Dresner, Stability of superconductors.2002
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Preface |
There is a strong need both for this book and for the series of which it is a part. The burgeoning knowledge of applied superconductivity is now spread through a voluminous literature that includes the international journal Cryogenics, the IEEE Transactions on Magnetics, the recently created IEEE Transactions on Applied Superconductivity, the series Advances in Cryogenic Engineering, and the proceedings of various conferences such as the biennial Applied Superconductivity Conferences, International Cryogenic Engineering Conferences, and Magnet Technology Conferences, to name only a few. These sources and others (e.g., special workshops such as the biennial US/Japan workshops on helium heat transfer and magnet stability) produce many thousands of pages of literature each year. The proceedings of the 1992 Applied Superconductivity Conference, for example, ran to almost 3000 pages, and this year Cryogenics will publish more than a thousand pages. Finding the present state of a specialized topic in this teeming literature is hard for the veteran and positively daunting for the novice. So the time is ripe for creating a smaller, better organized, didactic literature that can guide active researchers, newcomers, and students.
These remarks take care of the what, the who, and the why, but there still remains to be discussed the how, or what I call the tone of the book. By tone I mean how much mathematics is used and how rigorous it is. In establishing a comfortable tone, I follow the advice of R. W. Hamming, who chose for the motto of his book on numerical analysis the epigraph, “The purpose of computing is insight, not numbers.” In this book, as in Hamming’s, the purpose of computation (analytical here rather than numerical) is to provide insight. Accordingly, I defer rigor in favor of clarity and keep the mathematics as simple as possible, consistent with satisfactory understanding.
We cannot do without mathematics, however, because applied superconductivity is a quantitative science in which we must be able successfully to build magnets. Here I have been guided by the words of Philip Morrison, who said, “The aim of quantity in science is not mere maximum precision but approximations reliable enough to argue from.” (italics mine) Such approximations, based though they are on idealizations and simplifications, often serve us better than inclusive computer programs that solve the same problem. Approximate methods form the backbone of this book. The formulas that result allow relatively easy computation of quantities of interest and so are extremely useful in design. Furthermore, if the reader studies how they have been derived, he will, I hope, learn something of what E. P. Wigner called “the essence of creative . . . thought: methods of work, tools of argument.”
I should like to acknowledge here the generosity of Dr. John Sheffield, Director of the Fusion Energy Division of Oak Ridge National Laboratory, who made available to me the time to compose this book and put at my disposal the resources of Oak Ridge National Laboratory. Of those resources, none was more important than the graphics department of the Reports Office, and I would like here to express
Preface |
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my thanks to artists Margaret Eckerd, Judy Neeley, and Shirley Boatman for the excellence of their drawings. I should like to record here, too, my gratitude to Mr. M. S. Lubell, head of the Magnetics and Superconductivity Section, for his constant support and encouragement, not just in this endeavor, but over two decades of work together. Finally, I should also like to note a more diffuse kind of debt to four of my colleagues, Dr. J. R. Miller, Dr. J. W. Lue, Prof. S. W. van Sciver, and the late Mr. M. O. Hoenig. Over the years, close collaboration with these four colleagues has helped to determine the focus of my work in the field of applied superconductivity, as I think can be seen by the frequency with which their names appear among the references.
Lawrence Dresner
Oak Ridge, Tennessee
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Units
The units used in this book to specify physical quantities are those of the International System (meter, second, kilogram, ampere). The electrical units are rationalized. This means that factors of 4π do not appear in Maxwell’s equations.
The unit of magnetic field (H) in the International System is amperes per meter, but it has become customary in applied superconductivity to express fields in terms of the magnetic induction (B) they produce in free space, the unit of which is the tesla (abbr.: T). I will adhere to this custom.
In carrying out mathematical derivations, special units are often defined and used. A special system of units is one in which various quantities have been set equal to 1 for convenience. Only a limited number of quantities can be set equal to 1. The criterion for this is that no dimensionless combination of the quantities should exist because dimensionless quantities, which are not necessarily equal to 1, have the same value in all systems of units.
The introduction of special units simplifies the appearance of the equations and reduces the amount of writing in a mathematical derivation. However, when a derivation is complete, there will be missing from the final formula in various places powers of the quantities that have been set equal to 1. These must be inserted in the final formula to make it dimensionally homogeneous. There is only one way to do this, so nothing is ever lost by the introduction of special units. I call this reconstituting the formula to regular units.
Introducing special units is an alternative to creating appropriate dimensionless variables at the outset of the problem. It allows us to jump into the real work with a minimum of formal preliminaries. But when we are done, we must perform the work of reconstituting our results. How one proceeds is largely a matter of choice, and I prefer special units to dimensionless variables, having grown habituated to them by long practice.
