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The Dynamics Of Quantum Vortices.

Author Yuriy K. Krasnov
Edited by D. H. Delphenich

To my unforgettable beloved wife Elena.
Yuriy K. Krasnov.


» Preface (back to top)

This book contains the consecutive and uniform description of the quantum vortex dynamics in the non-charged and charged liquids. However, this book is not about superfluid or superconductors only; the many of considered topics will attract attention and will be useful for different specialists, who, one way or another, have an interest to understand the possible organization of the motions of a condensed matter.

This book is written for scientists and well educated students, because it assumes knowledge of hydrodynamics of an ideal liquid and a familiarity with the theory of superconductivity.

This book actually is based on the works made by author alone or in collaboration with colleagues. Some of them were published in well-known scientific editions, some - in not too much available for the western readers. But, a significant part of material, included in this book, is really original and is published for the first time. In any way, such a book was not yet published anywhere. There are same books on the quantum vortices, but, by mine best knowledge, this book is the first one about the quantum vortices, i.e., the one that is devoted to and consisting of the consecutive and uniform description of the basic principle of the quantum vortices dynamics.

Writing this book I consistently pursued the goal to give my readers not only a logical path of the development of basic ideas, but to provide them with the methods of accurate and resultative calculations also.

The major goal, the author has pursued, was to show that the quantum vortex dynamics is absolutely integral branch of the physics of liquids, having its own specific objects; that it is grounded on the specific original basic principles, and is developing of its own specific methods and approaches. Naturally, all those led to the significant results and ideas that not only are bringing up the understanding of the different phenomena of Nature connected with vortices, but that considerably have enriched our possibilities in many of unexpected areas of physics. Of course, the demonstration of the natural connection of the quantum vortex dynamics with the hydrodynamics of ideal liquids was the author's special concern.

"The Dynamics of Quantum Vortices" actually is a book about self-organizing none-dissipative structures in a continuous medium - the quantified excitations - the singular Rotons, and on their special state of self-collectivization - the quantum vortices.

Concrete "logical body" of this book one can observe by reading the Contents. Here it will be appropriate to notice that the seven Appendixes, added to the basic text, are essential not only to the understanding of causes, ideas and methods of general topics, but they have an independent original interest and meaning.

By some specific reasons, this book could not be issued in Russia in the appropriate time. So that its present publishing is one of wonderful benefits I am enjoying after my admission in the United States of America. However, the readers should pay for indulgences that America provides to own new residents. In the given case it will be some discomfort caused by my Russian English. I only hope that present book contains enough the scientific results to smooth over this discomfort due to the intellectual pleasure and practical usefulness of the exposed material.

Finishing this book, I was remembering the all of my co-authors that contributed their professional efforts to develop, step by step, the dynamics of quantum vortices during 25 years. Not always this journey was grateful and even unpunished one. But almost nobody of them betrayed the ideas that have roused us to the laborious researches. My great acknowledgments go to all of them. My special thanks must go to my former chief, friend and co-author Spartak Belyaev for understanding and sympathy to my difficult situation in that times, for his persistent demands to find out the basic principles and mechanisms of each feature of the quantum vortices, what was a main source of the many of our joint works.

And my special thank goes to my editor D. H. Delphenich, whose profound efforts to translate my Russian English in proper English have definitely "humanized" the text.

Y.K.K.

» Instead of Introduction (back to top)

§1 On the notion "a quantum vortex"   8
§2 The quantum vortices in the Nature.   11
§3 The dynamical properties of the quantum vortices   13
§4 The basic dynamical principle   16
§5 On the quantum vortices nature   17
§6 The systems of the quantum vortices.   18
§7 The practical use of quantum vortices.   19

» Part I: The quantum vortices. (back to top)

CHAPTER I: The hydrodynamic description of the quantum vortices.
§8 The singular-on-line quantum vortex: the quantum vortex thread.  

23

§9 The interaction of the vortex threads with an external flow.  

33

§10 The interaction of the vortex threads with a boundary wall.   35
§11 The planar flows: the method of the analytical functions.   43
§12 The interaction of the vortex threads with each other.   51
§13 The interaction of the vortex threads with a normal liquid.   54
§14 The vortex thread's effective mass.   55
     
CHAPTER II: The elementary hydrodynamic excitation in liquids.
§15 On the notion of "hydrodynamic excitation".   63
§16 The Landau rotons.   67
§17 The singular-in-point hydrodynamic excitation.   68
§18 The Rotons.   70
§19 The interaction of the Rotons with an external flow.   79
§20 The interaction of the Rotons with each other.   83
§21 The effective mass of Roton.   85
     
CHAPTER III: The basic dynamical principle.
§22 The external forces that act on a vortex.   88
§23 The Hall-Vinen principle.   89
§24 The equation of motion for a vortex thread.   90
§25 The vortex does not remember own dynamical history.   91
§26 Resistive and gyroscopic effects at the vortex motion.   92
§27 The tempoconformal invariance.   95
§28 The possible nature of the Hall-Vinen's principle.   97
     
