard equation 368 spin-singlet state 383 thermodynamic description 360– 364 superconductor critical current and critical magnetic field 401 free energy and entropy 361 ideal diamagnet 349 in magnetic field 352–359 infrared absorption 392 London penetration depth 365 order parameter 369 571 pinning centers 377 re-entrant behavior 421 Shubnikov phase 356 specific heat 362, 393 thermal conductivity 396 ultrasonic absorption 395 vortices 356 superfluid He 99–147 collective modes 139 107 determination of s / energy gap 114 flow experiments 106 fourth sound 143 heat transport 109 He-A 116 Josephson effects 133 Leggett equations 122 NMR experiments 104, 122 phase diagram 100 quantization of circulation 127 quantum states 110 second sound 140 sound propagation 139 specific heat 101, 137 spin dynamics 122–125 superfluidity 103 third sound 142 Viscosity 108 vortices 129 superfluid component of He experimental determination 107 superfluid density of He-A texture effects 121 superfluidity He 103 helium II 15, 59 superinsulation 468 superleak 31 temperature scales 507 textures He-A 118 He-B 117, 121 thermal conductivity crystals with tunneling systems 302 dielectric crystals 183–192 experimental determination 182 glasses 325–327 He 87 572 Index helium II 63 metals 221 one-dimensional systems 203 phonon–defect scattering 187–192 thermal conductivity crystals with tunneling systems 305 thermally activated process 292 thermomechanical effect 20, 31 thermometers primary 508–524 secondary 524–543 thermometry 507–543 third sound helium II 39 superfluid He 142 transition temperature 347 isotope effect 378 transport properties He/4 He mixtures 160–163 tricrystal experiment 445 tunnel junction, superconducting 396 tunnel parameter 287 tunnel splitting 287 tunneling model 320 tunneling systems 285–341 acoustic attenuation 295 acoustic loss 292 amorphous dielectrics 319–335 Bloch equations 336 complex susceptibility 290 coupling to electric and elastic fields 287 critcal intensity 330 crystals 296–319 deformation potential 288 dielectric constant 291, 295 dielectric loss 291, 295 double-well potential 286 echo experiments 336–341 electrical dipole moment 288 energy splitting 287 heat release 323 metallic glasses 335–336 one-phonon process 294 relaxation processes 288–294 relaxation-time approximation 289 relaxation times 292 resonant interaction 294–295 sound velocity 295 static susceptibility 289 tunnel parameter 287 tunnel splitting 287 two-level systems 285 velocity of sound 292 tunneling systems in amorphous dielectrics 319–335 dielectric constant 332–335 dipole gap 335 heat release 323 influence on the thermal conductivity 325–327 memory effect 334 nonequilibrium phenomena 334 relaxation absorption 327–328 resonant absorption 329–331 sound velocity 332–335 specific heat 320–325 tunneling systems in crystals 296–319 dielectric susceptibility 305–307 influence on the heat conduction 302 influence on the thermal conductivity 305 interaction 310–315 level crossing 304 level scheme 296–299 sound velocity 308–310 specific heat 300–302 two-dimensional solid 178 two-fluid model 24–44 two-level systems entropy 254 spins 252 tunneling systems 285 type I superconductor 352 critical magnetic field 352 Meissner effect 349 type II superconductor 356–359 Abrikosov lattice 358 flux line 356 mixed phase 356 type I superconductor 354 intermediate state 353 umklapp process 185, 186, 193 unconventional superconductors 415–448 Index ferromagnetically ordered rare-earth systems 423 fullerides 419 heavy-fermion systems 430 high-Tc superconductor 434 itinerant ferromagnets 426 magnetic ordered systems 420 organic superconductors 417 Van der Waals force 5, 178 Van der Waals potential Van’t Hoff law 157 Van-Vleck paramagnets 273, 500 vapor-pressure cell 509 vapor-pressure thermometers 508 variable-range hopping 527 velocity of vortex rings 68 virtual phonons 379 viscosity He/4 He mixtures 161 helium II 16, 27 normalfluid He 85 superfluid He 108 viscosity measurements flow through thin capillaries 27 oscillating-disc 17 oscillating-disc viscosimeter 27 rotary viscosimeter 17, 27 vortex rings 68 vortices superconductors 356 vortices with quantized circulation helium II 55 superfluid He 129 Wiedemann–Franz law Wilson ratio 240 WKB method 287 Wă urger model 313 zero sound 93, 94, 478 transversal 95 zero-field cooling 269 zero-point energy 222 573 Printing: Strauss GmbH, Mörlenbach Binding: Schäffer, Grünstadt ... He, could only be touched upon since a thorough discussion of this topic is beyond the scope of the book Chapters six to ten cover aspects of solids at low temperatures Naturally, superconductivity... arise at a certain critical velocity This, in turn, leads to a sudden increase of the thermal resistance 2.1.5 Second Sound Temperature waves, which propagate with a characteristic velocity, are... misconception was disproved in 1939 by Alvarez and Cornog in cyclotron accelerator experiments [8] To obtain He in quantities for use in low- temperature physics experiments, it has to be produced