To be sure that the matter is clear, I give here a short derivation of the formula connecting the length and the period of a pendulum. Let us introduce special units
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Units |
in which the mass m of the pendulum bob, the length h of the pendulum, and the acceleration of gravity g all have the value 1. Then when the pendulum is displaced slightly from the vertical by an angle θ, the tangential restoring force is -sin θ ~ -θ and the tangential acceleration is θ″ ≡ d 2θ/dt2. The differential equation of the pendulum in special units is then θ″ + θ = 0, the solution of which is θ = A sin t + B cos t. In special units, then, the period ∆t = 2π. To reconstitute this formula, we must add powers of m, h, and g to this last equation so that it is dimensionally homogeneous. Now the left-hand side has the dimensions of time T. The dimensions of m are mass M, those of h are length L, and those of g are LT-2. Then we must add a factor (h/g)1/2 to the right-hand side so that both sides have the dimensions T. Our result then becomes in regular units ∆t = 2π(h/g)1 / 2.
Numbering of Sections, Equations, etc.
The numbering of text sections is consecutive within each chapter. Accordingly, Section 3 of Chapter 4 is referred to as Section 4.3 in other parts of the text. Equations are numbered consecutively in each section, but the numbering begins anew in the next section. Thus if Section 4.3 contained five equations, the last would be Eq. (4.3.5). The next equation (if it occurred in the next section) would be Eq. (4.4.1). References are noted in the text using authors’ names and year. Otherwise identical references have the short title appended after the year. The references are listed at the end of the book in alphabetic order of authors, then by year, then by alphabetic order of title.
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Contents
Chapter 1. Introduction and Overview |
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1.1. |
Early Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
1 |
1.2. |
Critical Temperature and Critical Field . . . . . . . . . . . . . . . . . . . . . |
3 |
1.3. |
Type-II Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
4 |
1.4. |
Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
8 |
1.5. |
Composite Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
10 |
1.6. |
Quenching and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
12 |
1.7. |
The Phase Diagram of Helium . . . . . . . . . . . . . . . . . . . . . . . . . |
13 |
1.8. |
High-Temperature Superconductors. . . . . . . . . . . . . . . . . . . . . . |
15 |
Chapter 2. Material Properties |
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2.1. |
Specific Heat: The Debye Formula . . . . . . . . . . . . . . . . . . . . . . |
19 |
2.2. |
Specific Heats of Type-II Superconductors . . . . . . . . . . . . . . . . |
22 |
2.3. |
First Law of Thermodynamics for a MagnetizableBody . . . . . . . . . |
25 |
2.4. |
Gibbs Free Energy of a Magnetizable Body . . . . . . . . . . . . . . |
27 |
2.5. |
Specific Heat at Zero Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
28 |
2.6. |
Contribution of the Magnetic Field to the Specific Heat . . . . . . . |
29 |
2.7. |
Matrix Resistivity; Bloch-Grüneisen Formula . . . . . . . . . . . . . . |
30 |
2.8. Magnetoresistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
31 |
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2.9. |
The Wiedemann-Franz Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
33 |
Chapter 3. Flux Jumping |
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3.1. |
TheCritical-StateModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
35 |
3.2. |
The Three-Part Curve of Joule Power . . . . . . . . . . . . . . . . . . . |
36 |
3.3. |
Charging of a Superconductor: Critical-State Model . . . . . . . . . . |
38 |
3.4. |
Charging of a Superconductor: Power-Law Resistivity . . . . . . . . . |
39 |
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Contents |
3.5. |
Flux Jumping Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . . 42 |
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3.6. |
Validity of the Adiabatic Assumption . . . . . . . . . . . . . . . . . . |
. . . . 45 |
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3.7. |
Stability against an External Magnetic Field . . . . . . . . . . . . |
. . . . 46 |
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3.8. |
Twisted Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 48 |
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3.9. |
Self-Field Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 51 |
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3.10. A Final Word on Flux Jumping . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 52 |
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Chapter 4. |
Boiling Heat Transfer and Cryostability |
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4.1. |
Fundamentals of Boiling Heat Transfer . . . . . . . . . . . . . . . |
. . . . 53 |
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4.2. |
Additional Factors Affecting Boiling Heat Transfer . . . . . . . . |
. . . . 55 |
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4.3. |
Cryostability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 56 |
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4.4. |
Cold-EndRecovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 58 |
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4.5. |
Improving Boiling Heat Transfer . . . . . . . . . . . . . . . . . . . . . |
. . . . 62 |
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4.6. |
Minimum Propagating Zones (I) . . . . . . . . . . . . . . . . . . . . |
. . . . 64 |
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4.7. |
Minimum Propagating Zones (II) . . . . . . . . . . . . . . . . . . . . |
. . . . 68 |
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4.8. |
The Formation Energy of the Minimum Propagating Zone . . . |
. . . . 71 |
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4.9. The Maximum Allowable Resistive Fault . . . . . . . . . . . . . . |
. . . . 72 |
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4.10. Stability of Partly Covered Conductors . . . . . . . . . . . . . . . . |
. . . . 75 |
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4.11. Transient Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . 77 |
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Chapter 5. |
Normal Zone Propagation |
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5.1. |
Exact Calculation of the Propagation Velocity . . . . . . . . . . |
. . . . 83 |
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5.2. |
Approximate Calculation of the Propagation Velocity . . . |
. . . . 85 |
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5.3. |
Comparison with Experiments of Iwasa and Apgar . . . . . |
. . . . 87 |
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5.4. |