CHAPTER IV: The quantum vortex's birth and death.
§29 The problem of vortices formation.   99
§30 The formation of vortices in the bulk.   100
§31 The formation of vortices in front of a surface.   102
§32 The formation of vortices on a surface.   104
§33 The surface barrier.   107
§34 The role of boundary surface roughness.   108
§35 The dependence of the critical velocity for a vortex creation on temperature.   111

» Part II: Effects Of Vortex Motion. (back to top)

CHAPTER V: The oscillations of a quantum vortex thread.
§36 The proper oscillation of a solitary vortex thread.   118
§37 The impedance of type-II superconductors.   120
§38 The superconductor in the amplifier feedback network.   125
§39 One method of investigating the electro-magnetic properties of a superconductor.   128
§40 The oscillation of a quantum vortex associated with a filament.   129
     
CHAPTER VI: The quantum vortices motion in the channels.
§41 The motion of a quantum vortex ring.   132
§42 Quantum vortices in a chink.   134
§43 The equations of motion for a superfluid in a chink.   138
§44 The transition to the dissipative flow.   144
§45 The gravitational outflow of superfluid through a narrow channel.   146
§46 He-II into the systems of a cryostabilisation.   147
§47 The quantum vortices in the superconducting plane   148
     
CHAPTER VII: The motion of quantum vortices in superconductors.
§48 The dynamics of Abrikosov's vortex in the field of a pinning center.   153
§49 The case of an ideal gyroscopic effect.   155
§50 The case of the non-ideal gyroscopic effect.   160
§51 The focusing effect for quantum vortex trajectories into the field of the pinning-forces.   161
§52 Perturbations of the trajectories of the vortices scattered from a pinning center   162
§53 The interaction of a vortex with a cavity.   164
     
CHAPTER VIII: The static and dynamic effects of the pinning of the quantum vortices in superconductors.
§54 The basics of the stahastic-dynamical theory.   170
§55 The critical current in a superconductor.   172
§56 The role of the dynamical effects in the formation of VAC.   178
§57 The statistical theory of VAC.   180
§58 The Hall's effect in the real superconductors.   183
     
CHAPTER IX: The dynamics of the rotating superconductors.
§59 The method of torsion balance.   185
§60 The basic dynamical principle for the rotating superconductors.   187
§61 The critical angle: the Heise's angle.   188
§62 A non-electric method for studying the pinning forces counter-torque and the frictional counter-torque.   191

» Part III: Systems Of Quantum Vortices. (back to top)

CHAPTER X: The systems of the quantum vortices.
§63 The central theorem on the equilibrium of the system of the mutually repulsive particles.   194
§64 The saturated equilibrium vortex lattices.   197
§65 Unsaturated equilibrium vortex lattices.   201
§66 The quantum vortices into pulsars.   206
§67 The dynamics of the essentially unsaturated vortex system under a solid-body rotation of the normal component of the liquid.   211
     
CHAPTER XI: The dynamics of the systems of the quantum vortices for a solid-body rotation of the normal component of superfluid.
§68 The basics for the continual approach.   218
§69 The evolution of the vortex front.   220
§70 The energy and momentum.   223
§71 The system of equations of motion of the quantum vortices during a solid-body rotation of the normal component of superfluid.   226
§72 Azimuthally-symmetrical motion.   229
§73 The dynamics of the small perturbations.   234
§74 The waves in the system of quantum vortices.   238
§75 The edge waves.   239
     
CHAPTER XII: The dynamics of the systems of the quantum vortices during a viscous rotation of the normal component of superfluid.
§76 The Reppy group experiments.   242
§77 The system of equations of motion of the quantum vortices during a viscous rotation of the normal component of superfluid.   244
§78 The description of the Reppy group experiments.   250
     
CHAPTER XIII: The hydrodynamics of the quantum fluids.
§79 The basics for hydrodynamics of a quantum fluids.   254
§80 The Euler equation for the superfluid component of the quantum liquid.   257
§81 The derivation of the Euler's equation for the superfluid component of the quantum liquid from the dynamics of quantum vortices.   259

» Appendixes (back to top)

APPENDIX 1. The scattering of neutrons on superfluid Helium.   263
APPENDIX 2. The spectrum of the singular Rotons.   280
APPENDIX 3. Roton near a flat boundary surface.   284
APPENDIX 4. The Oseen problem's solution for the tornado-type flow of a viscous fluid.   288
APPENDIX 5. On the electrodynamics of a large superconductor.   296
APPENDIX 6. The streamlining of ditches.   306
APPENDIX 7. The planetary system around pulsars or the edge waves?   314
INDEX   324

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