Effect of Transient Heat Transfer . . . . . . . . . . . . . . . . . . . |
. . . . 88 |
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5.5. |
Traveling Normal Zones (I) . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 90 |
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5.6. |
Traveling Normal Zones (II) . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 92 |
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5.7. |
Traveling Normal Zones (III) . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 93 |
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5.8. |
The Excess Joule Heat Due to Current Redistribution . . . . . . . |
. . . 95 |
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5.9. |
The Special Case of a Cylindrical Conductor . . . . . . . . . . . . |
. . . 97 |
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5.10. Comparison with Experiment of Pfotenhauer et al . . . . . . . |
. . . 98 |
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Chapter 6. |
Uncooled Conductors |
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6.1. |
The Bifurcation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 101 |
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6.2. |
Group Analysis of the Bifurcation Energy . . . . . . . . . . . . . . . |
. . . 102 |
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6.3. |
Estimation of the Undetermined Constant . . . . . . . . . . . . . . . |
. . . 104 |
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6.4. |
Size of the Bifurcation Energy . . . . . . . . . . . . . . . . . . . . . . . |
. . . 105 |
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6.5. |
The Effect of Transverse Heat Conduction on the Bifurcation Energy 106 |
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6.6. |
Bifurcation Energies of High-Temperature Superconductors |
. . . . 107 |
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6.7. |
PropagationVelocities of UncooledSuperconductors . . . . . . . |
. . . . 110 |
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Contents |
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6.8. |
Propagation withTemperature-Dependent Material Properties . . . |
113 |
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6.9. |
The Effect of Current Sharing on the Propagation Velocity . . . . . |
115 |
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6.10. An Interesting Counterexample . . . . . . . . . . . . . . . . . . . . . . . . |
115 |
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6.11. The Approach to a Traveling Wave . . . . . . . . . . . . . . . . . . . . |
119 |
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6.12. The Effect of Heat Transfer to the Potting on the Propagation Velocity 121 |
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6.13. The Adiabatic Hot-Spot Formula . . . . . . . . . . . . . . . . . . . . . |
122 |
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6.14. Thermal Stresses during a Quench . . . . . . . . . . . . . . . . . . . . . . . |
124 |
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Chapter 7. |
Internally Cooled Superconductors |
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7.1. |
Stability Margins and Induced Flow . . . . . . . . . . . . . . . . . . . . |
129 |
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7.2. |
The One-Dimensional Equations of Compressible Flow . . . . . . |
132 |
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7.3. |
Induced Flow in a Long Hydraulic Path . . . . . . . . . . . . . . . . . . . |
134 |
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7.4. |
Induced Flow in the Experiments of Lue et al . . . . . . . . . . . . . . |
136 |
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7.5. |
MultipleStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
141 |
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7.6. |
The Limiting Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
146 |
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7.7. |
Discussion of the Isobaric Assumption . . . . . . . . . . . . . . . . . |
148 |
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7.8. |
The Lower Stability Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
151 |
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Chapter 8. Hydrodynamic Phenomena |
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8.1. |
Neglect of Fluid Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
153 |
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8.2. |
Maximum Quench Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
154 |
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8.3. |
Thermal Expulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
156 |
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8.4. |
Expulsion into an Unheated Part of the Conductor . . . . . . . . . |
158 |
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8.5. |
Short Initial Normal Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
158 |
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8.6. |
The Piston Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
159 |
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8.7. |
The Piston Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
161 |
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8.8. |
Slug Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
162 |
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8.9. |
The Propagation Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
163 |
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8.10. Thermal Hydraulic Quenchback . . . . . . . . . . . . . . . . . . . . . . . . |
164 |
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8.11. Hydrodynamic Quench Detection . . . . . . . . . . . . . . . . . . . . . . |
167 |
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8.12. Rational Design of cable-in-Conduit Conductors . . . . . . . . . . . |
168 |
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8.13. PerforatedJackets: Modified Hydrodynamic Equations . . . . . . . |
169 |
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8.14. Perforated Jackets: Reduction of the Quench Pressure . . . . . . |
172 |
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8.15. Perforated Jackets: Effect on the Stability Margin . . . . . . . . . . . |
173 |
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Chapter 9. |
Cooling with Superfluid Helium |
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9.1. |
The Superfluid Diffusion Equation . . . . . . . . . . . . . . . . . . |
175 |
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9.2. |
Superconductor Stability: The Method of Seyfert et al . . . . . . . |
176 |
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9.3. |
Similarity Solution in a Long Channel . . . . . . . . . . . . . . . . |
178 |
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9.4. |
The Temperature Dependence of the Properties of He-II . . . . . . |
180 |
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